--- schema_version: 1 title: 'Range-Dependent Attribute Weighting in Consumer Choice: An Experimental Test' authors: - Jason Somerville language: en edition: Econometrica author manuscript and appendix paper_url: https://jason-somerville.com/papers/range-dependent-attribute-weighting/ pdf_url: https://jason-somerville.com/files/range-dependent-attribute-weighting/paper.pdf format: Markdown with TeX mathematics --- # Range-Dependent Attribute Weighting in Consumer Choice: An Experimental Test Jason Somerville Econometrica author source · Main paper and online appendix ## Abstract This paper investigates whether the range of an attribute’s outcomes in the choice set alters its relative importance. I derive distinguishing predictions of two prominent theories of range-dependent attribute weighting: the focusing model of Kőszegi and Szeidl (2013) and the relative thinking model of Bushong et al. (2021). I test these predictions in a laboratory experiment in which I vary the prices of high- and low-quality variants of multiple products. The data provide clear evidence of choice-set dependence consistent with relative thinking: price increases that expand the range of prices in the choice set lead to *more* purchases. Structural estimates imply economically meaningful effect sizes: the average participant was willing to pay around 17% more when a seemingly irrelevant option is added to their choice set. ## Introduction Theories of range-dependent attribute weighting propose that an attribute’s importance depends on its range of outcomes in the choice set. For example, an individual deciding between two options can be swayed by the availability of an inferior third option if it alters the range of price and quality. Two recent theories make conflicting assumptions about how an attribute’s range affects its importance. Kőszegi and Szeidl (2013) (henceforth KS) propose a theory of “focusing” in which individuals overweight attributes that have the widest range of values in the choice set. In contrast, the “relative thinking” model of Bushong et al. (2021) (henceforth BRS) assumes that fixed differences between attribute values appear larger when compared to a relatively narrow range of possible values in the choice set. Both focusing and relative thinking offer plausible frameworks for explaining and predicting when violations of the independence of irrelevant alternatives (IIA) assumption might arise. In practice, little is known about whether range weighting generates systematic departures from IIA in simple consumer choice situations or whether any such deviations are driven by the focusing or relative thinking mechanism. This lack of empirical clarity is problematic given the broad economic implications of range weighting. For example, range weighting can shed light on why it may be optimal for firms to offer product lines that can be hard to rationalize from the neoclassical perspective. Firms can leverage premium or add-on options, such as a reserve bottle of wine or a first class airline upgrade, to decrease consumers’ price sensitivity. Even if these options are rarely chosen, they can boost sales by making standard options more appealing. In principal-agent settings, range weighting can explain why bonus schemes sometimes backfire: bonuses for exerting high effort might alter agents’ responsiveness to rewards and induce lower effort. More generally, demand estimation may be systematically biased if range weighting is not taken into account: price changes that alter the price range can attenuate or amplify the overall price sensitivity estimates. This paper tests the distinguishing predictions of focusing and relative thinking by investigating experimentally how the range of outcomes within the choice set impacts how individuals weigh different attributes. The experimental design focuses on situations in which individuals choose between a high- and low-quality variant of a product, with the option of not buying either. I set prices such that one option is clearly inferior but nonetheless influences the relative preference for the remaining two options by affecting the attribute weights. This creates three potential price regions in which range weighting is identified: each region is defined by which one of the three options is irrelevant from the perspective of surplus maximization. Intuitively, changes in the price range generate unique shifts in the agent’s willingness-to-pay when the surplus on offer is fixed. In a lab experiment, I individually calibrate and vary the prices of high- and low-quality variants of multiple products to generate independent variation in the relative attribute ranges and the surpluses on offer. The key innovation of the design is to use the range-weighting comparative statics to tailor the choice sets that participants face. For each decision that participants make, there is always a choice that is inconsistent with one of the two range theories. I show that this environment provides ideal variation to test the nonparametric predictions of focusing and relative thinking, and for estimating the structural parameters of the attribute-weighting function. The experimental results provide clear evidence of choice-set dependence consistent with relative thinking. First, I reject utility maximization under very general conditions: price increases that expand the price range in the choice set lead to *more* purchases. Second, I show that choice probabilities respond to price changes in manner consistent with the key assumptions of relative thinking. Expanding the price range makes participants more likely to buy the nondecoy option and less likely to not buy. Increasing the price level while holding the price difference between two options fixed causes substitution to higher-quality options. These patterns of demand are consistent with the idea that wider price ranges lead to lower price sensitivity. I complement this nonparametric analysis with structural estimation of the attribute-weighting function. While this approach requires some parametric assumptions, it allows me to quantify the impact of range weighting in terms of foregone surplus. My estimates imply large aggregate deviations from the neoclassical model due to range-dependent attribute weighing: on average, participants were willing to pay around a 17% premium relative to their intrinsic value due to the presence of seemingly irrelevant options in the choice set. Regardless of the empirical approach or subsample restriction, I find consistent evidence of relative thinking and stable parameter estimates. Using both nonparametric and structural approaches, I document that between 60 and 75% of participants are best characterized as relative thinkers. Participants that are not best classified as relative thinkers make noisier decisions and do not respond to price increases in a manner consistent with focusing. The structural estimates show that there is substantial individual heterogeneity in the magnitude but not the direction of range weighting. Overall, I find that the majority of participants make choices consistent with relative thinking and no evidence that a nontrivial share of respondents make choices consistent with focusing. I address three concerns that could affect the interpretation of my results. First, quality is uncertain and may have been misspecified. A particular concern is that participants interpreted prices as signals of quality. I show that both the nonparametric evidence and quantitative estimates of relative thinking are robust to controlling for price signaling. Second, the within-subject nature of the experimental price variation creates the potential for choice-set spillovers across trials. Using several approaches, I show that such spillovers are small and do not alter the results. Finally, an alternative approach to choice-set dependence called salience theory (Bordalo et al. 2013) shares some similar intuitions to relative thinking and might provide a better fit to the data. I show that for a subset of prices in which relative thinking and salience theory make distinguishing predictions, the data continue to support relative thinking and are inconsistent with salience theory. I also show that a generalization of the pairwise normalization model of Landry and Webb (2021) makes the same predictions as relative thinking in my experimental context—thus, my results are also consistent with this model. This paper makes several contributions to the literature. It is one of the first papers to test the predictions of focusing and relative thinking in simple consumer choice contexts, and the first to estimate the structural parameters of these models. A prior experiment by Andersson et al. (2021) found mixed evidence of range-dependent weighting in an intertemporal choice context: they show that expanding the range of immediate payments leads to more impatient choices, but expanding the range of future payments does not induce more patient choices. A contemporaneous study by Dertwinkel-Kalt et al. (2021) finds evidence for a bias toward payments concentrated in a single period—a behavior consistent with focusing.[^1] My results do not necessarily contradict this evidence. The choice task faced by participants in my experiment was relatively simple; they had to evaluate only three options, one of which was an outside option of not buying. Each option varied in only two attributes: quality and price. BRS suggest that relative thinking may apply in simple environments, while focusing may dominate in complex settings where attention is stretched and individuals must selectively hone in on certain attributes. My results support this idea: relative thinking is ubiquitous in perhaps the simplest conceivable choice setting in which range weighting is identified. In a more recent experiment, BRS test the qualitative predictions of relative thinking versus focusing in a real-effort context. They find that respondents prefer to work more when the range of effort costs are wide relative to the range of monetary payments. Their results provide further support for the key assumption of relative thinking: fixed differences appear larger when compared to a narrow range of outcomes in the choice set. My structural estimates quantify the implications of this assumption in terms of foregone surplus and find substantial heterogeneity across individuals. Finally, this paper contributes to a large literature that has documented choice-set effects in marketing and psychology. Traditionally, choice-set dependence has been demonstrated in forced choice amongst three options with a dominated or near-dominated “decoy” option (Huber et al.,1982; Huber and Puto, 1983). My experimental design generalizes the concept of a decoy option: in addition to demonstrating classic asymmetric dominance effects, I show that superior quality options and not buying can serve as decoys. More importantly, I also quantify the impact of decoy options in terms of a surplus premium and find economically meaningful changes in willingness-to-pay due to their presence. ## 1 Theoretical Framework This section describes a general model of range-dependent attribute weightings that nests focusing and relative thinking. I demonstrate that range-based attribute weights are identified in a common economic situation: choosing between a high- and a low-quality variant of a product, along with the option of not buying. I also derive the distinguishing predictions of the focusing and relative thinking models, and outline the key comparative statics that the experiment tests. ### 1.1 Basic Setup Consider an agent who is making a one-time decision about which of $j=1,...,J$ vertically differentiated options to choose. Let $q_j$ denote the quality of option $j$ in monetary terms; this can be thought of as the agent’s intrinsic value for $j$, absent any choice-set influences. A choice set is $\mathcal{C}=\{(q_j,p_j)_{j=1}^J\}$, where $p_j \in \mathbb{R}_+$ is the price of $j$. Utility maximization is based on on the following assumptions. - *(Utility Maximization). The utility of option $j$ is increasing in $q_j$, decreasing in $p_j$, and independent of $(q_k,p_k)$ for all $k \neq j$.* A1 states that agents like higher-quality options, dislike paying higher prices, and that their evaluation of each option $j$ is independent of other options in the choice set. In illustrative examples and parametric estimation, I will sometimes use the following specific form of utility maximization. - *(Surplus Maximization). The utility of option $j$ is $U_j = q_j - p_j$.