--- schema_version: 1 title: Distinguishing Common Ratio Preferences from Common Ratio Effects Using Paired Valuation Tasks authors: - Christina McGranaghan - Kirby Nielsen - Ted O’Donoghue - Jason Somerville - Charles D. Sprenger language: en edition: AER author manuscript and final online appendix paper_url: https://jason-somerville.com/papers/distinguishing-common-ratio/ pdf_url: https://jason-somerville.com/files/distinguishing-common-ratio/paper.pdf format: Markdown with TeX mathematics --- # Distinguishing Common Ratio Preferences from Common Ratio Effects Using Paired Valuation Tasks Christina McGranaghan, Kirby Nielsen, Ted O’Donoghue, Jason Somerville, Charles D. Sprenger Current local AER author source · Main paper and final online appendix ## Abstract Without strong assumptions about how noise manifests in choices, we can infer little from existing empirical observations of the common ratio effect (CRE) about whether there exists an underlying common ratio preference (CRP). We propose to solve this inferential challenge using paired valuations, which yield valid inference under common assumptions. Using this approach in an online experiment with 900 participants, we find no evidence of a systematic CRP. To reconcile our findings with existing evidence, we present the same participants with paired choice tasks and demonstrate how noise can generate a CRE even for individuals without an associated CRP. ## Introduction The common ratio effect (CRE) refers to an empirical observation that, when choosing between a smaller amount that is more likely and a larger amount that is less likely, scaling down the probabilities by a common ratio makes people more prone to choose the riskier option. Allais (1953) first proposed the CRE as a plausible hypothetical counterexample to expected utility (EU). Later, Kahneman and Tversky (1979) provided experimental evidence of the CRE and used it to motivate one of the key features of their probability weighting function: “subproportionality.’’ An extensive subsequent experimental literature has provided further empirical evidence of the CRE. Based on this evidence, the CRE is now commonly invoked as a violation of EU, and being able to explain it is a frequent litmus test for new models of choice under risk (for instance, see Loomes and Sugden 1982; Gul 1991; Bordalo et al. 2012; Cerreia-Vioglio et al. 2015). There is an important challenge, however, to the standard interpretation of the CRE as a manifestation of non-EU preferences: If choices are stochastic rather than deterministic, then inference from paired choice tasks becomes problematic. Prior researchers have noted that EU with i.i.d. additive utility noise can naturally generate a CRE in paired choice tasks (Ballinger and Wilcox 1997; Loomes 2005; Hey 2005; Wilcox 2008; Blavatskyy 2007, 2010; Bhatia and Loomes 2017). Building on this literature, we demonstrate that, without strong assumptions about how noise manifests in choices, we can infer little from the existence or absence of a CRE in paired choice tasks about whether there exists an underlying *common ratio preference* (CRP).[^1] This paper proposes a solution to this inference challenge using *paired valuation tasks* instead of *paired choice tasks*. We first demonstrate theoretically that, under the same common assumptions about the structure of noise, paired choice tasks can yield a biased test of the EU null, whereas paired valuation tasks can yield unbiased tests. We then implement this paired-valuation approach in an experiment with 900 participants. We find no evidence of a systematic CRP at the aggregate level, although we observe substantial heterogeneity. To reconcile our findings with existing CRE evidence that uses paired choice tasks, we present the same participants with standard paired choice tasks. Individual heterogeneity in CRP, as measured by paired valuations, is highly predictive of whether an individual exhibits a CRE in paired choices. In addition, we demonstrate how appropriately chosen experimental parameters can generate a CRE in paired choice tasks even for individuals without an associated CRP (the same is true for *reverse* common ratio effects (RCRE) and preferences (RCRP)). In Section I, we develop theoretical results for testing the EU null—or, more generally, the null of no CRP—using paired choice tasks and paired valuation tasks when both are subject to noise. Importantly, we assume that behavior results from the same underlying preference and noise structure across both types of tasks, so different assumptions do not drive the results. We focus on commonly used assumptions about the noise structure—specifically, we highlight two cases, one where the noise is a simple disturbance to an underlying value and one where the noise reflects additive utility noise in the spirit of McFadden (1974, 1981). To illustrate how noise can affect inference, consider the two choices below, which together form a *paired choice task*: $$ \begin{array}{lll} AB \text{ Choice:} & \text{Lottery $A$: 100 percent chance of \$12} & \text{ vs. } \\ & \text{Lottery $B$: 50 percent chance of \$30} \\ \\ CD \text{ Choice:} & \text{Lottery $C$: 20 percent chance of \$12} & \text{ vs. } \\ & \text{Lottery $D$: 10 percent chance of \$30} \end{array} $$ In the absence of noise, EU makes a strong prediction that individuals should prefer either lotteries $A$ and $C$ or lotteries $B$ and $D$, since lotteries $C$ and $D$ are just lotteries $A$ and $B$, respectively, scaled down by a common ratio of 0.2. Counter to this prediction, the CRE describes a systematic empirical pattern of people appearing more risk-tolerant in the scaled-down problem, resulting in aggregate choice frequencies where the share choosing $A$ over $B$ is larger than the share choosing $C$ over $D$. Blavatskyy et al. (2023) provide a recent meta-study and document that out of 143 paired-choice experiments, 78 percent find that the share choosing $A$ is greater than the share choosing $C$. The average difference in the shares across all 143 experiments is 22.0 percent. Many researchers have interpreted the empirical finding of a CRE in paired choice tasks as a rejection of EU; most notably, Kahneman and Tversky (1979) use it to justify their assumption of subproportionality, one of the key properties of their probability weighting function. However, this interpretation may be invalid in the presence of noise. To illustrate, consider first the case of EU with i.i.d. additive utility noise that prior work has explored (Ballinger and Wilcox 1997; Loomes 2005; Hey 2005; Wilcox 2008; Blavatskyy 2007, 2010; Bhatia and Loomes 2017). Suppose the person’s underlying EU preferences favor $A$ and $C$. Despite this underlying preference, the person might sometimes choose $B$ over $A$ or $D$ over $C$ due to noise. The probability of choosing their preferred alternative depends on the *strength* of their preference. Under EU, when the probabilities of the initial choice are scaled down by a common ratio, the strength of the preference is scaled down by precisely the same ratio—that is, for a person who prefers $A$ and $C$, their preference for $A$ over $B$ will be stronger than their preference for $C$ over $D$. As a result, noise that is i.i.d. across the two choice tasks will be more impactful for the latter choice and will drive that choice probability further from one. Such noise thus implies that the probability of choosing $A$ over $B$ should be larger than the probability of choosing $C$ over $D$—i.e., the person would exhibit a CRE even though they have underlying EU preferences and thus no CRP. An analogous intuition applies for a person whose EU preferences favor $B$ and $D$, for whom additive i.i.d. noise implies that the probability of choosing $A$ over $B$ should be smaller than that for choosing $C$ over $D$, indicating that they would exhibit an RCRE. In Section I, we generalize this logic to other noise structures besides i.i.d. additive utility noise and to permit heterogeneity both in preferences and in the impact of noise. While in principle the bias in paired choice tasks due to noise could go in either direction, there is evidence that, in practice, the prior literature’s choice of experimental parameters has led to a systematic bias in the direction of finding a CRE. Of the 143 CRE paired-choice experiments reviewed by Blavatskyy et al. (2023), more than 75 percent exhibit a share of $A$ choices greater than or equal to 50 percent, consistent with the case in which underlying preferences favor $A$ and $C$ and in which noise could thus yield a CRE even if there were no underlying CRP. In Section I.D, we further detail the use of selected parameter values in the prior literature. As an alternative to the paired choice tasks above, consider instead the two valuations below, which together form a *paired valuation task*: $$ \begin{array}{lllllll} AB \text{ Valuation: state an $m_{AB}$ such that} & \\ \\ & & & & & \text{100 percent chance of $m_{AB}$} \sim \text{50 percent chance of \$30} \\ \\ CD \text{ Valuation: state an $m_{CD}$ such that} & \\ \\ & & & & & \text{20 percent chance of $m_{CD}$} \sim \text{10 percent chance of \$30} \end{array} $$ EU again makes a strong prediction that, in the absence of noise, individuals should state valuations that satisfy $m_{AB}=m_{CD}$ or, equivalently, $\Delta m \equiv m_{CD}-m_{AB}=0$. In contrast, the logic of the CRE implies that individuals are more risk-tolerant for the $CD$ comparison than for the $AB$ comparison, and thus we would see $m_{CD}>m_{AB}$ or $\Delta m \equiv m_{CD}-m_{AB}>0$. Although valuation tasks are common in experiments, researchers have rarely used them in the context of the CRE. We demonstrate, however, that under the same commonly used assumptions about noise where paired choice tasks can yield biased tests of the null of no CRP, paired valuation tasks can yield unbiased tests. Specifically, if elicited valuations are unbiased measures of the underlying values, then we can test whether the mean of $\Delta m$ equals 0. Alternatively, even if elicited valuations are biased measures of underlying values—e.g., due to utility curvature—then as long as the noise is symmetric around its median, we can instead use a sign test to assess whether there are equal proportions of positive and negative instances of $\Delta m$. Notably, both tests are robust to noise having a differential impact across the $AB$ and $CD$ tasks. In Section II, we describe the details of our experimental design. We recruit 900 participants from Prolific for an online experiment. In stage 1 of the experiment, we elicit paired valuations. For each participant, we elicit the value of $m_{AB}$ that makes them indifferent between $(\$m_{AB}, 1)$ and $(\$30, p)$, and we separately elicit the value of $m_{CD}$ that makes them indifferent between $(\$m_{CD}, r)$ and $(\$30, rp)$. Each participant reports these paired valuations for five values of $p$, and between subjects we consider three different values of $r$; hence, in stage 1, we elicit paired valuations for 15 combinations of $(p,r)$. In stage 2 of the experiment, we present the same participants with paired choice tasks, with one paired choice task linked to each paired valuation task from stage 1. We use this connection between stages to validate the stage 1 valuations and to reconcile our findings with the prior literature.[^2] Section III describes our main results using data from the paired valuation tasks. We conduct our two tests for each of the 15 paired valuation tasks in stage 1. Out of the 15 means tests, we reject the null that the mean of $\Delta m$ is zero in eight comparisons at the 5 percent level. All eight rejections indicate an RCRP rather than the standard CRP, and the means are small in magnitude. Out of the 15 sign tests, we find seven significant deviations from the null of equal proportions at the 5 percent level. Six of these are consistent with an RCRP, and there is only one test in which the deviation from equal proportions is in the direction of a CRP. Beyond the formal tests, we also find that in 14 of 15 cases, the median value of $\Delta m$ is zero. Thus, our paired valuation tasks yield no evidence of a systematic CRP. Our failure to find a systematic CRP in the aggregate does not imply that our data are consistent with EU. While the data indicate an aggregate central tendency of no CRP, we document substantial CRP heterogeneity—specifically, we find significant within-individual correlations of $\Delta m$ across different $(p,r)$ combinations, so some individuals appear to have a stable CRP while others seem to have a stable RCRP. Moreover, our $m_{AB}$ valuations yield data consistent with probability weighting models and thus inconsistent with EU. Specifically, our $m_{AB}$ elicitations are equivalent to the tasks researchers commonly use to estimate probability weighting functions, and they yield an inverse-S-shaped probability weighting function that matches those typically found in the literature. However, the probability weighting function implied by our $m_{AB}$ valuations is wholly inconsistent with our elicited $m_{CD}$ valuations—indeed, it would predict $m_{CD}$ valuations consistent with a large CRP. In Section IV, we analyze the connections between the valuations elicited in stage 1 and the corresponding choices made in stage 2. For each paired valuation task from stage 1 (i.e., for each of a participant’s five $(p,r)$ combinations), we choose a random value of $M$ in stage 2 and then offer the participant a binary $AB$ choice between $(\$M, 1)$ and $(\$30, p)$, and a binary $CD$ choice between $(\$M, r)$ and $(\$30, rp)$. The connection between these linked valuations and choices allows us to assess whether there is differential noise across the $AB$ and $CD$ choices and to reconcile our main finding of no systematic CRP in paired valuation tasks with the vast literature that finds a CRE in paired choice tasks. Two key predictions link a person’s stage 1 valuations to their stage 2 choices. First, reflecting the impact of preferences, the stage 1 *value difference* $\Delta m$ should predict whether an individual exhibits a CRE or an RCRE at stage 2. That is, individuals who have a CRP as measured through their stage 1 valuations should be more likely to exhibit a CRE in their stage 2 choices. Second, the impact of differential noise depends on the difference between the randomly chosen amount $M$ and the stage 1 average indifference point; we refer to this measure as the *distance to indifference*. A sufficiently large positive distance to indifference means $M$ is large enough that preferences favor $A$ and $C$. If, in addition, the choice noise is more impactful for the $CD$ choice, then pattern $AD$ will be more likely than pattern $BC$; that is, we would observe a CRE. Analogously, a sufficiently large negative distance to indifference means $M$ is small enough that preferences favor $B$ and $D$; in this case, if the choice noise is more impactful for the $CD$ choice, then we would observe an RCRE. When we take these two predictions to the data, we find strong support for them at both the individual and experiment level—where an “experiment” refers to the aggregate behavior of a subset of participants who faced the same paired choice task at stage 2.[^3] Our finding that stage 1 value differences strongly predict stage 2 choices provides validation that our valuations are capturing underlying preferences. Our finding that the distance to indifference predicts stage 2 choices reveals the existence of differential noise. The latter finding further demonstrates how specific parameter combinations—in particular, those which induce a positive distance to indifference—can generate a CRE in paired choice tasks even if there is no underlying CRP. Indeed, in Section IV.D, we describe how our use of a broader and more balanced set of parameter combinations (relative to the prior literature) leads to little evidence of a CRE in our stage 2 paired choices. Our analysis has several connections to the prior literature. First, a small number of papers do use paired valuation tasks in the context of the CRE (see, e.g., Castillo and Eil 2014; Dean and Ortoleva 2019; Schneider and Shor 2017; Freeman et al. 2019). These papers address different research questions relative to ours, and none of them address the differential noise problem associated with paired choice tasks nor the fact that paired valuation tasks are robust to this problem. Our paper also connects to research highlighting how systematic behaviors that appear to contradict standard models can emerge from standard models combined with noise. Even within our context, prior work has demonstrated how an observed CRE in paired choice tasks could be due to EU with additive utility noise (Ballinger and Wilcox 1997; Loomes 2005; Hey 2005; Wilcox 2008; Blavatskyy 2007, 2010; Bhatia and Loomes 2017). In recent years, this theme has re-emerged in other domains. For instance, apparent nonlinear probability weighting can emerge from underlying expected value preferences combined with cognitive imprecision in the perception of probabilities (Khaw et al. 2021; Frydman and Jin 2023) or cognitive uncertainty about the optimal action choice (Enke and Graeber, forthcoming); an apparent preference for commitment could reflect the impact of noise on a binary choice whether to commit (Carrera et al. 2022); and recent work by Oprea (2022) and Enke et al. (2023) suggests that probability weighting and present bias might reflect heuristic reactions to the inherent complexity of the choice problems. Our work also relates more generally to literature that explicitly accounts for noise in experimental design and the analysis of risky choice data (Harless and Camerer 1994; Hey and Orme 1994; Ballinger and Wilcox 1997; Loomes and Sugden 1998; Stott 2006). More recently, a serious evaluation of noise in canonical experimental designs has led to a preference for new types of data collection, such as valuations (Bernheim and Sprenger 2020; Carrera et al. 2022), repeated choices (Gillen et al. 2019), decision times (Alós-Ferrer et al. 2021), or measures of individual uncertainty and confidence (Butler and Loomes, 2011; Enke and Graeber, forthcoming). This sort of work connecting theory and data collection efforts with an encompassing view of noise carries promise for improving the reliability of the inference drawn from experimental observations and our understanding of the forces driving behavioral anomalies. Our paper has two broad implications for future research that we expand on in our concluding Section V. First, our methodological contribution applies far beyond the domain of the CRE. Canonical tests for many behavioral-economic phenomena use reduced-form comparisons of pairs of choice probabilities, with examples ranging from risk preferences to time preferences to context dependence. Our theoretical results readily extend to these settings, and valuations may be similarly suited to overcome the underlying inference problem.[^4] Second, our core empirical findings present an important challenge for the literature that seeks to understand risk attitudes. Like prior research, we find clear evidence against EU as an accurate descriptive model of behavior. However, unlike prior research, our core empirical finding contradicts the received wisdom that most people exhibit a systematic underlying CRP. If this finding turns out to be robust, then it calls for a reassessment of non-EU models built around the CRP as a motivating feature of preferences. First and foremost are models of probability weighting in the tradition of prospect theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992), where a CRP (or subproportionality) is the central motivating concept behind the structure of probability weighting. Other models also use the CRE as a key motivating fact and thus build in a systematic CRP; these include disappointment aversion (Bell 1985; Gul 1991) and cautious expected utility (Cerreia-Vioglio et al. 2015), among others. ## I Underlying Theory and Proposed Tests ### A Paired Choices and Paired Valuations The standard common-ratio test presents participants with *paired choice tasks* that take the following form: $$ \begin{array}{lll} AB \text{ Choice Task: choose Lottery }A \equiv (M,1) & \text{or} & \text{Lottery }B \equiv (H,p) \\ CD \text{ Choice Task: choose Lottery }C \equiv (M,r) & \text{or} & \text{Lottery }D \equiv (H,rp), \end{array} $$ where $H>M>0$ and $p,r\in (0,1)$. The key feature is that the $CD$ choice task is derived from the $AB$ choice task by scaling down the probabilities for the non-zero outcomes by a common ratio $r$.[^5] As highlighted by Allais (1953), paired choice tasks of this type are interesting because EU makes a sharp prediction. Normalizing $u(0)=0$: $$ \begin{array}{rcl} EU(A)-EU(B) &>& 0\qquad \Leftrightarrow \qquad \quad u(M)-pu(H) \hspace{3.5pt}>0\qquad \text{and} \qquad \text{(1)} \\ EU(C)-EU(D) &>&0 \qquad \Leftrightarrow \qquad r\left[ u(M)-pu(H)\right] >0. \qquad \text{(2)} \end{array} $$ Hence, EU predicts that a person should prefer either lotteries $A$ and $C$ or lotteries $B$ and $D$. In contrast to this prediction, the empirical finding of a *common ratio effect* (CRE) involves deviations systematically in the direction of choosing lotteries $A$ and $D$—in other words, people appear more risk-tolerant in the scaled-down choice. More precisely, letting $\widehat{\Pr}(X)$ be the proportion of participants who choose $X$, the common finding is $\widehat{\Pr }(AD)>\widehat{\Pr}(BC)$ or, equivalently, $\widehat{\Pr }(A)>\widehat{\Pr}(C)$.[^6],[^7] We use the label *reverse common ratio effect* (RCRE) for the less common finding of deviations in the direction of choosing lotteries $B$ and $C$, or a finding of $\widehat{\Pr }(A)<\widehat{\Pr}(C)$. An alternative common-ratio test that researchers have used much less often presents participants with *paired valuation tasks*. Our main analysis will focus on $m$*-valuation tasks* in which we fix $(H,p,r)$ and present participants with the following tasks: $$ \begin{array}{l} AB \text{ Valuation Task: state an $m_{AB}\in [0,H]$ such that } (m_{AB},1) \sim (H,p) \\ CD \text{ Valuation Task: state an $m_{CD}\in [0,H]$ such that } (m_{CD},r) \sim (H,rp). \end{array} $$ For paired valuation tasks, a finding of $\Delta m \equiv m_{CD} - m_{AB} > 0$ reflects a CRE. Again, a CRE involves people acting more risk-tolerant in the scaled-down task. Thus, an individual will demand a higher premium to accept the safer option in the $CD$ task relative to what they demand in the $AB$ task, or $m_{CD}>m_{AB}$. Analogously, a finding of $\Delta m < 0$ reflects an RCRE, and a finding of $\Delta m = 0$ would be consistent with EU (among other models). A CRE, in choices or valuations, is an empirical finding; when interpreting this empirical finding in terms of underlying preferences, it is essential to account for noise. The following two subsections do so, first for the special case of EU and i.i.d. additive utility noise, and then for the more general case. ### B EU and i.i.d. Additive Utility Noise Prior work has noted that EU with i.i.d. additive utility noise can naturally generate a CRE in paired choice tasks (Ballinger and Wilcox 1997; Loomes 2005; Hey 2005; Wilcox 2008; Blavatskyy 2007, 2010; Bhatia and Loomes 2017). In this subsection, we illustrate the intuition behind this result. Suppose there exist i.i.d. distributed noise draws $\epsilon_{AB}$ and $\epsilon_{CD}$ such that a person chooses lottery $A$ over lottery $B$ when $EU(A)-EU(B) > \epsilon_{AB}$, and chooses lottery $C$ over lottery $D$ when $EU(C) - EU(D) > \epsilon_{CD}$. Applying equations (1) and (2), $EU(C)-EU(D) = r[EU(A)-EU(B)]$. Hence, if $F$ is the shared CDF for $\epsilon_{AB}$ and $\epsilon_{CD}$, then the likelihoods of choosing lottery $A$ over lottery $B$ and of choosing lottery $C$ over lottery $D$ are, respectively, $$ \begin{array}{rcl} \Pr(A) = F\left(EU(A)-EU(B)\right) & \text{and} & \Pr(C) = F\left(r\left[EU(A)-EU(B)\right]\right). \end{array} $$ Figure 1 illustrates the implications for the case when $F$ is a standard normal distribution. Panel A presents the choice probabilities. The solid line depicts $\Pr(A)$ as a function of $EU(A)-EU(B)$. If $EU(A)-EU(B)=0$, in which case the person’s underlying EU preferences are indifferent between lotteries $A$ and $B$, then $\Pr(A) = 1/2$. As the person’s underlying preference for lottery $A$ gets stronger—i.e., as $EU(A)-EU(B)$ gets more positive—$\Pr(A)$ grows toward 1. As the person’s underlying preference for lottery $B$ gets stronger—i.e., as $EU(A)-EU(B)$ gets more negative—$\Pr(A)$ shrinks toward 0. The dashed and dotted lines in Panel A depict $\Pr(C)$ as a function of $EU(A)-EU(B)$ for the cases of $r=0.5$ and $r=0.25$. Each line has the same qualitative pattern as $\Pr(A)$. However, when we scale down the probabilities by a common ratio $r$, we also scale down the strength of preference by the same factor. As a result, the i.i.d. noise will be more impactful for the $CD$ choice than for the $AB$ choice. Therefore, for any specific value of $EU(A)-EU(B)$, $\Pr(C)$ is closer to $1/2$ than $\Pr(A)$ is. Panel A. Choice Probabilities ![Figure 1: Predicted Choice Probabilities and CRE − RCRE. (panel 1)](assets/figure1a-f0e028.png) Panel B. Predicted $CRE-RCRE$ ![Figure 1: Predicted Choice Probabilities and CRE − RCRE. (panel 2)](assets/figure1b-1a5491.png) **Figure 1: Predicted Choice Probabilities and $CRE - RCRE$.** Predicted choice probabilities (panel A) and predicted $CRE - RCRE$ (panel B) as a function of $EU(A)-EU(B)$ under expected utility with additive i.i.d. noise. In both panels, the dashed red line reflects the case $r_1 = 0.5$ and the dotted green line reflects the case $r_2 = 0.25$. Both panels assume that the noise follows a standard normal distribution. Panel B of Figure 1 converts the choice probabilities from Panel A into a predicted difference $CRE-RCRE \equiv \Pr(A) - \Pr(C)$. Panel B illustrates that a person whose underlying EU preferences favor $A$ and $C$ will exhibit a CRE. Conversely, a person whose underlying EU preferences favor $B$ and $D$ will exhibit an RCRE. Panel B further highlights how the predicted magnitude of $CRE-RCRE$ depends on (i) the magnitude of the common ratio $r$ and (ii) the strength of the person’s underlying preference $EU(A)-EU(B)$. Finally, note that at extreme levels of $EU(A)-EU(B)$, $CRE-RCRE$ goes to zero as the preference component becomes so strong that it dominates the noise. Figure 1 illustrates a fundamental problem in using paired choice tasks to test for an underlying CRP. In the following subsection, we develop a more general model of underlying preferences and noise that permits us to expand on this problem and illustrate how paired valuation tasks can be a solution. ### C More General Model of Underlying Preferences and Noise We develop a framework to interpret data from both paired choice tasks and paired valuation tasks. Specifically, we assume a person has a realized indifference value that is determined from a combination of their underlying preferences and noise. The person then makes a choice or states a valuation implied by this realized indifference value. The case of EU with i.i.d. additive utility noise studied in Section I.B is a special case of this framework. Importantly, we impose the same structure of preferences and noise across both types of tasks. Hence, the different results we document are not due to different assumptions across the two types of tasks. Moreover, this framework implies a strong connection between paired valuation tasks and paired choice tasks. Motivated by this connection, our experiment collects data from the same participants on paired valuation tasks and linked paired choice tasks. We show in Section IV that participants’ valuations are predictive of their choices in a way that is consistent with this framework.[^8] Without loss of generality, we fix $(H,p,r)$ and focus on behavior as a function of $M$. Assuming preferences are monotonic and continuous, for each $(H,p,r)$ a person will have a pair of underlying indifference points $(m_{AB}^{\ast},m_{CD}^{\ast})$ such that their (noise-free) preferences satisfy: - Prefer $A\equiv (M,1)$ over $B\equiv (H,p)$ if and only if $M\geq m_{AB}^{\ast }$, and - Prefer $C\equiv (M,r)$ over $D\equiv (H,rp)$ if and only if $M\geq m_{CD}^{\ast }$.[^9] Given these underlying indifference points, we assume that noise impacts choices and valuations as follows: **Assumption 1:** Impact of Noise on Choices and Valuations A person’s *realized indifference points* $(m_{AB},m_{CD})$ are $m_{AB}\equiv \Gamma (m_{AB}^{\ast },\varepsilon _{AB})$ and $m_{CD}\equiv \Gamma (m_{CD}^{\ast },\varepsilon _{CD})$, where $(\varepsilon_{AB},\varepsilon_{CD})$ are noise draws from a continuous joint distribution with convex support, and $\Gamma$ is increasing in both arguments and has $\Gamma (m,0)=m$ for all $m$. Then: - In an $AB$ choice task, the person chooses $A\equiv (M,1)$ over $B\equiv (H,p)$ if and only if $M\geq m_{AB} \equiv \Gamma (m_{AB}^{\ast },\varepsilon _{AB})$, - In a $CD$ choice task, the person chooses $C\equiv (M,r)$ over $D\equiv (H,rp)$ if and only if $M\geq m_{CD} \equiv \Gamma (m_{CD}^{\ast },\varepsilon _{CD})$, - In an $AB$ valuation task, the person states valuation $m_{AB} \equiv \Gamma (m_{AB}^{\ast },\varepsilon _{AB})$, and - In a $CD$ valuation task, the person states valuation $m_{CD} \equiv \Gamma (m_{CD}^{\ast },\varepsilon _{CD})$. The distinction between underlying preferences and observed behaviors is integral to our analysis. To highlight this distinction, we say that a person has a *common ratio preference* (CRP) if they have $\Delta m^{\ast}\equiv m_{CD}^{\ast}-m_{AB}^{\ast}>0$, and a *reverse common ratio preference* (RCRP) if they have $\Delta m^{\ast}<0$. Assessing whether an observed CRE in choices or valuations is evidence of an underlying CRP is the key inferential challenge that we focus on in the remainder of this section.[^10] In Assumption 1, the function $\Gamma$ permits a variety of models for how a person’s underlying indifference points combine with choice noise to generate their realized indifference points. We highlight two special cases of Assumption 1: **Assumption 2a:** $\Gamma(m,\varepsilon)=m+\varepsilon$, $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$, and $E(\varepsilon _{AB})=E(\varepsilon _{CD})=0$. **Assumption 2b:** $\Gamma(m,\varepsilon)$ is potentially nonlinear in $m$ and $\varepsilon$, but $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$, and $\varepsilon_{AB}$ is symmetric about 0. Assumption 2a is consistent with assumptions researchers frequently use when analyzing valuations data, where they model noise as an additive disturbance to an underlying value. Assumption 2b is consistent with assumptions researchers frequently use when analyzing choice data, where they instead model noise as a symmetric additive perturbation of utility in the spirit of McFadden (1974, 1981). To illustrate, we describe when Assumptions 2a and 2b would hold under two prominent models of underlying preferences. **Example 1:** Expected Utility and Prospect Theory Suppose that a person evaluates a lottery $(x,q)$ with $x>0$ as $\pi(q)u(x)$. This formulation corresponds to both original prospect theory as in Kahneman and Tversky (1979) and cumulative prospect theory as in Tversky and Kahneman (1992), where $\pi(\cdot)$ is a probability weighting function and $u(\cdot)$ is a value function defined over gains and losses. This formulation also reduces to EU when $\pi(q)=q$ for all $q$ and $u(\cdot)$ is a Bernoulli utility function. Under this formulation, the underlying indifference points satisfy $$ \begin{array}{rcl} u(m_{AB}^{\ast})=\pi(p)u(H) & \qquad \Leftrightarrow \qquad & m_{AB}^{\ast}=u^{-1}\left( \pi(p)u(H) \right) \\ \pi(r)u(m_{CD}^{\ast})=\pi(rp)u(H) & \qquad \Leftrightarrow \qquad & m_{CD}^{\ast}=u^{-1}\left( \frac{\pi(rp)}{\pi(r)}u(H) \right). \end{array} $$ When working with valuations data, one might incorporate noise by assuming that observed valuations satisfy $m_{AB} = m_{AB}^{\ast} + \varepsilon_{AB}$ and $m_{CD} = m_{CD}^{\ast} + \varepsilon_{CD}$. This formulation satisfies Assumption 2a as long as $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$ and $E(\varepsilon _{AB})=E(\varepsilon _{CD})=0$—e.g., if $\varepsilon_{AB}$ and $\varepsilon_{CD}$ are both mean-zero normal or logistic distributions with possibly different variances.[^11] Alternatively, for either valuations or choice data, one might incorporate additive utility noise by instead assuming that the realized indifference points satisfy $$ \begin{array}{rcl} u(m_{AB})=\pi(p)u(H) + \epsilon_{AB} & \Leftrightarrow & m_{AB}=u^{-1}\left( u(m_{AB}^{\ast}) + \epsilon_{AB} \right) \\ \pi(r)u(m_{CD})=\pi(rp)u(H) + \epsilon_{CD} & \Leftrightarrow & m_{CD}=u^{-1}\left( u(m_{CD}^{\ast}) + \epsilon_{CD}/\pi(r) \right) \end{array} $$ where $\epsilon_{AB}$ and $\epsilon_{CD}$ reflect additive utility noise.[^12] This formulation fits Assumption 1 with $\Gamma (m,\varepsilon) = u^{-1}(u(m)+\varepsilon)$, $\varepsilon_{AB}=\epsilon_{AB}$, and $\varepsilon_{CD}=\epsilon_{CD}/\pi(r)$. This formulation further satisfies Assumption 2b as long as $\epsilon_{AB}$ is symmetric about 0 and $\epsilon_{CD}\,{\buildrel d \over =}\,k'\epsilon _{AB}$ for some $k'>0$—e.g., if the additive utility noise terms $\epsilon_{AB}$ and $\epsilon_{CD}$ are both mean-zero normal or logistic distributions.[^13] Finally, note that the case of EU with i.i.d. additive utility noise that we described in Section I.B fits Assumption 2b with $\varepsilon_{CD}=\varepsilon_{AB}/r$, and so $k=1/r$. The framework of Assumption 1, with Assumptions 2a and 2b as special cases, allows us to demonstrate the problems with using paired choice tasks, and when paired valuation tasks might be robust to those problems. Proposition 1 establishes conditions under which paired choice tasks yield a biased test of the null of $\Delta m^\ast = 0$. All proofs appear in Online Appendix A. ***Proposition 1** (Paired Choice Tasks Can Yield Biased Tests of $\Delta m^\ast=0$):* Consider a person who has $m_{AB}^{\ast }=m_{CD}^{\ast}\equiv m^{\ast }$ and thus $\Delta m^{\ast }=0$, and suppose that $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$, and define $Z \equiv \Pr(\varepsilon _{AB}<0) = \Pr(\varepsilon _{CD}<0)$. - If $M-m^{\ast }>0$ and thus the person prefers $A$ and $C$, then: - $k>1$ implies $\Pr(A)>\Pr(C)>Z$ (CRE); - $k<1$ implies $\Pr(C)>\Pr(A)>Z$ (RCRE); and - $k=1$ implies $\Pr(A)=\Pr(C)>Z$. - If $M-m^{\ast }<0$ and thus the person prefers $B$ and $D$, then: - $k>1$ implies $\Pr(A)<\Pr(C)1$ (see Example 1). Moreover, under EU, $M-m^\ast>0$ implies $EU(A)-EU(B)>0$ whereas $M-m^\ast<0$ implies $EU(A)-EU(B)<0$. Parts (1)(a) and (2)(a) of Proposition 1 therefore imply the same predictions as illustrated in Figure 1, reiterating the core intuition that the $AB$ preference is stronger than the $CD$ preference under EU, and thus i.i.d. noise will have a larger impact on the $CD$ choice. Hence, if EU preferences favor $A$ and $C$, then the prediction is $\Pr(A)>\Pr(C)$. In contrast, if EU preferences favor $B$ and $D$, then the prediction is $\Pr(A)<\Pr(C)$.[^14] Proposition 1 generalizes these conclusions in several ways. For the case of EU, Proposition 1 characterizes predictions under different noise structures besides i.i.d. additive noise, including when the noise is a disturbance to the underlying indifference values and when there is additive utility noise but with different variances across the two choices. Our framework and Proposition 1 also apply to any non-EU model that exhibits $\Delta m^\ast = 0$. The key feature that determines predictions is whether the noise is more impactful for the $CD$ choices (i.e., $k>1$), in which case the predictions are as in Section I.B. The predictions flip if the noise is more impactful for the $AB$ choices (i.e., $k<1$). The implication is that, when using paired choice tasks, a person with $\Delta m^{\ast}=0$ could exhibit either a CRE or an RCRE depending on the combination of whether (i) the offered $M$ leads to an underlying preference for $A$ and $C$ versus $B$ and $D$, and (ii) choice noise has a larger impact on the $AB$ choice versus the $CD$ choice. When using data from paired choice tasks to test the null of $\Delta m^\ast=0$, researchers typically assess whether $\widehat{\Pr}(A)=\widehat{\Pr}(C)$. Proposition 1 implies that the theoretical prediction of $\Pr(A)=\Pr(C)$ holds only when $k=1$. This knife-edge case seems unlikely to hold in practice, and thus paired choice tasks are likely to yield a biased test. Whereas paired choice tasks yield a biased test in all but this knife-edge case, Proposition 2 shows that unbiased tests are possible using paired valuation tasks under a much broader set of conditions. ***Proposition 2** (Paired Valuation Tasks Can Yield Unbiased Tests of $\Delta m^\ast=0$):* Consider a person who faces a paired valuation task, and let $m_{AB}$ and $m_{CD}$ be their stated valuations. - If $\Gamma (m,\varepsilon ) = m + \varepsilon$ and $E(\varepsilon_{AB})=E(\varepsilon _{CD})$, then $E(\Delta m) = \Delta m^{\ast}$. - If a person has $m_{AB}^{\ast}=m_{CD}^{\ast }\equiv m^{\ast }$, and if the joint distribution $(\varepsilon _{AB},\varepsilon _{CD})$ is symmetric around some median vector $(\varepsilon',\varepsilon')$,[^15] then $\Pr(\Delta m>0)=\Pr(\Delta m<0)=1/2$. Part (1) of Proposition 2 describes conditions under which we can test the null of $\Delta m^\ast = 0$ using a means test, specifically, testing whether $\widehat{E}(\Delta m) = 0$.[^16] When $E(\varepsilon_{AB}) = E(\varepsilon _{CD}) = 0$, as under Assumption 2a, $m_{AB}$ and $m_{CD}$ are unbiased measures of the underlying indifference points, and thus their difference is an unbiased measure of $\Delta m^\ast$. Furthermore, even if the errors have non-zero means, in which case $m_{AB}$ and $m_{CD}$ are biased measures of the underlying indifference points, their difference remains an unbiased measure of $\Delta m^\ast$ as long as the errors have the same mean.[^17] A test based on $\widehat{E}(\Delta m)$ becomes biased if $\Gamma$ is a nonlinear function of $m$, which can arise when the noise is modeled as additive utility noise (see Example 1). Indeed, in Online Appendix B.1 we show that, for the case of EU with additive i.i.d. utility noise, concave utility implies $E(\Delta m) > 0$. Thus a test based on $\widehat{E}(\Delta m)$ would be biased towards rejecting the null of $\Delta m^\ast =0$ in favor of a CRP. Given this potential concern, part (2) of Proposition 2 describes conditions under which we can test the null of $\Delta m^\ast = 0$ using a sign test. This test compares whether the observed proportions of $\Delta m > 0$ and $\Delta m < 0$ are the same. The intuition behind why this approach provides an unbiased test of $\Delta m^* = 0$ is easiest to see when noise is symmetric around zero (i.e., $\varepsilon'=0$). For this case, a person with $m_{AB}^{*} = m_{CD}^{*}$ will exhibit $\Delta m > 0$ if and only if $\varepsilon_{CD}>\varepsilon _{AB}$. The symmetry of the noise then implies that, for any $\bar{\varepsilon}$, it is equally likely to get $\Delta m<0$ due to $\varepsilon_{AB}=\bar{\varepsilon}$ and $\varepsilon_{CD}<\bar{\varepsilon}$ as it is to get $\Delta m>0$ due to $\varepsilon _{AB}=-\bar{\varepsilon}$ and $\varepsilon _{CD}>-\bar{\varepsilon}$. Averaging over all $\bar{\varepsilon}$, it is equally likely to get $\Delta m>0$ and $\Delta m<0$.[^18] Proposition 1 outlines scenarios under which paired choice tasks yield a biased test of the null of $\Delta m^\ast = 0$. In contrast, Proposition 2 outlines scenarios under which paired valuation tasks can yield an unbiased test. Corollary 1 highlights how Assumptions 2a and 2b, which reflect assumptions commonly made in the literature, satisfy both scenarios. ***Corollary 1:*** Under Assumption 2a, paired choice tasks can yield a biased test of $\Delta m^\ast = 0$, whereas paired valuations can yield an unbiased test based on the mean of $\Delta m$. Under Assumption 2b, paired choice tasks can yield a biased test of $\Delta m^\ast = 0$, whereas paired valuations can yield an unbiased test based on the sign of $\Delta m$. ### D Evaluating Prior Experimental Tests Figure 2 depicts the set of theoretical predictions for a population with $\Delta m^\ast =0$ under Assumption 2a together with data from existing experimental tests of the CRE. The theoretical predictions, which we formally derive in Online Appendix B.2, permit heterogeneity in preferences and in the impact of noise. Panel A. Paired Choices ![Figure 2: Predictions and Observations for Prior Experiments (panel 1)](assets/figure2a-3014b4.png) Panel B. Paired Valuations ![Figure 2: Predictions and Observations for Prior Experiments (panel 2)](assets/figure2b-49dc2b.png) **Figure 2: Predictions and Observations for Prior Experiments** Predictions and observations for paired-choice experiments (panel A) and paired-valuation experiments (panel B). In each panel, points below the 45-degree line are combinations that indicate a CRE, while points above the 45-degree line are combinations that indicate an RCRE. The shaded grey region in panel A denotes predicted $(\Pr(A),\Pr(C))$ combinations consistent with $\Delta m^\ast = 0$ under Assumption 2a; the solid grey line in panel B denotes predicted $(E(m_{AB}),E(m_{CD}))$ combinations consistent with $\Delta m^\ast = 0$ under Assumption 2a (see Online Appendix B.2 for derivations). The black circles denote empirical observations from previous experiments. Panel A depicts 143 CRE paired-choice experiments surveyed by Blavatskyy et al. (2023) scaled by the number of observations; panel B depicts six similarly scaled CRE paired-valuation experiments identified by us (see Online Appendix B.3 for details). Panel A focuses on data from paired-choice experiments, where observed behavior in a population is $\widehat{\Pr}(A)$ and $\widehat{\Pr}(C)$. If, for everyone, the impact of noise were the same for both the $AB$ and the $CD$ choices (i.e., $k=1$ in Proposition 1), then the set of predicted $(\Pr(A),\Pr(C))$ combinations consistent with a population in which everyone has $\Delta m^\ast=0$ would be represented by the 45-degree line. Once we allow for the possibility of differential noise, the set of predicted $(\Pr(A),\Pr(C))$ combinations consistent with $\Delta m^*=0$ expands considerably to become the gray shaded region.[^19] Panel B focuses on data from paired-valuation experiments, where observed behavior in a population is $\widehat{E}(m_{AB})$ and $\widehat{E}(m_{CD})$. Under Assumption 2a, the set of predicted $(E(m_{AB}),E(m_{CD}))$ combinations that are consistent with a population in which everyone has $\Delta m^\ast=0$ is represented by the grey bold-faced 45-degree line, even with differential noise. The black circles in panel A depict observed $(\widehat{\Pr}(A),\widehat{\Pr}(C))$ combinations from 143 CRE paired-choice experiments across 39 studies identified in the meta-study by Blavatskyy et al. (2023). The typical test for a CRE in paired choice tasks is to assess whether $\widehat{\Pr}(A)>\widehat{\Pr}(C)$, and panel A reveals that the vast majority of experiments (112 experiments, or 78 percent) exhibit this pattern—hence, the widespread perception that there exists a systematic CRE. At the same time, panel A also reveals that almost all experiments (129 experiments, or 90 percent) fall within the gray area and thus are consistent with $\Delta m^\ast = 0$, i.e., no CRP or RCRP, once one permits the possibility of differential noise.[^20] The black circles in panel B of Figure 2 depict observed $(\widehat{E}(m_{AB}),\widehat{E}(m_{CD}))$ combinations from six CRE paired-valuation experiments across two studies that we identified (see Online Appendix B.3 for details about the data). Panel B demonstrates that the data from paired valuation tasks do not fall far from the 45-degree line. Moreover, comparing panel B to panel A reveals that the literature has tested for a CRE using almost exclusively paired choice tasks. Another notable feature of Figure 2(A) is that more than 75 percent of prior paired-choice experiments found $\widehat{\Pr}(A) \ge 1/2$. Interpreted in terms of Proposition 1, if the noise is indeed more impactful for the CD choices, then the vast majority of prior studies reflect instances where noise would yield a CRE even if there were no underlying CRP. Of course, whether $\widehat{\Pr}(A) >1/2$ in any particular study depends largely on the experimenter’s choice of parameters $M$, $H$, and $p$. In other words, prior studies have focused on parameters that predominantly yield $\widehat{\Pr}(A) >1/2$. A closer look at the data from Blavatskyy et al. (2023) reveals that there has been relatively little variation in experimental parameters in the prior literature. Indeed, 33.6 percent of the experiments in Figure 2(A) use exactly the same combination $(p=0.8,r=0.25)$ that was used in Problems 3 and 4 in Kahneman and Tversky (1979). More notably, the ratio $M/(pH)$ is a natural measure of the likely preference for $A$ versus $B$: The larger this ratio, the more likely it is that participants prefer $A$ over $B$. 115 of the 143 experiments in Figure 2(A) have $M/(pH) \ge 0.75$; among these, the average (sample-weighted) $CRE - RCRE$ is 23.9 percent, compared to only 8.8 percent among the other 28 experiments. Unlike the prior literature, when we collect data on paired choice tasks in stage 2 of our study, we use a broader and more balanced set of experimental parameters. In Section IV.D, we investigate the impact of doing so on what one might conclude from paired choice tasks. ### E Robustness: $h$-Valuation Tasks Our primary analysis focuses on the $m$-valuation tasks described in Sections I.A and I.C in which we fix $(H,p,r)$ and elicit the $m$ that makes people indifferent. We prefer these tasks for two reasons. First, they have a natural bounded domain of $m\in [0,H]$ that is the same for all $p$ and $r$. Second, the $AB$ variants of the $m$-valuation tasks are equivalent to the valuation tasks that researchers typically use to estimate probability weighting functions; thus, one way to validate our approach is to compare our observed $AB$ valuations to what the prior literature has found. However, as a robustness check, we also consider $h$*-valuation tasks* in which we fix $(M,p,r)$ and elicit an $h_{AB} \geq M$ and an $h_{CD} \geq M$ such that $$ \begin{array}{rcl} (M,1) &\sim &(h_{AB},p) \\ \text{and\qquad }(M,r) &\sim &(h_{CD},rp). \end{array} $$ Online Appendix B.4 provides a full development of the theory underlying $h$-valuation tasks that is analogous to the theory developed in Section I.C for $m$-valuation tasks. Here, we highlight two important points that we use in our analysis. First, in terms of the underlying indifference values $(h_{AB}^{*},h_{CD}^{*})$, a CRP implies $h_{AB}^*>h_{CD}^*$. We therefore define $\Delta h^* \equiv h_{AB}^* - h_{CD}^*$ so that, analogous to $\Delta m^* >0$, a CRP is reflected by $\Delta h^*>0$. Our empirical object of interest is thus $\Delta h \equiv h_{AB} - h_{CD}$. Second, for a fixed $(p,r)$, the $m$- and $h$-valuation tasks measure approximately the same preference. Hence, for any given $(p,r)$, there should be a positive correlation between $m_{z}/H$ in an $m$-valuation task and $M/h_{z}$ in the corresponding $h$-valuation task as both measure a proportional risk premium.[^21] We assess this correlation empirically in Section III.B as one way to validate our valuation-task data. ## II Experimental Methodology Our experimental design closely mirrors our theoretical framework and consists of two main stages.[^22] At the beginning of the experiment, each participant is randomly assigned a common-ratio factor $r \in \lbrace 0.2, 0.4, 0.6 \rbrace$. This value remains constant throughout the experiment. In stage 1, participants complete ten paired valuation tasks for a total of 20 valuations. In stage 2, participants complete ten paired choice tasks for a total of 20 binary choices. Each paired choice task corresponds to one of the paired valuation tasks from stage 1. Participants complete all 20 valuations before proceeding to the 20 binary choices, and we randomize the order of questions within each stage. Figure 3 provides a high-level overview of the experiment timeline. ![Figure 3: Experiment Timeline](assets/figure3-d59f4e.png) **Figure 3: Experiment Timeline** ### A Stage 1: Paired Valuation Tasks Table 1 provides an example of one paired $m$-valuation task. Each valuation in the pair consists of a series of choices in a multiple-price-list format. For all $m$-valuations, we fix the larger outcome at $H=\$30$, and we elicit an indifference value $m$ by varying $M$ in \$1 increments from \$0 to \$30. In the $AB$ variant, the left-hand option remains fixed at a $p$ chance of \$30, while the right-hand option is a 100 percent chance of $M$ that starts with $M=\$0$ and increases by \$1 per row. The paired $CD$ variant has the same structure, except we reduce both probabilities by the participant’s common ratio $r$. For each variant, we take the average value of $M$ at the switching rows to be our measure of the realized indifference point, which we denote by $m_{AB}$ or $m_{CD}$. **Table 1: Paired $m$-Valuation Tasks**
Panel A. $AB$ Variant
Option $B$ Option $A$
$1-p$ CHANCE OF \$0, OR 100% CHANCE OF \$0
$p$ CHANCE OF \$30
$1-p$ CHANCE OF \$0, OR 100% CHANCE OF \$1
$p$ CHANCE OF \$30
... ... ...
$1-p$ CHANCE OF \$0, OR 100% CHANCE OF \$30
$p$ CHANCE OF \$30
Panel B. $CD$ Variant
Option $D$ Option $C$
$1-rp$ CHANCE OF \$0, OR $1-r$ CHANCE OF \$0
$rp$ CHANCE OF \$30 $r$ CHANCE OF \$0
$1-rp$ CHANCE OF \$0, OR $1-r$ CHANCE OF \$0
$rp$ CHANCE OF \$30 $r$ CHANCE OF \$1
... ... ...
$1-rp$ CHANCE OF \$0, OR $1-r$ CHANCE OF \$0
$rp$ CHANCE OF \$30 $r$ CHANCE OF \$30
Structure of multiple-price lists for the $AB$ and $CD$ variants of a paired $m$-valuation task. Each participant faces five such pairs, all with the same $r$, but with five different values for $p$. For the $h$-valuations, we fix the smaller outcome and elicit the larger outcome as the indifference point. Specifically, we fix the value of the smaller outcome at $M=\$(p\cdot30)$, and we elicit an indifference value $h$ by varying $H$ in \$1 increments from \$$(p\cdot30)$ to $\$(p\cdot30+30)$. For instance, for $p=0.2$, we fix $M=\$6$ and vary $H$ from \$6 to \$36. The left-hand option is again fixed: It offers \$$(p\cdot30)$ for sure in the $AB$ variant and an $r$ chance of \$$(p\cdot30)$ in the $CD$ variant. The right-hand option starts with an outcome of \$$(p\cdot30)$ and increases by \$1 per row. For each variant, we take the average value of $H$ at the switching rows to be our measure of the realized indifference point, which we denote by $h_{AB}$ or $h_{CD}$. For each price list, we enforce a unique switching point—which naturally corresponds to our theoretical framework in which individuals reveal a unique indifference point. For convenience, when participants click a row in the left panel, it highlights the left-hand option in that row and all rows above. Analogously, when they click a row in the right panel, it highlights the right-hand option in that row and all rows below. They can adjust their choices as much as they want before submitting their final choices for that valuation. Online Appendix Figures C.1 to C.5 provide example screenshots of the $m$- and $h$-valuations for both the $AB$ and $CD$ variants. We elicit valuations for five different probabilities, $p \in \{0.1,0.2,0.5,0.8,0.9\}$. For each $p$, participants complete the $AB$ and $CD$ variants for both the $m$- and $h$-valuation tasks. Participants therefore complete a total of 20 valuations in stage 1: five probabilities $(0.1,0.2,0.5,0.8,0.9)$ $\times$ two variants ($AB$ and $CD$) $\times$ two types of valuation tasks ($m$ and $h$). Thus, for each participant, we elicit the realized indifference points $(m_{AB},m_{CD},h_{AB},h_{CD})$ for the five different levels of $p$. We randomize the order in which participants complete these 20 valuations. Hence, while the valuations are paired from our perspective, there is no obvious sense in which they are paired from the participants’ perspective.[^23] ### B Stage 2: Paired Choice Tasks For each of the ten paired valuation tasks from stage 1, each participant faces a corresponding paired choice task that isolates one specific row from stage 1. Table 2 provides an overview of the ten $AB$ binary choices that participants see in stage 2. For the $m$-choices (in panel A), the larger outcome is always $H=\$30$ as in the corresponding price list, and we randomly draw a value for $M$ corresponding to a randomly selected row of the price list. Each participant sees one $AB$ variant for each value of $p$; and, for each value of $p$, we randomly draw a value of $M$ from the values listed in the table. For the $h$-choices, the smaller outcome is always $M=\$(p\cdot30)$, and we randomly draw a value for $H$. Again, each participant sees one $AB$ variant for each value of $p$, and for each, we randomly draw a value of $H$ from the values listed in the table. We chose the possible values for $M$ and $H$ based on pilot data, with the aim that one extreme would yield a majority choosing lottery $A$ while the other extreme would yield a majority choosing lottery $B$.[^24] For each of the ten $AB$ variants shown in Table 2, the participant also sees the paired $CD$ variant in which we hold fixed the values for $M$ and $H$ but scale down the probabilities by that participant’s common ratio $r$. Thus, each participant makes a total of 20 binary choices: five probabilities $(0.1,0.2,0.5,0.8,0.9)$ $\times$ two variants ($AB$ and $CD$) $\times$ two choice tasks ($m$ and $h$). We randomize the order in which participants see the 20 binary choices and the relative position on the screen (left or right) of the two options within each binary choice. Online Appendix Figures C6 to C10 provide example screenshots of the $m$- and $h$-choices for both the $AB$ and $CD$ variants.[^25] **Table 2: Summary of Binary Choices**
Panel A. $m$-Choices
100% chance of $M\in \left\lbrace\$1,\$3,\$5,\$8\right\rbrace$ $\;$ or $\;$ 10% chance of \$30
100% chance of $M\in \left\lbrace\$1,\$4,\$7,\$10\right\rbrace$ $\;$ or $\;$ 20% chance of \$30
100% chance of $M\in \left\lbrace\$5,\$8,\$11,\$14\right\rbrace$ $\;$ or $\;$ 50% chance of \$30
100% chance of $M\in \left\lbrace\$8,\$12,\$16,\$20\right\rbrace$ $\;$ or $\;$ 80% chance of \$30
100% chance of $M\in \left\lbrace\$10,\$14,\$18,\$22\right\rbrace$ $\;$ or $\;$ 90% chance of \$30
Panel B. $h$-Choices
100% chance of \$3 $\;$ or $\;$ 10% chance of $H\in \left\lbrace\$30, \$25,\$20,\$13\right\rbrace$
100% chance of \$6 $\;$ or $\;$ 20% chance of $H\in \left\lbrace\$35, \$30,\$25,\$20\right\rbrace$
100% chance of \$15 $\;$ or $\;$ 50% chance of $H\in \left\lbrace\$45, \$40,\$35,\$30\right\rbrace$
100% chance of \$24 $\;$ or $\;$ 80% chance of $H\in \left\lbrace\$52, \$45,\$38,\$33\right\rbrace$
100% chance of \$27 $\;$ or $\;$ 90% chance of $H\in \left\lbrace\$54, \$47,\$40,\$35\right\rbrace$
Summary of all possible $AB$ variants of the $m$- and $h$-choices. A participant faces one binary choice from each row, where we randomly draw a value of $M$ for each row in panel A and a value of $H$ for each row in panel B. The $CD$ variant of each row keeps the same $M$ and $H$ values but scales all probabilities down by the participant’s common ratio $r$. ### C Additional Design Details Before beginning stage 1 of the experiment, participants complete an unincentivized attention check and quiz about the payment mechanism. After stages 1 and 2 of the experiment, participants complete two incentivized comprehension checks to gauge their understanding of the multiple-price-list format and the binary-choice tasks. The first comprehension check tests whether individuals can correctly fill out a price list given a specified indifference value. The second comprehension check tests whether participants can correctly answer a binary-choice question when given another person’s responses to a price list. Online Appendix Figures C.11 and C.12 provide example screenshots of these comprehension checks.[^26] To break up the tasks and reduce fatigue, we present participants with an unincentivized visual puzzle after every fifth question in both stages of the experiment. Online Appendix Figure C.13 provides an example. ### D Recruiting We recruited 900 participants through Prolific who had at least a high school education, were between the ages of 18 and 30, were living in the United States or Western Europe, and had a high approval rating on Prolific (see Online Appendix Table D.1 for summary statistics about participants). We focused on this sample for comparison to prior common-ratio studies, the bulk of which have used undergraduate samples in the United States and Western Europe. We recruited an equal number of male and female participants.[^27] The experiment took place in August 2021. Participants received a \$5 payment upon completion. We also randomly selected one in five participants to receive an additional bonus payment based on their decisions in the study. Each of the 42 questions (20 valuations, 20 binary choices, and two incentivized comprehension checks) was equally likely to determine the bonus payment amount. If we randomly selected a valuation, then we randomly selected one row of the price list and paid the participant based on the option they selected in that row. If we randomly selected a binary choice, then we paid the participant based on the option they selected. If we randomly selected a comprehension check, then we paid the participant \$5 if they answered correctly. The experiment took 27 minutes to complete on average, and participants earned an average total payment of \$6.51.[^28] ## III Analysis of Paired Valuation Tasks Stage 1 of the experiment implements the paired valuation tasks needed to conduct our proposed valuations-based tests for a CRP. Our primary focus is an analysis of the $m$-valuations; however, we also use the $h$-valuations as a robustness check and to validate our approach. ### A Main Results Figure 4 provides an initial visualization of our data for both $m$-valuations (in panel A) and $h$-valuations (in panel B). This figure is analogous to panel B of Figure 2; each dot denotes the mean valuations for the $AB$ task and the $CD$ task for a fixed $(p,r)$. For paired $m$-valuation tasks, a CRE corresponds to $m_{CD}>m_{AB}$, and is therefore consistent with observations below the 45-degree line. For paired $h$-valuation tasks, a CRE corresponds to $h_{AB}>h_{CD}$, and thus, given the change in axes, is again consistent with observations below the 45-degree line. Figure 4 reveals little evidence of a systematic CRE. Panel A. Paired $m$-valuations ![Figure 4: Mean Valuations for Paired Valuation Tasks (panel 1)](assets/figure4a-971cb6.png) Panel B. Paired $h$-valuations ![Figure 4: Mean Valuations for Paired Valuation Tasks (panel 2)](assets/figure4b-f79e68.png) **Figure 4: Mean Valuations for Paired Valuation Tasks** Mean valuations for each of the 15 $(p,r)$ combinations for the paired $m$-valuation task (panel A) and the paired $h$-valuation task (panel B). In each panel, points below the 45-degree line are combinations that indicate a CRE, while points above the 45-degree line are combinations that indicate an RCRE. More formally, we focus on $m$-valuations and conduct the two tests developed in Section I.C based on Proposition 2. For both, we focus on $\Delta m\equiv m_{CD}-m_{AB}$ at the individual level, and test the null of $\Delta m^{\ast}=0$. Consider first the test based on the mean of $\Delta m$, which is valid under Assumption 2a. Columns (2) and (3) of Table 3 present the mean value of $\Delta m$ along with the $p$-value for the corresponding means test. Out of the 15 means tests, we reject the null hypothesis of $\Delta m^{\ast}=0$ in eight comparisons at the 5 percent level. All eight rejections indicate an RCRP rather than a CRP. Moreover, even the statistically significant means are relatively small in magnitude. (See Online Appendix Table D.2 for complete summary statistics on the $m$-valuations.) As discussed in Section I.C, the means test may be biased if the function $\Gamma$ in Assumption 1 is nonlinear; hence, we also consider a test based on the sign of $\Delta m$, which is appropriate under Assumption 2b. Columns (4)–(6) report the raw frequency data: For each combination of $r$ and $p$, the table reports the number of participants who exhibit $\Delta m>0$ (consistent with CRP), $\Delta m<0$ (consistent with RCRP), and $\Delta m=0$. Column (7) reports the $p$-value from the two-sided sign test that we proposed in Section I.C (see footnote 18). Out of the 15 sign tests, we find seven significant deviations from the null of equal proportions at the 5 percent level. Six of these are consistent with an RCRP, and there is only one test in which the deviation from equal proportions is in the direction consistent with a CRP. Beyond the formal sign test, we also note that in 14 of 15 cases, the median value of $\Delta m$ shown in column (8) is zero, indicating a strong central tendency toward $\Delta m = 0$. **Table 3: Testing the Null Hypothesis of $\Delta m^{\ast}=0$**
(1) (2) (3) (4) (5) (6) (7) (8)
Number of Cases
4-6 $\Delta m$ Means Test $\Delta m>0$ $\Delta m=0$ $\Delta m<0$ Sign Test $\Delta m$
(Mean) ($p$-value) $(CRP)$ $(RCRP)$ ($p$-value) (Median)
Panel A. $r=0.2$
$p = 0.1$ $-$1.55 0.000 79 75 144$^\dagger$ 0.000 0
$p = 0.2$ $-$1.29 0.003 80 73 145$^\dagger$ 0.000 0
$p = 0.5$ 0.04 0.932 123 60 115 0.650 0
$p = 0.8$ 1.00 0.052 140$^\dagger$ 54 104 0.025 0
$p = 0.9$ $-$1.47 0.014 127 42 129 0.950 0
Panel B. $r=0.4$
$p = 0.1$ $-$0.63 0.152 103 71 129 0.101 0
$p = 0.2$ $-$1.14 0.003 97 65 141$^\dagger$ 0.005 0
$p = 0.5$ $-$1.22 0.007 104 62 137$^\dagger$ 0.039 0
$p = 0.8$ $-$0.60 0.262 127 41 135 0.665 0
$p = 0.9$ $-$0.16 0.782 124 52 127 0.900 0
Panel C. $r=0.6$
$p = 0.1$ $-$0.49 0.158 94 90 115 0.166 0
$p = 0.2$ 0.14 0.692 111 84 104 0.682 0
$p = 0.5$ $-$2.05 0.000 89 65 145$^\dagger$ 0.000 0
$p = 0.8$ $-$1.26 0.008 113 57 129 0.335 0
$p = 0.9$ $-$2.03 0.000 79 60 160$^\dagger$ 0.000 $-$1