* According to A2, agents simply compare the difference between quality and price for each option, and choose the one with the largest surplus. In contrast to utility maximization, models of choice-set dependence propose that any option $k \neq j$ in the choice set can directly affect the utility of $j$ by altering how agents weight each attribute. The focusing model of Kőszegi and Szeidl (2013) and the relative thinking model of Bushong et al. (2021) both assume that an attribute’s weight depends on its range of outcomes within the choice set. While both focusing and relative thinking assume that the range determines attribute importance, they make opposite assumptions about the direction of the effect. Focusing assumes that attributes with the widest range of values in the choice set receive the largest decision weights. A psychological intuition for focusing is that people have limited attention and therefore focus on the attributes that vary the most across options. Motivation for this assumption comes from Schkade and Kahneman (1998), who examined life satisfaction ratings and found that people “focus on salient differences” (p. 344) such as the weather in California versus the Midwest. Conversely, relative thinking assumes that a fixed amount appears larger when compared to a relatively narrow range. Thaler (1999) conveys the intuition for this assumption by drawing on insights from psychophysics: “\$50 will appear larger by itself than in the context of a much larger bill” (p. 193). - *(Range Weighting): The utility of option $j$ is $V_j = g(\Delta_q;\gamma) \cdot q_j -g(\Delta_p;\gamma) \cdot p_j$ where $\Delta_x \equiv \max_j x_j - \min_j x_j$ is the range of attribute $x\{q,p\}$ in choice set $\mathcal{C}$ and the weighting function $g$ satisfies the following properties:* - $g$ is continuously differentiable in both $\Delta$ and $\gamma$ and $\lim_{\Delta \rightarrow 0} \Delta \cdot g(\Delta;\gamma) = 0$ - $\Delta g(\Delta;\gamma)$ is strictly increasing in $\Delta$ - For any $\Delta_1>\Delta_2$, $g(\Delta_1;\gamma)/g(\Delta_2;\gamma)$ is strictly increasing in $\gamma$ A3 provides a general characterization of range-dependent attribute weighting. The weighting function $g$ can be increasing, decreasing, or constant, thereby nesting focusing, relative thinking, and surplus maximization respectively. Part (ii) imposes some limit on the extent of range-weighting in the case where $g$ is decreasing. It implies, for example, that agents must prefer \$2 evaluated against a range of \$2 over \$1 evaluated against a range of \$1. This is an implication of focusing, whereas BRS impose it as an additional assumption. Part (iii) of A3 establishes that a single parameter $\gamma$ indexes the direction and magnitude of range weighting. A larger $\gamma$ leads to a relative increase in the weight attached to the attribute with the wider range. Without loss of generality, I normalize $\gamma=0$ to represent the case of surplus maximization. A negative value for $\gamma$ then implies range weighting consistent with relative thinking, with the limiting case of $\gamma=\underline{\gamma}$ characterized by a preference for ratios: $g(\Delta_1;\underline{\gamma})/g(\Delta_2;\underline{\gamma})=\Delta_2/\Delta_1$.[^2] Positive $\gamma$ values indicate focusing, and, in the limiting case of $\gamma=\overline{\gamma}$, individuals only pay attention to the attribute with the widest range within the choice set: $g(\Delta_1;\overline{\gamma})/g(\Delta_2;\overline{\gamma}) = \infty$. Part (i) provides mild technical conditions necessary for establishing identification of the range-weighting parameter $\gamma$. To illustrate the predictions of range weighting and to facilitate structural estimation, I will sometimes adopt the following parametric weighting function. - *(Parametric Range Weighting): $g(\Delta_x;\gamma) = \left(\Delta_x\right)^\gamma$ where $\gamma\in(-1,\infty)$* The weighting function A4 provides a simple characterization of range weighting. Surplus maximization is nested in this specification by $\gamma=0$, focusing by $\gamma \in (0,\infty)$, and relative thinking by $\gamma \in (-1,0)$. In Section 1.2, I use this parametric form to illustrate graphically the key predictions of range weighting and my approach to identification. My structural estimation in Section 4 also uses this parametric form. However, I demonstrate in Section 1.3 that identification of $\gamma$ holds for the more general specification of the attribute-weighting function in A3. Moreover, in Section 1.4 I derive the main comparative-statics predictions of utility maximization and range weighting using only A1 and A3. Therefore my primary tests are based on the nonparametric specifications in A1 and A3 and do not rely on the specific functional forms in A2 and A4. ### 1.2 Predictions and Identification I derive the predictions of range-dependent attribute weighting in two simple contexts: two- and three-option choice sets. In each context, I focus on situations in which individuals choose between a high-quality $h$ and a low-quality $l$ variant of a product, with fixed qualities $q_h$ and $q_l$ and where $q_h>q_l$. The contexts differ in whether the option to not buy is available. I assume that not buying has zero quality and is costless. I then investigate how predicted choices depend on the price pairs $(p_h,p_l)$ that people face. Roughly speaking, two forces motivate range-weighting agents: obtaining surplus and the influence of overweighted attributes in choice sets. In practice, $\gamma$ is identified when these two forces come into sufficient conflict, such that surplus maximization and range weighting make different predictions. Formally, identification requires that changes in $\gamma$ lead to unique changes in predicted choices. The remainder of this section illustrates which choice sets and prices lead to distinct predictions and formalizes the identification argument. I assume the parametric forms A2 and A4 hold for surplus maximization and range weighting respectively. In Sections 1.3 and 1.4, I show that the predictions and identification hold nonparametrically. Consider first a binary choice between $h$ and $l$. In this context, changes in $\gamma$ do not lead to changes in choices, and therefore $\gamma$ cannot be identified. To see this, note that option $h$ will be preferred whenever $\left( \Delta_q \right)^\gamma q_h - \left(\Delta_p \right)^\gamma p_h \geq \left( \Delta_q \right)^\gamma q_l - \left( \Delta_p \right)^\gamma p_l$. For the non-trivial case where $p_h>p_l$, this condition can be rewritten as $\left(\Delta_q\right)^{1+\gamma}>\left(\Delta_p\right)^{1+\gamma}$, which holds if and only if the surplus on $h$ exceeds that on $l$. Therefore, surplus maximization and range weighting make the same predictions in this context.[^3] It is therefore possible to obtain proxies for quality in this environment. In particular, when evaluating a product with quality $q$ in isolation, range-weighting agents will buy if $pq$, and will be indifferent at $p = q$. Now suppose that the option of not buying either product, $o$, is also available and therefore the choice set is $\mathcal{C}(p_h,p_l) = \{(q_h,p_h),(q_l,p_l),(0,0)\}$. Note that because the qualities are fixed, the prices of $h$ and $l$ define the choice set. Identification of $\gamma$ therefore hinges on how predicted choices vary as a function of these prices. Figure 1 depicts optimal choices for a surplus maximizer ($\gamma=0$) facing choice sets of the form $\mathcal{C}(p_h,p_l)$ as a function of $p_h$ and $p_l$. The solid black lines depict indifference boundaries: that is, price pairs such that the agent is indifferent between the two adjacent options. A surplus maximizer will choose $l$ whenever it has a positive surplus $(p_lq_h-q_l$). This occurs in the top-left region of the $(p_h,p_l)$ space. In the bottom-right region, surplus maximizers prefer $h$ as it has a positive surplus ($p_hq_l$ and $p_h>q_h$), and hence not buying is optimal. ![Figure 1: Optimal choices as a function of (p_(h), p_(l)) for a surplus-maximizing agent choosing between a high- and low-quality variant of a product along with the option to not buy. The qualities of the high- and…](assets/composition-1.png) **Figure 1: Optimal choices as a function of $(p_h,p_l)$ for a surplus-maximizing agent choosing between a high- and low-quality variant of a product along with the option to not buy. The qualities of the high- and low-variants are fixed and denoted by $q_h$ and $q_l$.** Changes in $\gamma$ will impact the location of these boundaries in the price space. The range of quality is set solely by the high-quality option, $\Delta_q = q_h$, while the price range is determined by whichever price is highest, $\Delta_p=\max\{p_h,p_l\}$. Figure 2 depicts the indifference boundaries under focusing in panel (a) and relative thinking in panel (b). First, consider how $\gamma$ impacts the boundary between buying $l$ and not buying. In the area surrounding this boundary, the price range ($\Delta_p=p_h$) exceeds the quality range ($\Delta_q=q_h$). According to focusing, this causes agents to overweight prices relative to quality. As not buying is cheaper than $l$, focusing agents become more likely to not buy: hence, the indifference set shifts to the left as depicted in the top-left area of panel (a). Conversely, relative thinking assumes that the wider price range makes agents less price sensitive, and therefore more likely to purchase $l$. This causes the indifference boundary to shift to the right, as depicted in panel (b). ![Figure 2: Optimal choices for focusing (left panel) and relative-thinking (right panel) agents given a choice between a high- and low-quality variant of a product along with the option to not buy. The subjective…](assets/composition-2.png) ![Figure 2: Optimal choices for focusing (left panel) and relative-thinking (right panel) agents given a choice between a high- and low-quality variant of a product along with the option to not buy. The subjective…](assets/composition-3.png) **Figure 2: Optimal choices for focusing (left panel) and relative-thinking (right panel) agents given a choice between a high- and low-quality variant of a product along with the option to not buy. The subjective values of the high- and low-quality variants are fixed and denoted by $q_h$ and $q_l$. The weight on attribute $x \in \{q,p\}$ is $\left(\Delta_x\right)^\gamma$ where $\Delta_q=q_h$ and $\Delta_p = \max\{p_h,p_l\}$. The left panel depicts the indifference set for $\gamma=1$ and the right panel depicts the indifference set for $\gamma=-0.6$. The dotted lines denote the surplus-maximizing indifference set ($\gamma=0$). The highlighted points are examples of choice reversals that are inconsistent with surplus maximization.** A similar intuition extends to the region surrounding the boundary between buying $h$ and not buying. This boundary will shift when option $l$ sets the price range, $p_l>p_h$, and the price range ($\Delta_p=p_l$) exceeds the quality range ($\Delta_q=q_h$). According to focusing, this leads to heightened price sensitivity and a downward shift in the $h$-$o$ boundary. Relative thinking predicts that agents become less price sensitive and therefore more likely to buy $h$. This causes an upward shift in the $h$-$o$ boundary. Finally, consider the area surrounding the boundary between buying $l$ and buying $h$. Around this boundary, both prices are lower than $q_h$ and thus $\Delta_q>\Delta_p$. Focusing predicts that agents will overweight quality, shifting preferences toward $h$ and leading to an upward shift in the $l$-$h$ boundary. Relative thinking predicts the opposite: increased price sensitivity that favors $l$ and causes a downward shift in the $l$-$h$ boundary. Figure 2 highlights the source of identification for $\gamma$: the observed choice boundaries identify its sign and magnitude. More formally, consider an agent with some $\gamma_1$ making choices in the area surrounding the $l$-$o$ boundary. The threshold prices for which this agent is indifferent between $l$ and $o$ is defined by the equation $p_l = q_l \left(q_h/p_h\right)^{\gamma_1}$. Using this structure, it is possible to use observed choices at different prices to map out the indifference boundary and hence calculate the value of $\gamma_1$. The same intuition extends to the $h$-$o$ and $l$-$h$ boundaries. The experiment will create variation around all three boundaries, and my main estimation of $\gamma$ will use all the data. However, in principle, $\gamma$ could be estimated from any one boundary. As a validation exercise, I also estimate $\gamma$ separately using data around each boundary and check whether I get similar estimates. ### 1.3 Nonparametric Identification The identification argument in Section 1.2 generalizes beyond the parametric form A4. The following proposition establishes identification of the range-weighting parameter $\gamma$ with ideal data for choice sets of the form $\mathcal{C}(p_h,p_l)$. Ideal data in this context consist of observing choices at every price pair $(p_h,p_l) \in \mathbb{R}_+^2$. Let $j(p_h,p_l)$ denote the observed choice correspondence. Let $j^*(p_h,p_l|\gamma) \equiv \arg\max_j \left[ g(\Delta_q;\gamma) \; q_j - g(\Delta_p;\gamma) \; p_j\right]$ define the model-implied choice correspondence, which maps prices to choices for a given $\gamma$. The data set contains the observed choice $j(p_h,p_l)$ at all price pairs $(p_h,p_l) \in \mathbb{R}_+^2$. Proposition 1: Suppose that A3 holds. Then for choice set $\mathcal{C}(p_h,p_l)$, there exists either - No $\gamma$ such that $j^*(p_h,p_l|\gamma)=j(p_h,p_l)$ for all $(p_h,p_l) \in \mathbb{R}_+^2$, or - A unique $\gamma$ such that $j^*(p_h,p_l|\gamma)=j(p_h,p_l)$ for all $(p_h,p_l) \in \mathbb{R}_+^2$. Proposition 1 states that the parameter $\gamma$ can be identified for any family of functions admitting a single-index representation, subject to mild regularity conditions.