Means test and sign test for paired $m$-valuations for all 15 combinations of $(p,r)$. $\Delta m$ denotes the difference between the $CD$ and $AB$ $m$-valuations. We conduct a two-sided t-test for the difference in means. We also conduct a two-sided sign test, where we exclude all ties (instances of $\Delta m = 0$). A $\dagger$ indicates the larger group when the sign test rejects the null of equal proportions at the 5 percent level. That the sign test sometimes rejects the null of equal proportions even though the median of $\Delta m$ is zero is partly because there are many observations of $\Delta m = 0$. In conducting the sign tests in Table 3, we adopt the conventional approach of ignoring these ties—that is, we exclude all $\Delta m=0$ observations from our calculations. Including these ties can lead to changes in the $p$-values; however, there are multiple ways to incorporate ties, and there is no agreement in the literature about which is best (for discussions, see (Coakley and Heise 1996) and (Randles 2001)). Here, we discuss two approaches that offer some bounds on our results. First, we could split the ties evenly between $\Delta m>0$ and $\Delta m<0$. This approach would increase all $p$-values and thus make it less likely that we reject the null of equal proportions. Second, we could split the ties between $\Delta m>0$ and $\Delta m<0$ using the same proportions we observe in the non-ties. This approach would decrease all the $p$-values and thus make it more likely that we reject the null of equal proportions. In Online Appendix Table D.3, we present the sign-test results using both approaches and show that the overall message is almost identical. **Result 1**. *We find no evidence of systematic common ratio preferences.* A possible challenge to Result 1 is that a price-list effect might mask an underlying CRP. Indeed, one of the most common criticisms of using multiple-price lists to elicit valuations is that valuations may get pulled to the center of the list. In principle, asymmetric pull-to-the-center effects could obfuscate an underlying CRP. For instance, let $m^o$ denote the row to which these effects pull valuations. If $m_{AB}^\ast < m_{CD}^\ast < m^o$ (e.g., for low values of $p$) but the pull effect is stronger for the elicited $m_{AB}$ than the $m_{CD}$, then we might observe $m_{AB}=m_{CD}$ despite there being an underlying CRP. There are several reasons to believe such list effects do not drive Result 1. First and foremost, we document in Section IV.D that we do not observe a systematic CRE in our paired choice tasks when we use a broader and more balanced set of parameters relative to the prior literature. In other words, our stage 2 choice data generate a conclusion akin to Result 1, although the impact of noise on paired choice tasks complicates inference. Second, there is no apparent reason for the pull-to-the-center effect to take the asymmetric form necessary to generate Result 1 despite an underlying CRP. In addition, for such asymmetric pull-to-the-center effects to mask an underlying CRP, the asymmetry would need to be in the opposite direction—i.e., stronger for the elicited $m_{CD}$—for high $p$, where an underlying CRP would manifest as $m^o < m_{AB}^\ast < m_{CD}^\ast$. There is no apparent reason to assume that this is the case. More generally, the strong link between participants’ stage 1 valuations and their stage 2 choices that we document in Section IV is further evidence against this (or some other) list effect obfuscating an underlying CRP. A second possible challenge to Result 1 is whether the absence of a CRE in our valuation data is due to our choosing experimental parameter values for which models featuring a CRP would predict very small magnitudes for $\Delta m$. To assess this issue, we interpret our data within the prospect theory (PT) structure from Example 1. Figure 5 presents the mean $m$-valuations in a different way from Figure 4(A): For each of the five values of $p$, blue dots denote the mean $m_{AB}$ valuations and red diamonds denote the mean $m_{CD}$ valuations. The three panels separate results by the three different values for $r$. We then use the fact that our $AB$ valuation tasks are identical to those that researchers frequently use to estimate a PT probability weighting function. Following that literature, we use participants’ five $m_{AB}$ values to estimate a probability weighting function for each value of $r$. Online Appendix E.1 provides the details of this structural estimation and reports the parameter estimates; the $AB$ valuations that these estimates predict are depicted in Figure 5 by the dashed blue lines. In each case, the estimated probability weighting function takes the familiar inverse-S shape that the prior literature has typically found—e.g., the dashed blue lines all look quite similar to Figure 2 in the Barberis (2013) review of the prospect-theory literature.[^29] Panel A. $r=0.2$ ![Figure 5: Mean Valuations by Probability p and Common Ratio r (panel 1)](assets/figure5a-fcc913.png) Panel B. $r=0.4$ ![Figure 5: Mean Valuations by Probability p and Common Ratio r (panel 2)](assets/figure5b-0e83bd.png) Panel C. $r=0.6$ ![Figure 5: Mean Valuations by Probability p and Common Ratio r (panel 3)](assets/figure5c-4f3138.png) **Figure 5: Mean Valuations by Probability $p$ and Common Ratio $r$** Mean valuations by probability $p$ and common ratio $r$. The blue dots denote mean $AB$ valuations, and the red diamonds denote mean $CD$ valuations. The dashed blue lines represent the $AB$ valuations predicted by PT with parameters estimated from the $AB$-valuation data. The dashed-and-dotted red lines denote the $CD$ valuations predicted by PT given the parameter estimates represented by the blue lines. We next use the probability weighting functions estimated from the $AB$ valuations to predict the $CD$ valuations. The red dashed-and-dotted lines in Figure 5 depict these predictions. The difference between the two lines reflects the predicted magnitude of CRP, $\Delta m$, given the observed $AB$ valuations. Figure 5 reveals that, under PT, we should see substantial positive values of $\Delta m$; across the 15 combinations of $p$ and $r$, the predicted $\Delta m$ ranges from 3.15 to 8.63, with an average of 6.30.[^30] Hence, we observe small $\Delta m$ values in the data *despite* the fact that PT predicts much larger differences. A further implication of Figure 5 is that our data are inconsistent with models of probability weighting in the tradition of Kahneman and Tversky’s (1979, 1992) prospect theory. In such models, a CRP (or subproportionality) is perhaps the central motivating fact behind the structure of probability weighting. Our failure to find a systematic CRP is thus problematic for such models. Indeed, our discussion above highlights how a model in which individuals apply a single probability weighting function to both types of comparisons cannot explain our combined $m_{AB}$ and $m_{CD}$ data.[^31] ### B Validation and Robustness Result 1 stands in stark contrast to the widely accepted belief that there exists a systematic CRP. Hence, it is important to validate that our valuation measures reflect underlying preferences, and that our finding of $\Delta m \approx 0$ is not an artifact of our experimental task. We outline several features of our data that alleviate this concern. First, we reiterate that our $m_{AB}$ valuation tasks are identical to the valuation tasks used in the extensive literature that estimates probability weighting functions, and our data yield probability weighting functions that look very similar to those estimated in the literature (as seen in Figure 5 and discussed in footnote 29). It is therefore reassuring that part of our task aligns so closely with the broader literature on probability weighting, and supports our use of valuation data to test for CRP. Second, participants’ valuations respond sensibly to changes in $p$ and hence the expected value of the lottery: Increases in $p$ lead participants to report higher valuations. We see this pattern even though we present 20 tasks in random order that differ along multiple dimensions: an $m$- or $h$-valuation, an $AB$ or $CD$ variant, and the value of $p$. Third, we designed our $h$-valuation tasks for both validation and robustness. In terms of robustness, the $h$-valuation tasks present a different choice structure to participants, yet permit analogous means and sign tests. Table 4 reports the results for the $h$-valuation tasks, and they are similar to the results for the $m$-valuation tasks in Table 3 (see Online Appendix Table D.5 for complete summary statistics on the $h$-valuations). While the mean $\Delta h$ is often significantly different from zero, and the magnitudes are slightly larger than those for the $m$-valuation tasks, there are roughly equal numbers of positive and negative instances of $\Delta h$. Out of the 15 sign tests, we find nine significant deviations from the null of equal proportions at the 5 percent level: Five are consistent with an RCRP, and four are consistent with a CRP.[^32] Finally, the median $\Delta h$ is zero in 12 of the 15 cases. Hence, Table 4 reaffirms Result 1: There is no evidence of a systematic CRP in valuations.[^33] **Table 4: Testing the Null Hypothesis of $\Delta h^{\ast} = 0$**
(1) (2) (3) (4) (5) (6) (7) (8)
Number of Cases
4-6 $\Delta h$ Mean Test $\Delta h>0$ $\Delta h=0$ $\Delta h<0$ Sign Test $\Delta h$
(Mean) ($p$-value) $(CRP)$ $(RCRP)$ ($p$-value) (Median)
Panel A. $r=0.2$
$p = 0.1$ $-$1.67 0.006 100 60 138$^\dagger$ 0.016 0
$p = 0.2$ $-$1.94 0.001 94 53 151$^\dagger$ 0.000 $-$1
$p = 0.5$ 1.64 0.001 136$^\dagger$ 81 81 0.000 0
$p = 0.8$ 4.31 0.000 174$^\dagger$ 45 79 0.000 3
$p = 0.9$ 2.11 0.000 143$^\dagger$ 64 91 0.001 0
Panel B. $r=0.4$
$p = 0.1$ $-$2.53 0.000 82 59 162$^\dagger$ 0.000 $-$1
$p = 0.2$ $-$1.59 0.002 92 65 146$^\dagger$ 0.001 0
$p = 0.5$ $-$1.05 0.036 101 70 132$^\dagger$ 0.049 0
$p = 0.8$ 1.84 0.002 148$^\dagger$ 47 108 0.015 0
$p = 0.9$ 1.13 0.055 138 47 118 0.235 0
Panel C. $r=0.6$
$p = 0.1$ $-$0.73 0.192 100 71 128 0.074 0
$p = 0.2$ 0.83 0.108 131 65 103 0.077 0
$p = 0.5$ $-$0.72 0.130 93 85 121 0.065 0
$p = 0.8$ 0.84 0.146 136 47 116 0.231 0
$p = 0.9$ 0.76 0.173 126 54 119 0.702 0
Means test and sign test for paired $h$-valuations for all 15 combinations of $(p,r)$. $\Delta h$ denotes the difference between the $AB$ and $CD$ $h$-valuations. We conduct a two-sided t-test for the difference in means. We also conduct a two-sided sign test, where we exclude all ties (instances of $\Delta h = 0$). A $\dagger$ indicates the larger group when the sign test rejects the null of equal proportions at the 5 percent level. We also use the $h$-valuation data to validate the $m$-valuation data. As described in Section I.E, the two valuations approximately measure the same preference, and thus the proportional risk premium $m_{z}/H$ for an $m$-valuation should be strongly correlated with the proportional risk premium $M/h_{z}$ for the corresponding $h$-valuation. In Online Appendix Table D.7, we report the rank correlations between $m_{AB}/H$ and $M/h_{AB}$ and the rank correlations between $m_{CD}/H$ and $M/h_{CD}$ for each of the 15 combinations of $(p,r)$. All 30 rank correlations are significantly positive at the 5 percent level, and the average is 0.28. Overall, these positive correlations confirm that our valuations capture meaningful information about underlying preferences and are not merely driven by some artifact of our experimental task.[^34] ### C Heterogeneous Preferences and Noise in Stage 1 Data Our aggregate tests provide no evidence of a systematic CRP. However, we also observe significant variation in participants’ stage 1 responses. This variation could merely reflect the impact of choice noise; indeed, the premise of our analysis is that choice noise can generate idiosyncratic variation in responses around underlying values. However, this variation could also be due to heterogeneity in participants’ underlying preferences. There are two relevant forms of heterogeneity in preferences that we investigate. First, there is heterogeneity in the degree of risk aversion, as reflected in the levels of participants’ $m_{AB}^*$ and $m_{CD}^*$. Second, there is heterogeneity in the degree to which people have an underlying CRP or RCRP, as reflected by heterogeneity in $\Delta m^*$. We begin by assessing the degree of heterogeneity in risk aversion. For each value of $p$, our stage 1 data contain two valuations that reflect an individual’s risk aversion: $m_{AB}$ and $m_{CD}$. As a benchmark, risk neutrality would lead a person to state $m_{AB}=m_{CD}=pH$. Hence, we define $\bar{m}\equiv (m_{AB}+m_{CD})/2$ and then use $pH-\bar{m}$ as a measure of a person’s risk aversion. With this measure, positive values reflect risk aversion and negative values reflect risk seeking. Panel A of Figure 6 presents the distribution of the observed $pH-\bar{m}$ across all 15 combinations of $(p,r)$. There is substantial variation in the magnitude of $pH-\bar{m}$. While some of this variation is surely due to noise, there is also evidence that some of it is due to heterogeneity in participants’ underlying $\bar{m}^*$. First, in Section III.B, we highlighted the strong rank correlations between participants’ $m_x / H$ and $M/h_x$ for the same $(p,r)$, which indicate the existence of heterogeneity in underlying preferences. Second, panel A of Online Appendix Table D.8 documents substantial rank correlations of $pH-\bar{m}$ across different values of $p$: Across the 30 possible combinations of $(p,p',r)$, all rank correlations are positive, ranging from 0.06 to 0.61, with an average of 0.31.[^35] Panel A. Risk Premia ![Figure 6: Distributions of Risk Premia and Value Differences (panel 1)](assets/figure6a-69d87f.png) Panel B. Value Differences ![Figure 6: Distributions of Risk Premia and Value Differences (panel 2)](assets/figure6b-aa6225.png) **Figure 6: Distributions of Risk Premia and Value Differences** Distributions of risk premia (panel A) and value differences (panel B) in $m$-valuations. The risk premium is $pH-\bar{m}$ where $\bar{m} = (m_{AB}+m_{CD})/2$ and $H=30$. Positive risk premia indicate risk aversion, and negative risk premia indicate risk seeking. The value difference is $\Delta m = m_{CD} - m_{AB}$. Positive value differences indicate CRP, and negative value differences indicate RCRP. For each panel, the data include all 15 combinations of $(p,r)$. We next assess the degree of heterogeneity in underlying CRP versus RCRP. Panel B of Figure 6 presents the distribution of the observed $\Delta m$ across all 15 combinations of $(p,r)$. Again, there is substantial variation, and while some of it is undoubtedly due to noise, there is also evidence that some of it is due to heterogeneity in participants’ underlying $\Delta m^*$. First, analogous to our assessment of the rank correlations between $m_x / H$ and $M/h_x$ for the same $(p,r)$, we study rank correlations between the differences $m_{CD}/H-m_{AB}/H$ and $M/h_{CD}-M/h_{AB}$. Online Appendix Table D.9 reports this rank correlation for all 15 combinations of $(p,r)$. Twelve are significantly positive at the 5 percent level, and the average across all 15 rank correlations is 0.15. Second, panel B of Online Appendix Table D.8 documents substantial rank correlations of $\Delta m$ across different values of $p$: Across the 30 possible combinations of $(p,p',r)$, ten are significantly positive at the 5 percent level, and none are significantly negative. The average across all 30 rank correlations is 0.11.[^36] Hence, while we fail to find a systematic CRP in the aggregate, the variation in participants’ stage 1 valuations reflects a combination of idiosyncratic noise and systematic heterogeneity in underlying preferences. Heterogeneity in both the levels of risk attitudes and underlying CRP or RCRP drives the positive correlations observed between measures. Choice noise attenuates these correlations toward zero, even for values that should measure the same construct. Recognizing both heterogeneity in preferences and a substantial role for noise will have important implications for our stage 2 paired choice tasks. We turn to these connections next. ## IV Analysis of Paired Choice Tasks This section analyzes empirical results from our paired choice tasks, making explicit the connections between the valuations elicited in stage 1 and the corresponding choices made in stage 2. Doing so allows us to assess whether there is differential noise across the $AB$ and $CD$ choices and to reconcile our main finding of no systematic CRP in paired valuation tasks with the vast literature that finds a CRE in paired choice tasks. ### A Connections Between Paired Valuations and Paired Choices Under Assumption 1 from Section I.C, the same underlying preferences drive behavior for both paired valuation tasks and paired choice tasks, and thus there should be a strong connection between the two. To illustrate, consider the special case of Assumption 2a where a person with underlying indifference values $(m_{AB}^{\ast },m_{CD}^{\ast })$ would choose $A$ over $B$ when $M\geq m_{AB}^{\ast}+\varepsilon _{AB}$ and would choose $C$ over $D$ when $M\geq m_{CD}^{\ast}+\varepsilon _{CD}$, where $\varepsilon _{CD}\,{\buildrel d \over =}\,k\varepsilon_{AB}$. For this case, the probability of making CRE choices ($A$ and $D$) minus the probability of making RCRE choices ($B$ and $C$) is $$ \begin{array}{rcl} CRE-RCRE & \equiv & \Pr (A)-\Pr (C) \\ & = & \Pr\left(\varepsilon_{AB}<(M-m_{AB}^{\ast})\right) - \Pr\left(\varepsilon_{CD}<(M-m_{CD}^{\ast})\right) \\ & = & \Pr\left(\varepsilon_{AB}<(M-m_{AB}^{\ast})\right) - \Pr\left(\varepsilon_{AB} <\frac{1}{k}(M-m_{CD}^{\ast })\right)\text{.} \end{array} $$ Defining $\Psi \equiv (M-m_{CD}^{\ast })/k$, and substituting $\bar{m}^{\ast }\equiv (m_{AB}^{\ast}+m_{CD}^{\ast })/2$ and $\Delta m^{\ast }\equiv m_{CD}^{\ast }-m_{AB}^{\ast }$, we can rewrite this as: $$ \begin{multline} CRE-RCRE = \\ \Pr\left(\varepsilon_{AB} < \Psi + 0.5\left(1+\frac{1}{k}\right)\Delta m^{\ast} + \left(1-\frac{1}{k}\right)(M-\bar{m}^{\ast })\right) - \Pr\left(\varepsilon_{AB} < \Psi\right). \end{multline} $$ This formulation links the extent of $CRE-RCRE$ in a paired choice task to two terms: (i) a *scaled value difference* term $0.5(1+1/k) \Delta m^{\ast}$, and (ii) a *scaled distance to indifference* term $(1-1/k)(M-\bar{m}^{\ast })$. To illustrate the implications of equation (3), Figure 7 presents the predicted $CRE-RCRE$ as a function of the value difference term (panel A) and the distance to indifference term (panel B) when $\varepsilon_{AB}$ is distributed $N(0,7^2)$ and $k =2.5$, so that the noise is more impactful for the $CD$ choice. Panel A. Value Difference ![Figure 7: Predicted CRE − RCRE by Value Difference and Distance to Indifference (panel 1)](assets/figure7a-5330d1.png) Panel B. Distance to Indifference ![Figure 7: Predicted CRE − RCRE by Value Difference and Distance to Indifference (panel 2)](assets/figure7b-72b58e.png) **Figure 7: Predicted $CRE-RCRE$ by Value Difference and Distance to Indifference** Predicted $CRE-RCRE$ as a function of the value difference $0.5(1+1/k)\Delta m^*$ (panel A) or distance to indifference $(1-1/k)(M-\bar{m}^*)$ (panel B). Panel A depicts predicted $CRE-RCRE$ as a function of value difference for three cases: (i) a positive distance to indifference (green dashed), (ii) a negative distance to indifference (red dotted), and (iii) a zero distance to indifference (black solid). Panel B depicts predicted $CRE-RCRE$ as a function of distance to indifference for three cases: (i) a positive ($CRP$) value difference (green dashed), (ii) a negative ($RCRP$) value difference (red dotted), and (iii) a zero value difference (black solid). Both panels assume $\varepsilon_{AB} \sim N(0,7^2)$ and $k=2.5$. The value difference captures the strength of a person’s underlying CRP (when $\Delta m^\ast>0$) or RCRP (when $\Delta m^\ast<0$). The impact of the value difference on $CRE-RCRE$ seen in panel A therefore reflects the impact of underlying preferences on choice-based measures of the CRE. Indeed, in taking empirically observed instances of $CRE-RCRE>0$ in paired choice tasks as evidence of an underlying CRP, the prior literature has effectively assumed that this effect is the dominant effect. Panel A highlights that the relationship between $CRE-RCRE$ and the value difference is straightforward: The larger the value difference, the larger $CRE-RCRE$. The distance to indifference has its most natural interpretation under EU, when $m^\ast_{AB} = m^\ast_{CD} \equiv m^\ast$ and thus $\bar{m}^\ast = m^\ast$. Under EU, the magnitude of the distance to indifference captures the strength of a person’s underlying EU preference for $A$ and $C$ (when $M-m^\ast > 0$) or for $B$ and $D$ (when $M-m^\ast < 0$). The solid line in panel B illustrates the predictions highlighted in Section I.B: When EU preferences favor $A$ and $C$, noise generates a prediction of $CRE-RCRE>0$, and when EU preferences favor $B$ and $D$, noise generates a prediction of $CRE-RCRE<0$. The dashed and dotted lines in panel B illustrate that the same pattern holds even for people with $\Delta m^* \neq 0$, where equation 3 establishes that $M-\bar{m}^\ast$ is the natural analogue for $M-m^\ast$. Hence, an individual with an underlying RCRP may exhibit a CRE if the offered $M$ implies a large and positive distance to indifference, and an individual with an underlying CRP may exhibit an RCRE if the offered $M$ implies a large and negative distance to indifference.[^37] The scaling factors in equation (3) depend on $k$, which captures the extent of differential noise across $AB$ versus $CD$ choices. When $k=1$, there is no differential noise impact, and the scaled distance to indifference term disappears—that is, the person will exhibit a CRE if and only if they have an underlying CRP. In contrast, when there is differential noise across the $AB$ and $CD$ choices ($k \ne 1$), the scaled distance to indifference will impact whether people exhibit a CRE or an RCRE. Our empirical analysis in Sections IV.B and IV.C will provide clear evidence that the distance to indifference has a positive impact on $CRE-RCRE$, implying that $k>1$ (as depicted in Figure 7). In other words, we find evidence that noise has a larger impact on the $CD$ choices than on the $AB$ choices. In the following subsections, we test these predictions relating the value difference and the distance to indifference to $CRE-RCRE$. We emphasize three points as we transition from theoretical predictions to empirical results. First, we have framed the predictions above in terms of a person’s underlying value difference $\Delta m^\ast$ and distance to indifference $\bar{m}^\ast$, both of which are unobserved. In our empirical analysis, we replace these quantities with their empirical counterparts $\Delta m$ and $\bar{m}$, recognizing that there is measurement error and attempting to correct for it when possible. Second, our empirical analysis combines data for the three different values of $r$ to increase the power of our statistical tests. However, $r$ may impact the extent of differential noise (i.e., $k$)—e.g., EU with additive utility noise implies $k=1/r$. Moreover, a change in $k$ would change the slopes in Figure 7, where a $k$ closer to one would yield that Figure 7(A) is steeper while Figure 7(B) is flatter (in the range around $M-\bar{m}^\ast=0$). To account for this, and motivated by the EU case, our empirical analysis uses $0.5(1+r) \Delta m$ for the scaled value difference and $(1-r)(M-\bar{m})$ for the scaled distance to indifference.[^38] Finally, when we connect people’s stage 1 valuations to their stage 2 choices, we assume that both reflect the same underlying preferences as stated in Assumption 1. However, we do not impose that the error distributions for valuations are the same as those for choices. Our analysis considers stage 1 valuations as noisy measures of underlying preferences, which motivates an instrumental variables approach. However, this approach does not require consistency between the noise in valuations and noise in the choices. ### B Individual-Level Stage 2 Data We first analyze stage 2 data at the individual level. As in Section III, we focus on the paired $m$-choice tasks and use the paired $h$-choice tasks as a robustness check. There are 60 different paired $m$-choice tasks, which reflect different combinations of $p$, $r$, and $M$. Each participant faces five of these, one for each value of $p$. Consider a graphical analysis of stage 2 behavior analogous to the theoretical predictions in Figure 7. Figure 8 plots the relationship between observed $CRE-RCRE$ and the scaled value difference (panel A) and the scaled distance to indifference (panel B). We create 20 equally sized bins of the value difference and the distance to indifference, respectively, and report the average $CRE-RCRE$ within each bin. Panel A. Value Difference ![Figure 8: Stage 2 Individual-Level CRE − RCRE (panel 1)](assets/figure8a-6401c3.png) Panel B. Distance to Indifference ![Figure 8: Stage 2 Individual-Level CRE − RCRE (panel 2)](assets/figure8b-8e8cd1.png) **Figure 8: Stage 2 Individual-Level $CRE-RCRE$** Stage 2 individual-level $CRE-RCRE$ as a function of individual-level stage 1 value difference (panel A) and distance to indifference (panel B). For each, we divide the stage 1 individual-level data into 20 equally sized bins, and the dots in each graph denote the average $CRE-RCRE$ for a given bin (we join four value difference bins together at zero due to the large number of observations at this point). The solid lines represent the best-fitting linear prediction for $CRE-RCRE$ as a function of the value difference (panel A) or the distance to indifference (panel B). Panel A of Figure 8 shows a tight connection between individuals’ stage 1 value differences and their stage 2 choices: Consistent with value differences capturing people’s underlying CRP or RCRP, individuals with more positive value differences are more likely to exhibit a CRE, and those with more negative value differences are more likely to exhibit an RCRE. Panel B shows that there is also a clear relationship between an individual’s distance to indifference and whether we observe a CRE or an RCRE. As predicted by models with noise that is more impactful for the $CD$ choice, individuals with greater distances to indifference are more likely to exhibit a CRE relative to an RCRE. To provide a quantitative assessment of these relationships, Table 5 reports the results from linear regressions using the outcome variable $CRE-RCRE \in \lbrace -1, 0, 1 \rbrace$ at the individual level. The regressions control for the values of $p$ and $r$ associated with the stage 2 paired choice task and individual characteristics. The regressors of interest are the scaled value difference and the scaled distance to indifference. Columns (1) and (2) include each regressor by itself, while column (3) includes both together. The coefficient estimates are quite stable across columns. They imply that a \$10 increase in the scaled value difference implies an 11 percentage-point increase in $CRE-RCRE$. At the same time, a \$10 increase in the scaled distance to indifference implies an eight percentage-point increase in $CRE-RCRE$. Moreover, these numbers imply that a person with a scaled value difference of $-\$7$ (i.e., a strong RCRP) would exhibit a CRE if their scaled distance to indifference were larger than $\$10$. These results show that distance to indifference, through its interaction with choice noise, plays an important role in whether people exhibit a CRE or an RCRE in their stage 2 behavior.[^39] **Table 5: Predicting Individual-Level $CRE-RCRE$**
(1) (2) (3) (4)
Outcome: $CRE - RCRE \in \{-1,0,1\}$
OLS OLS OLS 2SLS
Scaled Value Difference
$\frac{1 + r}{2} \Delta m$ 1.12 1.14 6.98
(0.14) (0.14) (1.07)
Scaled Distance to Indifference
$(1$ $-$ $r$)($M$ $-$ $\overline{m})$ 0.78 0.82 0.83
(0.19) (0.19) (0.40)
Outcome Mean 2.64 2.64 2.64 2.64
Individuals 900 900 900 900
Observations 4,500 4,500 4,500 4,500
OLS regressions using individual-level $m$-task data with dependent variable $CRE-RCRE \in \{-1,0,1\}$. Specifications include $p$ and $r$ fixed effects, as well as controls for gender, education, age, language, student status, employment, and the number of previous Prolific approvals. All numbers reported in percentage points; individual-cluster-robust standard errors in parentheses. For column (4), instruments are $(1-r)p\overline{h}$, $0.5(1+r)p\Delta h$, and $(1-r)M$. Since our stage 1 values for $m_{AB}$, $m_{CD}$, and $\Delta m$ reflect a combination of preference and noise, the coefficients in Table 5 might be attenuated by measurement error. To account for this, column (4) of Table 5 pursues an instrumental-variables approach using the $h$-valuation tasks and the exogenous value $(1-r)M$ as instruments. The coefficient on the scaled distance to indifference is unaffected, which is not surprising given that a large share of the variation in this variable is due to our random variation in $M$. In contrast, the coefficient on the scaled value difference is substantially larger, which again is not surprising given that the variation in this variable is entirely endogenous and thus strongly influenced by choice noise. While it is difficult to compare the magnitudes of the two effects under the IV specification, the latter result provides further support for our conclusion that the stage 1 valuations include a substantial preference component, especially when using information across both the $m$- and $h$-valuation tasks.[^40] ### C Experiment-Level Stage 2 Data We next analyze stage 2 data at the experiment level—that is, we analyze the share of CRE choices minus the share of RCRE choices among participants who saw the same pair of choice tasks. As described in Section IV.B, there are 60 combinations of $(M,p,r)$ linked to stage 1 $m$-valuations, which yields 60 experiments. To expand the number of experiment-level observations, we also include the 60 analogous experiments linked to stage 1 $h$-valuations—that is, for 60 combinations of $(H,p,r)$. Each participant completed ten of these 120 experiments, and each experiment had between 57 and 101 observations with an average sample size across experiments of 75. We present the summary data for all 120 experiments in Online Appendix Tables D.11 and D.12. Our goal is to demonstrate that we can predict which experiments will likely yield a CRE or an RCRE in choices based on our linked stage 1 data on valuations. Specifically, we calculate for each experiment (i) the average scaled value difference ($0.5(1+r)\Delta m$ for $m$-choice tasks or $0.5(1+r)p\Delta h$ for $h$-choice tasks) and (ii) the average scaled distance to indifference ($(1-r)(M-\bar{m})$ or $(1-r)p(\bar{h}-H)$).[^41] We then investigate whether these averages predict $CRE-RCRE$ at the experiment level. Figure 9 provides visual evidence and is the experiment-level analogue to the individual-level analysis in Figure 8. Panel A presents the relationship between $CRE-RCRE$ and the scaled value difference, where the variables on both axes are residualized by the distance to indifference. Panel B presents the relationship between $CRE-RCRE$ and the scaled distance to indifference, where the variables on both axes are residualized by the value difference. In both cases, each point is one experiment, and the size of each point is proportional to the sample size of each experiment. Similar to the individual-level data, we see that both variables predict whether we observe a CRE or an RCRE. Panel A. Value Difference ![Figure 9: Stage 2 Experiment-Level CRE − RCRE (panel 1)](assets/figure9a-f42eb3.png) Panel B. Distance to Indifference ![Figure 9: Stage 2 Experiment-Level CRE − RCRE (panel 2)](assets/figure9b-d6f6b3.png) **Figure 9: Stage 2 Experiment-Level $CRE-RCRE$** Stage 2 experiment-level $CRE-RCRE$ as a function of experiment-level stage 1 value differences (panel A) and distance to indifference (panel B). Each dot denotes an experiment, and the size is proportional to the number of observations in that experiment. Panel A observations are residualized by the average distance to indifference; panel B observations are residualized by the average value difference. The solid and dashed lines represent the best fitting OLS estimates from columns (3) and (4) of Table 6; the latter excludes experiments with values in the bottom quintile of the distance to indifference variable. In Table 6, we re-conduct the analysis of Table 5 at the experiment level. The dependent variable is now the continuous variable $CRE-RCRE$ for each experiment, and the regressions all control for the values of $p$ and $r$ along with whether the experiment involves $m$ or $h$ choices. The regressors of interest are the average scaled value difference and the average scaled distance to indifference. Columns (1) and (2) include each by themselves, while column (3) includes both together; once again, the coefficients are quite stable across columns. **Table 6: Predicting Experiment-Level $CRE-RCRE$**
(1) (2) (3) (4)
Outcome: $CRE - RCRE \in [-100,100]$
Scaled Value Difference
$\frac{1 + r}{2} \Delta m$ or $\frac{1 + r}{2} p \Delta h$ 3.94 4.22 4.13
(1.06) (1.02) (1.15)
Scaled Distance to Indifference
$(1$ $-$ $r)(M$ $-$ $\overline{m})$ or $(1$ $-$ $r)p(\overline{h}$ $-$ $H)$ 0.80 0.90 1.96
(0.30) (0.28) (0.53)
Mean CRE $-$ RCRE 2.69 2.69 2.69 1.92
Data Exclusion: Distance to Indifference None None None Bottom Quintile
Number of Experiments 120 120 120 96