[^4] Intuitively, any feasible $\gamma$ maps out unique indifference boundaries in the $(p_h,p_l)\in \mathbb{R}_+^2$ space. Observing choices for every possible price pair maps out the boundaries generated by $\gamma$. Qualitatively, these boundaries are similar to those depicted in Figures 2 and 3: they shift smoothly in response to changes in $\gamma$. Therefore, each partition corresponds to a distinct $\gamma$, and so revealing these boundaries is sufficient to identify $\gamma$. The proof of Proposition 1 defines three mutually exclusive regions that contain the $l$-$o$, $h$-$o$, and $l$-$h$ indifference boundaries. Within each of these regions, either of the two relevant choices can be rationalized by some $\gamma \in (\underline{\gamma},\overline{\gamma})$. This formalizes the intuition that there are three distinct sources of identification for $\gamma$: one for each indifference boundary. ### 1.4 Comparative-Statics Predictions In this section, I derive the comparative-statics predictions that the experiment tests. In the previous section, I assumed that there was a perfect mapping from utilities to choice. In practice, choices are noisy which can lead to random deviations from the model predictions. To account for such errors, I assume that choice is based on utility plus an additive i.i.d. error term that captures choice noise. This gives rise to choice probabilities and my comparative-static predictions focus on how those probabilities respond to changes in prices under each model. First, consider what random utility maximization predicts for choice sets of the form $\mathcal{C}(p_h,p_l)$. Under this approach, agents evaluate each option $\mathcal{U}_j = U_j + \varepsilon_j$ and choose $j^* \equiv \arg\max_j \mathcal{U}_j$. Let $\widehat{\varepsilon}_{jk} \equiv \varepsilon_j - \varepsilon_k$ denote the error difference between options $j$ and $k$. The only restriction I place on this error difference is that the joint distribution of error differences is strictly positive at all values; in other words, that there is always a non-zero probability of choosing each option. With minimal assumptions on $U_j$ and $\varepsilon_j$, random utility maximization has clear testable predictions. *Proposition 2: Suppose A1 holds, the choice set is $\mathcal{C}(p_h,p_l)$ and choice is determined by $j^*$. Then, the probability of choosing $h$ or $l$ is strictly decreasing in the prices of $h$ and $l$.* Proposition 2 states an obvious prediction of utility maximization: agents are less likely to buy $h$ or $l$ when prices increase. Crucially, range-dependent attribute weighting suggests that there are certain price regions in which this prediction does not hold. Around each of the indifference boundaries in Figure 2, one of three options serves as a “decoy”: this option is never chosen, yet it affects preferences for the two nondecoy options by altering the relative weights on quality versus price. Framed in this way, there are three types of choice sets contained in $\mathcal{C}(p_h,p_l)$ that are of interest: $h$-decoy, $l$-decoy, and $o$-decoy.[^5] Prices from these decoy regions provide a test of utility maximization and allow for simple comparative-statics tests of whether $\gamma$ is positive, negative, or zero. Prices around the $l$-$o$ boundary imply that $h$ is the decoy option. While $h$ has superior quality, agents will not purchase it due to its high price. However, increasing the price of $h$ in this region can cause demand to shift toward either $l$ or $o$ depending on the sign of $\gamma$. Intuitively, because $p_h$ determines the price range, changes in $p_h$ alter the relative weight on price. For example, consider points $a$ and $a^\prime$ in Figure 2 panel (a). At point $a$, $l$ is preferred to not buying. At point $a^\prime$, the higher price of $h$ expands the price range while $p_l$ remains unchanged. According to focusing, this leads to greater price sensitivity and a choice reversal from buying $l$ to not buying. Conversely, relative thinking predicts that the higher $p_h$ can shift demand toward buying $l$. Intuitively, a wider price range reduces price sensitivity, boosting the appeal of $l$ relative to not buying. For example, moving from point $x$ to $x^\prime$ in panel (b) leads to a choice reversal from not buying to buying $l$. Prices around the $h$-$o$ boundary imply that $l$ is the decoy option. This is a more traditional decoy situation because $h$ asymmetrically dominates $l$. Once again, the price range exceeds the quality range on $l$-decoy decisions. As a result, focusing predicts that demand can shift from buying $h$ to not buying (for example, $b$ to $b^\prime$) and relative thinking predicts demand shifts from not buying to buying $h$ (for example, $y$ to $y^\prime$). Prices from around the $l$-$h$ boundary lead to the most novel type of decoy situations. Here, the prices of both $l$ and $h$ are relatively low, and hence one of these two options is always chosen. Not buying serves as a decoy because it expands the quality range by more than the price range, but it is never chosen. As a result, increasing the price level brings the attribute ranges, and hence their relative weights, closer to parity. Therefore, range-dependent attribute weighting predicts that both relative prices and the absolute price level matter when deciding between $h$ and $l$. Specifically, for a fixed price difference, focusing predicts that higher prices can cause demand to shift toward $l$ (for example, point $c$ to $c^\prime$), while relative thinking predicts that demand can shift toward $h$ (for example, point $z$ to $z^\prime$). The predictions outlined above do not account for choice noise. In practice, random errors in evaluation will imply that the decoy option is sometimes chosen. An implication of this behavior is that choice probabilities may not be monotonic in the utility difference between the nondecoy options due to substitution to and from the decoy option. For example, suppose that probability of choosing $l$ is increasing in the price of $h$ in the $h$-decoy region. This is consistent with relative thinking but could also be consistent with focusing if there is enough substitution from $h$ to $l$. However, focusing would also predict that the probability of choosing $o$ is increasing in the price of $h$, while relative thinking predicts the opposite. As a result, the full pattern of substitution needs to be taken into account when testing for focusing versus relative thinking. Formally, suppose that range-weighting agents evaluate each option $j$ as $\mathcal{V}_j = V_j + \epsilon_j$, and choose $j^*_\gamma \equiv \arg\max_j \mathcal{V}_j$. The following proposition establishes how choice probabilities respond to changes in prices for different $\gamma$ values. Proposition 3: Suppose A3 holds, the choice set is $\mathcal{C}(p_h,p_l)$, and choice is determined by $j^*_\gamma$. - If the price of $h$ increases in the $h$-decoy region, then the probability of choosing: - $l$ is increasing, $h$ is decreasing, and $o$ is increasing if $\gamma = 0$ - $l$ is increasing or decreasing, $h$ is decreasing, and $o$ is increasing if $\gamma > 0$ - $l$ is increasing, $h$ is decreasing, and $o$ is increasing or decreasing if $\gamma < 0$. - If the price of $l$ increases in the $l$-decoy region, then the probability of choosing: - $l$ is decreasing, $h$ is increasing, and $o$ is increasing if $\gamma = 0$ - $l$ is decreasing, $h$ is increasing or decreasing, and $o$ is increasing if $\gamma > 0$ - $l$ is decreasing, $h$ is increasing, and $o$ is increasing or decreasing if $\gamma < 0$ - If the price of $h$ increases in the $o$-decoy region for a fixed $p_h-p_l$, then the probability of choosing: - $l$ is decreasing, $h$ is decreasing, and $o$ is increasing if $\gamma = 0$ - $l$ is increasing or decreasing, $h$ is decreasing, and $o$ is increasing if $\gamma > 0$ - $l$ is decreasing, $h$ is increasing or decreasing, and $o$ is increasing if $\gamma < 0$ Proposition 3 provides a guide for mapping empirical choice probabilities to an implied $\gamma$. In each price region, the three models make distinguishing predictions for how the likelihood of choosing each option responds price increases. These comparative statics are main nonparametric tests explored in this paper. Crucially, they show that it is possible to distinguish the predictions of surplus maximization, focusing, and relative thinking using only data on prices and choices. ### 1.5 From Theory to Design The experiment is designed to generate ideal variation for identifying range-based attribute weighting. I focus on the context in which participants are asked to choose between a high- and low-quality variant of a product along with the option of not buying either. It is possible to identify $\gamma$ in this common economic environment, with the practical advantage of requiring only two known qualities per product. Mapping the theoretical predictions to actual choice settings requires further assumptions. First, I assume that each option is summarized by two attributes, price and a generic measure of quality that can be expressed in monetary terms. The meaning of quality will differ across products and is not directly comparable: for example, the “quality” of chocolate will differ from the “quality” of a television. In practice, some measures of quality will be easier to put on a scale than others; it might be harder to place two options varying in brand or functionality on a quality ladder than two options varying in quantity or duration. In the experiment, I use a wide range of products to enhance the generalizability of the results. Section 2.2 provides further details about the products used in the experiment. Second, in the theoretical framework I assume that not buying is treated like any other option in the choice set and that it has zero quality. In the experiment, I also treat not buying like any other option in the choice set: it is given equal prominence in the display and is labeled with a price tag of \$0.00. In real markets, not buying might not feature so prominently in consumers’ choice sets. If participants place a nonzero quality value on not buying for some options, then the willingness-to-pay measures that I elicit can simply be reinterpreted as the difference in quality between buying and not buying. The theoretical predictions are based on observing choices at all possible prices. In reality, the number of choices I can elicit from participants is limited by practical considerations. The identification argument in Section 1.2 suggests an efficient way to sample prices for the parametric form $\left(\Delta\right)^\gamma$. For example, suppose the experimenter wanted to detect some $\gamma \in (0,1)$. Then, sampling prices from the shaded areas in panel (a) of Figure 2 is sufficient to identify any $\gamma\in(0,1)$. Similarly, sampling prices from the shaded areas in panel (b) of Figure 2 is sufficient to identify any $\gamma\in(-0.6,0)$. Intuitively, the experimenter only needs to sample prices from regions that contain the indifference boundaries. Figure 3 depicts seven indifference boundaries generated A4 for the set of $\gamma$ values $\{2,1,1/2,0,$ $-1/3,-2/3,-1\}$. Each $\gamma$ value corresponds to a unique indifference boundary in each of the $l$-$o$, $h$-$o$, and $l$-$h$ regions. This generates six price zones around each indifference boundary from which I draw experimental prices. Observing choice patterns across these zones is sufficient to identify any $\gamma\in(-1,2)$. In the experimental design, I generate prices for each subject by randomly drawing one $(p_h,p_l)$ from each of these 18 price zones. Defining the experimental price zones in Figure 3 requires knowledge of the individual-specific qualities, $q_h$ and $q_l$. One approach to measuring quality is to vary the scale of quality dimension and estimate the shape of the entire quality ladder. An advantage of this method is that it permits richer experimental variation in quality. However, this method requires many observations per product and additional structural assumptions. My approach to identification treats quality as fixed and leverages price variation. Furthermore, Section 1.2 shows that for the range-weighting model, reservation values reveal quality when individuals choose whether or not to buy a product in isolation. I use this result to obtain direct proxies for each individual’s quality for both a high- and a low-quality variant of multiple products. Specifically, I implement a two-stage experimental design: an online survey designed to elicit proxies for quality and an in-person choice task in which prices are calibrated using the first stage responses.[^6] This approach to measuring quality has the practical advantage of requiring only two questions per product and allows me to consider a broader range of products. One drawback of eliciting a single measure of quality per product variant is measurement error. Recent work has highlighted that using multiple, independent measures can help to reduce this error (Gillen et al. 2019). Building on this intuition, I elicit two measures of quality: one prior to the choice task and one after. I also use two different elicitation techniques: free response and a multiple price list. While both approaches have some well-publicized shortcomings (e.g. Cason and Plott (2014) and Andersen et al. (2006)), both measures should provide some signal value about underlying quality. By obtaining two signals of quality using different elicitation methods, I can combine them optimally in the structural analysis and perform additional robustness tests. Section 2 describes the full details of the design. ![Figure 3: Depiction of the zones from which experimental prices were drawn. The solid black lines denote indifference boundaries for {γ̄₁, γ̄₂, γ̄₃, γ̄₄, γ̄₅, γ̄₆, γ̄₇} = {2, 1, 1/2, 0, −1/3, −2/3, −1} and are…](assets/composition-4.png) **Figure 3: Depiction of the zones from which experimental prices were drawn. The solid black lines denote indifference boundaries for $\{\bar{\gamma}_1,\bar{\gamma}_2,\bar{\gamma}_3,\bar{\gamma}_4,\bar{\gamma}_5,\bar{\gamma}_6,\bar{\gamma}_7\}=\{2,1,1/2,0,-1/3,-2/3,-1\}$ and are constructed for the attribute weighting function $\left(\Delta\right)^\gamma$. Agents are modeled as choosing between a high- and low-quality variant of a product along with the option to not buy. The subjective values of the high- and low-quality variants are fixed and denoted by $q_h$ and $q_l$.** ## 2 Experimental Design This section describes the implementation of my laboratory experiment. Appendix B contains all instructions to participants and additional screenshots of the experiment. Online surveys took place on the Qualtrics platform. The main experimental interface was programmed in Matlab and Psychtoolbox. In-person sessions took place at Cornell’s Lab for Experimental Economics and Decision Research (LEEDR) in nine sessions spread over five days in February 2018. ### 2.1 Method and Implementation I recruited 190 non-students for the experiment from an online participant pool administered by LEEDR. Participants enrolled in this system are non-faculty Cornell staff and non-affiliated local residents.[^7] The experiment consisted of two parts: an online valuation task and an in-person choice task. *Stage 1: The Online Valuation Task* The valuation task consisted of decisions of the following form: “Please enter the amount that makes you indifferent between the following two options. - **Option A:** Do not purchase the product. - **Option B:** Spend \$ and receive \[product description\].” The product was depicted below this text and participants could choose to reveal detailed product information by clicking a link. Each participant completed 24 decisions like this (referred to hereafter as trials)—one for a “high”-quality variety ($h$) and one for a “low”-quality variety ($l$) for each of 12 products (see Section 2.2 for product details). The trials for the $h$ and $l$ varieties of each product appeared back-to-back in random order. Participants could enter any positive amount they wished and could state a higher valuation for whichever of the two varieties they preferred. Upon completion of the trials for a given product, a confirmation screen displayed the stated valuation for each variety side-by-side and asked participants to confirm their responses. The order in which each of the 12 products appeared was also randomized. All participants received a flat \$20 payment for taking part in the study and had the opportunity to win additional bonuses. To incentivize responses, participants had an approximately one-in-five chance of winning an additional cash payment of \$80 during the second stage (see details below).[^8] If participants won this additional \$80, there was a 20% chance that one of their choices from the first stage would be implemented for real using the BDM mechanism.[^9] An appealing feature of this bonus-payment structure is that it incentivizes participants to treat each decision independently. The independence of each trial was repeatedly emphasized in the instructions. Participants were emailed a link to the valuation task seven days before the in-person choice task and asked to complete it in one sitting. The deadline for completing the online survey was 72 hours after the link was received. This meant there were at least 96 hours between completion of the first stage and the beginning of the second stage. This gap reduced the possibility that subjects anchored on their first-stage valuations when making choices during the second stage.[^10] I imposed certain eligibility criteria on the first-stage reservation values. First, reservation values for the two varieties of the product had to differ. Second, valuations had to be non-trivial to allow scope for price variation. Third, reservation values could not be so high that the participation payment could not cover the purchase cost. Finally, blatant non-monotonicities, such as participants stating higher valuations for $l$ than $h$, were removed. To ensure these four conditions were satisfied, participants had to value the high-quality variant strictly more than the low-quality variant, and their valuations had to lie within the interval $[\$4,\$100]$ for at least nine of the products. If fewer than nine products satisfied these criteria, participants were ineligible for second stage. If more than nine product valuations met the eligibility criteria, any products valued at the upper limit of \$100 were dropped to enhance the scope for price variation in the second stage. If the final set of eligible products was ten or more, then nine were selected at random. Of the 190 participants who completed the valuation task, 26 were ineligible for the second stage. An additional 14 participants did not show up or cancelled, leaving a final sample size of 150. The final sample was 72.7% female, 81.3% white, and 54% married. The median participant was 47 years old, had a bachelor’s degree, and lived in a household with annual income between \$75,000 and \$99,999. *Stage 2: The Choice Task* At the in-person lab session, participants made a series of choices from bundles containing two varieties of the same product with the option to not purchase either. In each trial, participants encountered one of nine products from the first stage. Participants completed a total of 18 trials (two per product), presented in two blocks of nine (one trial per product). Trials in each block were randomized subject to the proviso that the last good in block 1 was not the first good of block 2. The location of each option was random in each trial. Figure 4 depicts a typical trial. Each option was displayed onscreen along with a price tag. The option of not buying was included explicitly; it was represented by a red circle with a line through it and a price tag labaled \$0.00. The onscreen text instructed participants to consider the three options for the first ten seconds. Once this initial viewing time elapsed, additional text appeared onscreen asking participants to choose their preferred option. Participants responded by pressing one of three keys labeled “A”, “B”, or “C” on the keyboard in front of them. They could also reveal detailed product information by pressing (and holding) one of three “INFO” buttons on the keyboard. The average time spent completing each trial was 18 seconds. ![Figure 4: Screenshot of a second-stage choice task. Participants respond by pressing one of three keys labeled “A”, “B”, and “C” on the keyboard in front of them. Detailed product information is displayed by pressing…](assets/example-trial-3883c1.png) **Figure 4: Screenshot of a second-stage choice task. Participants respond by pressing one of three keys labeled “A”, “B”, and “C” on the keyboard in front of them. Detailed product information is displayed by pressing (and holding) one of three “INFO” keys. This example trial is generated for an individual who valued the Fire 8 tablet at \$59.99 and the Fire 7 Tablet at \$39.99 in the first stage.** The key experimental manipulation in each trial was the prices of the high- and low-quality variants. In each trial, the prices of $h$ and $l$ were drawn from one of the 18 subregions visualized in Figure 3.[^11] These subregions were chosen to ensure that there was sufficiently granular price variation for estimation while not fatiguing participants with too many trials. I imposed two further restrictions on prices. First, I set prices so that the decoy option in each region had the smallest surplus by a margin of at least 25%. In practice, the minimum price of $h$ and $l$ was set to $1.25 q_h$ in the $h$-decoy and $l$-decoy price regions, respectively. For prices in the $o$-decoy price region, the maximum price of $l$ was $0.75 q_l$. Second, I imposed a universal maximum price of $\max\{2 q_h,\$ 100\}$ and a universal minimum price of $\$0.50$. Participants had the opportunity to win an \$80 bonus payment after each decision. They chose a number between one and 144, which was then highlighted in green on one of 144 red balls onscreen. An animation then highlighted one of the balls in yellow every 0.01 seconds and participants could stop the animation by pressing the space key (see Appendix B for screenshots of this animation).[^12] If the animation stopped on their chosen number, they received the bonus payment and their previous choice was implemented for real; in other words, they would purchase the chosen option at the posted price.[^13] Participants could win multiple bonuses and spent an average of 13.6 seconds completing each ball draw. The motivation for this incentive structure was twofold. The first was to incentivize participants to respond as if each decision were real; making each choice *potentially* real was the most feasible option given the relatively large stakes involved. The second motive was to reduce any potential contamination of the choice set across trials by providing psychological distance between each choice. Therefore, the incentive implementation was designed to be distracting and somewhat entertaining. To encourage participants to treat each decision independently, the following text was displayed on completion of the incentive animation: “your previous choice is now final and will have no bearing on future decisions.” Once participants had completed all 18 choice trials, they faced a valuation task similar to that in stage 1 for the nine products used in the second stage. The purpose of this task was to obtain a second estimate of quality to aid estimation and allow for robustness checks. The procedure was the same as the first stage, but with the following modifications. Rather than inputting a value directly, participants decided whether or not to purchase the displayed product at the posted price. The price started at \$0 and rose in increments of \$5, up to \$100. On average, participants spent 29.5 seconds making choices for each product variety. The average price at the switching point was taken as the reservation value for that product. If more than one crossing point was indicated, the non-monotonicity was explained to participants and they repeated their choices for that product variety. At the end of the second stage, participants had a further opportunity to win a bonus of \$80 and have one of their valuation trials (24 online + 18 in person) implemented for real. Participants could choose nine numbers from one to 144. Then, the random-ball-draw animation was implemented as before. If the animation stopped on one of the chosen numbers, the participant received a bonus of \$80. Further, one of their valuation trials from either the online survey or in-person trials was chosen at random and implemented for real using the BDM procedure. Participants proceeded through the task at their own pace and all instructions appeared onscreen. Once the task was completed, participants were given a brief post-study questionnaire to complete before receiving their payments. All products were delivered electronically or mailed to participants. ### 2.2 Products The products used in the experiment were chosen to achieve several objectives. First, reservation values needed to be non-trivial. To achieve this, the menu of products had an average retail value of approximately \$60 and was chosen to be widely appealing. Furthermore, the exact products participants saw were tailored to their tastes: before completing any trials, each participant selected 12 products that they would most like to purchase from a list of 20. The chosen products were then used throughout the remainder of the experiment. Second, the experiment featured many products to prevent potential contamination of the choice sets due to repeated encounters with the same product. Nine out of the 12 products chosen by participants in stage 1 appeared in stage 2. Participants never saw the same product back-to-back and made no more than two choices per product. Third, the products provide a clear mapping from the theoretical framework