OLS regressions using experiment-level data with dependent variable $CRE-RCRE \in [-100,100]$. Specifications include $p$ and $r$ fixed effects, as well as task ($m$ versus $h$) fixed effects. All numbers reported in percentage points. For column (4), data restricted to experiments with an average distance-to-indifference value in the top four quintiles. Across the 120 experiments, the average value difference has a range of about \$4. The estimates suggest that this range corresponds to a roughly 16 percentage-point change in the aggregate $CRE-RCRE$. Across the 120 experiments, the average distance to indifference has a range of about \$15. This range corresponds to a roughly 14 percentage-point change in $CRE-RCRE$. Hence, as in our individual-level analysis, we see that, in practice, the distance to indifference has an impact similar in magnitude to that of the value difference. Thus the distance to indifference—which is largely determined by the experimenter’s choice of parameter values—plays an essential role in determining whether an experiment will have a CRE or an RCRE. Panel B of Figure 9 reveals that our experimental variation in $M$ and $H$ leads to some large negative values of the average distance to indifference. Given that Figure 7 implies that the relationship between $CRE-RCRE$ and distance to indifference could invert for sufficiently large magnitudes, it is natural to exclude observations with extreme magnitudes. In column (4) of Table 6, we re-run the regression after dropping the bottom 20 percent of observations in terms of distance to indifference. The effect of distance to indifference on $CRE-RCRE$ within this sub-sample more than doubles. Hence, our numbers above perhaps understate the importance of distance to indifference in generating observed CRE or an RCRE. **Result 2**. *Both value difference and distance to indifference significantly predict whether standard paired choice tasks will reveal a common ratio effect.* ### D Stage 2 Paired Choices Relative to the Prior Literature In Section I.D, we highlighted how the prior literature has used a limited set of experimental parameters. In contrast, we use a broader and more balanced set of experimental parameters in our stage 2 paired-choice experiments. We now investigate the implications of doing so, further highlighting the inferential challenges inherent to paired-choice data. In fact, our stage 2 paired-choice data yield the same broad conclusion as our stage 1 paired-valuations data: There is little evidence of a systematic CRP. Looking across all 9000 observations in our data at the individual level, we observe $\widehat{\Pr}(A) = 49.14$ percent and $\widehat{\Pr}(C) = 46.46$ percent.[^42] Hence, we observe that $CRE-RCRE = 2.7$ percent, which is substantially smaller than the (sample-weighted) 22.0 percent observed across all prior studies. At the experiment level, if we follow Blavatskyy et al. (2023) and use a Conlisk z-test with a 5 percent significance level, then, among our 120 experiments, $16.7$ percent find a CRE while $9.2$ percent find an RCRE, which is substantially more balanced than the 57.3 percent and 9.1 percent among prior experiments. Online Appendix Figure D.1 provides a visualization of the latter point by depicting the analogue of Figure 2(A) for our 120 experiments. Hence, based on our stage 2 paired-choice data, at best one might conclude that there is evidence for a mild underlying CRP.[^43] Moreover, we can document that the different patterns in our experiments relative to prior experiments derive from our use of a broader and more balanced set of experimental parameters (see Online Appendix B.8 for full details). Specifically, we develop a measure of whether an experiment is more representative of prior studies or more representative of our study based on the experimenter-chosen values for $p$, $r$, and $M/(pH)$. To do so, we first create a combined data set of 263 observations consisting of our own and prior experiments. We then regress an indicator for an experiment coming from a prior study on $p$, $r$, and $M/(pH)$. Finally, we use the estimated coefficients from this regression to generate for each experiment a predicted likelihood that it comes from a prior study. Importantly, this predicted likelihood depends on only an experiment’s experimenter-chosen values for $p$, $r$, and $M/(pH)$, and is independent of the experiment’s observed realization for $CRE-RCRE$. We next compare experiments based on whether they are more representative of prior studies (predicted likelihood larger than 0.50) or more representative of our study (predicted likelihood smaller than 0.50). Among the 143 prior experiments, 112 are more representative of prior studies and have a (sample-weighted) average $CRE-RCRE$ of 24.7 percent, while the other 31 have an average of 4.5 percent. Among our 120 experiments, 40 are more representative of prior studies and have an average $CRE-RCRE$ of 8.4 percent, while the other 80 have an average of $-0.1$ percent. In other words, when we (or prior studies) use experimental parameters that are more representative of prior studies, we find more CRE; in contrast, when we (or prior studies) use experimental parameters that are less representative of prior studies, we find much less CRE. While our stage 2 paired-choice data are suggestive that, at best, there might be a mild underlying CRP, it is not obvious how to turn this into a rigorous test. This point highlights the key limitation of using paired-choice data to test for an underlying CRP: The underlying null hypothesis of no CRP implies that we should see a CRE for some parameters, an RCRE for other parameters, and neither for yet other parameters. Given these predictions for paired-choice data, it is difficult to create a test for an underlying CRP without making structural assumptions about both preferences and noise. In contrast, with paired valuations, our two simple tests rely on relatively weak assumptions only about the nature of the noise. ## V Discussion Our paper has two main contributions. First, methodologically, we demonstrate the limitations of using paired choice tasks to make inferences about preferences and illustrate how paired valuations can overcome these limitations. Second, empirically, our paired valuation data yield no evidence of a systematic CRP but are also inconsistent with EU. We conclude by discussing the implications of our paper for future research. Methodologically, our paper argues that valuation data may be better suited than choice data to reveal features of underlying preferences in the presence of noise. We do not claim valuations are always superior, even when working with paired decisions. For example, when estimating a fully specified structural model of underlying preferences and noise, there is no inherent advantage to using valuation versus choice data (beyond the fact that observing one valuation is effectively equivalent to observing many choices). The benefits of valuation data emerge in model-free tests of features of preferences that are akin to axiom testing. Our methodological point relates to, but is distinct from, contemporary work that highlights the usefulness of valuation data. Carrera et al. (2022) point out how noise can bias choice data when studying *single decisions* and how valuations may be immune to that bias. Specifically, they investigate whether people exhibit a preference for commitment in the context of intertemporal choice. For a binary choice of whether to take up a commitment contract, they note how mean-zero error on underlying valuations can distort the take-up rate relative to what noise-free valuations would imply. They then point out that if one instead elicits valuations directly, the same mean-zero error does not create a bias. Our contribution is distinct in that it revolves around how noise creates bias when comparing *pairs of decisions*. Proposition 1 establishes that, if there is differential noise across the two decisions, then a systematic bias emerges even in populations with homogeneous preferences. Moreover, unobserved heterogeneity in underlying preferences exacerbates the inference problem. At the same time, Proposition 2 establishes that using valuations can be a useful solution under both mean-zero valuation error as in Carrera et al. (2022) (i.e., under Assumption 2a) and mean-zero utility error (i.e., under Assumption 2b). We further highlight that our methodological contribution applies to many other domains. Canonical tests for a large number of behavioral-economic phenomena use paired choice tasks: dynamic inconsistency and present bias (comparing choices with different time stamps), loss aversion (comparing choices between mixed and non-mixed gambles), endowment effects (comparing choices of owners and non-owners), decoy effects and compromise effects (comparing choices for two-option versus three-option choice sets), ambiguity attitudes, rank dependence, and common consequence effects (comparing choices between particular sets of acts or lotteries). All of these paradigms are potentially subject to the central critique embodied in our main theoretical result and are similarly suited to the use of paired valuations to overcome the underlying inference problem. Our core empirical contribution presents an important challenge for the literature that seeks to understand risk attitudes. Like prior research, we find clear evidence against EU as an accurate descriptive model of behavior, thus supporting the need for alternative models. However, many prominent non-EU models are built around a systematic CRP as a motivating feature of preferences—indeed, admission of a CRP is often axiomatic. Perhaps most prominent are models of probability weighting in the tradition of prospect theory. Based on their evidence, Kahneman and Tversky (1979) take a global CRP to be a property of preferences and design their model of probability weighting to accommodate this property. Later, Prelec (1998) takes an axiomatic approach to probability weighting; his key axiom of “subproportionality” is equivalent to assuming a global CRP.[^44] The models of disappointment aversion in Bell (1985) and Gul (1991) are also motivated by evidence on the CRE, as is the model of cautious expected utility in Cerreia-Vioglio et al. (2015). The latter two papers pursue axiomatic approaches that modify the standard EU axioms to permit a CRP. Our core empirical finding of no aggregate CRP calls for a reassessment of these models. Because our $m_{AB}$ valuations are consistent with the standard S-shaped probability weighting function found in the literature, it might be fruitful for future work to understand how to retain the appealing psychology of probability-weighting models without requiring a built-in systematic CRP. Moreover, while we find no systematic CRP on average, we do find that some individuals seem to exhibit a reliable CRP while others seem to exhibit a reliable RCRP. Hence, it may be important to develop models that permit heterogeneity in whether people exhibit a CRP versus an RCRP. More generally, our paper calls for more data to develop a solid empirical foundation upon which to build descriptively accurate models of risk attitudes. It is critical to assess the robustness of the findings obtained here (while still being responsive to the central identification problems discussed in this manuscript), and to further explore the space of parameters for common-ratio problems. Moreover, future work should investigate a richer space of decision problems. For instance, in ongoing work (McGranaghan et al. 2023a), we use a valuations approach to gather data on both common-ratio problems and common-consequence problems over a much wider range of parameters than previously studied for either problem. Interestingly, in some circumstances, a CRP emerges. More importantly, we highlight the natural connection that exists between the two types of problems, and studying connected common-ratio and common-consequence problems reveals novel information on the shape of risk preferences. As more studies of this type emerge, the literature will be better positioned to develop more descriptively accurate models of risk attitudes. **Distinguishing Common Ratio Preferences from\ Common Ratio Effects Using Paired Valuation Tasks** Christina McGranaghan, Kirby Nielsen, Ted O’Donoghue,\ Jason Somerville, and Charles D. Sprenger September 24, 2023 **Online Appendix** ## A Proofs Proof of Proposition 1: By Assumption 1, a person with $m_{AB}^{\ast }=m_{CD}^{\ast }\equiv m^{\ast }$ chooses $A$ over $B$ when $M\geq \Gamma (m^{\ast },\varepsilon _{AB})$, and chooses $C$ over $D$ when $M\geq \Gamma (m^{\ast },\varepsilon _{CD})$. Define $\bar{\varepsilon}(M)$ such that $\Gamma (m^{\ast },\bar{\varepsilon} (M))=M$. Then $\Gamma (m,0)=m$ for all $m$ and $\Gamma$ increasing in $\varepsilon$ together imply $\bar{\varepsilon}(M)=0$ when $M=m^{\ast }$ and $\bar{\varepsilon}$ is increasing in $M$. Finally, using $\bar{\varepsilon}(M)$ and the fact that $\varepsilon _{CD} \,{\buildrel d \over =}\, k\varepsilon _{AB}$, the choice probabilities as a function of $M$ are $\Pr (A)=\Pr(\varepsilon _{AB}<\bar{\varepsilon}(M))$ and $\Pr (C)=\Pr(\varepsilon _{CD}<\bar{\varepsilon}(M))=\Pr (\varepsilon_{AB}<\bar{\varepsilon}(M)/k)$. - $M-m^{\ast }>0$ implies $\bar{\varepsilon}(M)>0$ and thus, given $\Pr(\varepsilon _{AB}<0)=Z$, $\Pr (A)>Z$ and $\Pr (C)>Z$. Moreover, $k>1$ implies $\bar{\varepsilon}(M)/k<\bar{\varepsilon}(M)$ and thus $\Pr (A)>\Pr(C)$; $k<1$ implies $\bar{\varepsilon}(M)/k>\bar{\varepsilon}(M)$ and thus $\Pr (A)<\Pr (C)$; and $k=1$ implies $\bar{\varepsilon}(M)/k=\bar{\varepsilon}(M)$ and thus $\Pr (A)=\Pr (C)$. - $M-m^{\ast }<0$ implies $\bar{\varepsilon}(M)<0$ and thus, given $\Pr(\varepsilon _{AB}<0)=Z$, $\Pr (A)1$ implies $\bar{\varepsilon}(M)/k>\bar{\varepsilon}(M)$ and thus $\Pr (A)<\Pr(C)$; $k<1$ implies $\bar{\varepsilon}(M)/k<\bar{\varepsilon}(M)$ and thus $\Pr (A)>\Pr (C)$; and $k=1$ implies $\bar{\varepsilon}(M)/k=\bar{\varepsilon}(M)$ and thus $\Pr (A)=\Pr (C)$. - $M-m^{\ast }=0$ implies $\bar{\varepsilon}(M)=\bar{\varepsilon}(M)/k=0$ for all $k>0$, and thus $\Pr (A)=\Pr (C)=Z$ for all $k>0$. $\blacksquare$ Proof of Proposition 2: Proof of part (1): Because $m_{AB}=\Gamma(m_{AB}^\ast,\varepsilon_{AB})=m_{AB}^\ast + \varepsilon_{AB}$, we have $E(m_{AB})=m_{AB}^\ast + E(\varepsilon_{AB}).$ Analogously, $E(m_{CD})=m_{CD}^\ast + E(\varepsilon_{CD}).$ Then $E(\varepsilon_{AB})=E(\varepsilon_{CD})$ implies $E(\Delta m)=E(m_{CD}-m_{AB})=m_{CD}^\ast - m_{AB}^\ast = \Delta m^\ast$. Proof of part (2): By assumption, the joint distribution of $(\varepsilon_{AB},\varepsilon_{CD})$ has continuous PDF $f$ that satisfies $f(\varepsilon'+z_{AB},\varepsilon'+z_{CD})=f(\varepsilon'-z_{AB},\varepsilon'-z_{CD})$ for all $(z_{AB},z_{CD})$. Define $\nu_{AB} \equiv \varepsilon_{AB}-\varepsilon'$ and $\nu_{CD} \equiv \varepsilon_{CD}-\varepsilon'$. Then the joint distribution of $(\nu_{AB},\nu_{CD})$ has continuous PDF $g$ that satisfies $g(\nu _{AB},\nu _{CD})=g(-\nu_{AB},-\nu _{CD})$ for all $(\nu _{AB},\nu _{CD})$. The marginal distribution for $\nu _{CD}$ is $g_{\nu_{CD}}(\nu _{CD})\equiv \int_{\nu_{AB}=-\infty }^{\infty} g(\nu _{AB},\nu _{CD})d\nu _{AB}$, and symmetry around zero implies\ $\int_{\nu _{CD} = -\infty }^{0}g_{\nu_{CD}}(\nu _{CD})d\nu_{CD}=\int_{\nu_{CD}=0}^{\infty }g_{\nu _{CD}}(\nu _{CD})d\nu_{CD}=1/2$. If $m_{AB}^{\ast }=m_{CD}^{\ast }\equiv m^{\ast }$, then $m_{AB}=\Gamma(m^{\ast },\varepsilon _{AB})$ and $m_{CD}=\Gamma (m^{\ast },\varepsilon_{CD})$, and thus, given that $\Gamma$ is increasing in its second argument, $m_{CD}>m_{AB}$ if and only if $\varepsilon _{CD}>\varepsilon _{AB}$, which is equivalent to $\nu_{CD}>\nu_{AB}$. Hence: $$ \begin{array}{rcl} \Pr (\Delta m>0)&=&\Pr (\nu _{CD}>\nu _{AB})=\int_{\nu _{CD}=-\infty }^{\infty }\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \\ &=&\int_{\nu _{CD}=-\infty }^{0}\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \\ & & +\int_{\nu _{CD}=0}^{\infty }\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD}. \end{array} $$ Note that $\int_{\nu _{AB}=-\infty }^{\nu_{CD}}g(\nu _{AB},\nu _{CD})d\nu_{AB}=g_{\nu _{CD}}(\nu _{CD})-\int_{\nu_{AB}=\nu _{CD}}^{\infty }g(\nu_{AB},\nu_{CD})d\nu _{AB}$ (since\ $\int_{\nu _{AB}=-\infty }^{\infty}g(\nu _{AB},\nu _{CD})d\nu _{AB}=g_{\nu_{CD}}(\nu _{CD})$), and note that $$ \int_{\nu _{AB}=\nu _{CD}}^{\infty }g(\nu _{AB},\nu _{CD})d\nu _{AB}=\int_{\nu _{AB}=-\infty }^{-\nu _{CD}}g(-\nu _{AB},\nu _{CD})d\nu _{AB}=\int_{\nu _{AB}=-\infty }^{-\nu _{CD}}g(\nu _{AB},-\nu _{CD})d\nu _{AB} $$ (the first equality uses a simple change in variables replacing $\nu_{AB}$ with $-\nu _{AB}$, and the second follows from symmetry about zero). Hence, $$ \begin{array}{rcl} &&\int_{\nu _{CD}=-\infty }^{0}\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \\ &=&\int_{\nu _{CD}=-\infty }^{0}\left( g_{\nu _{CD}}(\nu _{CD})-\int_{\nu _{AB}=\nu _{CD}}^{\infty }g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \\ &=&\frac{1}{2}-\int_{\nu _{CD}=-\infty }^{0}\left( \int_{\nu _{AB}=-\infty }^{-\nu _{CD}}g(\nu _{AB},-\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \\ &=&\frac{1}{2}-\int_{\nu _{CD}=0}^{\infty }\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \end{array} $$ (the first equality merely substitutes from above, the second equality uses another substitution plus the fact that symmetry around zero implies $\int_{\nu _{CD}=-\infty }^{0}g_{\nu_{CD}}(\nu_{CD})d\nu _{CD}=1/2$, and the third equality uses a simple change in variables replacing $\nu _{CD}$ with $-\nu _{CD}$). Combining terms yields $$ \begin{array}{rcl} \Pr (\Delta m>0) &=& \frac{1}{2}-\int_{\nu _{CD}=0}^{\infty }\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} \\ & & + \int_{\nu _{CD}=0}^{\infty }\left( \int_{\nu _{AB}=-\infty }^{\nu _{CD}}g(\nu _{AB},\nu _{CD})d\nu _{AB}\right) d\nu _{CD} = \frac{1}{2}. \end{array} $$ An analogous argument can be used to prove that $\Pr (\Delta m<0)=\Pr(\nu _{CD}<\nu _{AB})=1/2$. $\blacksquare$ ## B Additional Analysis ### B.1 Predictions for $E(\Delta m)$ when $\Gamma$ is Non-Linear In Section I.C, we discuss the possibility of testing the null of $\Delta m^\ast = 0$ by testing whether the mean of $\Delta m$ is zero. This test is valid under Assumption 2a where the function $\Gamma$ is linear in both $m$ and $\varepsilon$, because then $E(\Delta m)=\Delta m^\ast$. In contrast, if $\Gamma$ is nonlinear, then it need not be the case that $E(\Delta m)=\Delta m^\ast$, and thus a test based on the mean of $\Delta m$ is potentially biased. To further explore this point, consider the EU model with additive utility noise. Under EU, $m_{AB}^\ast=m_{CD}^\ast \equiv m^\ast$ and therefore $\Delta m^\ast = 0$. From Example 1, $m_{AB}=u^{-1}\left( u(m^{\ast}) + \epsilon_{AB} \right)$ and $m_{CD}=u^{-1}\left( u(m^{\ast}) + \epsilon_{CD}/r \right)$, and thus $\Gamma (m,\varepsilon) = u^{-1}(u(m)+\varepsilon)$, $\varepsilon_{AB}=\epsilon_{AB}$, and $\varepsilon_{CD}=\epsilon_{CD}/r$. Suppose further that $E(\epsilon_{AB})=0$ and $\epsilon_{CD}\,{\buildrel d \over =}\,k'\epsilon _{AB}$ for some $k'>0$, and thus $\varepsilon_{CD}\,{\buildrel d \over =}\,k\epsilon _{AB}$ where $k=k'/r$. Then $E(\Delta m)$ depends on $k$ and the curvature of the utility function $u$ as follows: - If $u(x)$ is linear, then $E(m_{AB})=E(m_{CD})=m^{\ast }$, and thus $E(\Delta m)=0$. - If $u(x)$ is strictly concave, then $E(m_{AB})>m^{\ast}$ and $E(m_{CD})>m^{\ast}$, and moreover $k=1$ implies $E(\Delta m)=0$, $k>1$ implies $E(\Delta m)>0$, and $k<1$ implies $E(\Delta m)<0$. - If $u(x)$ is strictly convex, then $E(m_{AB})1$ implies $E(\Delta m)<0$, and $k<1$ implies $E(\Delta m)>0$. Proof: Given a $m^\ast$, define a function $m(\varepsilon) = u^{-1}(u(m^{\ast })+\varepsilon)$, and note $m(0)=m^{\ast }$. Then $m_{AB}=m(\varepsilon_{AB})$ and $m_{CD}=m(\varepsilon_{CD})$. Proof of part (1): If $u(x)$ is linear—that is, $u(x)=\alpha x+\beta$ for some $\alpha >0$ and any $\beta$—then $m(\varepsilon)=m^{\ast }+\varepsilon/\alpha$. The results then follow from the same logic as the proof of Proposition 2(1). Proof of part (2): If $u(x)$ is strictly concave, then $u^{-1}(x)$ is strictly convex and therefore $m(\varepsilon)$ is also strictly convex. By Jensen’s inequality, $E_{\varepsilon_{AB} }\left[ m(\varepsilon_{AB} )\right] >m(E_{\varepsilon_{AB} }\left[\varepsilon_{AB} \right] )=m^{\ast }$ and $E_{\varepsilon_{CD} }\left[m(\varepsilon_{CD} )\right] >m(E_{\varepsilon_{CD} }\left[ \varepsilon_{CD}\right])=m^{\ast }$. If $k=1$ then $\varepsilon_{CD} \,{\buildrel d \over =}\, \varepsilon_{AB}$, and thus $E_{\varepsilon_{CD} }\left[ m(\varepsilon_{CD} )\right] =E_{\varepsilon_{AB} } \left[ m(\varepsilon_{AB} )\right]$. It follows that $E\left[ \Delta m\right] = 0$. If instead $k>1$, then $\varepsilon_{CD}$ is a mean-preserving spread of $\varepsilon_{AB}$, and thus, since $m(\varepsilon)$ is convex, $E_{\varepsilon_{CD} }\left[ m(\varepsilon_{CD} )\right] > E_{\varepsilon_{AB} } \left[ m(\varepsilon_{AB} )\right]$. It follows that $E\left[ \Delta m\right] >0$. An analogous logic can be used to show that $k<1$ implies $E\left[ \Delta m\right] < 0$. Proof of part (3): Analogous to the proof of part (2), and so omitted. $\blacksquare$ Finally, for the case frequently discussed in the literature of EU with i.i.d. additive utility noise, we have $k'=1$ and thus $k=1/r>1$. Hence, the typical assumption of concave utility would imply $E(\Delta m)>0$, and thus a test based on the mean of $\Delta m$ is biased towards rejecting the null of $\Delta m^\ast =0$ in favor of a CRP. ### B.2 Predictions for Figure 2 We derive predictions for Figure 2 assuming that every individual $i$ has $m_{AB,i}^\ast=m_{CD,i}^\ast=m_i^*$ and thus $\Delta m_i^\ast = 0$. We assume that everyone satisfies Assumption 2a and thus has $m_{AB,i}=m_i^\ast + \varepsilon_{AB,i}$ and $m_{CD,i}=m_i^\ast + \varepsilon_{CD,i}$, where $E(\varepsilon_{AB,i})=E(\varepsilon_{CD,i})=0$. We further assume that everyone has median-zero noise, so $\Pr(\varepsilon_{AB,i}<0)=\Pr(\varepsilon_{CD,i}<0)=1/2$. However, we permit heterogeneity in $m_i^\ast$ and in the distributions of $\varepsilon_{AB,i}$ and $\varepsilon_{CD,i}$.[^45] #### B.2.1 Predictions for Paired Choice Tasks (Panel A) For a specific paired choice task, observed behaviors in a population are the observed proportions choosing $A$ over $B$ and $C$ over $D$, which we denote by $\widehat{\Pr}(A)$ and $\widehat{\Pr}(C)$, respectively. Hence, we need predictions for $\Pr(A)$ and $\Pr(C)$. Starting at the individual level, a person with $m_i^\ast < M$ has noise-free preferences that favor $A$ and $C$, and thus, given the assumption of median-zero noise, must have $1 \ge \Pr(A) \ge 1/2$ and $1 \ge \Pr(C) \ge 1/2$. But given the flexibility to choose different distributions for $\varepsilon_{AB,i}$ and $\varepsilon_{CD,i}$, there are no further restrictions on $\Pr(A)$ and $\Pr(C)$. Analogously, a person with $m_i^\ast>M$ must have $1 \ge \Pr(B) \ge 1/2$ and $1 \ge \Pr(D) \ge 1/2$, but there are no further restrictions. At the population level, then, we can parse the population into those who prefer $A$ and $C$ ($AC$ types) and those who prefer $B$ and $D$ ($BD$ types), and characterize the population by five parameters: | | | | |:---|:---|:---| | $q$ | = | the proportion of the population who are $AC$ types. | | $\lambda_{A}$ | = | the proportion of $AC$ types who actually choose $A$ over $B$. | | $\lambda_{C}$ | = | the proportion of $AC$ types who actually choose $C$ over $D$. | | $\lambda_{B}$ | = | the proportion of $BD$ types who actually choose $B$ over $A$. | | $\lambda_{D}$ | = | the proportion of $BD$ types who actually choose $D$ over $C$. | As a function of these five parameters, the population’s $(\Pr (A),\Pr (C))$ will be $$ \begin{array}{rcl} \Pr (A) &=&q\lambda _{A}+(1-q)(1-\lambda _{B}) \\ \Pr (C) &=&q\lambda _{C}+(1-q)(1-\lambda _{D}). \end{array} $$ The only constraints on these parameters are that $q \in [0,1]$ and that each $\lambda_A,\lambda_C,\lambda_B,\lambda_D \in [1/2,1]$. One can then show (see proof below) that the set of possible predictions for the population include any $(\Pr(A),\Pr(C))$ combinations that satisfy $$ \Pr (A) \leq \frac{1}{2} \text{ and } 0 \leq \Pr (C)\leq \Pr (A)+\frac{1}{2} \text{ or} $$ $$ \Pr(A) \geq \frac{1}{2} \text{ and } \Pr (A)-\frac{1}{2} \leq \Pr (C) \leq 1. $$ These combinations are depicted by the gray shaded region in panel A of Figure 2. To provide some intuition, consider one possible extreme point where $\Pr(A)=75\%$ and $\Pr(C)=25\%$. This outcome occurs when the population is equally split between $AC$ types and $BD$ types (i.e., $q=1/2$), where the $AC$ types make no errors in the $AB$ choice but respond randomly in the $CD$ choice (i.e., $\lambda_A=1$ while $\lambda_C=1/2$), while the $BD$ types make no errors in the $CD$ choice but respond randomly in the $AB$ choice (i.e., $\lambda_B=1/2$ while $\lambda_D=1$). This example illustrates how the full gray shaded region takes advantage of being able to have $AC$ types be more impacted by noise on one type of choice (above, on the $CD$ choice) while $BD$ types are more impacted by noise on the other type of choice (above, on the $AB$ choice). However, significant deviations from the 45-degree line are still possible even if we impose that both types must be more impacted by noise in the same choice. For instance, suppose we impose—consistent with the logic of EU with i.i.d. additive utility noise—that both types must be more impacted by noise in the $CD$ choice. This restriction creates additional constraints that $\lambda_A \ge \lambda_C$ and $\lambda_B \ge \lambda_D$. Even with this additional constraint, one can still show that the set of possible predictions for the population include any $(\Pr(A),\Pr(C))$ combinations that satisfy[^46] $$ \frac{\Pr(A)}{2} \leq \Pr(C) \leq \frac{1}{2} + \frac{\Pr(A)}{2} $$ Hence, significant deviations from the 45-degree line are still possible. For instance, one possibility is $\Pr(A)=2/3$ and $\Pr(C)=1/3$. This outcome occurs when the population has $2/3$ $AC$ types and $1/3$ $BD$ types (i.e., $q=2/3$), where the $AC$ types make no errors in the $AB$ choice but respond randomly in the $CD$ choice (i.e., $\lambda_A=1$ while $\lambda_C=1/2$), while the $BD$ types make no errors in either choice (i.e., $\lambda_B=\lambda_C=1$). Proof of conditions for gray area in Figure 2 panel A: For any fixed $\Pr (A)=\omega$, we find the parameters $(q,\lambda _{A},\lambda _{C},\lambda _{B},\lambda _{D})$ that minimize $\Pr (C)$ and those that maximize $\Pr (C)$. The constraints are $q\in \lbrack 0,1]$ and $\lambda _{A},\lambda_{C},\lambda _{B},\lambda _{D}\in \lbrack 1/2,1]$. To minimize $\Pr (C)$: First note that, for $\omega \leq 1/2$, setting $q=0$, $\lambda _{B}=1-\omega$, and $\lambda _{D}=1$ implies $\Pr (A)=\omega$ and $\Pr (C)=0$, so the minimum $\Pr (C)=0$. For $\omega \geq 1/2$, we clearly want $\lambda _{C}$ to be as small as possible and $\lambda _{D}$ to be as large as possible, and thus we set $\lambda _{C}=1/2$ and $\lambda _{D}=1$. Then $\Pr (C)=q/2$, so we choose $\lambda _{A}$ and $\lambda _{B}$ to minimize $q$. Because $\Pr (A)=\omega$ implies $q=(\omega -(1-\lambda _{B}))/(\lambda_{A}-(1-\lambda _{B}))$, we minimize $q$ by setting $\lambda _{A}=1$ and $\lambda _{B}=1/2$. Hence, the parameters that minimize $\Pr (C)$ are $q=2\omega -1$, $\lambda _{A}=\lambda _{D}=1$, and $\lambda _{B}=\lambda _{C}=1/2$, which imply the minimum $\Pr (C)=(2\omega -1)/2=\omega -1/2$. To maximize $\Pr (C)$: First note that, for $\omega \geq 1/2$, setting $q=1$, $\lambda _{A}=\omega$, and $\lambda _{C}=1$ implies $\Pr (A)=\omega$ and $\Pr (C)=1$, so the maximum $\Pr (C)=1$. For $\omega \leq 1/2$, we clearly want $\lambda _{C}$ to be as large as possible and $\lambda_{D}$ to be as small as possible, and thus we set $\lambda _{C}=1$ and $\lambda _{D}=1/2$. Then $\Pr (C)=q+(1-q)/2$, so we choose $\lambda _{A}$ and $\lambda _{B}$ to maximize $q$. Because $\Pr (A)=\omega$ implies $q=(\omega -(1-\lambda _{B}))/(\lambda_{A}-(1-\lambda_{B}))$, we maximize $q$ by setting $\lambda _{A}=1/2$ and $\lambda_{B}=1$. Hence, the parameters that maximize $\Pr(C)$ are $q=2\omega$, $\lambda _{A}=\lambda _{D}=1/2$, and $\lambda _{B}=\lambda_{C}=1$, which imply the maximum $\Pr (C)=2\omega +(1-2\omega)/2=\omega +1/2$. It follows that (i) for any fixed $\Pr (A)\leq 1/2$, we must have $0\leq \Pr (C)\leq \Pr (A)+1/2$, and (ii) for any fixed $\Pr (A)\geq 1/2$, we must have $\Pr (A)-1/2\leq \Pr (C)\leq 1$. $\blacksquare$ #### B.2.2 Predictions for Paired Valuation Tasks (Panel B) For a specific paired valuation task, observed behaviors in a population are the population averages for the two reported valuations, which we denote by $\widehat{E}(m_{AB})$ and $\widehat{E}(m_{CD})$, respectively. Hence, we need predictions for $E(m_{AB})$ and $E(m_{CD})$. Under Assumption 2a, individual $i$ states valuations $m_{AB,i}=m_i^\ast + \varepsilon_{AB,i}$ and $m_{CD,i}=m_i^\ast + \varepsilon_{CD,i}$, where $E(\varepsilon_{AB,i})=E(\varepsilon_{CD,i})=0$. It follows immediately that, if we let $\bar{m}^\ast$ denote the population average for $m_i^\ast$, then the predicted population averages for the two reported valuations are $E(m_{AB})=\bar{m}^\ast$ and $E(m_{CD})=\bar{m}^\ast$. Hence, the set of possible predictions for the population include any $(E(m_{AB}),E(m_{CD}))$ such that $E(m_{AB})=E(m_{CD})$, which is equivalent to the 45-degree line in panel B of Figure 2. ### B.3 Data for Figure 2 #### B.3.1 Data for Panel A of Figure 2 For the data in panel A, we rely on the meta-study by (Blavatskyy et al. 2023). In their Table 1, they provide information on 143 CRE paired-choice experiments taken from 39 studies. Specifically, for each experiment, they provide the total number of participants ($N$) along with the number of participants that chose each of the four possible choice patterns ($N_{AC}$, $N_{BD}$, $N_{AD}$, and $N_{BC}$). We use this data to construct the empirical choice ratios as $$ \widehat{\Pr}(A) = \frac{N_{AC}+N_{AD}}{N} \hspace{.25in} \text{and} \hspace{.25in} \widehat{\Pr}(C) = \frac{N_{AC}+N_{BC}}{N}. $$ We then depict each of these 143 experiments as one circle in panel A, where the location of the circle is given by its $(\widehat{\Pr}(A),\widehat{\Pr}(C))$ and the size of the circle is proportional to its $N$. #### B.3.2 Data for Panel B of Figure 2 For panel B, we are not aware of any analogous meta-study, and Blavatskyy et al. (2023) do not mention any CRE experiments that use valuations. We therefore conducted our own search. We identified only six studies that use valuations in the context of the CRE: - Freeman et al. (2019) use a modified form of valuations in the context of the CRE. Specifically, they elicit probability equivalents in which they hold fixed outcomes $H$ and $M$ and vary the probability $p$. Thus, the valuations they elicit do not fit into the framework that we depict in panel B. - Schneider and Shor (2017) elicit (hypothetical) minimum prices at which participants would be willing to sell binary lotteries $A$, $B$, $C$, and $D$ using the Kahneman-Tversky parameters $(M=\$3000,H=\$4000,p=0.8,r=.25)$. While this pricing task is a type of valuation, it does not generate data that fit into the framework that we depict in panel B. - Dean and Ortoleva (2019) conduct two paired $h$-valuations, using $(M=\$4,p=0.8,r=0.25)$ and $(M=\$8,p=0.8,r=0.25)$. The paper does not report statistics that would permit us to calculate the average valuations. However, we contacted the authors and they provided the average valuations: For the first pair, $\widehat{E}(h_{AB}) = \$5.80$ and $\widehat{E}(h_{CD}) = \$5.27$; for the second pair, $\widehat{E}(h_{AB}) = \$10.66$ and $\widehat{E}(h_{CD}) = \$9.97$. Because these are $h$-valuations while panel B depicts $m$-valuations, for presentation purposes we transform these values by setting $\widehat{E}(m_{AB}) = M$ and $\widehat{E}(m_{CD}) = M( \widehat{E}(h_{AB})/\widehat{E}(h_{CD}))$. Hence, these two experiments appear in panel B at the locations $(\widehat{E}(m_{AB})=\$4,\widehat{E}(m_{CD})=\$4.40)$ and $(\widehat{E}(m_{AB})=\$8,\widehat{E}(m_{CD})=\$8.55)$. - Castillo and Eil (2014) conduct three $m$-valuations, an $AB$ valuation for $(H=\$10,p=0.4)$ and $CD$ valuations for $(H=\$10,p=0.4,r=0.5)$ and $(H=\$10,p=0.4,r=0.25)$. We treat these as two paired $m$-valuation experiments, one for $(H=\$10,p=0.4,r=0.5)$ and one for $(H=\$10,p=0.4,r=0.25)$ (these are not independent experiments because they use the same $AB$ valuation, but this is not important for our illustrative purposes). The paper does not report any statistics, but the authors provided the average valuations: In panel B, the first experiment appears at $(\widehat{E}(m_{AB})=\$3.95,\widehat{E}(m_{CD})=\$3.98)$, and the second experiment appears at $(\widehat{E}(m_{AB})=\$3.95,\widehat{E}(m_{CD})=\$4.41)$. They also conduct three $h$-valuations, an $AB$ valuation for $(M=\$4,p=0.4)$ and $CD$ valuations for $(M=\$4,p=0.4,r=0.5)$ and $(M=\$4,p=0.4,r=0.25)$. We treat these as two paired $h$-valuation experiments. The average valuations provided by the authors for the three valuations are $\$9.77$, $\$9.95$, and $\$10.22$. Using the same transformation as we do for the paired $h$-valuations in Dean and Ortoleva (2019), these two experiments appear in panel B at the locations $(\widehat{E}(m_{AB})=\$4,\widehat{E}(m_{CD})=\$3.93)$ and $(\widehat{E}(m_{AB})=\$4,\widehat{E}(m_{CD})=\$3.82)$. - Chapman et al. (2022) conduct two paired $h$-valuations, using $(M=\$2.50,p=0.8,r=0.25)$ and $(M=\$4,p=0.75,r=0.2)$. The paper does not report any statistics that would permit us to calculate the four average valuations. We contacted the authors but thus far they have not provided us with any additional information. - Agranov and Ortoleva (forthcoming) conduct two paired $h$-valuations, using $(M=\$14,p=0.8,r=0.25)$ and $(M=\$16,p=0.8,r=0.25)$. Because the authors use these valuations primarily as control variables while studying something else, the paper does not report any statistics for these valuations. We contacted the authors but thus far they have not provided us with any additional information. ### B.4 Development for $h$-Tasks In Section I.C, we fix $(H,p,r)$ and focus on behavior as a function of $M$, which links directly to our $m$-tasks. Here, we revisit some of the analysis from Section I.C when we instead fix $(M,p,r)$ and focus on behavior as a function of $H$, which links directly to our $h$-tasks. Note that, while we use some of the same notation below as we use in Section I.C, we are now referring to different (though analogous) objects. Assuming underlying preferences are monotonic and continuous, for each $(M,p,r)$ a person will have an underlying pair of indifference points $(h_{AB}^{\ast},h_{CD}^{\ast})$ such that their underlying (noise-free) preferences satisfy: - Prefer $A\equiv (M,1)$ over $B\equiv (H,p)$ if and only if $H\leq h_{AB}^{\ast }$, and - Prefer $C\equiv (M,r)$ over $D\equiv (H,rp)$ if and only if $H\leq h_{CD}^{\ast }$. Here, EU implies $h_{AB}^{\ast }=h_{CD}^{\ast }$, whereas a CRP would mean $h_{AB}^{\ast }>h_{CD}^{\ast }$ (an individual would prefer combination $AD$ for any $H\in (h_{CD}^{\ast },h_{AB}^{\ast })$), and an RCRP would mean $h_{AB}^{\ast }0$ reflects a CRP, we define $\Delta h^{\ast }=h_{AB}^{\ast }-h_{CD}^{\ast }$ so that $\Delta h^{\ast }>0$ reflects a CRP while $\Delta h^{\ast }<0$ reflects an RCRP. Given these underlying preferences, Assumption 1h is the analogue for Assumption 1: **Assumption 1h:** Impact of Noise on Choices and Valuations A person’s *realized indifference points* $(h_{AB},h_{CD})$ are $h_{AB}\equiv \Gamma (h_{AB}^{\ast },\varepsilon _{AB})$ and $h_{CD}\equiv \Gamma (h_{CD}^{\ast },\varepsilon _{CD})$, where $(\varepsilon_{AB},\varepsilon_{CD})$ are noise draws from a continuous joint distribution with convex support, and where $\Gamma$ is increasing in both arguments and has $\Gamma (h,0)=h$ for all $h$. Then: - In an $AB$ choice task, the person chooses $A\equiv (M,1)$ over $B\equiv (H,p)$ if and only if $H\leq h_{AB} \equiv \Gamma (h_{AB}^{\ast },\varepsilon _{AB})$, - In a $CD$ choice task, the person chooses $C\equiv (M,r)$ over $D\equiv (H,rp)$ if and only if $H\leq h_{CD} \equiv \Gamma (h_{CD}^{\ast },\varepsilon _{CD})$, - In an $AB$ valuation task, the person states valuation $h_{AB} \equiv \Gamma (h_{AB}^{\ast },\varepsilon _{AB})$, and - In a $CD$ valuation task, the person states valuation $h_{CD} \equiv \Gamma (h_{CD}^{\ast },\varepsilon _{CD})$. Assumptions 2ah and 2bh are the analogues for Assumptions 2a and 2b: **Assumption 2ah:** $\Gamma(h,\varepsilon)=h+\varepsilon$, $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$, and $E(\varepsilon _{AB})=E(\varepsilon _{CD})=0$. **Assumption 2bh:** $\Gamma(h,\varepsilon)$ is potentially nonlinear in $h$ and $\varepsilon$, but $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$, and $\varepsilon_{AB}$ is symmetric about 0. In EU and PT, Assumptions 2ah and 2bh apply under the same assumptions as Assumptions 2a and 2b: **Example 1h:** Expected Utility and Prospect Theory If a person evaluates a lottery $(x,q)$ with $x>0$ as $\pi(q)u(x)$, the underlying indifference points satisfy $$ \begin{array}{rcl} u(M)=\pi(p)u(h_{AB}^{\ast}) & \qquad \Leftrightarrow \qquad & h_{AB}^{\ast}=u^{-1}\left( \frac{1}{\pi(p)}u(M) \right) \\ \pi(r)u(M)=\pi(rp)u(h_{CD}^{\ast}) & \qquad \Leftrightarrow \qquad & h_{CD}^{\ast}=u^{-1}\left( \frac{\pi(r)}{\pi(rp)}u(M) \right) \end{array} $$ One way to incorporate noise is by assuming that $h_{AB} = h_{AB}^{\ast} + \varepsilon_{AB}$ and $h_{CD} = h_{CD}^{\ast} + \varepsilon_{CD}$. This formulation satisfies Assumption 2ah as long as $\varepsilon_{CD}\,{\buildrel d \over =}\,k\varepsilon _{AB}$ for some $k>0$ and $E(\varepsilon _{AB})=E(\varepsilon _{CD})=0$. Alternatively, one might incorporate additive utility noise by assuming that the realized indifference points satisfy $$ \begin{array}{rcl} u(M)=\pi(p)u(h_{AB}) + \epsilon_{AB} & \qquad \Leftrightarrow \qquad & h_{AB}=u^{-1}\left( u(h_{AB}^{\ast}) - \epsilon_{AB}/\pi(p) \right) \\ \pi(r)u(M)=\pi(rp)u(h_{CD}) + \epsilon_{CD} & \qquad \Leftrightarrow \qquad & h_{CD}=u^{-1}\left( u(h_{CD}^{\ast}) - \epsilon_{CD}/\pi(rp) \right) \end{array} $$ where $\epsilon_{AB}$ and $\epsilon_{CD}$ reflect additive utility noise.[^47] This formulation fits Assumption 1h with $\Gamma (h,\varepsilon) = u^{-1}(u(h)+\varepsilon)$, $\varepsilon_{AB}=-\epsilon_{AB}/\pi(p)$, and $\varepsilon_{CD}=-\epsilon_{CD}/\pi(rp)$. This formulation further satisfies Assumption 2bh as long as $\epsilon_{AB}$ is symmetric about 0 and $\epsilon_{CD}\,{\buildrel d \over =}\,k'\epsilon _{AB}$ for some $k'>0$. Finally, EU with additive utility noise that is i.i.d. across the $AB$ and $CD$ choice tasks implies $\varepsilon_{CD}=\varepsilon_{AB}/r$. Because this formulation is exactly parallel to our formulation for $m$-tasks, analogues for Propositions 1 and 2 and for Corollary 1 follow straightforwardly, and so are omitted. Finally, consider analogues to the predictions for stage 2 behavior from Section IV.A. As in the text, consider the case of Assumption 2ah where a person with underlying indifference values $(h_{AB}^{\ast },h_{CD}^{\ast })$ would choose $A$ over $B$ when $H\leq h_{AB}^{\ast}+\varepsilon _{AB}$ and would choose $C$ over $D$ when $H\leq h_{CD}^{\ast}+\varepsilon _{CD}$, where $\varepsilon _{CD}\,{\buildrel d \over =}\,k\varepsilon_{AB}$. For this case, the probability of making CRE choices ($A$ and $D$) minus the probability of making RCRE choices ($B$ and $C$) is $$ CRE-RCRE\equiv \Pr (A)-\Pr (C) = \Pr\left(-\varepsilon_{AB}<(h_{AB}^{\ast}-H)\right) - \Pr\left(-\varepsilon_{AB} <\frac{1}{k}(h_{CD}^{\ast } - H)\right)\text{.} $$ Defining $\Psi \equiv (h_{CD}^{\ast }-H)/k$, and substituting $\bar{h}^{\ast }\equiv (h_{AB}^{\ast }+h_{CD}^{\ast })/2$ and $\Delta h^{\ast } = h_{AB}^{\ast }-h_{CD}^{\ast }$, we can rewrite this as: $$ CRE-RCRE = \Pr\left(-\varepsilon_{AB} < \Psi + 0.5\left(1+\frac{1}{k}\right)\Delta h^{\ast} + \left(1-\frac{1}{k}\right)(\bar{h}^{\ast }-H)\right) - \Pr\left(-\varepsilon_{AB} < \Psi\right) \text{.} $$ This equation is analogous to equation 3 from Section IV.A. It yields similar intuitions, and could also be used to construct a figure analogous to Figure 7. For some of our empirical analysis in Section IV, we increase power by (i) combining data across different values for $p$ and $r$ and (ii) combining data for both $h$-tasks and $m$-tasks. Because these differences in parameters and the type of task may impact the extent of differential noise (i.e., $k$), we make a correction to the scaled value difference and the scaled distance to indifference. To do so in a disciplined way, we use the correction that would be valid under EU with additive i.i.d. utility noise.[^48] If we let $\epsilon$ reflect the additive utility noise, then for the $m$-tasks $\varepsilon_{AB} \,{\buildrel d \over =}\,\epsilon$ and $\varepsilon_{CD} \,{\buildrel d \over =}\,\epsilon/r$ (see Example 1), which motivates using $0.5(1+r) \Delta m$ for the scaled value difference and $(1-r)(M-\bar{m})$ for the scaled distance to indifference (as discussed in Section IV.A). For the $h$-tasks, and using the same $\epsilon$ reflect the additive utility noise, we have instead that $\varepsilon_{AB} \,{\buildrel d \over =}\,-\epsilon/p$ and $\varepsilon_{CD} \,{\buildrel d \over =}\,-\epsilon/(rp)$ (see Example 1h). We can then define $\Psi' \equiv rp(h_{CD}^\ast-H)$ write: $$ \begin{aligned} CRE-RCRE & = \Pr(-\varepsilon_{AB}\epsilon_{AB}$, and chooses lottery $C$ over lottery $D$ when $EV(C)-EV(D)>\epsilon_{CD}$. Suppose further that $\epsilon_{AB}$ and $\epsilon_{CD}$ are i.i.d., and specifically each takes on the values $1$ and $-1$ with equal probability. Table B.1 illustrates how this person would behave for six rows in an experimental task with $H=\$29$ (we use $H=\$29$ instead of $H=\$30$ as in our study to eliminate indifference), $p=0.5$, and $r=0.5$. Columns (2) and (3) present six paired choice tasks (one in each row) for $M$ varying from \$12 to \$17. Column (4) presents the expected-value differences for each row. Columns (5)-(8) present combined behavior for the paired choice task in each row as a function of the four possible realizations of $(\epsilon_{AB},\epsilon_{CD})$. Based on these, column (9) presents the prediction for $CRE-RCRE=\Pr(A)-\Pr(C)$ for the paired choice task in each row if it were presented in an isolated paired choice task. This last column reveals that, despite there being no CRP or RCRP, if we happen to choose experimental parameters such that the participant moderately prefers $A$ and $C$—as in row 5—then we will observe a CRE, and if we happen to choose experimental parameters such that the participant moderately prefers $B$ and $D$—as in row 2—then we will observe an RCRE. The bottom panel of the Table B.1 illustrates how valuations can solve the problem. Applying our approach of taking the average value of $M$ at the switching rows to be our measure of the realized indifference point, the table presents for each of the four possible realizations of $(\epsilon_{AB},\epsilon_{CD})$ the measured indifference points $m_{AB}$ and $m_{CD}$ along with $\Delta m \equiv m_{CD}-m_{AB}$. The table illustrates that, while the noise leads to positive and negative realizations of $\Delta m$, these realizations average out to zero. This stylized example highlights how a full understanding of behavior requires observing outcomes across all six rows of the table. However, the paired-choice-task approach in the prior literature typically studies behavior from only one row, thus yielding only a partial view of behavior. Moreover, as discussed in Section I.D, the prior literature has focused on a selected set of parameter configurations that are more like row 5 in Table B.1. In contrast, our paired-valuation-task approach reveals a more complete view of behavior. **Table B.1: Stylized Example With Additive Noise** lllllllll & & & & &\ Row & & & & & & & &\ ... & ... & ...& ... & ... & ... &... &... & ...\ & & & & $BD$ & $BD$ & $BD$ & $BD$ & $0$\ & & & & $BC$ & $BD$ & $BC$ & $BD$ & $-\frac{1}{2}$\ & & & & $AC$ & $AD$ & $BC$ & $BD$ & $0$\ & & & & $AC$ & $AD$ & $BC$ & $BD$ & $0$\ & & & & $AC$ & $AD$ & $AC$ & $AD$ & $\frac{1}{2}$\ & & & & $AC$ & $AC$ & $AC$ & $AC$ & $0$\ ... & ... & ... & ... &... &... &... & ... & ...\ & $13.5$ & $13.5$ & $15.5$ & $15.5$ &\ & $12.5$ & $16.5$ & $12.5$ & $16.5$ &\ & $-1$ & $3$ & $-3$ & $1$ &\ \ *Note:* The pair of choices in row 5 will exhibit a CRE, with $\Pr(A) > \Pr(C)$. The pair of choices in row 2 will exhibit an RCRE, with $\Pr(A) < \Pr(C)$. ### B.6 Impact of Distance to Indifference Without Noise Our analysis in Section IV focuses on the impact of distance to indifference in the presence of choice noise, where we present theoretical predictions in Figure 7, and we plot empirical relationships in Figures 8 and 9 that confirm those theoretical predictions. In this section, we support the claims made in footnote 37 that predictions for Figures 8 and 9 would be very different in the absence of choice noise, that is, when all variation in the data is due to heterogeneity in preferences. Suppose that there is heterogeneity in $(m_{AB}^{\ast },m_{CD}^{\ast })$, where we focus on heterogeneity in $\bar{m}^{\ast }\equiv (m_{AB}^{\ast }+m_{CD}^{\ast })/2$ and $\Delta m^{\ast }=m_{CD}^{\ast }-m_{AB}^{\ast }$. The development below assumes that $\bar{m}^{\ast }$ and $\Delta m^{\ast }$ are independently distributed, motivated by the fact that we observe limited empirical correlations between the $\bar{m}$’s and $\Delta m$’s elicited in stage 1 of our experiment—across the 15 combinations of $(p,r)$, these correlations range from $-0.04$ to $0.10$, with a mean of $0.04$. Hence, we let $Q_{\bar{m}^{\ast }}(\bar{m}^{\ast })$ denote the population distribution of $\bar{m}^{\ast }$, and $Q_{\Delta m^{\ast }}(\Delta m^{\ast })$ denote the population distribution of $\Delta m^{\ast }$, and assume $Q_{\bar{m}^{\ast }}$ and $Q_{\Delta m^{\ast }}$ are independent of each other. Consider first the behavior of an individual characterized by $(m_{AB}^{\ast },m_{CD}^{\ast })$ with $\Delta m^{\ast }>0$ (i.e., with a CRP) as a function of an offered $M$ at stage 2. In the absence of noise, this individual will exhibit a CRE if $m_{AB}^{\ast }2|M-\bar{m}^{\ast }|$. Notice the symmetry around a zero distance to indifference: Whether the person exhibits a CRE does not depend on whether $M-\bar{m}^{\ast }$ is positive or negative; all that matters is whether the magnitude of $\Delta m^{\ast }$ is larger than the magnitude of $2(M-\bar{m}^{\ast })$. Next consider the behavior of an individual characterized by $(m_{AB}^{\ast },m_{CD}^{\ast })$ with $\Delta m^{\ast }<0$ (i.e., with an RCRP). By an analogous logic, in the absence of noise, the person will exhibit an RCRE when $\Delta m^{\ast }<2(M-\bar{m}^{\ast })<-\Delta m^{\ast }$, or, equivalently, $\Delta m^{\ast }<-2|M-\bar{m}^{\ast }|$. Again, note the symmetry around a zero distance to indifference. Moreover, note the symmetry around a zero value difference: For a fixed distance to indifference, a person with $\Delta m^{\ast }=\delta>0$ exhibits a CRE if and only if a person with $\Delta m^{\ast }=-\delta$ exhibits an RCRE. Now consider the behavior of a population as a function of the distance to indifference $M-\bar{m}^{\ast }$, that is, a prediction to compare to Figure 8. Because this essentially controls for $\bar{m}^{\ast }$, and because the distribution of $\Delta m^{\ast }$ is independent of $\bar{m}^{\ast }$, the distribution $Q_{\bar{m}^{\ast }}$ of $\bar{m}^{\ast }$ is irrelevant for this prediction. Given an $M-\bar{m}^{\ast }=d$, anyone with $\Delta m^{\ast }>2|d|$ will exhibit a CRE while anyone with $\Delta m^{\ast }<-2|d|$ will exhibit an RCRE, and thus $CRE-RCRE=(1-Q_{\Delta m^{\ast }}(2|d|))-Q_{\Delta m^{\ast }}(-2|d|)$. Simplifying, the prediction is $$ CRE-RCRE=1-Q_{\Delta m^{\ast }}(2d)-Q_{\Delta m^{\ast }}(-2d)\equiv C(d). $$ Hence, predicted behavior for this population depends on the nature of the distribution $Q_{\Delta m^{\ast }}$. Various possibilities can arise; but we highlight two points. First, if the distribution $Q_{\Delta m^{\ast }}$ is symmetric around zero—so that $1-Q_{\Delta m^{\ast }}(2d)=Q_{\Delta m^{\ast }}(-2d)$ for all $d$—then $C(d)=0$ for all $d$. Hence, if all variation in the data is due to heterogeneity in preferences, then $CRE-RCRE$ can depend on the distance to indifference only if the distribution of $\Delta m^{\ast }$ is asymmetric, which is not what we see in Figure 6(B). Second, even when $Q_{\Delta m^{\ast }}$ is asymmetric, $C(d)$ must still be symmetric around $d=0$. In other words, if all variation in the data is due to heterogeneity in preferences, then whatever $CRE-RCRE$ we see for some positive value of $M-\bar{m}^{\ast }$, we ought to see the same $CRE-RCRE$ for that same negative value of $M-\bar{m}^{\ast }$. This is not what we see in Figure 8. Finally, consider the behavior of a population as a function of the average distance to indifference $M-E(\bar{m}^{\ast })$, that is, a prediction to compare to Figure 9. Define $z=\bar{m}^{\ast }-E(\bar{m}^{\ast })$, $H(z)\equiv Q_{\bar{m}^{\ast }}(E(\bar{m}^{\ast })+z)$, and assume that distribution $H$ has a PDF $h$. Suppose $M-E(\bar{m}^{\ast })=d$, in which case all people with $\bar{m}^{\ast }$ have $M-\bar{m}^{\ast }=(d+E(\bar{m}^{\ast }))-(z+E(\bar{m}^{\ast }))=d-z$, and thus that group will have $CRE-RCRE=C(d-z)$. Integrating over $z$, the overall population will have $$ CRE-RCRE=\int_{z=-\infty }^{\infty }C(d-z)h(z)dz\equiv \bar{C}(d). $$ If we then assume $Q_{\bar{m}^{\ast }}$ is symmetric around $\bar{m}^{\ast }=E(\bar{m}^{\ast })$, which implies $H$ is symmetric around $z=0$, we have $$ \begin{array}{rcl} \bar{C}(-d) &=&\int_{z=-\infty }^{\infty }C(-d-z)h(z)dz=\int_{z^{\prime }=-\infty }^{\infty }C(-d+z^{\prime })h(-z^{\prime })dz \\ &=&\int_{z^{\prime }=-\infty }^{\infty }C(d-z^{\prime })h(z^{\prime })dz= \bar{C}(d), \end{array} $$ where the second equality uses a change of variables with $z^{\prime }=-z$ and the third equality uses $C(x)=C(-x)$ and $h(-z^{\prime })=h(z^{\prime })$ given the symmetry of $H$ around $z=0$. It follows that, if all variation in the data is due to heterogeneity in preferences, and if the distribution of $\bar{m}^{\ast }$ is symmetric about $\bar{m}^{\ast }=E(\bar{m}^{\ast })$, then whatever $CRE-RCRE$ we see for some positive value of $M-E(\bar{m}^{\ast })$, we ought to see the same $CRE-RCRE$ for that same negative value of $M-E(\bar{m}^{\ast })$. This is not what we see in Figure 9. Hence, under the conditions described above, a model in which all variation in the data is due to heterogeneity in preferences would generate very different predictions from what we see in Figures 8 and 9. Of course, we make some simplifying assumptions above, most notably the assumption that the distributions of $\bar{m}^{\ast }$ and $\Delta m^{\ast }$ are independent (used for predictions for Figure 8), and the additional assumption that the distribution of $\bar{m}^{\ast }$ is symmetric around $\bar{m}^{\ast }=E(\bar{m}^{\ast })$ (used for predictions for Figure 9). It is possible that, with the appropriate assumptions about correlated heterogeneity and asymmetric distributions, one might be able to generate predictions closer to Figures 8 and 9. ### B.7 Assessment of Corrected Regressors in Tables 5, 6, and D.10 In Table 5, to increase statistical power, we combine data for the three different values of $r$. However, we need to correct for the fact that $r$ may impact the magnitude of the coefficients on $\Delta m$ and $M-\bar{m}$ (via its impact on $k$ in equation 3). Motivated by the EU case where $k=1/r$, we use the corrected regressors $0.5(1+r) \Delta m$ and $(1-r)(M-\bar{m})$. As we note in footnote 38, this correction is not perfect. To assess the impact of this correction, Appendix Table B.2 reports the equivalent of Column (3) of Table 5 broken out by $r$. Panel A merely uses the regressors $\Delta m$ and $M-\bar{m}$. The qualitative results are much the same as in Table 5—both regressors have a significant positive impact for each $r$, except for $M-\bar{m}$ when $r=0.6$. Moreover, consistent with the need to correct for $r$ and the directional effects in equation 3, the coefficient on $\Delta m$ increases with $r$ while the coefficient on $M-\bar{m}$ decreases with $r$. Panel B instead uses our corrected regressors, still running separate regressions for each $r$. Here, the estimated coefficients are more stable in magnitude across $r$, and they are similar to those in column (3) of Table 5 but with larger standard errors. On net, then, it appears that the corrected regressors perform as intended: They increase precision without changing the qualitative results. We separately perform analogous analyses for Appendix Table D.10 and Table 6 (although we do not report the results here). For the former, which is the analogue of Table 5 for $h$ choice tasks, this analysis reaches much the same conclusions. For the latter, which is the analogue of Table 5 for the experiment-level analysis, running separate regressions for each $r$ leads to estimates with large standard errors due to the low sample sizes ($N=40$ for each $r$); hence, while a similar message seems to emerge, we are cautious in concluding too much. **Table B.2: Predicting Individual-Level $CRE-RCRE$ Separately by $r$**
(1) (2) (3)
Outcome: $CRE - RCRE \in \{-1,0,1\}$
$r = 0.2$ $r = 0.4$ $r = 0.6$
Panel A: Unscaled Estimates
Value Difference: $\Delta m$ 0.64 0.78 0.90
(0.18) (0.15) (0.19)
Distance to Indifference: $M - \bar{m}$ 0.76 0.61 0.06
(0.23) (0.22) (0.23)
Panel B: Scaled Estimates
Scaled Value Difference: $\frac{1 + r}{2} \Delta m$ 1.06 1.11 1.12
(0.31) (0.21) (0.24)
Scaled Distance to Indifference: $(1$ $-$ $r$)($M$ $-$ $\overline{m})$ 0.90 0.89 0.41
(0.26) (0.32) (0.49)
Outcome Mean 3.02 3.76 1.14
Individuals 298 303 299
Observations 1,490 1,515 1,495
*Note:* OLS regressions using individual-level $m$-task data with the dependent variable $CRE-RCRE \in \{-1,0,1\}$ separately for each common ratio $r\in\{0.2,0.4,0.6\}$. Panel A presents estimates using the unscaled regressors, and panel B presents estimates using the scaled regressors. All specifications include $p$ fixed effects, as well as controls for gender, education, age, language, student status, employment, and the number of previous Prolific approvals. All numbers are reported in percentage points; individual-cluster-robust standard errors in parentheses. ### B.8 Our Experiments and Prior Experiments In Section IV.D, we compare behavior in our stage 2 experiments to that observed in the prior literature by developing a measure of whether an experiment is more representative of prior studies or more representative of our study based on the experimenter-chosen values for $p$, $r$, and $M/(pH)$. This section provides details for this analysis. We first create a combined data set of 263 observations consisting of our own and prior experiments. We then regress an indicator for an observation coming from a prior study on $p$, $r$, and $M/(pH)$. Based on inspection, we expect a nonlinear impact of $M/(pH)$ because the vast majority of prior experiments have $M/(pH) \in [0.75,1]$ while our experiments have more representation for $M/(pH)<0.75$ and $M/(pH)>1$. Hence, letting $y_i$ be a dummy variable for whether experiment $i$ comes from a prior study, we run a logistic regression based on the following specification: $$ \begin{array}{rcl} y^*_i & = & \beta_0 + \beta_1 p_i + \beta_2 r_i + \beta_3 \mathbb{1}\left[ M/(pH)_i \in [0.75,1] \right] + \beta_4 \mathbb{1}\left[M/(pH)_i > 1\right] +\varepsilon_i \\ y_{i} &= & \left\{ \begin{array}{cl} 1 & \text{if }y_{i}^{\ast }>0 \\ 0 & \text{otherwise} \end{array} \right. \end{array} $$ Panel A of Table B.3 presents the estimates. An experiment is more likely to come from a prior study if it has a larger $p$, a smaller $r$, or an $M/(pH) \in [0.75,1]$. Using the estimated coefficients from this regression, we can assign to each experiment a predicted likelihood that it comes from a prior study. Specifically, the predicted likelihood for experiment $i$ is $$ \Pr(-\varepsilon_i < \hat{\beta}_0 + \hat{\beta}_1 p_i + \hat{\beta}_2 r_i + \hat{\beta}_3 \mathbb{1}\left[ M/(pH)_i \in [0.75,1] \right] + \hat{\beta}_4 \mathbb{1}\left[M/(pH)_i > 1 \right] ) $$ using a standard logistic distribution for $\varepsilon_i$. This predicted likelihood indicates how representative an experiment is of prior studies. Importantly, this predicted likelihood depends on only an experiment’s experimenter-chosen values for $p$, $r$, and $M/(pH)$, and is independent of the experiment’s observed realization for $CRE-RCRE$. We next compare experiments based on whether they are more representative of prior studies (predicted likelihood larger than 0.50) or more representative of our study (predicted likelihood smaller than 0.50). Panel B of Table B.3 reports the sample-weighted average $CRE-RCRE$ for experiments grouped by different predicted likelihoods and by experiment type. As discussed in Section IV.D, among the 143 prior experiments, 112 are more representative of prior studies and have an average $CRE-RCRE$ of 25.8 percent, while the other 31 have an average of 4.5 percent. Among our 120 experiments, 40 are more representative of prior studies and have an average $CRE-RCRE$ of 8.4 percent, while the other 80 have an average of $-0.1$ percent. In other words, when we (or prior studies) use experimental parameters that are more representative of prior studies, we find more CRE; in contrast, when we (or prior studies) use experimental parameters that are less representative of prior studies, we find much less CRE. **Table B.3: Comparing Prior Experiments to Our Experiments Using Predicted Likelihoods** llllll & & & & &\ \ \ Variable: & $p$ & $r$ & $\mathbb{1}[M/(pH)$ $\in [0.75,1]]$ & $\mathbb{1}[M/(pH) > 1]$ & Constant\ & & & & &\ Coefficient & 3.516& $-$1.463& 1.744& 0.583& $-$2.527\ Standard Error & (0.738)& (0.832)& (0.327)& (0.676)& (0.694)\ \ \ & & & & &\ &\ & $N$ & $CRE-RCRE$ & & $N$ & $CRE-RCRE$\ | | | | |:-----------------|:----|:------| | Prior Literature | 112 | 24.7% | | Our Experiments | 40 | 8.4% | | | | | |:-----------------|:----|:----------------| | Prior Literature | 31 | 4.5% | | Our Experiments | 80 | $-$0.1% | *Note:* Panel A reports results from logistic regression using the specification in the text. Panel B presents the number of experiments and the average $CRE-RCRE$ for the subset of experiments that are more representative of prior studies or our study based on the predicted likelihoods. All average $CRE-RCRE$ are calculated by weighting by the number of observations in the experiments. ## C Screenshots from the Online Experiment ### C.1 Screenshots from Stage 1: Valuation Tasks #### C.1.1 $m$-Valuation Tasks ![Figure C.1: Example Price List for Stage 1 AB m-Valuation Task with p = 0.2](assets/part1-mab-empty-142868.png) **Figure C.1: Example Price List for Stage 1 $AB$ $m$-Valuation Task with $p=0.2$** ![Figure C.2: Example Price List for Stage 1 AB m-Valuation Task with p = 0.2 (with example completion)](assets/part1-mab-completed-82ce47.png) **Figure C.2: Example Price List for Stage 1 $AB$ $m$-Valuation Task with $p=0.2$ (with example completion)** ![Figure C.3: Example Price List for Stage 1 CD m-Valuation Task with p = 0.2 and r = 0.6](assets/part1-mcd-empty-64d33c.png) **Figure C.3: Example Price List for Stage 1 $CD$ $m$-Valuation Task with $p=0.2$ and $r=0.6$** #### C.1.2 $h$-Valuation Tasks ![Figure C.4: Example Price List for Stage 1 AB h-Valuation Task with p = 0.2](assets/part1-hab-empty-44b320.png) **Figure C.4: Example Price List for Stage 1 $AB$ $h$-Valuation Task with $p=0.2$** ![Figure C.5: Example Price List for Stage 1 CD h-Valuation Task with p = 0.2 and r = 0.6](assets/part1-hcd-empty-34eeb0.png) **Figure C.5: Example Price List for Stage 1 $CD$ $h$-Valuation Task with $p=0.2$ and $r=0.6$** ### C.2 Screenshots from Stage 2: Binary-Choice Tasks #### C.2.1 $m$-Choice Tasks ![Figure C.6: Example Stage 2 Binary AB(M) Choice Task with p = 0.2 and M = $4](assets/part2-mab-empty-c78861.png) **Figure C.6: Example Stage 2 Binary $AB(M)$ Choice Task with $p=0.2$ and $M=\$4$** ![Figure C.7: Example Stage 2 Binary CD(M) Choice Task with p = 0.2, r = 0.6, and M = $4](assets/part2-mcd-empty-f40cd4.png) **Figure C.7: Example Stage 2 Binary $CD(M)$ Choice Task with $p=0.2$, $r=0.6$, and $M=\$4$** #### C.2.2 $h$-Choice Tasks ![Figure C.8: Example Stage 2 Binary AB(H) Choice Task with p = 0.2 and H = $20](assets/part2-hab-empty-744be3.png) **Figure C.8: Example Stage 2 Binary $AB(H)$ Choice Task with $p=0.2$ and $H=\$20$** ![Figure C.9: Example Stage 2 Binary AB(H) Choice Task with p = 0.2, and H = $20 (with example completion)](assets/part2-hab-completed-70fcc4.png) **Figure C.9: Example Stage 2 Binary $AB(H)$ Choice Task with $p=0.2$, and $H=\$20$ (with example completion)** ![Figure C.10: Example Stage 2 Binary CD(H) Choice Task with p = 0.2, r = 0.6, and H = $20](assets/part2-hcd-empty-f047df.png) **Figure C.10: Example Stage 2 Binary $CD(H)$ Choice Task with $p=0.2$, $r=0.6$, and $H=\$20$** ### C.3 Screenshots from Comprehension Checks and Visual Puzzle Task ![Figure C.11: Incentivized Comprehension Check 1: Multiple Price List](assets/quiz-question1-c90f74.png) **Figure C.11: Incentivized Comprehension Check 1: Multiple Price List** ![Figure C.12: Incentivized Comprehension Check 2: Binary-Choice Task](assets/quiz-question2-4710f4.png) **Figure C.12: Incentivized Comprehension Check 2: Binary-Choice Task** ![Figure C.13: Example: Camouflaged Animal Task](assets/camouflaged-animal-043bb8.png) **Figure C.13: Example: Camouflaged Animal Task** ## D Supplementary Figures and Tables **Table D.1: Participant Demographics** | | Full Sample | $r = 0.2$ | $r = 0.4$ | $r = 0.6$ | |:-----------------------------|:------------|:----------|:----------|:----------| | Number of Participants | 900 | 298 | 303 | 299 | | Time Taken (in minutes) | 27.3 | 27.3 | 27.8 | 26.9 | | Age | 22.4 | 22.5 | 22.2 | 22.5 | | Prolific Score | 99.9 | 99.9 | 99.9 | 99.9 | | Number of Approvals | 32.2 | 30.3 | 34.5 | 31.9 | | Female | 50.0 | 50.7 | 49.8 | 49.5 | | Current Student | 64.8 | 61.1 | 71.6 | 61.5 | | College Degree | 49.3 | 48.0 | 49.2 | 50.8 | | Working (full- or part-time) | 44.2 | 45.0 | 40.9 | 46.8 | | English First Language | 52.0 | 48.7 | 49.5 | 57.9 | | *Attention Checks* | | | | | | Incentive Question Correct | 92.3 | 93.3 | 92.1 | 91.6 | | Passed Attention Check | 83.7 | 82.2 | 85.1 | 83.6 | | *Comprehension Questions* | | | | | | MPL Question Correct | 83.9 | 81.5 | 84.8 | 85.3 | | Bin Question Correct | 86.2 | 85.9 | 86.1 | 86.6 | | Both Questions Correct | 74.0 | 70.8 | 75.2 | 75.9 | | *Current Residency* | | | | | | United States | 51.4 | 47.7 | 49.2 | 57.5 | | United Kingdom | 6.3 | 7.0 | 5.9 | 6.0 | | Portugal | 22.1 | 22.1 | 24.4 | 19.7 | | Spain | 5.4 | 6.4 | 4.0 | 6.0 | | Germany | 4.7 | 5.4 | 4.6 | 4.0 | *Note:* Participant demographics for all 900 participants. Each participant assigned to a single value of $r$. **Table D.2: Summary Statistics: $m$-Valuations**
Percentile
5-9 $p$ Mean SD 10th 25th 50th 75th 90th
Panel A: $r$ = 0.2 (298 participants)
$m_{AB}$ 0.1 8.86 6.56 0.50 4.50 9.50 10.50 15.50
$m_{CD}$ 0.1 7.31 6.67 0.50 2.50 5.50 9.50 15.50
$m_{AB}$ 0.2 10.75 6.59 2.50 5.50 9.50 14.50 19.50
$m_{CD}$ 0.2 9.46 7.05 0.50 4.50 9.50 11.50 19.50
$m_{AB}$ 0.5 15.95 6.34 9.50 13.50 15.00 19.50 24.50
$m_{CD}$ 0.5 15.99 7.01 7.50 10.50 15.50 20.50 24.50
$m_{AB}$ 0.8 19.54 7.06 9.50 15.50 19.50 24.50 29.50
$m_{CD}$ 0.8 20.55 7.46 9.50 15.50 22.50 25.50 29.50
$m_{AB}$ 0.9 22.78 7.32 10.50 19.50 24.50 29.50 29.50
$m_{CD}$ 0.9 21.31 8.00 9.50 14.50 24.50 28.50 29.50
Panel B: $r$ = 0.4 (303 participants)
$m_{AB}$ 0.1 7.29 5.83 0.50 3.50 5.50 9.50 14.50
$m_{CD}$ 0.1 6.66 6.15 0.50 2.50 4.50 9.50 14.50
$m_{AB}$ 0.2 9.33 6.12 0.50 4.50 9.50 13.50 15.50
$m_{CD}$ 0.2 8.20 5.88 0.50 4.50 7.50 10.50 15.50
$m_{AB}$ 0.5 14.48 6.37 6.50 10.50 14.50 19.50 20.50
$m_{CD}$ 0.5 13.27 6.71 4.50 9.50 14.50 15.50 20.50
$m_{AB}$ 0.8 19.22 7.43 9.50 14.50 19.50 24.50 29.50
$m_{CD}$ 0.8 18.63 7.41 9.50 14.50 19.50 24.50 27.50
$m_{AB}$ 0.9 21.55 7.59 9.50 19.50 24.50 26.50 29.50
$m_{CD}$ 0.9 21.39 8.10 9.50 18.50 24.50 27.50 29.50
Panel C: $r$ = 0.6 (299 participants)
$m_{AB}$ 0.1 6.76 5.46 0.50 2.50 5.50 9.50 14.50
$m_{CD}$ 0.1 6.27 5.61 0.50 2.50 4.50 9.50 12.50
$m_{AB}$ 0.2 8.57 5.59 0.50 4.50 9.50 10.50 15.50
$m_{CD}$ 0.2 8.70 5.99 0.50 4.50 9.50 10.50 15.50
$m_{AB}$ 0.5 14.36 6.37 5.50 10.50 14.50 18.50 20.50
$m_{CD}$ 0.5 12.32 6.25 4.50 9.50 13.50 15.50 19.50
$m_{AB}$ 0.8 19.36 7.12 9.50 14.50 19.50 24.50 29.50
$m_{CD}$ 0.8 18.11 7.05 9.50 13.50 19.50 23.50 25.50
$m_{AB}$ 0.9 23.05 6.94 11.50 19.50 24.50 28.50 29.50
$m_{CD}$ 0.9 21.01 7.04 10.50 15.50 22.50 26.50 29.50
*Note:* Summary statistics for all 30 $m$-valuations. Each participant was assigned a single $r$, and completed all 10 $m$-valuations for that value of $r$. Each valuation is equal to the average value of $M$ at the participant’s switching rows in a multiple-price list (MPL). Each MPL permits the valuations to range from $-0.50$ to $30.50$. **Table D.3: Adjusting the Sign Test for Ties ($m$-Valuation Tasks)**
(1) (2) (3) (4) (5) (6) (7)
Number of Cases Sign Tests
2-4 $\Delta m>0$ $\Delta m=0$ $\Delta m<0$ Default Equal Split Prop. Split
$(CRE)$ $(RCRE)$ ($p$-value) ($p$-value) ($p$-value)
Panel A: $r=0.2$ (298 participants)