to the experimental design. Products conformed to a natural quality ladder, with each product differentiated by one of four broad features: quantity (more vs. less), duration (longer vs. shorter), functionality (more vs. less), and brand (known high quality vs. private label). Each product had two vertically differentiated variants. Examples of these product categories include quantity of chocolate truffles, duration of a wine club membership, a premium versus basic streaming device, and branded versus private label reusable batteries. The complete list of products and variants used in the experiment can be found in Appendix B. ## 3 Nonparametric Evidence First, I assess whether the aggregate demand shares are consistent with random utility maximization. Proposition 2 establishes that utility maximization implies demand for $h$ or $l$ is decreasing in the prices of $h$ and $l$. However, range-dependent attribute weighting suggests that this prediction may not hold for the price of the decoy option in the associated price region. To test this prediction, I estimate the probability of choosing $h$ or $l$ as function of the prices of $h$ and $l$. **Table 1: Testing Utility Maximization** | | | | |:-----------------------------|:--------------------|:--------------------| | | \(1\) | \(2\) | | Decoy Region: | $h$ | $l$ | | Dependent Variable: $\qquad$ | Buy *low* or *high* | Buy *low* or *high* | | | | | | Price of $h$ | 0.28 | $-$0.83 | | | (0.10) | (0.11) | | Price of $l$ | $-$0.60 | 0.61 | | | (0.15) | (0.10) | | | | | | Mean of DV | 59.1 | 59.3 | | Individuals | 150 | 150 | | Observations | 900 | 900 | Average marginal effects derived from logistic regression estimates. The choice set consists of a high-quality ($h$) and low-quality ($l$) variant of a product, along with the option to not buy. The decoy option was priced such that it is clearly inferior and it alters the attribute ranges within the choice set. The specification includes participant demographics (age, gender, education, marital status and ethnicity) and round fixed effects. Individual cluster-robust standard errors in parentheses. All numbers reported in percentages. Columns (1) and (2) of Table 1 present the estimates from the $h$- and $l$-decoy regions. The estimates are inconsistent with utility maximization in both regions: higher decoy prices lead to an increase in buying. A \$1 increase in the price of the $h$-decoy option made participants 0.28 percentage points more likely to buy $h$ or $l$, while a \$1 increase in the price of the $l$-decoy option made participants 0.61 percentage points more likely to buy $h$ or $l$. The main hypothesis I assess is whether the data are consistent with focusing or relative thinking. My identification argument relies on observing shifts in choice probabilities due to changes in the prices of decoy options. Table 2 presents the aggregate demand shares for each experimental price region. In all three experimental price regions, participants saw an equal number of prices above and below their stated indifference values for the two nondecoy options. If the first-stage reservation values are unbiased proxies for quality, then surplus maximization predicts equal demand shares for these two options. In contrast, range-dependent attribute weighting predicts higher demand for one of the two nondecoy options, with focusing and relative thinking making opposite predictions. **Table 2: Demand Shares by Experimental Price Region**
(1) (2) (3)
Fraction Choosing: $\quad$ high ($h$) low ($l$) don’t buy ($o$)
$\boldsymbol h$-decoy region 10.9% 48.2% 40.9%
H$_0$: (2) = (3) $\quad \left[p<0.01\right]$
$\boldsymbol l$-decoy region 57.7% 1.7% 40.7%
H$_0$: (1) = (3) $\quad \left[p<0.001\right]$
$\boldsymbol o$-decoy region 36.7% 49.1% 14.2%
H$_0$: (1) = (2) $\quad\left[p<0.001\right]$
Aggregate demand shares for each option across the three experimental price regions. The price of the decoy option was always above the participant’s stated willingness-to-pay. For the two nondecoy options, participants saw an equal number of prices above and below their stated indifference values. In all three regions, I test the null hypothesis that the demand shares of the two nondecoy options are equal. In the $h$-decoy region, participants were 7.3% more likely to choose $l$ over not buying ($p<0.01$). This is consistent with a rightward shift in the indifference boundary as predicted by relative thinking and depicted in Figure 2 panel (b). In the $l$-decoy region, the fraction of participants choosing $h$ was 17.0% higher than the fraction choosing to not buy ($p<0.001$). This implies an upward shift in the indifference boundary in this region consistent with the prediction of relative thinking. Finally, in the $o$-decoy region participants preferred $l$ over $h$ by a margin of 12.4% ($p<0.001$). This implies a downward shift in the indifference boundary, which is again consistent with relative thinking. Participants sometimes chose the decoy option in each region, with some differences across regions. The pattern of decoy choices is consistent with choice noise. Participants were least likely to choose the $l$ decoy, which had the greatest surplus difference relative to the next-best option. Participants were most likely to choose the $o$ decoy, which had the smallest absolute surplus difference compared to the next-best option. Moreover, participants whose first- and second-stage valuations were more highly correlated were less likely to choose the decoy option. To formally test for range dependence, I assess comparative-statics predictions derived in Section 1.4. Columns (i), (ii), and (iii) of Table 3 summarize the predictions stated in Proposition 3 for surplus maximization, focusing, and relative thinking, respectively. This analysis does not rely on the first-stage quality perceptions elicited in the experiment. To the extent that experimental prices were miscalibrated due to inaccurate quality proxies, the estimates will be biased towards finding a null effect. **Table 3: Change in Choice Probabilities in Response to Price Increases**
(i) (ii) (iii) (iv)
Surplus Maximization Focusing Relative Thinking Data
$\gamma=0$ $\gamma>0$ $\gamma<0$
(a) $\uparrow \; \boldsymbol p_h$ in the $\boldsymbol h$-decoy region
Buy high $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
Buy low $\uparrow$ $\uparrow$ or $\downarrow$ $\uparrow$ $\uparrow$
Don’t buy $\uparrow$ $\uparrow$ $\uparrow$ or $\downarrow$ $\downarrow$
(b) $\uparrow \; \boldsymbol p_l$ in the $\boldsymbol l$-decoy region
Buy high $\uparrow$ $\uparrow$ or $\downarrow$ $\uparrow$ $\uparrow$
Buy low $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$$^\dagger$
Don’t buy $\uparrow$ $\uparrow$ $\uparrow$ or $\downarrow$ $\downarrow$
(c) $\uparrow \; \boldsymbol p_h$ in the $\boldsymbol o$-decoy region$^*$
Buy high $\downarrow$ $\downarrow$ $\uparrow$ or $\downarrow$ $\uparrow$
Buy low $\downarrow$ $\uparrow$ or $\downarrow$ $\downarrow$ $\downarrow$
Don’t buy $\uparrow$ $\uparrow$ $\uparrow$ $\uparrow$$^\dagger$
$^*$For a fixed $p_h - p_l$. $^\dagger$not statistically significant at conventional levels. Summary of the predictions of surplus maximization, focusing, and relative thinking, along with the choice probability estimates from Table 4. The choice set consists of a high- and low-quality variant of a product, along with the option to not buy. The decoy option is priced such that it is clearly inferior and it alters the attribute ranges within the choice set. The predictions in panels (i), (ii), and (iii) are stated in Proposition 3 in Section 1.4. Panel IV summarizes the direction of the estimated coefficients in Table 4. Table 3 provides a guide for mapping choice probability estimates to a value of $\gamma$. The key to identifying range weighting is to focus on deviations from surplus maximization. Panels (a), (b), and (c) summarize how choice probabilities respond to expansions in the price range for surplus maximization, focusing, and relative thinking. According to focusing, expanding the price range makes individuals more price sensitive. This implies that individuals will become less likely to buy $l$ in the $h$-decoy region, less likely to $h$ in the $l$-decoy region, and more likely to buy $l$ in the $o$-decoy region. Conversely, relative thinking assumes that wider price ranges decrease price sensitivity. This implies that individuals will become less likely to not buy in the $h$- and $l$-decoy regions, and more likely to $h$ in the $o$-decoy region. To test these predictions, I estimate the probability of choosing $h$, $l$, and $o$ as a function of the prices of $h$ and $l$ using a logistic regression for each of the three decoy choice sets. Panel (a) of Table 4 presents the results for the $h$-decoy choice sets. The estimates are consistent with the law of demand: participants were less likely to buy $l$ or $h$ when its own price was higher, as predicted by surplus maximization, focusing, and relative thinking. However, higher prices of the decoy option $h$ led to an increase in the likelihood of choosing $l$ and a decrease in the likelihood of not buying. Consistent with relative thinking, a \$1 increase in the price of $h$ increased the probability of buying $l$ by 0.44 percentage points and decreased in the probability not buying by 0.28 percentage points. **Table 4: Testing Focusing versus Relative Thinking**
(1) (2) (3)
Dependent Variable: Buy high Buy low Don’t Buy
Panel (a): $\boldsymbol h$-decoy region
Price of $h$ $-$0.16 0.44 $-$0.28
(0.07) (0.09) (0.10)
Price of $l$ 0.42 $-$1.11 0.60
(0.09) (0.14) (0.15)
Panel (b): $\boldsymbol l$-decoy region
Price of $h$ $-$0.80 $-$0.01 0.83
(0.11) (0.03) (0.11)
Price of $l$ 0.60 $-$0.01 $-$0.61
(0.10) (0.03) (0.10)
Panel (c): $\boldsymbol o$-decoy region
Price of $h$ 0.52 $-$0.74 0.22
(0.21) (0.24) (0.15)
Price Difference $-$1.48 1.75 $-$0.35
($h$ minus $l$) (0.30) (0.32) (0.20)
Individuals 150 150 150
Observations 900 900 900
Average marginal effects derived from logistic regression estimates. The choice set consists of a high-quality ($h$) and low-quality ($l$) variant of a product, along with the option to not buy. The decoy option was priced such that it is clearly inferior and it alters the attribute ranges within the choice set. The specification includes participant demographics (age, gender, education, marital status and ethnicity) and round fixed effects. Demographic and round fixed are excluded in specification (2) for the $l$-decoy regressions due to the lack of variation in the outcome variable. Individual cluster-robust standard errors in parentheses. All numbers reported in percentages. Panel (b) presents the estimates for the $l$-decoy choice sets. The probability of choosing $h$ is again decreasing in its own price but increasing in the price of the decoy option, $l$: a \$1 increase in the price of $l$ increased the probability of buying $h$ by 0.60 percentage points. Furthermore, the probability of not buying is decreasing in the price of the decoy option: A \$1 increase in the price of $l$ reduces the likelihood of not buying by 0.61 percentage points. This pattern of results is again consistent with relative thinking, but inconsistent with focusing and surplus maximization. To assess the impact of parallel price increases in the $o$-decoy region, I estimate the likelihood of choosing each option as a function of both the price difference between $h$ and $l$ and the absolute price level of $h$. The results, presented in panel (c), show that the absolute price level matters: for a fixed price difference, higher prices of $h$ are associated with increased demand for $h$. This result is only possible under relative thinking, and is not permitted under focusing or surplus maximization. Panel IV of Table 3 summarizes the evidence from Table 4. Across all three price regions, expanding the price range through decoy options increases the probability of buying the highest quality nondecoy option. The overall pattern of results is consistent with relative thinking, which predicts that wider price ranges reduce price sensitivity. In all cases in which surplus maximization, focusing, and relative thinking make distinguishing predictions, the data support relative thinking. ### 3.1 Individual Heterogeneity Evidence of relative thinking in the aggregate may mask substantial individual heterogeneity in range-dependent attribute weighting. This is particularly relevant in my experimental context because relative thinking and focusing make opposite predictions. It is possible that many participants made choices consistent with focusing despite aggregate demand patterns consistent with relative thinking. To investigate individual heterogeneity in range-dependent attribute weighting, I measure how each participant’s choices align with the predictions of focusing and relative thinking. The price variation that permits identification of $\gamma$ in the aggregate data also allows me to classify each participant as a focuser or relative thinker. In each trial, there is always a choice that is consistent with focusing, relative thinking, and surplus maximization. For