3-6

0.1

 79 75 144 0.000 0.000 0.000
0.2 80 73 145 0.000 0.000 0.000
0.5 123 60 115 0.650 0.685 0.524
0.8 140 54 104 0.025 0.042 0.013
0.9 127 42 129 0.950 0.954 0.862
Panel B: $r=0.4$ (303 participants)

3-6

0.1

 103 71 129 0.101 0.135 0.051
0.2 97 65 141 0.005 0.011 0.001
0.5 104 62 137 0.039 0.066 0.016
0.8 127 41 135 0.665 0.646 0.566
0.9 124 52 127 0.900 0.909 0.818
Panel C: $r=0.6$ (299 participants)

3-6

0.1

 94 90 115 0.166 0.247 0.083
0.2 111 84 104 0.682 0.729 0.563
0.5 89 65 145 0.000 0.001 0.000
0.8 113 57 129 0.335 0.355 0.247
0.9 79 60 160 0.000 0.000 0.000
*Note:* Columns (2)-(4) report raw frequencies of $\Delta m>0$, $\Delta m=0$, and $\Delta m<0$ (identical to those reported in columns (4)-(6) in Table 3). Column (5) reports the $p$-values for the default sign tests (identical to those reported in column (7) in Table 3) that exclude all ties (instances of $\Delta m = 0$). The adjusted sign tests in column (6) split ties equally between $\Delta m>0$ and $\Delta m<0$. The adjusted sign tests in column (7) split ties in proportion to the observed share of $\Delta m>0$ and $\Delta m<0$. **Table D.4: Predictions for $\Delta m$ under PT Versus Observed $\Delta m$ in Data**
(1) (2) (3) (4)
Current Estimates:
2-4 Lower Bound Point Estimate Upper Bound Data
Panel A: $r = 0.2$
$p$ = 0.1 4.07 4.76 5.45 $-$1.55
$p$ = 0.2 5.66 6.40 7.15 $-$1.29
$p$ = 0.5 7.82 8.54 9.26 0.04
$p$ = 0.8 7.20 7.84 8.48 1.00
$p$ = 0.9 5.78 6.36 6.94 $-$1.47
Panel B: $r = 0.4$
$p$ = 0.1 4.02 4.69 5.36 $-$0.63
$p$ = 0.2 5.62 6.34 7.06 $-$1.14
$p$ = 0.5 7.94 8.63 9.31 $-$1.22
$p$ = 0.8 7.56 8.19 8.82 $-$0.60
$p$ = 0.9 6.17 6.75 7.33 $-$0.16
Panel C: $r = 0.6$
$p$ = 0.1 2.66 3.15 3.64 $-$0.49
$p$ = 0.2 3.89 4.45 5.01 0.14
$p$ = 0.5 5.89 6.49 7.09 $-$2.05
$p$ = 0.8 5.89 6.47 7.05 $-$1.26
$p$ = 0.9 4.84 5.37 5.90 $-$2.03
*Note:* Columns (1)-(4) present predictions for $\Delta m^{\ast}\equiv m_{CD}^{\ast}-m_{AB}^{\ast}$ under a PT model with $\pi (q)=q^{\gamma }/\left[ q^{\gamma }+(1-q)^{\gamma }\right]^{1/\gamma }$ and $v(x)=x^{\alpha }$. Columns (1)-(3) use parameter estimates based on our stage 1 $m_{AB}$-valuations reported in Appendix Table E.1, with separate estimates for each $r$. Column (2) reports predictions using the point estimates, while columns (1) and (3) report lower and upper bounds of the 95 percent confidence interval computed using the delta method. Column (4) reports mean $\Delta m$ values in our data from the $m$-valuation tasks. **Table D.5: Summary Statistics: $h$-Valuations**
Percentile
5-9 $p$ Mean SD 10th 25th 50th 75th 90th
Panel A: $r$ = 0.2 (298 participants)
$h_{AB}$ 0.1 19.99 9.50 7.50 10.50 19.50 29.50 32.50
$h_{CD}$ 0.1 21.66 9.72 9.50 12.50 23.50 30.50 33.50
$h_{AB}$ 0.2 22.54 8.26 11.50 15.50 21.50 29.50 34.50
$h_{CD}$ 0.2 24.49 8.78 12.50 18.50 25.50 30.50 36.50
$h_{AB}$ 0.5 30.59 8.26 19.50 24.50 29.50 35.50 44.50
$h_{CD}$ 0.5 28.95 7.68 19.50 24.50 29.50 32.50 39.50
$h_{AB}$ 0.8 36.54 8.74 26.50 29.50 34.50 40.50 51.50
$h_{CD}$ 0.8 32.22 6.83 24.50 28.50 30.00 35.50 41.50
$h_{AB}$ 0.9 35.92 8.54 27.50 29.50 34.50 39.50 49.50
$h_{CD}$ 0.9 33.82 7.43 27.50 28.50 30.50 36.50 44.50
Panel B: $r$ = 0.4 (303 participants)
$h_{AB}$ 0.1 21.48 9.37 9.50 14.50 20.50 29.50 33.50
$h_{CD}$ 0.1 24.01 8.86 9.50 18.50 26.50 31.50 33.50
$h_{AB}$ 0.2 24.46 7.95 13.50 19.50 24.50 30.50 36.50
$h_{CD}$ 0.2 26.04 8.48 13.50 19.50 29.50 32.50 36.50
$h_{AB}$ 0.5 30.34 7.34 20.50 25.50 29.50 33.50 40.50
$h_{CD}$ 0.5 31.39 7.06 22.50 28.50 30.50 35.50 40.50
$h_{AB}$ 0.8 36.07 8.57 26.50 29.50 34.50 39.50 49.50
$h_{CD}$ 0.8 34.22 7.21 27.50 29.50 31.50 39.50 45.50
$h_{AB}$ 0.9 35.66 8.66 27.50 29.50 33.50 39.50 50.50
$h_{CD}$ 0.9 34.53 7.37 27.50 29.50 32.50 37.50 43.50
Panel C: $r$ = 0.6 (299 participants)
$h_{AB}$ 0.1 22.54 9.09 9.50 14.50 24.50 30.50 33.50
$h_{CD}$ 0.1 23.27 9.51 9.50 14.50 26.50 30.50 33.50
$h_{AB}$ 0.2 25.07 8.00 14.50 19.50 25.50 30.50 36.50
$h_{CD}$ 0.2 24.24 8.47 12.50 17.50 24.50 30.50 36.50
$h_{AB}$ 0.5 30.16 7.20 20.50 24.50 29.50 33.50 40.50
$h_{CD}$ 0.5 30.88 7.43 20.50 25.50 29.50 35.50 40.50
$h_{AB}$ 0.8 35.13 8.18 24.50 29.50 33.50 39.50 48.50
$h_{CD}$ 0.8 34.29 7.67 26.50 29.50 31.50 39.50 45.50
$h_{AB}$ 0.9 35.29 8.38 27.50 29.50 32.50 39.50 49.50
$h_{CD}$ 0.9 34.52 6.82 27.50 29.50 32.50 38.50 44.50
*Note:* Summary statistics for all 30 $h$-valuations. Each participant was assigned a single $r$, and completed all 10 $h$-valuations for that value of $r$. Each valuation is equal to the average value of $H$ at the participant’s switching rows in a multiple-price list (MPL). Each MPL permits the valuations to range from $p30-0.50$ to $p30+30.50$. **Table D.6: Adjusting the Sign Test for Ties ($h$-Valuation Tasks)**
(1) (2) (3) (4) (5) (6) (7)
Number of Cases Sign Tests
2-4 $\Delta h>0$ $\Delta h=0$ $\Delta h<0$ Default Equal Split Prop. Split
$(CRE)$ $(RCRE)$ ($p$-value) ($p$-value) ($p$-value)
Panel A: $r=0.2$ (298 participants)

3-6

0.1

 100 60 138 0.016 0.032 0.006
0.2 94 53 151 0.000 0.001 0.000
0.5 136 81 81 0.000 0.001 0.000
0.8 174 45 79 0.000 0.000 0.000
0.9 143 64 91 0.001 0.003 0.000
Panel B: $r=0.4$ (303 participants)

3-6

0.1

 82 59 162 0.000 0.000 0.000
0.2 92 65 146 0.001 0.002 0.000
0.5 101 70 132 0.049 0.085 0.021
0.8 148 47 108 0.015 0.021 0.006
0.9 138 47 118 0.235 0.251 0.168
Panel C: $r=0.6$ (299 participants)