example, when $l$ is priced below a participant’s reservation value in the $h$-decoy region, choosing $l$ is consistent with all three models provided $\gamma$ is not too large. However, there is also a choice that can falsify one of the range models and surplus maximization: for example, choosing $o$ is inconsistent with relative thinking and surplus maximization in the previous example. By the same logic, when $l$ is priced above a participant’s reservation value in the $h$-decoy region, choosing $o$ or $l$ is consistent with relative thinking but only choosing $o$ is consistent with focusing. For each participant, I score how many of their 18 experimental choices are consistent with focusing or with relative thinking. Based on these scores, 58.7% of participants are most consistent with relative thinking, 32.7% are most consistent with focusing, and 8.7% are equally consistent with both models. The average consistency scores for the relative thinkers, focusers, and equally consistent types were 15.0, 14.0, and 13.7, respectively. As a benchmark, a participant choosing at random would make nine choices consistent with both focusing and relative thinking in expectation. To test whether these scores are statistically different from random choice, I compute the random-choice null distribution of consistency scores. Noting that the consistency score for each participant is simply the sum of two binomial distributions under this null, it follows that a score of 12 or more is inconsistent with random choice at the conventional 5% significance threshold. Using this cutoff value, I can reject the random-choice benchmark for 97% of relative thinkers, 94% of focusers, and 92% the equally consistent types. I also score each participant’s choices for consistency with surplus maximization. On average, participants made 10 choices that were consistent with surplus maximization. As a benchmark, a participant choosing at random would make six surplus maximizing choices. A score of nine or more is inconsistent with the randomization null at the conventional 5% significance threshold. I can reject this benchmark for 72% of participants. Overall, relative thinking improves the consistency score versus surplus maximization by at least one score for 86.7% participants. While almost one third of participants were best classified as focusers, they also appear to make noisier decisions. First, focusers had significantly lower consistency scores on average compared to relative thinkers ($p<0.01$). Second, the quality valuations in the first and second stage were less correlated for focusers compared to relative thinkers. The average correlation was 0.51 for focusers versus 0.63 for relative thinkers, a difference that is statistically significant from zero ($p<0.01$). Furthermore, focusers were more likely to choose the decoy option compared to relative thinkers. On average, focusers chose the decoy option 2.1 times versus 1.3 times for relative thinkers ($p<0.01$). To assess whether participants are classified as focusers or relative thinkers due to noise or genuine choice-set dependence, I re-estimate the choice probabilities as a function of prices for both groups. Columns (1) to (3) of Table 5 present the results for the relative thinkers. The estimates are similar to the aggregate results: wider price ranges are associated with buying more and cause substitution to higher quality options. However, for the focusers, the estimates in Panel (b) do not support the comparative-statics predictions of focusing. In the $h$- and $o$-decoy regions, the estimated coefficients are most consistent with noisy surplus maximization. In the $l$-decoy region, the estimates are consistent with relative thinking. Overall, the apparent focusers do not respond to price changes in a manner consistent with focusing. **Table 5: Choice Probability Estimates by Type**
Relative Thinkers Focusers
(1) (2) (3) (4) (5) (6)
Outcome: Buy high Buy low Don’t Buy Buy high Buy low Don’t Buy
$\boldsymbol h$-decoy region $\boldsymbol h$-decoy region
Price of $h$ $-$0.18 0.49 $-$0.30 $-$0.16 0.31 $-$0.15
(0.09) (0.13) (0.13) (0.14) (0.13) (0.17)
$\boldsymbol l$-decoy region $\boldsymbol l$-decoy region
Price of $l$ 0.46 0.03 $-$0.53 0.62 $-$0.06 $-$0.57
(0.13) (0.03) (0.13) (0.16) (0.05) (0.17)
$\boldsymbol o$-decoy region $\boldsymbol o$-decoy region
Price of $h$ 0.73 $-$1.02 0.35 0.22 $-$0.29 0.08
(0.27) (0.30) (0.14) (0.34) (0.35) (0.32)
Observations 528 528 528 288 288 288
Individuals 88 88 88 48 48 48
Choice-probability estimates by type classification. Participants are classified as relative thinkers or focusers based on how well their choices aligned with the theoretical predictions of each model. The numbers reported are average marginal effects derived from logistic regression estimates. The specification includes participant demographics (age, gender, education, marital status and ethnicity) and round fixed effects. Demographic and round fixed are excluded from specifications (2) and (5) for the $l$-decoy regressions due to the lack of variation in the outcome variable. The regressions either control for the price of $l$ (panel a), the price of $h$ (panel b), or the price difference between $h$ and $l$ (panel c). Individual cluster-robust standard errors in parentheses. All numbers reported in percentages. Finally, I implement a parametric approach to estimating individual heterogeneity in range-dependent attribute weighting in Appendix C.1. The results confirm that the majority of participants are relative thinkers: I estimate negative $\gamma$ for around three-quarters of participants. For the remaining participants, I estimate large choice noise and imprecise $\gamma$ parameters: only two participants in my sample have a $\gamma$ parameter that is positive and significantly different from zero. Furthermore, the best fitting range-weighting distribution has a unimodal shape, implying a single relative-thinking type that differs in the intensity but not the direction of range weighting. Overall, the majority of participants made decisions consistent with relative thinking and there is no evidence that a nontrivial share of participants made choices consistent with focusing. ## 4 Estimating Structural Parameters This section provides estimates for the range-dependent attribute weighting parameter $\gamma$. This approach allows me to quantify choice-set dependence in terms of surplus foregone and to conduct other counterfactual analyses. This addresses a key motivation of this paper: does range weighting generate economically meaningful departures from surplus maximization? This approach also allows for a formal statistical test of whether relative thinking improves fit versus surplus maximization and for other robustness tests. For individual $i=1,...,I$ on trial $t=1,...,T$, I model the overall utility of option $j$ as $$ \mathcal{V}_{ijt} = g(\Delta_{q_t}; \gamma_i) \cdot q_{ijt} - g(\Delta_{p_t}; \gamma_i) \cdot p_{ijt} + \lambda_i \varepsilon_j \tag{1} $$ where $\varepsilon_j$ follows a type-I extreme-value distribution and $\lambda_i$ is the scale parameter for individual $i$. I assume that the attribute weighting function takes the parametric form A4: $g(\Delta;\gamma_i)=\left(\Delta\right)^{\gamma_i}$. The parameter of interest is $\gamma_i$, which indexes the direction and magnitude of range-based attribute weighting for individual $i$. Recall that $\gamma_i<0$ indicates relative thinking, while $\gamma_i>0$ implies focusing.[^14] Conditional on $\gamma_i$ and the scale parameter $\lambda_i$, specification (1) gives choice probabilities defined by the standard logit formula. To estimate equation (1), I need to specify quality. The experimental design elicited reservation values for each good twice. I assume that the quality of each product variety, $q_{ijt}$, is the average of these reservation values. I explore alternative specifications of quality in Section C.4 and find that taking the average reservation value produces a relatively conservative estimate of the magnitude of $\gamma$. In this section, I focus on estimating a homogeneous-agent model where $(\gamma_i,\lambda_i)=(\gamma,\lambda)$ for all $i$. In Appendix C.1, I estimate $\gamma_i$ for each individual separately. Table 6 presents the parameter estimates obtained via maximum likelihood estimation. The full sample estimates are presented in column (1). A $\hat{\gamma}$ coefficient of $-0.34$ \[95% CI: $-$0.41,$-$0.25\] implies that an attribute’s decision weight is inversely related to its range within the choice set, supporting the key assumption of the relative thinking model. Surplus maximization ($\gamma=0$) lies well outside the 95% confidence interval and a likelihood-ratio test provides support for the range-weighting specification versus the null of surplus maximization ($\chi_1^2=126.08, p<0.001$). **Table 6: Homogeneous Model Estimates** | | | | | | |:---|:---|:---|:---|:---| | | \(1\) | \(2\) | \(3\) | \(4\) | | **Trials:** | All | $h$-decoy | $l$-decoy | $o$-decoy | | $\hat{\gamma}$ | $-$0.34 | $-$0.33 | $-$0.39 | $-$0.19 | | | \[$-$0.41,$-$0.25\] | \[$-$0.46,$-$0.20\] | \[$-$0.48,$-$0.29\] | \[$-$0.35,$-$0.01\] | | $\hat{\lambda}$ | 9.19 | 9.44 | 9.95 | 8.22 | | | \[8.27,10.28\] | \[8.01,11.13\] | \[8.43,11.39\] | \[6.95,9.69\] | | $LL(\hat{\gamma},\hat{\lambda})$ | $-$2158 | $-$757 | $-$598 | $-$795 | | \# Obs. | 2,700 | 900 | 900 | 900 | | \# Ind. | 150 | 150 | 150 | 150 | Individual-cluster-robust bootstrapped 95% confidence intervals in brackets. The attribute-weighting function is defined by $g(\Delta_x;\gamma)= \left(\Delta_x\right)^{\gamma}$ for attribute $x \in \{q,p\}$. $\lambda$ is the scale parameter of the type-1 extreme value error. Quality is defined as the average of the first- and second-stage reservation values. The scale parameter estimate $\hat{\lambda}$ provides an indication of the average choice noise. While it is not of direct interest, it helps to rationalize random choice deviations from the predictions of the model. The normalization of the attribute weights means that it can be interpreted roughly in surplus terms. A coefficient of 9.19 \[95% CI: 8.27, 10.28\] implies a standard deviation in monetized terms of $\hat{\lambda} \pi / \sqrt{6} \approx \$12$. To put this in context, the average surplus differential between the first- and second-ranked options was around \$9, while for the second- and third-ranked options, the average gap was around \$21. Recall that in principle, $\gamma$ is identified in any one of the three price regions. As a validation check, I assess the consistency of the parameter estimates across the three price regions. Table 6 columns (2) to (4) present the estimates. The qualitative results are similar for each of the experimental price regions, but the magnitude of range weighting is smaller in the $o$-decoy region. A likelihood-ratio test rejects the null of constant $\gamma$ and $\lambda$ across the three regions ($p<0.01)$. One possibility is that because prices are more visually salient than quality, range weighting is stronger when price is the dominant attribute range. However, while there is some variation in the magnitude of $\hat{\gamma}$ across the types of choice sets, it is negative and statistically significant in all three cases. Relative thinking is therefore supported across all three experimental decoy regions and offers a coherent explanation for the demand patterns documented in Section 3. ### 4.1 Economic Implications The structural estimate of $\gamma$ reported in the previous section is the first in the literature. In this section, I consider the economic implications of this estimate. Specifically, I quantify the willingness-to-pay premium for each of the three experimental decoy environments. Suppose an individual has the quality perceptions implied by the average reservation values for the high- and low-quality variants: \$41 and \$24, respectively. Column (1) of Table 7 reports willingness-to-pay for the nondecoy options assuming surplus maximization, $\gamma=0$. Column (3) reports willingness-to-pay conditional on the average realized price ranges for $\hat{\gamma}=-0.34$. Columns (2) and (4) report willingness-to-pay at the upper and lower bounds of the 95% confidence interval for this estimate: $-$0.25 and $-$0.41, respectively. **Table 7: Quantifying Range-Dependent Attribute Weighting**
(1) (2) (3) (4)
Surplus Max Lower Bound Estimate Upper Bound
Range weighting: $\gamma =$ 0 $-$0.41 $-$0.34 $-$0.25
Quality: $(q_h, q_l)$ = (\$41,\$24)
$\boldsymbol h$-decoy region ($p_h$=\$62)
Willingness-to-pay for $l$ \$24.0 \$28.4 \$27.6 \$26.6
% premium vs. surplus max – 18.5% 15.1% 10.9%
$\boldsymbol l$-decoy region ($p_l$=\$66)
Willingness-to-pay for $h$ \$41.0 \$49.8 \$48.2 \$46.2
% premium vs. surplus max – 21.6% 17.6% 12.6%
$\boldsymbol o$-decoy region ($p_h$=\$28)
Willingness-to-pay for $l$ \$11.0 \$15.0 \$13.1 \$12.5
% premium vs. surplus max – 22.4% 18.8% 14.1%