3-6

0.1

 100 71 128 0.074 0.105 0.037
0.2 131 65 103 0.077 0.105 0.037
0.5 93 85 121 0.065 0.105 0.021
0.8 136 47 116 0.231 0.247 0.165
0.9 126 54 119 0.702 0.729 0.644
*Note:* Columns (2)-(4) report raw frequencies of $\Delta h>0$, $\Delta h=0$, and $\Delta h<0$ (identical to those reported in columns (4)-(6) in Table 4). Column (5) reports the $p$-values for the default sign tests (identical to those reported in column (7) in Table 4) that exclude all ties (instances of $\Delta h = 0$). The adjusted sign tests in column (6) split ties equally between $\Delta h>0$ and $\Delta h<0$. The adjusted sign tests in column (7) split ties in proportion to the observed share of $\Delta h>0$ and $\Delta h<0$. **Table D.7: Correlations of Risk Premia across Corresponding $m$- and $h$-Valuations**
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Variant $r$ Panel A: Pearson’s Correlation Panel B: Spearman’s Rank Correlation
$p = 0.1$ $p = 0.2$ $p = 0.5$ $p = 0.8$ $p = 0.9$ $p = 0.1$ $p = 0.2$ $p = 0.5$ $p = 0.8$ $p = 0.9$
$AB$ 0.2 0.19$^{*}$ 0.30$^{*}$ 0.42$^{*}$ 0.31$^{*}$ 0.34$^{*}$ 0.30$^{*}$ 0.28$^{*}$ 0.36$^{*}$ 0.34$^{*}$ 0.35$^{*}$
$CD$ 0.2 0.21$^{*}$ 0.26$^{*}$ 0.25$^{*}$ 0.21$^{*}$ 0.30$^{*}$ 0.23$^{*}$ 0.26$^{*}$ 0.18$^{*}$ 0.22$^{*}$ 0.30$^{*}$
$AB$ 0.4 0.19$^{*}$ 0.25$^{*}$ 0.34$^{*}$ 0.34$^{*}$ 0.25$^{*}$ 0.34$^{*}$ 0.41$^{*}$ 0.30$^{*}$ 0.30$^{*}$ 0.27$^{*}$
$CD$ 0.4 0.03 0.11 0.04 0.28$^{*}$ 0.16$^{*}$ 0.23$^{*}$ 0.30$^{*}$ 0.14$^{*}$ 0.22$^{*}$ 0.20$^{*}$
$AB$ 0.6 0.15$^{*}$ 0.25$^{*}$ 0.44$^{*}$ 0.27$^{*}$ 0.33$^{*}$ 0.33$^{*}$ 0.33$^{*}$ 0.37$^{*}$ 0.27$^{*}$ 0.33$^{*}$
$CD$ 0.6 0.19$^{*}$ 0.16$^{*}$ 0.12$^{*}$ 0.30$^{*}$ 0.22$^{*}$ 0.40$^{*}$ 0.32$^{*}$ 0.17$^{*}$ 0.27$^{*}$ 0.22$^{*}$
*Note:* Correlations between $m_z/H$ and $M/h_z$ (where $H=30$ and $M=p30$) for each of the 30 combinations of $(p,r)$ and $x\in\{AB,CD\}$. Panel A reports Pearson’s correlations; panel B reports Spearman’s rank correlations. $^*$ denotes that a correlation is statistically significant at the 5 percent level. **Table D.8: Correlations across $p$ for Paired $m$-Valuation Tasks**
Panel A: Correlations of $pH - \overline{m}$ across $p$
$p=$
$0.1$ $0.2$ $0.5$ $0.8$ $0.9$
(i) $r=0.2$
$p = 0.1$ 1.00
$p = 0.2$ 0.61$^{*}$ 1.00
$p = 0.5$ 0.42$^{*}$ 0.43$^{*}$ 1.00
$p = 0.8$ 0.12 0.20$^{*}$ 0.33$^{*}$ 1.00
$p = 0.9$ 0.09 0.11 0.26$^{*}$ 0.53$^{*}$ 1.00
(ii) $r=0.4$
$p = 0.1$ 1.00
$p = 0.2$ 0.56$^{*}$ 1.00
$p = 0.5$ 0.33$^{*}$ 0.37$^{*}$ 1.00
$p = 0.8$ 0.08 0.15$^{*}$ 0.37$^{*}$ 1.00
$p = 0.9$ 0.05 0.15 0.32$^{*}$ 0.54$^{*}$ 1.00
(iii) $r=0.6$
$p = 0.1$ 1.00
$p = 0.2$ 0.61$^{*}$ 1.00
$p = 0.5$ 0.40$^{*}$ 0.49$^{*}$ 1.00
$p = 0.8$ 0.09 0.26$^{*}$ 0.40$^{*}$ 1.00
$p = 0.9$ 0.06 0.13 0.28$^{*}$ 0.49$^{*}$ 1.00
Panel B: Correlations of $\Delta m$ across $p$
$p=$
$0.1$ $0.2$ $0.5$ $0.8$ $0.9$
(i) $r=0.2$
1.00
0.16$^{*}$ 1.00
0.01 0.15$^{*}$ 1.00
0.06 0.11 0.11 1.00
0.03 0.13 0.16$^{*}$ 0.25$^{*}$ 1.00
(ii) $r=0.4$
1.00
0.05 1.00
0.10 0.16$^{*}$ 1.00
0.08 0.16$^{*}$ 0.31$^{*}$ 1.00
0.08 0.07 0.21$^{*}$ 0.28$^{*}$ 1.00
(iii) $r=0.6$
1.00
0.09 1.00
0.04 $-$0.04 1.00
0.10 0.00 0.04 1.00
0.10 $-$0.09 0.09 0.19$^{*}$ 1.00
*Note:* Spearman’s rank correlations of $pH-\overline{m}$ across $p$ (panel A) and of $\Delta m$ across $p$ (panel B) for the 15 paired $m$-valuation tasks. $^*$ denotes that a correlation is statistically significant at the 5 percent level. **Table D.9: Correlations of Value Differences across Corresponding $m$- and $h$-Valuations**
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
$r$ Panel A: Pearson’s Correlation Panel B: Spearman’s Rank Correlation
$p = 0.1$ $p = 0.2$ $p = 0.5$ $p = 0.8$ $p = 0.9$ $p = 0.1$ $p = 0.2$ $p = 0.5$ $p = 0.8$ $p = 0.9$
0.2 0.10 0.16$^{*}$ 0.19$^{*}$ 0.29$^{*}$ 0.28$^{*}$ 0.13$^{*}$ 0.10 0.12$^{*}$ 0.32$^{*}$ 0.30$^{*}$
0.4 $-$0.01 0.08 0.17$^{*}$ 0.26$^{*}$ 0.16$^{*}$ 0.12$^{*}$ 0.15$^{*}$ 0.15$^{*}$ 0.24$^{*}$ 0.14$^{*}$
0.6 $-$0.02 0.01 0.28$^{*}$ 0.20$^{*}$ 0.20$^{*}$ 0.02 $-$0.00 0.24$^{*}$ 0.14$^{*}$ 0.14$^{*}$
*Note:* Correlations between $(m_{CD}/H-m_{AB}/H)$ and $(M/h_{CD}-M/h_{AB})$ (where $H=30$ and $M=p30$) for each of the 15 combinations of $(p,r)$. Panel A reports Pearson’s correlations; panel B reports Spearman’s rank correlations. $^*$ denotes that a correlation is statistically significant at the 5 percent level. **Table D.10: Predicting Individual-Level $CRE-RCRE$ ($h$ tasks)**
(1) (2) (3) (4)
Outcome: $CRE - RCRE \in \{-1,0,1\}$
OLS OLS OLS 2SLS
Value Difference
$\frac{1 + r}{2}p\Delta h$ 2.02 1.96 7.62
(0.22) (0.22) (1.35)
Distance to Indifference
$(1$ $-$ $r$)$p$($\overline{h}$ $-$ $H)$ 1.06 0.84 0.21
(0.24) (0.24) (0.44)
Outcome Mean 2.73 2.73 2.73 2.73
Individuals 900 900 900 900
Observations 4,500 4,500 4,500 4,500
*Note:* OLS regressions using individual-level $h$-task data with dependent variable $CRE-RCRE \in \{-1,0,1\}$. Specifications include $p$ and $r$ fixed effects, as well as controls for gender, education, age, language, student status, employment, and the number of previous Prolific approvals. All numbers reported in percentage points; individual-cluster-robust standard errors in parentheses. For column (4), instruments are $(1-r)\overline{m}$, $0.5(1+r)\Delta m$, and $(1-r)pH$. **Table D.11: Summary of Choice Patterns: Experiments Linked to $m$-Valuations** | | | | | | | | | | | | | |:---|:---|:---|:---|:---|:---|:---|:---|:---|:---|:---|:---| | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | | $r$ | $p$ | $M$ | Mean $\overline{m}$ | Mean $\Delta m$ | $AC$ | $AD$ $(CRE)$ | $BC$ $(RCRE)$ | $BD$ | $CRE$ $-$ $RCRE$ | Conlisk $p$-value | $N$ | | 0.2 | 0.1 | 1 | 8.41 | $-$0.46 | 9.0 | 7.7 | 6.4 | 76.9 | 1.3 | 0.764 | 78 | | 0.2 | 0.1 | 3 | 7.93 | $-$2.27 | 30.7 | 16.0 | 9.3 | 44.0 | 6.7 | 0.250 | 75 | | 0.2 | 0.1 | 5 | 8.38 | $-$1.15 | 70.6 | 11.8 | 10.3 | 7.4 | 1.5 | 0.798 | 68 | | 0.2 | 0.1 | 8 | 7.64 | $-$2.30 | 87.0 | 6.5 | 3.9 | 2.6 | 2.6 | 0.481 | 77 | | 0.2 | 0.2 | 1 | 10.26 | $-$1.49 | 2.6 | 6.5 | 2.6 | 88.3 | 3.9 | 0.256 | 77 | | 0.2 | 0.2 | 4 | 9.80 | $-$1.68 | 31.6 | 7.9 | 27.6 | 32.9 | $-$19.7 | 0.002 | 76 | | 0.2 | 0.2 | 7 | 10.25 | $-$0.83 | 40.3 | 9.7 | 27.8 | 22.2 | $-$18.1 | 0.009 | 72 | | 0.2 | 0.2 | 10 | 10.12 | $-$1.10 | 84.9 | 4.1 | 8.2 | 2.7 | $-$4.1 | 0.317 | 73 | | 0.2 | 0.5 | 5 | 16.42 | 0.69 | 3.4 | 10.3 | 9.2 | 77.0 | 1.1 | 0.809 | 87 | | 0.2 | 0.5 | 8 | 15.68 | 0.19 | 10.2 | 18.6 | 16.9 | 54.2 | 1.7 | 0.829 | 59 | | 0.2 | 0.5 | 11 | 16.57 | 0.09 | 24.7 | 13.6 | 21.0 | 40.7 | $-$7.4 | 0.256 | 81 | | 0.2 | 0.5 | 14 | 14.98 | $-$0.93 | 39.4 | 22.5 | 21.1 | 16.9 | 1.4 | 0.858 | 71 | | 0.2 | 0.8 | 8 | 20.49 | $-$0.16 | 1.5 | 9.0 | 13.4 | 76.1 | $-$4.5 | 0.440 | 67 | | 0.2 | 0.8 | 12 | 19.83 | 0.58 | 9.9 | 22.5 | 4.2 | 63.4 | 18.3 | 0.002 | 71 | | 0.2 | 0.8 | 16 | 19.95 | 2.46 | 14.8 | 18.5 | 11.1 | 55.6 | 7.4 | 0.219 | 81 | | 0.2 | 0.8 | 20 | 19.96 | 0.89 | 25.3 | 39.2 | 15.2 | 20.3 | 24.1 | 0.002 | 79 | | 0.2 | 0.9 | 10 | 21.70 | $-$0.91 | 6.3 | 8.9 | 6.3 | 78.5 | 2.5 | 0.565 | 79 | | 0.2 | 0.9 | 14 | 23.20 | $-$1.61 | 2.4 | 13.1 | 7.1 | 77.4 | 6.0 | 0.224 | 84 | | 0.2 | 0.9 | 18 | 21.08 | $-$2.41 | 7.1 | 24.3 | 14.3 | 54.3 | 10.0 | 0.175 | 70 | | 0.2 | 0.9 | 22 | 22.02 | $-$0.94 | 16.9 | 36.9 | 9.2 | 36.9 | 27.7 | 0.000 | 65 | | 0.4 | 0.1 | 1 | 7.71 | $-$0.80 | 11.4 | 13.9 | 7.6 | 67.1 | 6.3 | 0.224 | 79 | | 0.4 | 0.1 | 3 | 6.53 | $-$0.27 | 63.6 | 6.5 | 6.5 | 23.4 | 0.0 | 1.000 | 77 | | 0.4 | 0.1 | 5 | 6.69 | $-$0.62 | 75.3 | 9.0 | 6.7 | 9.0 | 2.2 | 0.594 | 89 | | 0.4 | 0.1 | 8 | 7.02 | $-$0.90 | 93.1 | 0.0 | 5.2 | 1.7 | $-$5.2 | 0.078 | 58 | | 0.4 | 0.2 | 1 | 8.86 | $-$0.61 | 9.0 | 10.1 | 3.4 | 77.5 | 6.7 | 0.080 | 89 | | 0.4 | 0.2 | 4 | 8.95 | $-$2.20 | 32.9 | 14.5 | 18.4 | 34.2 | $-$3.9 | 0.550 | 76 | | 0.4 | 0.2 | 7 | 9.74 | $-$0.78 | 70.1 | 7.5 | 11.9 | 10.4 | $-$4.5 | 0.406 | 67 | | 0.4 | 0.2 | 10 | 7.52 | $-$1.00 | 88.7 | 8.5 | 2.8 | 0.0 | 5.6 | 0.154 | 71 | | 0.4 | 0.5 | 5 | 13.89 | 0.48 | 17.9 | 4.8 | 16.7 | 60.7 | $-$11.9 | 0.015 | 84 | | 0.4 | 0.5 | 8 | 13.52 | $-$1.26 | 13.8 | 12.5 | 27.5 | 46.2 | $-$15.0 | 0.030 | 80 | | 0.4 | 0.5 | 11 | 14.90 | $-$2.32 | 32.2 | 13.6 | 27.1 | 27.1 | $-$13.6 | 0.098 | 59 | | 0.4 | 0.5 | 14 | 13.46 | $-$2.14 | 46.2 | 11.2 | 21.2 | 21.2 | $-$10.0 | 0.113 | 80 | | 0.4 | 0.8 | 8 | 19.13 | $-$1.78 | 2.5 | 16.5 | 1.3 | 79.7 | 15.2 | 0.001 | 79 | | 0.4 | 0.8 | 12 | 19.15 | $-$0.77 | 6.3 | 19.0 | 12.7 | 62.0 | 6.3 | 0.317 | 79 | | 0.4 | 0.8 | 16 | 18.47 | $-$0.97 | 10.3 | 26.9 | 12.8 | 50.0 | 14.1 | 0.044 | 78 | | 0.4 | 0.8 | 20 | 18.94 | 1.45 | 31.3 | 40.3 | 7.5 | 20.9 | 32.8 | 0.000 | 67 | | 0.4 | 0.9 | 10 | 21.75 | 0.54 | 5.6 | 13.5 | 2.2 | 78.7 | 11.2 | 0.006 | 89 | | 0.4 | 0.9 | 14 | 22.30 | 0.12 | 4.6 | 12.3 | 4.6 | 78.5 | 7.7 | 0.128 | 65 | | 0.4 | 0.9 | 18 | 20.51 | $-$0.29 | 4.0 | 17.3 | 6.7 | 72.0 | 10.7 | 0.055 | 75 | | 0.4 | 0.9 | 22 | 21.37 | $-$1.09 | 36.5 | 32.4 | 13.5 | 17.6 | 18.9 | 0.013 | 74 | | 0.6 | 0.1 | 1 | 6.82 | $-$0.25 | 15.5 | 15.5 | 9.9 | 59.2 | 5.6 | 0.346 | 71 | | 0.6 | 0.1 | 3 | 6.68 | $-$0.94 | 51.5 | 8.8 | 11.8 | 27.9 | $-$2.9 | 0.595 | 68 | | 0.6 | 0.1 | 5 | 6.23 | $-$1.17 | 72.7 | 7.8 | 5.2 | 14.3 | 2.6 | 0.529 | 77 | | 0.6 | 0.1 | 8 | 6.37 | 0.30 | 78.3 | 9.6 | 8.4 | 3.6 | 1.2 | 0.797 | 83 | | 0.6 | 0.2 | 1 | 8.39 | 1.15 | 6.1 | 7.6 | 1.5 | 84.8 | 6.1 | 0.098 | 66 | | 0.6 | 0.2 | 4 | 9.12 | $-$1.15 | 37.7 | 11.5 | 16.4 | 34.4 | $-$4.9 | 0.469 | 61 | | 0.6 | 0.2 | 7 | 8.68 | 0.17 | 72.8 | 10.9 | 6.5 | 9.8 | 4.3 | 0.317 | 92 | | 0.6 | 0.2 | 10 | 8.41 | 0.24 | 88.8 | 5.0 | 2.5 | 3.8 | 2.5 | 0.415 | 80 | | 0.6 | 0.5 | 5 | 14.54 | $-$2.62 | 14.3 | 5.2 | 22.1 | 58.4 | $-$16.9 | 0.003 | 77 | | 0.6 | 0.5 | 8 | 12.79 | $-$2.14 | 15.3 | 16.7 | 30.6 | 37.5 | $-$13.9 | 0.082 | 72 | | 0.6 | 0.5 | 11 | 13.27 | $-$2.13 | 38.9 | 13.3 | 24.4 | 23.3 | $-$11.1 | 0.083 | 90 | | 0.6 | 0.5 | 14 | 12.57 | $-$1.07 | 45.0 | 10.0 | 21.7 | 23.3 | $-$11.7 | 0.104 | 60 | | 0.6 | 0.8 | 8 | 18.29 | $-$1.49 | 4.3 | 7.6 | 5.4 | 82.6 | 2.2 | 0.565 | 92 | | 0.6 | 0.8 | 12 | 18.37 | $-$1.35 | 10.1 | 16.5 | 8.9 | 64.6 | 7.6 | 0.177 | 79 | | 0.6 | 0.8 | 16 | 19.33 | $-$2.19 | 11.8 | 19.1 | 17.6 | 51.5 | 1.5 | 0.843 | 68 | | 0.6 | 0.8 | 20 | 19.23 | 0.28 | 45.0 | 21.7 | 15.0 | 18.3 | 6.7 | 0.395 | 60 | | 0.6 | 0.9 | 10 | 20.74 | $-$3.40 | 8.3 | 6.9 | 2.8 | 81.9 | 4.2 | 0.256 | 72 | | 0.6 | 0.9 | 14 | 22.65 | $-$0.14 | 7.6 | 11.4 | 8.9 | 72.2 | 2.5 | 0.619 | 79 | | 0.6 | 0.9 | 18 | 21.99 | $-$2.23 | 7.7 | 26.9 | 9.0 | 56.4 | 17.9 | 0.006 | 78 | | 0.6 | 0.9 | 22 | 22.70 | $-$2.54 | 17.1 | 34.3 | 15.7 | 32.9 | 18.6 | 0.024 | 70 | **Table D.12: Summary of Choice Patterns: Experiments Linked to $h$-Valuations** | | | | | | | | | | | | | |:---|:---|:---|:---|:---|:---|:---|:---|:---|:---|:---|:---| | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | | $r$ | $p$ | $H$ | Mean $\overline{h}$ | Mean $\Delta h$ | $AC$ | $AD$ $(CRE)$ | $BC$ $(RCRE)$ | $BD$ | $CRE$ $-$ $RCRE$ | Conlisk $p$-value | $N$ | | 0.2 | 0.1 | 13 | 20.99 | $-$1.85 | 63.5 | 14.9 | 13.5 | 8.1 | 1.4 | 0.828 | 74 | | 0.2 | 0.1 | 20 | 21.10 | $-$1.31 | 38.6 | 13.3 | 16.9 | 31.3 | $-$3.6 | 0.550 | 83 | | 0.2 | 0.1 | 25 | 20.57 | $-$0.85 | 39.7 | 19.1 | 10.3 | 30.9 | 8.8 | 0.177 | 68 | | 0.2 | 0.1 | 30 | 20.59 | $-$2.67 | 35.6 | 9.6 | 17.8 | 37.0 | $-$8.2 | 0.177 | 73 | | 0.2 | 0.2 | 20 | 22.54 | $-$2.57 | 52.2 | 23.2 | 20.3 | 4.3 | 2.9 | 0.717 | 69 | | 0.2 | 0.2 | 25 | 23.35 | $-$1.18 | 49.3 | 8.2 | 20.5 | 21.9 | $-$12.3 | 0.045 | 73 | | 0.2 | 0.2 | 30 | 23.58 | $-$1.65 | 51.4 | 8.3 | 18.1 | 22.2 | $-$9.7 | 0.104 | 72 | | 0.2 | 0.2 | 35 | 24.41 | $-$2.35 | 35.7 | 13.1 | 22.6 | 28.6 | $-$9.5 | 0.141 | 84 | | 0.2 | 0.5 | 30 | 30.80 | 3.46 | 57.7 | 22.5 | 11.3 | 8.5 | 11.3 | 0.098 | 71 | | 0.2 | 0.5 | 35 | 29.90 | 1.60 | 37.2 | 14.1 | 14.1 | 34.6 | 0.0 | 1.000 | 78 | | 0.2 | 0.5 | 40 | 29.65 | 0.83 | 12.5 | 23.6 | 19.4 | 44.4 | 4.2 | 0.592 | 72 | | 0.2 | 0.5 | 45 | 28.79 | 0.77 | 15.6 | 10.4 | 23.4 | 50.6 | $-$13.0 | 0.046 | 77 | | 0.2 | 0.8 | 33 | 34.58 | 5.23 | 30.0 | 41.7 | 13.3 | 15.0 | 28.3 | 0.001 | 60 | | 0.2 | 0.8 | 38 | 33.72 | 5.70 | 26.9 | 32.8 | 11.9 | 28.4 | 20.9 | 0.008 | 67 | | 0.2 | 0.8 | 45 | 34.86 | 3.74 | 11.9 | 31.7 | 8.9 | 47.5 | 22.8 | 0.000 | 101 | | 0.2 | 0.8 | 52 | 34.15 | 3.01 | 10.0 | 17.1 | 8.6 | 64.3 | 8.6 | 0.154 | 70 | | 0.2 | 0.9 | 35 | 33.31 | 0.17 | 22.2 | 30.6 | 18.1 | 29.2 | 12.5 | 0.125 | 72 | | 0.2 | 0.9 | 40 | 35.30 | 2.26 | 9.2 | 19.7 | 15.8 | 55.3 | 3.9 | 0.565 | 76 | | 0.2 | 0.9 | 47 | 34.70 | 4.03 | 6.7 | 25.3 | 5.3 | 62.7 | 20.0 | 0.001 | 75 | | 0.2 | 0.9 | 54 | 36.10 | 1.89 | 4.0 | 17.3 | 8.0 | 70.7 | 9.3 | 0.105 | 75 | | 0.4 | 0.1 | 13 | 21.38 | $-$2.97 | 72.7 | 10.2 | 5.7 | 11.4 | 4.5 | 0.285 | 88 | | 0.4 | 0.1 | 20 | 23.86 | $-$0.48 | 61.7 | 13.3 | 8.3 | 16.7 | 5.0 | 0.407 | 60 | | 0.4 | 0.1 | 25 | 23.56 | $-$3.84 | 56.7 | 8.9 | 11.1 | 23.3 | $-$2.2 | 0.639 | 90 | | 0.4 | 0.1 | 30 | 22.45 | $-$2.00 | 33.8 | 12.3 | 16.9 | 36.9 | $-$4.6 | 0.493 | 65 | | 0.4 | 0.2 | 20 | 25.97 | $-$1.75 | 73.9 | 10.1 | 10.1 | 5.8 | 0.0 | 1.000 | 69 | | 0.4 | 0.2 | 25 | 25.68 | $-$1.54 | 57.9 | 10.5 | 15.8 | 15.8 | $-$5.3 | 0.440 | 57 | | 0.4 | 0.2 | 30 | 25.81 | $-$2.37 | 62.7 | 8.4 | 12.0 | 16.9 | $-$3.6 | 0.468 | 83 | | 0.4 | 0.2 | 35 | 23.97 | $-$0.80 | 51.1 | 12.8 | 11.7 | 24.5 | 1.1 | 0.836 | 94 | | 0.4 | 0.5 | 30 | 30.67 | $-$3.69 | 65.4 | 12.8 | 17.9 | 3.8 | $-$5.1 | 0.415 | 78 | | 0.4 | 0.5 | 35 | 30.14 | $-$0.57 | 41.5 | 17.1 | 22.0 | 19.5 | $-$4.9 | 0.481 | 82 | | 0.4 | 0.5 | 40 | 31.00 | $-$0.24 | 41.2 | 13.2 | 25.0 | 20.6 | $-$11.8 | 0.113 | 68 | | 0.4 | 0.5 | 45 | 31.76 | 0.44 | 33.3 | 9.3 | 29.3 | 28.0 | $-$20.0 | 0.003 | 75 | | 0.4 | 0.8 | 33 | 35.02 | 1.99 | 43.2 | 31.1 | 2.7 | 23.0 | 28.4 | 0.000 | 74 | | 0.4 | 0.8 | 38 | 35.37 | 4.00 | 16.9 | 46.5 | 8.5 | 28.2 | 38.0 | 0.000 | 71 | | 0.4 | 0.8 | 45 | 35.96 | 1.08 | 14.9 | 24.1 | 19.5 | 41.4 | 4.6 | 0.518 | 87 | | 0.4 | 0.8 | 52 | 34.05 | 0.48 | 11.3 | 15.5 | 9.9 | 63.4 | 5.6 | 0.346 | 71 | | 0.4 | 0.9 | 35 | 35.76 | 0.41 | 26.0 | 28.8 | 8.2 | 37.0 | 20.5 | 0.002 | 73 | | 0.4 | 0.9 | 40 | 33.84 | 1.95 | 13.2 | 26.3 | 7.9 | 52.6 | 18.4 | 0.004 | 76 | | 0.4 | 0.9 | 47 | 35.18 | 0.39 | 4.2 | 23.6 | 9.7 | 62.5 | 13.9 | 0.037 | 72 | | 0.4 | 0.9 | 54 | 35.59 | 1.65 | 8.5 | 23.2 | 6.1 | 62.2 | 17.1 | 0.003 | 82 | | 0.6 | 0.1 | 13 | 21.05 | 0.34 | 73.5 | 4.4 | 10.3 | 11.8 | $-$5.9 | 0.204 | 68 | | 0.6 | 0.1 | 20 | 22.85 | $-$1.01 | 60.6 | 9.9 | 11.3 | 18.3 | $-$1.4 | 0.798 | 71 | | 0.6 | 0.1 | 25 | 23.33 | $-$0.56 | 49.4 | 13.6 | 13.6 | 23.5 | 0.0 | 1.000 | 81 | | 0.6 | 0.1 | 30 | 24.09 | $-$1.57 | 54.4 | 12.7 | 10.1 | 22.8 | 2.5 | 0.639 | 79 | | 0.6 | 0.2 | 20 | 25.94 | 0.79 | 74.6 | 11.3 | 7.0 | 7.0 | 4.2 | 0.406 | 71 | | 0.6 | 0.2 | 25 | 23.64 | 1.07 | 66.3 | 8.4 | 12.0 | 13.3 | $-$3.6 | 0.468 | 83 | | 0.6 | 0.2 | 30 | 25.51 | 0.80 | 60.6 | 15.2 | 7.6 | 16.7 | 7.6 | 0.194 | 66 | | 0.6 | 0.2 | 35 | 23.85 | 0.63 | 48.1 | 17.7 | 13.9 | 20.3 | 3.8 | 0.550 | 79 | | 0.6 | 0.5 | 30 | 30.22 | $-$1.77 | 64.2 | 17.3 | 11.1 | 7.4 | 6.2 | 0.297 | 81 | | 0.6 | 0.5 | 35 | 31.13 | $-$2.06 | 50.7 | 6.0 | 29.9 | 13.4 | $-$23.9 | 0.000 | 67 | | 0.6 | 0.5 | 40 | 30.57 | 0.47 | 32.9 | 8.9 | 27.8 | 30.4 | $-$19.0 | 0.004 | 79 | | 0.6 | 0.5 | 45 | 30.23 | 0.40 | 36.1 | 8.3 | 20.8 | 34.7 | $-$12.5 | 0.045 | 72 | | 0.6 | 0.8 | 33 | 34.55 | 2.20 | 53.3 | 16.0 | 18.7 | 12.0 | $-$2.7 | 0.697 | 75 | | 0.6 | 0.8 | 38 | 33.66 | 1.13 | 33.9 | 12.9 | 27.4 | 25.8 | $-$14.5 | 0.067 | 62 | | 0.6 | 0.8 | 45 | 35.81 | 0.32 | 14.8 | 12.5 | 19.3 | 53.4 | $-$6.8 | 0.256 | 88 | | 0.6 | 0.8 | 52 | 34.44 | $-$0.15 | 9.5 | 17.6 | 8.1 | 64.9 | 9.5 | 0.104 | 74 | | 0.6 | 0.9 | 35 | 34.92 | 0.57 | 25.3 | 24.1 | 21.7 | 28.9 | 2.4 | 0.747 | 83 | | 0.6 | 0.9 | 40 | 35.09 | 0.89 | 13.6 | 18.5 | 9.9 | 58.0 | 8.6 | 0.142 | 81 | | 0.6 | 0.9 | 47 | 34.42 | 1.49 | 5.5 | 16.4 | 9.6 | 68.5 | 6.8 | 0.250 | 73 | | 0.6 | 0.9 | 54 | 35.21 | 0.00 | 3.2 | 11.3 | 6.5 | 79.0 | 4.8 | 0.366 | 62 | **Panel A.** Prior Literature ![Figure D.1: Prior experiment-level observations for paired choice tasks (panel A) and our experiment-level results (panel B). In each panel, points below the 45-degree line exhibit a CRE, while points above the…](assets/figured1a-7f3372.png) **Panel B.** Our Experiments ![Figure D.1: Prior experiment-level observations for paired choice tasks (panel A) and our experiment-level results (panel B). In each panel, points below the 45-degree line exhibit a CRE, while points above the…](assets/figured1b-edaa62.png) **Figure D.1: Prior experiment-level observations for paired choice tasks (panel A) and our experiment-level results (panel B). In each panel, points below the 45-degree line exhibit a CRE, while points above the 45-degree line exhibit an RCRE. The shaded grey regions in both panels denote predicted $(\Pr(A),\Pr(C))$ combinations consistent with $\Delta m^\ast = 0$ under Assumption 2a. The black circles in panel A depict the 143 experiments surveyed by Blavatskyy et al. (2023) scaled by the number of observations; the black circles in panel B depict the 120 experiments that we run: 60 combinations of $(p,r,M)$ in the $m$-choice tasks and 60 combinations of $(p,r,H)$ for the $h$-choice tasks.** ## E Estimating a PT Model In this section, we develop a structural PT model and estimate its key parameters. Following our development in Example 1, under PT a person will have underlying indifference valuations $m_{AB}^*$ and $m_{CD}^*$ that satisfy $$ \begin{aligned} u(m_{AB}^*) &= \pi(p)u(H) \quad \text{ and } \\ \pi(r)u(m_{CD}^*) &= \pi(rp)u(H). \end{aligned} $$ For our estimation, we use the functional forms from Tversky and Kahneman (1992): $\pi(q)=q^{\gamma }/\left[ q^{\gamma }+(1-q)^{\gamma }\right]^{1/\gamma}$ and $u(x) = x^\alpha$. The goal is to estimate $\alpha$ and $\gamma$. ### E.1 Estimating a PT Model Using Stage 1 $m$ Valuations Given the functional forms, the underlying indifference valuations are given by: $$ \begin{aligned} (m_{AB}^*)^\alpha &= \left[\dfrac{p^\gamma}{\left(p^\gamma + (1-p)^\gamma\right)^\frac{1}{\gamma}}\right](H)^\alpha &\Leftrightarrow \qquad \dfrac{m_{AB}^*}{H} &= \left[\dfrac{p^\gamma}{\left(p^\gamma + (1-p)^\gamma\right)^\frac{1}{\gamma}}\right]^\frac{1}{\alpha} \\ \\ (m_{CD}^*)^\alpha &= \left[\dfrac{\dfrac{(rp)^\gamma}{\left((rp)^\gamma + (1-rp)^\gamma\right)^\frac{1}{\gamma}}} {\dfrac{r^\gamma}{\left(r^\gamma + (1-r)^\gamma\right)^\frac{1}{\gamma}}}\right](H)^\alpha &\Leftrightarrow \qquad \dfrac{m_{CD}^*}{H} &= \left[p^\gamma \left( \dfrac{r^\gamma + (1-r)^\gamma}{(rp)^\gamma + (1-rp)^\gamma} \right)^\frac{1}{\gamma} \right]^\frac{1}{\alpha}. \end{aligned} $$ Incorporating noise in a way that permits using the standard approach of nonlinear least squares estimation, we model the observed valuations of individual $i$ on trial $t$ as $$ \begin{aligned} \dfrac{m_{AB,it}}{H} &= \left[\dfrac{p_{it}^\gamma}{\left(p_{it}^\gamma + (1-p_{it})^\gamma\right)^\frac{1}{\gamma}}\right]^\frac{1}{\alpha} + \varepsilon_{it} \qquad \text{(E.1)}\\ \dfrac{m_{CD,it}}{H} &= \left[p_{it}^\gamma \left( \dfrac{r_i^\gamma + (1-r_i)^\gamma}{(r_i p_{it})^\gamma + (1-r_i p_{it})^\gamma} \right)^\frac{1}{\gamma} \right]^\frac{1}{\alpha} +\varepsilon_{it} \qquad \text{(E.2)} \end{aligned} $$ where $r_i$ is the common ratio for individual $i$, $p_{it}$ is the probability that individual $i$ faces on trial $t$, and $\varepsilon_{it}$ is a least-squares error term. **Table E.1: PT Estimates Using Data on $m$-Valuations** | | | | | | |:---|:---|:---|:---|:---| | | \(1\) | \(2\) | \(3\) | \(4\) | | | Overall | $r$ = 0.2 | $r$ = 0.4 | $r$ = 0.6 | | | | | | | | **Probability Weighting:** $\widehat{\gamma}$ | 0.600 | 0.580 | 0.587 | 0.636 | | | (0.008) | (0.014) | (0.014) | (0.014) | | | | | | | | **Utility Curvature:** $\widehat{\alpha}$ | 1.209 | 1.351 | 1.179 | 1.112 | | | (0.019) | (0.040) | (0.030) | (0.028) | *Note:* Nonlinear least squares estimation. The model assumes functional forms $\pi (q)=q^{\gamma }/\left[ q^{\gamma }+(1-q)^{\gamma }\right]^{1/\gamma }$ and $u(x)=x^{\alpha}$. Individual-cluster-robust standard errors in parentheses. Panel A estimates use data on $m_{AB}$-valuations and the structural equation (E.1). The typical approach in the literature is to use data on $m_{AB}$ valuations and equation (E.1) to estimate the parameters $(\widehat{\alpha},\widehat{\gamma})$ (Tversky and Kahneman 1992). Table E.1 presents estimates using our data on $m_{AB}$ valuations. Column (1) contains parameter estimates when using all $m_{AB}$-valuations and imposing the same $(\widehat{\alpha},\widehat{\gamma})$ for all $r$. Our estimate of $\widehat{\gamma}=0.60$ implies strong overweighting of low probabilities and underweighting of large probabilities. This estimate is in line with the typical values in the literature and is similar in magnitude to the estimate of $\widehat{\gamma}=0.61$ in Tversky and Kahneman (1992). Our estimate of $\widehat{\alpha}=1.209$ is significantly greater than one, which implies risk seeking in the absence of any probability distortions. Columns (2)-(4) present separate estimates for each common-ratio factor $r$. We find qualitatively similar estimates of $(\widehat{\alpha},\widehat{\gamma})$ across the three values for $r$, which is reassuring given that $r$ does not enter into equation (E.1). We use the estimates in columns (2)-(4) to construct the PT predictions denoted by the dashed blue and dashed-and-dotted red lines in Figure 5 of the main text. To formally test for differences in probability weighting between the $m_{AB}$ valuations versus the $m_{CD}$ valuations, we estimate the following joint specification: $$ \begin{aligned} m_{jit} &= \mathbb{1}(j = AB)\left[\dfrac{p_{it}^{\gamma_{AB}}}{\left(p_{it}^{\gamma_{AB}} + (1-p_{it})^{\gamma_{AB}}\right)^\frac{1}{\gamma_{AB}}}\right]^\frac{1}{\alpha_{AB}} \notag \\ &+ \mathbb{1}(j = CD)\left[p_{it}^{\gamma_{CD}} \left( \dfrac{r_i^{\gamma_{CD}} + (1-r_i)^{\gamma_{CD}}}{(r_i p_{it})^{\gamma_{CD}} + (1-r_i p_{it})^{\gamma_{CD}}} \right)^\frac{1}{\gamma_{CD}} \right]^\frac{1}{\alpha_{CD}} + \varepsilon_{it}, \qquad \text{(E.3)} \end{aligned} $$ where $j \in \{AB,CD\}$ denotes the valuation type. Table E.2 presents the results under different parameter restrictions. Columns (2) and (3) show that we reject the null of a stable $\gamma$ across the $m_{AB}$ and $m_{CD}$ valuations. **Table E.2: Testing for a Stable Probability Weighting Function**
(1) (2) (3)
Restrictions:
2-4 $\gamma_{AB} = \gamma_{CD} = \gamma$, $\alpha_{AB} = \alpha_{CD} = \alpha$ $\alpha_{AB} = \alpha_{CD} = \alpha$ None
Probability Weighting
$\gamma$ 0.773
(0.007)
$\gamma_{AB}$ 0.603 0.600
(0.008) (0.008)
$\gamma_{CD}$ 1.162 0.368
(0.017) (0.006)
Utility Curvature
$\alpha$ 0.916 1.198
(0.014) (0.019)
$\alpha_{AB}$ 1.209
(0.019)
$\alpha_{CD}$ 0.193
(0.009)
F-Test: $\gamma_{AB} = \gamma_{CD}$ $p<0.001$ $p<0.001$
Individuals 900 900 900
Observations 9000 9000 9000
*Note:* Nonlinear least squares estimation. The model assumes functional forms $\pi (q)=q^{\gamma }/\left[ q^{\gamma }+(1-q)^{\gamma }\right]^{1/\gamma }$ and $u(x)=x^{\alpha}$. Individual-cluster-robust standard errors in parentheses. The estimation uses data on both $m_{AB}$ and $m_{CD}$ valuations and the structural equation (E.3). ### E.2 Estimating a PT Model Using Stage 2 Choice Data To estimate the model use stage 2 choice data, we assume additive utility noise. Analogous to our approach in the prior subsection, we posit a model with differential probability weighting across the $AB$ and $CD$ choices, and then test whether they are the same. Hence, a person will choose $A$ over $B$ when $u(M)-\pi(p)u(H) > \epsilon_{AB}$, which becomes $$ \epsilon_{AB} < M^{\alpha_{AB}} - \left[\dfrac{p^{\gamma_{AB}}}{\left(p^{\gamma_{AB}} + (1-p)^{\gamma_{AB}}\right)^\frac{1}{\gamma_{AB}}}\right](H)^{\alpha_{AB}} \equiv D_{AB}(M,H,p). $$ Similarly, the person will choose $C$ over $D$ when $\pi(r)u(M)-\pi(rp)u(H) > \epsilon_{CD}$, which becomes $$ \epsilon_{CD} < \left[\dfrac{r^{\gamma_{CD}}}{\left(r^{\gamma_{CD}} + (1-r)^{\gamma_{CD}}\right)^\frac{1}{\gamma_{CD}}}\right]M^{\alpha_{CD}} - \left[\dfrac{(rp)^{\gamma_{CD}}}{\left((rp)^{\gamma_{CD}} + (1-rp)^{\gamma_{CD}}\right)^\frac{1}{\gamma_{CD}}}\right](H)^{\alpha_{CD}} \equiv D_{CD}(M,H,p,r). $$ As in the prior subsection, we first estimate the parameters $(\alpha_{AB},\gamma_{AB})$ using only the $AB$ choice data. We denote an $AB$ observation by $d_{AB,i}\equiv (a_i;M_i,H_i,p_i)$, where $a_i \in \{A,B\}$ is the person’s choice. We further assume that $\epsilon_{AB} \sim N(0,\sigma_{AB}^2)$. Then given parameter vector $\theta \equiv (\alpha_{AB},\gamma_{AB},\sigma_{AB})$, the likelihood of observation $d_{AB,i}$ is $$ \ell_{AB}(d_{AB,i};\theta) \equiv \mathbb{1}(a_i = A)\Phi\left(\frac{D_{AB}(M_i,H_i,p_i)}{\sigma_{AB}}\right) + \mathbb{1}(a_i = B)\left(1-\Phi\left(\frac{D_{AB}(M_i,H_i,p_i)}{\sigma_{AB}}\right)\right), $$ and the overall likelihood function is $$ L(\theta) \equiv \sum_{d_{AB,i}} log (\ell_{AB}(d_{AB,i};\theta)). $$ Column (1) of Table E.3 presents the parameter estimates. Much as for our estimates using valuations data, our estimate of $\widehat{\gamma_{AB}}=0.71$ implies strong overweighting of low probabilities and underweighting of large probabilities, and is similar in magnitude to that in column (1) of Table E.1.[^49] We next test for differences in probability weighting between the $AB$ choices and the $CD$ choices. We denote a $CD$ observation by $d_{CD,i}\equiv (a_i;M_i,H_i,p_i,r_i)$, where $a_i \in \{C,D\}$ is the person’s choice. We further assume that $\epsilon_{CD} \sim N(0,\sigma_{CD}^2)$, and thus the parameter vector is now $\theta \equiv (\alpha_{AB},\gamma_{AB},\alpha_{CD},\gamma_{CD},\sigma_{AB},\sigma_{CD})$. The likelihood of a $CD$ observation $d_{CD,i}$ is $$ \ell_{CD}(d_{CD,i};\theta) \equiv \mathbb{1}(a_i = C)\Phi\left(\frac{D_{CD}(M_i,H_i,p_i,r_i)}{\sigma_{CD}}\right) + \mathbb{1}(a_i = D)\left(1-\Phi\left(\frac{D_{CD}(M_i,H_i,p_i,r_i)}{\sigma_{CD}}\right)\right). $$ The overall likelihood function is then $$ L(\theta) \equiv \sum_{d_{AB,i}} log (\ell_{AB}(d_{AB,i};\theta)) + \sum_{d_{CD,i}} log (\ell_{CD}(d_{CD,i};\theta)). $$ **Table E.3: PT Estimates Using Stage 2 Choice Data**