Quality is set to the average reservation values reported in the experiment—\$41 and \$24 for the $h$ and $l$ variety. The price in each region is set to the average realized prices in each decoy region, and each region is defined using the average reservation values. The reported premiums are measured in percentage terms relative to the surplus maximization amount in column (1). The $\gamma$ values are taken from column (1) of Table 6. Willingness-to-pay is (i) $WTP(\gamma) = q_l (q_h/p_h)^\gamma$ in the $h$-decoy region, (ii) $WTP(\gamma) = q_h (q_h/p_l)^\gamma$ in the $l$-decoy region, and (iii) defined implicitly by $(q_h/p_h)^\gamma = (p_h - WTP(\gamma))/(q_h-q_l)$ in the $o$-decoy region. In the $h$-decoy region, the price of $h$ sets the price range and exceeds the quality range. A negative $\gamma$ coefficient implies that agents underweight price relative to quality, boosting willingness-to-pay for $l$ relative to surplus maximization. For choice sets in this region, the average realized $p_h$ was around \$62. This implies that an agent with $\hat{\gamma}=-0.34$ will be indifferent between buying $l$ and not buying at a price of approximately \$27.6—a premium of around 15% relative to surplus maximization. For prices in the $l$-decoy region, the price of $l$ sets the price range and ensures that the price range exceeds the quality range. At the average realized $p_l$ of \$66 in the data, an agent with $\hat{\gamma}=-0.34$ is willing to pay up to approximately \$48.2 to buy $h$—a premium of around 18% relative to surplus maximization. In the $o$-decoy region, the low prices of both $h$ and $l$ mean that the quality range is greater than the price range. A surplus maximizer is willing to pay up to $p_h - (q_h-q_l) = \$28-(\$41-\$24) = \$11$ to purchase $l$ over $h$. A negative $\gamma$ will cause agents to underweight prices and boost willingness-to-pay for $l$ relative to $h$. Given an average realized $p_h$ of \$28, an agent with $\hat{\gamma}=-0.34$ will be indifferent between $h$ and $l$ at $p_l =\$13.1$—a premium of around 19% relative to the surplus difference between $h$ and $l$. On average, participants were willing to pay an average premium of around 17% due to range weighting across the three decoy regions. In Table C.7 of Appendix C.4, I calculate the willingness-to-pay premium using alternative definitions of quality. Using the first-stage reservation values leads to a slightly lower average premium of 16%, while using the second-stage reservation values leads to substantially larger average premium of 26%. ## 5 Robustness and Alternative Theories This section addresses three concerns. First, quality is uncertain and may be subject to bias or measurement error. A particular concern is that participants may have interpreted prices as signals of quality, which could generate demand shifts consistent with the data. Second, factors other than the immediate choice set may have influenced attribute weights; such as previous choice sets, expectations, or other reference points. Third, I explore whether the salience theory of Bordalo et al. (2013) or pairwise normalization model Landry and Webb (2021) can provide an alternative explanation for the estimated choice probabilities. ### 5.1 Price as a Signal of Quality Economists have explored the possibility that prices can signal quality. For example, Wolinsky (1983) shows that quality uncertainty can lead to an equilibrium in which prices credibly signal quality. In my experimental context, it is possible that higher prices boosted demand by signaling higher quality. This alternative form of choice-set dependence can generate predictions that are qualitatively similar to relative thinking under certain conditions. Three conditions must hold for price signaling to explain my results. First, the perceived quality of option $j \in \{h,l\}$ must be increasing in the price of $k \neq j$. This leads to the prediction that higher decoy prices will boost buying in the $h$- and $l$-decoy price regions. Second, participants must judge the signal value of prices relative to their initial quality perceptions: for example, prices below their initial quality perceptions will lower their quality perceptions. This leads to a downward shift in the indifference boundary in the $o$-decoy price region. Finally, prices must provide a stronger signal of quality for $h$ than for $l$. This ensures that parallel price increases in the $o$-decoy region can cause individuals to shift from $l$ to $h$. The data allow for direct tests of these three conditions. The experiment elicited two measures of participants’ quality perceptions for both varieties of each good: one before seeing any prices and one after seeing two pairs of prices for each good. Participants were randomly assigned to $h$-, $l$-, or $o$-decoy prices on both encounters with a good, leading to nine possible price experiences. This creates exogenous variation in the price signals that participants saw for each good. To test whether these price signals affected quality perceptions, let $s_{ig} = \left(\overline{p}_{ig}-\overline{q}_{ig}\right)$ be the signal value for good $g$ for individual $i$, where $\overline{q}_{ig}$ is the average of the initial $h$- and $l$-quality perceptions and $\overline{p}_{ig}$ is the average of the $h$- and $l$ prices in the experiment.[^15] To test whether observing higher price signals increased stated valuations, I regress the change in reservation value for individual $i$ and good $g$ on the signal index $s_{ig}$. The estimated coefficient implies that observing higher prices did lead to higher second-stage reservation values: a \$1 increase in the average prices increased quality perceptions by \$0.20 on average ($p<0.001$). To test whether price signaling was equal for the $h$- and $l$ varieties, I then interact the signal index $s_{ig}$ with an indicator for the $l$ variety. The estimated coefficient is small and not statistically different from zero—thereby rejecting a necessary condition for the price signaling model to provide a coherent explanation for my experimental results. Even if price signals altered participants’ quality perceptions, the key question is whether evidence of relative thinking is robust to controlling for these price signals. To separately identify the impact of relative thinking from price signaling, I exploit the panel structure of the data: specifically, that participants made two decisions involving each good. Consider a participant who first observes prices for some good $g$ in the $o$-decoy region and receives a price signal of $-$\$15. Now suppose that on the next encounter with this product, the participant observes a price signal of +\$15, and therefore $s_{ig} = 0$. In this example, price signaling would predict no demand shifts while relative thinking continues to predict the demand shifts outlined in Section 1.4. Table 8 builds on this intuition by analyzing data from participants’ second encounter with each good. I repeat the analysis from Section 3 but restrict the data to the second set of decisions and control for the price signal index $s_{ig}$. If relative thinking is a robust phenomenon, then the prices of the decoy options should continue to provide explanatory power above and beyond their signal value. Columns (1) to (3) of Table 8 confirm that the choice-probability estimates documented in Section 3 are robust to controlling for price signals. The estimated coefficients are similar in both direction and magnitude to the estimates in Table 4. In contrast, the price signal index $s_{ig}$ has no significant effect on choice probabilities in any of the nine specifications. **Table 8: Choice Probability Estimates – Second Half of Trials**
(1) (2) (3)
Dependent Variable: $\qquad$ Buy high Buy low Don’t Buy
Panel (a): $\boldsymbol h$-decoy region
Price of $h$ $-$0.20 0.48 $-$0.30
(0.08) (0.12) (0.13)
$s_{ig}$ $-$0.04 0.25 $-$0.23
(0.15) (0.21) (0.24)
Panel (b): $\boldsymbol l$-decoy region
Price of $l$ 0.62 0.02 $-$0.68
(0.13) (0.04) (0.13)
$s_{ig}$ $-$0.01 $-$0.01 0.01
(0.23) (0.08) (0.22)
Panel (c): $\boldsymbol o$-decoy region
Price of $h$ 0.88 $-$1.18 0.23
(0.26) (0.31) (0.14)
$s_{ig}$ $-$0.27 0.39 $-$0.09
(0.22) (0.22) (0.16)
Sample: Second half of experimental trials. Average marginal effects derived from logistic regression estimates. The specification includes participant demographics (age, gender, education, marital status and ethnicity) and round fixed effects. Demographic and round fixed are excluded from specification (2) for the $l$-decoy regressions due to the lack of variation in the outcome variable. The regressions either control for the price of $l$ (panel a), the price of $h$ (panel b), or the price difference between $h$ and $l$ (panel c). All numbers reported in percentages. $s_{ig}$ measures the difference between the average prices individual $i$ saw during the experiment for good $g$ and their average first-stage reservation values for good $g$. There are 450 observations per region across 150 participants. Individual cluster-robust standard errors in parentheses. The results in Table 8 rely on a specific parametric form of price-signaling index, $s_{ig}$. While this index does predict changes in valuations between the first and second stages of the experiment, it is possible that it is misspecified and only provides a noisy measure of the signal value contained in prices. In Appendix C.2, I implement an alternative approach in which I only consider the choices of participants who had previously saw a negative price signal on their first encounter with a given product. This approach makes no assumptions about the specific form of price signaling but does assume that participants at least partially remember previous price signals.[^16] The results are qualitatively similar to those in Table 8 and show that relative thinking is robust even in situations in which price signaling predicts null or opposite effects. In Appendix C.2, I also incorporate price signaling into my structural estimation. Specially, I estimate the equations: $$ \begin{aligned} q^\prime_{ijt}=& \; q_{ijt} + \alpha_j (\overline{p}_{it} - \overline{q}_{it}) \qquad \text{(2)} \\ \mathcal{V}_{ijt} =& \; 2\frac{\left(\Delta_{q^\prime}\right)^{\gamma}}{\left(\Delta_{q^\prime}\right)^{\gamma}+\left(\Delta_p\right)^{\gamma}} \; q^\prime_{ijt} -2\frac{\left(\Delta_{p}\right)^{\gamma}}{\left(\Delta_{q^\prime}\right)^{\gamma}+\left(\Delta_p\right)^{\gamma}} \; p_{ijt} + \lambda \varepsilon_j \qquad \text{(3)} \end{aligned} $$ where $\overline{p}_{it}=(p_{iht}+p_{ilt})/2$, $\overline{q}_{it}=(q_{iht}+q_{ilt})/2$ and $\alpha_o=0$. The parameters $\alpha_h$ and $\alpha_l$ measure the impact of price signaling on the $h$ and $l$ variants. I find that (i) the structural estimates are robust to incorporating price signaling and (ii) relative thinking provides the best fit to the data. Overall, while there is some evidence that higher prices led to an increase in quality perceptions, the patterns of demand documented in Section 3 are not driven by price signals. These results are consistent with Heffetz and Shayo (2009) who show that stated reservation values are sensitive to price signals, but estimated demand is not. Crucially, evidence supporting relative thinking is robust to controlling for price signals. ### 5.2 Features Beyond the Immediate Choice Set In the main estimation in Section 4, I assume that agents evaluate each decision in isolation. However, external factors may have influenced participants due to the within-subjects design. Tversky and Simonson (1993) explicitly model these external factors as the “background context”; BGS distinguish the choice set from the “evoked set”. A particular concern in my experiment is that features of previous choice sets may impact subsequent decisions. While the product on offer (and hence the quality dimension) varied from trial to trial, price was a constant attribute across decisions, creating the potential for spillovers. For example, if an individual faces a relatively wide price range followed by a narrow one, they might not fully acclimate to the new price range. In Appendix C.3, I develop and estimate an alternative structural model that incorporates lagged prices into the evaluation of the current price range. The results show that previous price ranges have a statistically significant but small impact on attribute weights. More importantly, $\gamma$ estimates are robust to controlling for this lagged price dependency. Observing two trials per product may have induced spillovers from one trial to another—for example, by creating an expectation regarding future prices. Recall that each of the first nine trials featured a unique product. To control for any sequential effects due to repeatedly observing a single product, I restrict the sample to the first nine observations per individual. Columns (2) and (3) of Table 9 show that estimated range weighting is more pronounced in the first half of trials, $\hat{\gamma}=-0.44$, than for the second half of trials, $\hat{\gamma}=-0.24$. One possibility is that repeated exposure to products may have weakened the influence of the immediate choice set—for example, participants may have internalized the experiment’s global price range of $[\$0.5,\$100]$ over time. This is consistent with the finding that previous price ranges had a small but statistically significant impact on the attribute weights. However, while the magnitude of the estimated coefficient is sensitive to whether the sample contains the first or the second half of trials, both estimates imply statistically significant range-dependent attribute weighting consistent with relative thinking. **Table 9: Restricted Sample Estimates**