(1) (2) (3) (4)
Restrictions:
2-5
Only $\gamma_{AB} = \gamma_{CD} = \gamma$, $\alpha_{AB} = \alpha_{CD} = \alpha$ $\alpha_{AB} = \alpha_{CD} = \alpha$ None
Probability Weighting
$\gamma$ 0.809
(0.011)
$\gamma_{AB}$ 0.710 0.737 0.710
(0.013) (0.014) (0.013)
$\gamma_{CD}$ 0.886 0.461
(0.020) (0.014)
Utility Curvature
$\alpha$ 0.615 0.653
(0.008) (0.010)
$\alpha_{AB}$ 0.697 0.697
(0.011) (0.011)
$\alpha_{CD}$ 0.252
(0.015)
Utility Noise
$\sigma_{AB}$ 2.250 1.974 2.000 2.250
(0.110) (0.088) (0.086) (0.110)
$\sigma_{CD}$ 0.906 1.069 0.104
(0.044) (0.058) (0.013)
$H_0$: $\gamma_{AB} = \gamma_{CD}$ $p<0.001$ $p<0.001$
Individuals 900 900 900 900
Observations 4500 9000 9000 9000
*Note:* Maximum likelihood estimation using stage 2 choice data. The model assumes functional forms $\pi (q)=q^{\gamma }/\left[ q^{\gamma }+(1-q)^{\gamma }\right]^{1/\gamma }$ and $u(x)=x^{\alpha}$. The estimation in column (1) uses data on $m_{AB}$ choices, and the estimation in columns (2)-(4) uses data on both $m_{AB}$ and $m_{CD}$ choices. In columns (3) and (4), the null hypothesis of $\gamma_{AB} = \gamma_{CD}$ is tested via a Wald test. Columns (2)-(4) of Table E.3 presents estimates analogous to those in columns (1)-(3) in Table E.2. Columns (3) and (4) show that we again reject the null of a stable $\gamma$ across the $m_{AB}$ and $m_{CD}$ valuations.[^50] ## References Allais, Maurice. 1953. “Le Comportement de l’homme Rationnel Devant Le Risque: Critique Des Postulats Et Axiomes de l’École Américaine.” *Econometrica* 21 (4): 503–46. Alós-Ferrer, Carlos, Ernst Fehr, and Nick Netzer. 2021. “Time Will Tell: Recovering Preferences When Choices Are Noisy.” *Journal of Political Economy* 129 (6): 1828–77. Ballinger, T. Parker, and Nathaniel T. Wilcox. 1997. “Decisions, Error and Heterogeneity.” *The Economic Journal* 107 (443): 1090–105. Barberis, Nicholas C. 2013. “Thirty Years of Prospect Theory in Economics: A Review and Assessment.” *Journal of Economic Perspectives* 27 (1): 173–96. Bell, David E. 1985. “Disappointment in Decision Making Under Uncertainty.” *Operations Research* 33 (1): 1–27. Bernheim, B. Douglas, and Charles D. Sprenger. 2020. “On the Empirical Validity of Cumulative Prospect Theory: Experimental Evidence of Rank-Independent Probability Weighting.” *Econometrica* 88 (4): 1363–409. Bhatia, Sudeep, and Graham Loomes. 2017. “Noisy Preferences in Risky Choice: A Cautionary Note.” *Psychological Review* 124 (5): 678–87. Blavatskyy, Pavlo R. 2007. “Stochastic Expected Utility Theory.” *Journal of Risk and Uncertainty* 34 (3): 259–86. Blavatskyy, Pavlo R. 2010. “Reverse Common Ratio Effect.” *Journal of Risk and Uncertainty* 40 (3): 219–41. Blavatskyy, Pavlo, Valentyn Panchenko, and Andreas Ortmann. 2023. “How Common Is the Common-Ratio Effect?” *Experimental Economics* 26 (2): 253–72. Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer. 2012. “Salience Theory of Choice Under Risk.” *The Quarterly Journal of Economics* 127 (3): 1243–85. Brown, Alexander L., and Paul J. Healy. 2018. “Separated Decisions.” *European Economic Review* 101: 20–34. Bruhin, Adrian, Helga Fehr-Duda, and Thomas Epper. 2010. “Risk and Rationality: Uncovering Heterogeneity in Probability Distortion.” *Econometrica* 78 (4): 1375–412. Camerer, Colin F., and Teck-Hua Ho. 1994. “Violations of the Betweenness Axiom and Nonlinearity in Probability.” *Journal of Risk and Uncertainty* 8 (2): 167–96. Carrera, Mariana, Heather Royer, Mark Stehr, Justin Sydnor, and Dmitry Taubinsky. 2022. “Who Chooses Commitment? Evidence and Welfare Implications.” *Review of Economic Studies* 89 (3): 1205–44. Castillo, Marco, and David Eil. 2014. “Taring the Multiple Price List: Imperceptive Preferences and the Reversing of the Common Ratio Effect.” *Working Paper*. Cerreia-Vioglio, Simone, David Dillenberger, and Pietro Ortoleva. 2015. “Cautious Expected Utility and the Certainty Effect.” *Econometrica* 83 (2): 693–728. Chapman, Jonathan, Mark Dean, Pietro Ortoleva, Erik Snowberg, and Colin Camerer. 2022. “Econographics.” *Journal of Political Economy Microeconomics*. Coakley, Clint W., and Mark A. Heise. 1996. “Versions of the Sign Test in the Presence of Ties.” *Biometrics* 52 (4): 1242–51. Dean, Mark, and Pietro Ortoleva. 2019. “The Empirical Relationship Between Nonstandard Economic Behaviors.” *Proceedings of the National Academy of Sciences* 116 (33): 16262–67. Enke, Benjamin, Thomas Graeber, and Ryan Oprea. 2023. “Complexity and Time.” *Working Paper*. Freeman, David J., Yoram Halevy, and Terri Kneeland. 2019. “Eliciting Risk Preferences Using Choice Lists.” *Quantitative Economics* 10 (1): 217–37. Freeman, David J., and Guy Mayraz. 2019. “Why Choice Lists Increase Risk Taking.” *Experimental Economics* 22 (1): 131–54. Frydman, Cary, and Lawrence J. Jin. 2023. “On the Source and Instability of Probability Weighting.” *Working Paper*. Gillen, Ben, Erik Snowberg, and Leeat Yariv. 2019. “Experimenting with Measurement Error: Techniques with Applications to the Caltech Cohort Study.” *Journal of Political Economy* 127 (4): 1826–63. Gul, Faruk. 1991. “A Theory of Disappointment Aversion.” *Econometrica* 59 (3): 667–86. Harless, David W., and Colin F. Camerer. 1994. “The Predictive Utility of Generalized Expected Utility Theories.” *Econometrica* 62 (6): 1251–89. Hey, John D. 2005. “Why We Should Not Be Silent about Noise.” *Experimental Economics* 8 (4): 325–45. Hey, John D., and Chris Orme. 1994. “Investigating Generalizations of Expected Utility Theory Using Experimental Data.” *Econometrica* 62 (6): 1291–326. Kahneman, Daniel, and Amos Tversky. 1979. “Prospect Theory: An Analysis of Decision Under Risk.” *Econometrica* 47 (2): 263–92. Khaw, Mel Win, Ziang Li, and Michael Woodford. 2021. “Cognitive Imprecision and Small-Stakes Risk Aversion.” *Review of Economic Studies* 88 (4): 1979–2013. Loomes, Graham. 2005. “Modelling the Stochastic Component of Behaviour in Experiments: Some Issues for the Interpretation of Data.” *Experimental Economics* 8 (4): 301–23. Loomes, Graham, and Robert Sugden. 1982. “Regret Theory: An Alternative Theory of Rational Choice Under Uncertainty.” *The Economic Journal* 92 (368): 805–24. Loomes, Graham, and Robert Sugden. 1998. “Testing Different Stochastic Specifications of Risky Choice.” *Economica* 65 (260): 581–98. McFadden, Daniel. 1974. “Conditional Logit Analysis of Qualitative Choice Behavior.” In *Frontiers in Econometrics*, edited by Paul Zarembka. Academic Press. McFadden, Daniel. 1981. “Econometric Models of Probabilistic Choice Behavior.” Chap. 5 in *Structural Analysis of Discrete Data and Econometric Applications*, edited by Charles F. Manski and Daniel McFadden. MIT Press. McGranaghan, Christina, Kirby Nielsen, Ted O’Donoghue, Jason Somerville, and Charles D. Sprenger. 2021. *A Reconsideration of Common Ratio Effect and Probability Weighting*. McGranaghan, Christina, Kirby Nielsen, Ted O’Donoghue, Jason Somerville, and Charles D. Sprenger. 2023a. “Connecting Common Ratio and Common Consequence Preferences.” *Work in Progress*. McGranaghan, Christina, Kirby Nielsen, Ted O’Donoghue, Jason Somerville, and Charles D. Sprenger. 2023b. “Replication Data for: Distinguishing Common Ratio Preferences from Common Ratio Effects Using Paired Valuation Tasks.” *American Economic Association \[Publisher\]*. Oprea, Ryan. 2022. “Simplicity Equivalents.” *Working Paper*. Prelec, Drazen. 1998. “The Probability Weighting Function.” *Econometrica* 66 (3): 497–527. Quiggin, John. 1982. “A Theory of Anticipated Utility.” *Journal of Economic Behavior and Organization* 3 (4): 323–43. Randles, Ronald H. 2001. “On Neutral Responses (Zeros) in the Sign Test and Ties in the Wilcoxon–Mann–Whitney Test.” *The American Statistician* 55 (2): 96–101. Schneider, Mark, and Mikhael Shor. 2017. “The Common Ratio Effect in Choice, Pricing, and Happiness Tasks.” *Journal of Behavioral Decision Making* 30 (4): 976–86. Stott, Henry P. 2006. “Cumulative Prospect Theory’s Functional Menagerie.” *Journal of Risk and Uncertainty* 32 (2): 101–30. Tversky, Amos, and Daniel Kahneman. 1992. “Advances in Prospect Theory: Cumulative Representation of Uncertainty.” *Journal of Risk and Uncertainty* 5 (4): 297–323. Wilcox, Nathaniel T. 2008. “Stochastic Models for Binary Discrete Choice Under Risk: A Critical Primer and Econometric Comparison.” In *Risk Aversion in Experiments (Research in Experimental Economics, Vol. 12)*, edited by James C. Cox and Glenn W. Harrison. Emerald Group Publishing Limited. Wu, George, and Richard Gonzalez. 1996. “Curvature of the Probability Weighting Function.” *Management Science* 42 (12): 1676–90. [^1]: Throughout, we reserve the term “CRE” for empirical observations in experimental data, and we introduce the term “CRP” to refer to the underlying preference that these empirical observations are often taken to reveal. Our focus is whether one can empirically identify the existence of a CRP without making any assumptions about its source. [^2]: Each participant sees five paired valuation tasks and five paired choice tasks of the form described here. For robustness, we also present each participant with another five paired valuation tasks and another five paired choice tasks that use a different structure, as we describe in Sections I and II. [^3]: In other words, each combination of $(M,p,r)$ used at stage 2 generates a different “experiment.” Overall, we have 120 different experiments, with an average of 75 participants in each. [^4]: For a recent example, see Bernheim and Sprenger (2020), who use valuations to test the assumption of rank dependence (Quiggin 1982). While not a central part of their analysis, they discuss how choice noise presents a challenge to research that tests axioms using pairs of choice tasks. [^5]: To simplify notation, we adopt the convention of omitting the zero outcomes from lotteries. For example, Lottery $B$ yields $H$ with probability $p$ and zero with the remaining probability of $1-p$. Most experimental implementations of paired choice tasks set the low outcome equal to zero as we do; however, the key theoretical points also hold for a non-zero low outcome. [^6]: From here onward, we use $\widehat{\Pr }$ to denote empirically observed proportions, and $\Pr$ to denote model-predicted proportions. To ease notation, we suppress the choice set because it is typically self-explanatory; for example, $\widehat{\Pr}(A)$ is the proportion who choose $A$ from the choice set $\{A,B\}$, and $\widehat{\Pr}(AD)$ is the proportion who choose combination $AD$ from the choice set $\{AC,AD,BC,BD\}$. [^7]: Note that $\widehat{\Pr }(A)=\widehat{\Pr }(AD)+\widehat{\Pr }(AC)$ and $\widehat{\Pr }(C) = \widehat{\Pr }(BC) + \widehat{\Pr }(AC)$, and thus $\widehat{\Pr }(AD)>\widehat{\Pr }(BC)$ is equivalent to $\widehat{\Pr }(A)>\widehat{\Pr }(C)$. [^8]: Some researchers have debated whether valuations or choices provide better insight into people’s underlying preferences. For instance, based on discrepancies between behavior in valuations versus binary choices, combined with an assumption that binary choices reflect true preferences, some have argued that valuations are unreliable (Brown and Healy 2018; Freeman et al. 2019). Others have argued that binary choices, themselves, may be unreliable (Freeman and Mayraz 2019). The systematic connections that we document in Section IV suggest that neither conclusion is warranted. [^9]: To simplify the exposition, the text assumes that the person prefers and chooses the safer option when indifferent, but given we also assume continuous noise, this assumption is immaterial. [^10]: We use the term “preference” to capture all stable drivers of behavior (as opposed to transient confusion or noise); in addition to reflecting “true” tastes, these drivers might also include stable heuristics. We also note that, by CRP and RCRP, we do not mean that people care directly about common-ratio transformations of probabilities. Instead, we use these terms to capture the idea that underlying risk attitudes react systematically to these transformations. Based on the existing CRE evidence, many theories—most notably, models of probability weighting—assume that underlying preferences have a systematic CRP. [^11]: For instance, this approach is used by Tversky and Kahneman (1992) and Bruhin et al. (2010). [^12]: To help clarify our exposition, we use $\epsilon$ to denote additive perturbations to utility, and $\varepsilon$ to denote the noise described in Assumption 1. The latter equations use $\pi(p)u(H)=u(m_{AB}^{\ast})$ and $\pi(rp)u(H)=\pi(r)u(m_{CD}^{\ast})$. [^13]: For instance, this approach is used by Camerer and Ho (1994), Hey and Orme (1994), and Wu and Gonzalez (1996). [^14]: By assuming a standard normal distribution, Figure 1 also has the feature that either $\Pr(A)>\Pr(C)>1/2$ or $\Pr(A)<\Pr(C)<1/2$. Proposition 1 establishes that the $1/2$ threshold is replaced by $Z \equiv \Pr(\varepsilon_{AB}<0)$ for more general noise distributions. [^15]: Formally, if $(\varepsilon _{AB},\varepsilon _{CD})$ has joint distribution $F$ with $PDF$ $f$, then symmetry around $(\varepsilon',\varepsilon')$ implies $f(\varepsilon'+z_{AB},\varepsilon'+z_{CD})=f(\varepsilon'-z_{AB},\varepsilon'-z_{CD})$ for all $(z_{AB},z_{CD})$. This property holds, for instance, for a bivariate normal with mean $(\varepsilon',\varepsilon')$ and any correlation. [^16]: Analogous to our use of $\widehat{\Pr}$ to denote empirically observed proportions and $\Pr$ to denote model-predicted proportions, we use $\widehat{E}$ to denote empirically observed averages and $E$ to denote model-predicted averages. [^17]: Some researchers have raised concerns that certain methods for eliciting valuations, such as price lists, may lead to systematic errors of over- or under-valuation. Our results imply that a test based on $\hat{E}(\Delta m) = 0$ remains valid provided these systematic errors have the same mean across the pair of valuations. [^18]: Our formal test uses the following logic. If $\Pr(\Delta m>0)=\Pr(\Delta m<0)=1/2$ for every observation, the likelihood of observing at most $n$ instances of $\Delta m>0$ out of $N$ observations is equal to $G(n,N)$, where $G$ denotes the cumulative distribution function for a binomial distribution with a 50 percent success rate. Hence, if we observe $n_{+}$ instances of $\Delta m>0$ and $n_{-}$ instances of $\Delta m<0$, the $p$-value for a two-sided sign test under the null of $\Delta m^{\ast}=0$ is $2\ast G(\min \{n_{+},n_{-}\},n_{+}+n_{-})$. [^19]: The gray shaded region reflects the set of possible $(\Pr(A),\Pr(C))$ predictions that one can construct when one permits any distribution of preferences for $A$ versus $B$ and imposes no restrictions on noise other than it being median zero. The area below (above) the 45-degree line and above (below) $\Pr(C) = 1/2$ can be constructed using homogeneous preferences that favor $A$ and $C$ ($B$ and $D$) and noise that is more impactful for the $CD$ choice. The mirror images of these regions can be constructed using noise that is more impactful for the $AB$ choice. The remaining shaded area can be constructed using heterogeneity in the preferences for $A$ versus $B$, effectively permitting convex combinations of the above areas. Prior researchers (Ballinger and Wilcox 1997; Wilcox 2008) have provided examples that combine heterogeneity and noise to generate a population outcome with $\Pr(A)>1/2>\Pr(C)$; our analysis in Online Appendix B.2 characterizes the complete set. [^20]: While the existence of a CRE is typically assessed by testing whether $\widehat{\Pr}(A)-\widehat{\Pr}(C) > 0$, a more stringent test is whether $\widehat{\Pr}(AD) > 1/2$. Such a finding would be inconsistent with Proposition 1 if the noise has a median of zero and thus $Z=1/2$. Among the 143 experiments in panel A, there are 20 experiments with $\widehat{\Pr}(AD) \ge 1/2$. While this provides some evidence against the no-CRP null, one would probably not conclude from this evidence that there is a systematic CRP. [^21]: For instance, under EU with $u(x)=x^\alpha$, $m_{z}/H=M/h_{z}=p^{1/\alpha}$ for both $z=AB$ and $z=CD$. More generally, there need not be an exact equivalence, but there should be a positive correlation. [^22]: Our experiment was preregistered on the AEA RCT Registry (McGranaghan et al. 2021). The data and code are deposited with the AEA Data and Code Repository at ICPSR (McGranaghan et al. 2023b). [^23]: Given our use of multiple-price lists, a possible concern is that, for each list, participants are not providing an (effective) indifference point as our model assumes, but rather are choosing their switching rows based on the resulting overall compound lottery for the full list. Moreover, given our use of many tasks with only one chosen for payment, a second possible concern is that participants are not treating each task in isolation, but rather are assembling their preferred compound lottery for all tasks combined. Like most of the prior experimental literature—including the literature that estimates probability weighting from within-subject measures of certainty equivalents—we proceed assuming that neither of these is an issue. [^24]: Our pre-analysis plan specified five values of $M$ and $H$, but our implementation code had a small error and only implemented four of the five values in each case. [^25]: Online Appendix B.5 provides an alternative way to visualize, within the context of our specific experimental tasks, the bias in paired choice tasks and how paired valuation tasks are immune to that bias. [^26]: For each comprehension check, roughly 85 percent of participants answer correctly. While our analysis uses the full sample, restricting the sample to those who answer both comprehension checks correctly does not materially change our results. [^27]: We did not preregister a gender-balanced sample. After preregistering, we learned that Prolific had very recently experienced a large increase in young female participants due to a social media trend. To better approximate the typical college population, we recruited 450 men and 450 women. [^28]: This was double the \$6.50/hour minimum wage on Prolific. [^29]: In Online Appendix E.1, we use the functional forms from (Tversky and Kahneman 1992) and corresponding parameter labels, and the typical non-linear least-squares approach to estimate those parameters. The specific estimates for utility curvature ($\alpha$) and probability weighting ($\gamma$) are: $\alpha =1.351$ and $\gamma=0.580$ for $r=0.2$; $\alpha=1.179$ and $\gamma =0.587$ for $r=0.4$; and $\alpha =1.112$ and $\gamma =0.636$ for $r=0.6$. [^30]: Online Appendix Table D.4 reports the complete range of predicted $\Delta m$ values and confidence intervals for each $(p,r)$ combination. [^31]: For a more formal test, see Online Appendix E.1, where we estimate a structural model that permits separate probability weighting functions for the $AB$ and $CD$ valuations, and strongly reject the null of there being no difference. [^32]: Online Appendix Table D.6 shows that the message is almost identical no matter how we treat observations of $\Delta h = 0$. [^33]: Castillo and Eil (2014) propose and test a theory of status quo bias in which holding fixed the safe option in a paired valuation task (as in our $h$ valuations) leads to an RCRP, while holding fixed the risky option (as in our $m$-valuations) yields a CRP. While they find limited support in their data, the consistent pattern that we find across our $m$- and $h$-valuations runs counter to their theory. [^34]: Here and in subsequent sections, we provide interpretations of correlations in behavior across different tasks. These interpretations implicitly assume that any correlation in noise draws across tasks is limited; the interpretations would be different if there were sizeable correlations in noise draws across tasks. [^35]: These correlations are weakest when comparing $pH-\bar{m}$ for a high $p$ (0.8 or 0.9) with that for a low $p$ (0.1 or 0.2), which is perhaps not surprising given that some models can predict a negative correlation for such comparisons. For instance, under PT with heterogeneity in the extent of probability weighting, people with more pronounced probability weighting (smaller $\gamma$) would have a smaller (more negative) $pH-\bar{m}$ for low $p$ and a larger (more positive) $pH-\bar{m}$ for high $p$. [^36]: Dean and Ortoleva (2019) use paired valuation tasks to obtain a continuous measure of the CRE. They find that their valuations-based measure of the CRE is strongly correlated with other risk-preference-related behaviors, which parallels our finding of reliable heterogeneity in underlying CRP versus RCRP. [^37]: Online Appendix B.6 explores the impact of distance to indifference in the absence of choice noise, that is, when all variation in the data is due to heterogeneity in preferences. Two key predictions are: (i) If the population distribution of $\Delta m^*$ is symmetric, then the distance to indifference has no impact on $CRE-RCRE$; and (ii) otherwise, the distance to indifference can have an impact on $CRE-RCRE$, but that impact must be symmetric around a zero distance to indifference. Both of these predictions are inconsistent with our data. [^38]: Online Appendix B.4 derives analogous corrected regressors for $h$-tasks. Basing these corrections on the EU case is not perfect, as we have already documented that our data are inconsistent with that case. However, using that case as motivation, instead of choosing a correction in a more ad hoc way, imposes some discipline on our analysis. In Online Appendix B.7, we assess this approach and show that it works as intended. [^39]: The magnitude of distance to indifference also predicts stage 2 decision timing. When measured distance to indifference $|M- m_{XY}| = 0$, participants take around 7 seconds to complete their stage 2 choice. For every \$10 increase in $|M-m_{XY}|$, stage 2 choices occur around 1.4 seconds ($\approx$ 20 percent) faster. These results are consistent with the interpretation that easier choices occur faster, as suggested by some neuroscience models. [^40]: As a robustness check, we can perform an analogous individual-level analysis of the links between the stage 1 $h$-valuations and the stage 2 $h$-choices. Online Appendix Table D.10 is the $h$-task analogue to Table 5 and yields similar conclusions. [^41]: See Online Appendix B.4 for a discussion the terms we use for $h$-choice tasks. [^42]: The observed proportions of the four combinations are $\widehat{\Pr}(BD) = 38.0$ percent, $\widehat{\Pr}(AC) = 33.6$ percent, $\widehat{\Pr}(AD) = 15.6$ percent, and $\widehat{\Pr}(BC) = 12.9$ percent. [^43]: Our stage 2 choice data also yield the same broad conclusion as our stage 1 valuations data when interpreted in terms of probability weighting: The probability weighting function estimated from the $AB$ choice data takes the familiar inverse-S shape, but our combined $AB$ and $CD$ choice data are inconsistent with a stable probability weighting function. See Online Appendix E.2 for details. [^44]: In our notation, Kahneman and Tversky (1979) and Prelec (1998) both assume that, for any $H > M > 0$ and $0 < p < q \le 1$, $(H,p) \sim (M,q)$ implies $(H,rp) \succ (M,rq)$ for any $r \in (0,1)$. Note that this axiom covers a broader set of decisions than we study in our experiments since we restrict attention to the case with $q=1$. [^45]: The assumptions of mean-zero and median-zero noise serve to limit the set of possible outcomes; if we were to relax these assumptions, even more outcomes would be possible even under the null of everyone having $\Delta m_i^\ast=0$. [^46]: We omit the proof of this condition since we do not use it in the main text, but the approach is similar to the proof of the prior condition. [^47]: The latter equations use $u(M)/\pi(p)=u(h_{AB}^{\ast})$ and $\pi(r)u(M)/\pi(rp)=u(h_{CD}^{\ast})$. [^48]: We reiterate the point from footnote 38 that this correction is not perfect because our data are inconsistent with EU with additive i.i.d. utility noise; nonetheless, we use that case to impose some discipline on what we use for this correction. See also our discussion in Appendix B.7 [^49]: Our estimate of $\widehat{\alpha_{AB}}=0.70$, in contrast to that in column (1) of Table E.1, is significantly less than one. While not reported here, we also conduct both estimations while imposing that $\alpha=1$, and both estimated $\gamma$ parameters are still less than one, again consistent with inverse-S-shaped probability weighting. [^50]: While not reported here, we also conduct the estimations in both Tables E.2 and E.3 while imposing that $\alpha=1$, and we again reject the null of a stable $\gamma$ across the $m_{AB}$ and $m_{CD}$ valuations.