(1) (2) (3) (4)
Sample: Full First Half Second Half First Choice
Only
$\widehat{\gamma}$ $-0.34$ $-0.44$ $-0.24$ $-0.30$
[$-0.42$,$-0.26$] [$-0.53$,$-0.35$] [$-0.34$,$-0.13$] [$-$0.49,$-0.08$]
$\widehat{\lambda}$ 9.19 9.13 9.13 8.03
[8.18,10.27] [8.13,10.29] [7.79,10.71] [5.80,11.09]
Individuals 150 150 150 150
Observations 2,700 1,350 1,350 150
Individual-cluster-robust bootstrapped 95% confidence intervals in brackets. The attribute weighting function is $\left(\Delta_x\right)^{\gamma}$ for attribute $x \in \{q,p\}$. $\lambda$ is the scale parameter of the type-1 extreme value error. First half refers to trials 1 to 9. Second half refers to trials 10 to 18. Alternatively, repeated choices of any kind may have influenced participants. To completely remove any concerns owing to the within-subjects design, column (4) estimates the preferred specification using only the first choice from each individual. This pseudo between-subjects design eliminates any confounds related to exposing participants to repeated choice sets at the expense of statistical power. Column (4) of Table 9 presents the results. The point estimate of $-0.30$ is close to the full sample estimate. ### 5.3 Alternative Theories of Choice-Set Dependence #### 5.3.1 Salience Theory Bordalo et al. (2012, 2013) (henceforth BGS) propose a closely related approach to choice-set dependence that differs from range theories in two important regards. First, they assume that attribute weights are specific to each option: for each alternative $j$ agents compare quality and price to their average values in the choice set. Second, they posit that an attribute’s extremity, rather than its range, determines its decision weight. They label this property “ordering”: the more an attribute’s value deviates from its average in the choice set, the greater its decision weight. The ordering property of salience theory generates predictions that often line up with focusing: expanding an attribute’s range in the choice set is often equivalent to increasing its difference with the mean. However, salience theory imposes a second, counteracting force: deviations from the mean are subject to diminishing sensitivity. This second motive induces a preference for high quality-to-price ratios, similar to relative thinking. For a subset of prices in the $h$-decoy region, the ordering effect will unambiguously dominate, and thus the predictions of salience theory will align with focusing.[^17] To test whether the predictions of focusing and salience theory hold for this subset of prices, I re-estimate the choice probability models in Table 4 using only these trials. The estimated coefficients are again consistent with the predictions of relative thinking. For choice sets in which salience theory and relative thinking make distinguishing predictions, the data support relative thinking and are inconsistent with salience theory. #### 5.3.2 Pairwise Normalization Landry and Webb (2021) propose a model of choice-set dependence in which individuals evaluate an option through a series of pairwise attribute comparisons. The value attached to each attribute comparison is normalized by the magnitude of the attributes under consideration. Both relative thinking and pairwise normalization are motivated by a central insight from perceptual neuroscience: neurons have a bounded response range and thus some form of normalization is necessary. In my experimental context, this shared intuition translates to shared predictions. Specifically, pairwise normalization reduces to a model in which attribute $x \in \{q,p\}$ is weighted by $1/(x_h+x_l)$. This formulation leads to the same predictions as the limiting case of relative thinking: agents choose $h$ when $p_h/p_l< q_h/q_l$, $l$ when $p_h/p_l> q_h/q_l$, and are indifferent between all three options when $p_h/p_l=q_h/q_l$. Therefore, pairwise normalization predicts that agents will always choose $l$ in the $h$-decoy region, $h$ in the $l$-decoy region, and $l$ in the $o$-decoy region. This prediction is clearly inconsistent with the high proportion of not buying decisions made by participants in the $h$- and $l$-decoy regions and the high fraction of $h$ choices in the $o$-decoy region (see Table 2). Landry and Webb (2021) propose a generalization of their model in which a single parameter, $\sigma\geq0$, indexes the strength of attribute normalization. The model reduces to the basic specification above when $\sigma=0$ and converges to surplus maximization as $\sigma \rightarrow \infty$. For $0<\sigma<\infty$, the predictions of the model are qualitatively similar to relative thinking in each of the three decoy regions. Increasing the price of the decoy option decreases price sensitivity in the $h$- and $l$-decoy regions, boosting demand for $l$ and $h$, respectively. In the $o$-decoy region, increasing the price level while holding price differences fixed causes demand to shift toward $h$. Therefore, the nonparametric evidence presented in Section 3 is consistent with both relative thinking and the one-parameter generalization of pairwise normalization. ## 6 Discussion This paper provides a nonparametric test of relative thinking versus focusing and the first structural estimates of range-dependent attribute weighting. In a lab experiment, I find that expanding the price range causes participants to become less price sensitive and more likely to buy. I find little heterogeneity in attribute weighting across individuals; the majority of participants’ behavior was consistent with relative thinking and few made choices consistent with focusing. The data are also inconsistent with utility maximization and existing formulations of salience theory. The consistency of the results is particularly noteworthy: the predictions of relative thinking hold across individuals, trials, products, and price regions. Of course, the generalizability of any laboratory findings requires further research to establish the broader relevance of range weighting in real-world economic situations. However, the consistency of the results, and the psychological underpinnings of the theoretical predictions, bolster confidence in the mechanism underlying range-based attribute weighting. Models of choice-set dependence are still developing and this paper only considers a subset of prominent theories. Similar models include Cunningham (2013) and Arad and Maltz (2019); related approaches include limited attention models (Gabaix 2014; Dahremöller and Fels 2015), computational models (Tsetsos et al. 2010), informational asymmetries (Kamenica 2008), and models of bargain utility (Jahedi 2016). The approaches I focus on in this paper were chosen primarily for their structural similarity and their amenability to experimental testing. However, my primary empirical results show that models of choice-set-dependent preferences are well-founded and empirically relevant, and also offer potential guidance on refinements to these models. While the data do not support existing formulations of salience theory, Bordalo et al. (2020) have recently proposed an alternative version of their model that might be more consistent with the experimental data. Bordalo et al. (2020) assume that deviations from memory-based norms, rather than the average in the choice set, generate departures from surplus-maximizing behavior. While my experiment was not designed to test for memory effects, some of the results are qualitatively consistent with norm-based expectations. For example, the low prices of both the low- and the high-quality options in the $o$-decoy region may come as a surprise relative to the participant’s memory of the first stage. This price shock can cause agents to overweight prices in the Bordalo et al. (2020) model and is therefore consistent with the experimental data from this region. However, the high prices of the high-quality good in the $h$-decoy region and the low-quality good in the $l$-decoy region should also induce heightened price sensitivity. Thus, the qualitative predictions of memory-based salience are inconsistent with data from the $h$- and $l$-decoy regions. While my data are not consistent with specific form of memory effects posited by Bordalo et al. (2020), understanding how memory interacts with choice-set dependence is a promising avenue for future work. Despite robust evidence of relative thinking documented in this paper, there are many unresolved issues that limit the application of range weighting models. First, the cognitive mechanisms underlying range weighting are poorly understood. Relative thinking is motivated by perceptual encoding constraints, while the intuition of focusing emphasizes the importance of limited attention. How these forces interact with the complexity of the choice set is an open question. This has important economic implications in light of conflicting evidence on range weighting; while I document consistent evidence for relative thinking in a simple choice environment, other work has supported the predictions of focusing in more complex settings, such as intertemporal choice (Dertwinkel-Kalt et al. 2021). Whether a trip to the grocery store, buying car insurance, or online shopping are simple or complex decisions is unclear. A deeper understanding of what factors drive range weighting can help define the boundaries of relative thinking and focusing, and provide empirical guidance on how and when to apply these models. Second, a major degree of freedom in most models of choice-set dependence is specifying the set of factors that influence the attribute weights. My experiment was designed to maximize the likelihood that participants considered each decision in isolation. However, I also present evidence that external factors entered into subjects’ considerations. For example, previous prices had a statistically significant but small impact on how attributes were weighted. Furthermore, attribute weighting was less pronounced in the second half of the experiment, which may suggest that the local range set by the choice set was offset by a more global reference point. A more thorough understanding of which features of the choice environment influence attribute importance is needed, especially in real-world applications. My results also highlight a form of choice-set dependence that has received less attention in the literature. I find evidence that prices may shape quality perceptions: participants who see higher prices during the choice task report higher second-stage valuations but the demand estimates are robust to controlling for these signals. These results are consistent with Heffetz and Shayo (2009) but are not easily accommodated by existing models of choice-set dependence. While I show that evidence of relative thinking is robust to price signaling, I cannot fully rule out this potential confound without making some assumptions about the learning process. To disentangle the effects of relative thinking and price signaling, future work could focus on situations in which the two models make opposite predictions. For example, a higher wage rate might signal that a task is more profitable, yet relative thinking would predict decreased sensitivity to wages. Alternatively, researchers could remove price signaling concerns entirely and control for visual salience by examining non-monetary situations, such as receiving products in exchange for effort provision. Finally, further empirical guidance is required on which features of the choice set matter for attribute weighting. While I have shown that range is a useful proxy for choice-set dependence in simple choice environments, I cannot rule out similar determinants such as variance, maximum sensitivity, or divisive normalization. A systematic understanding of how price elasticities respond to exogenous changes in the features of the choice set would provide much-needed clarity on the fundamentals of choice-set dependence. **Online Appendix** ## A Proofs and Additional Results *Proof of Proposition 1*\ The proof of Proposition 1 follows from the following two results. The first states that the price space $(p_h,p_l) \in \mathbb{R}_+^2$ can be partitioned into five mutually exclusive and collectively exhaustive regions. In three of these regions, the theory predicts only two of the three options are ever chosen: $l$ or $o$, $h$ or $o$, and $l$ or $h$, respectively. I refer to these regions as the $h$-decoy, $l$-decoy, and $o$-decoy regions in the main text, and they are formally defined below by the sets $A$, $B$, and $C$. The second result builds on the first and establishes existence and uniqueness of the $l$-$o$, $h$-$o$, and $l$-$h$ indifference boundaries within these three regions. **Result 1 (Pairwise Choice Bounds):** For choice set $\mathcal{C}(p_h,p_l)$, the price space $(p_h,p_l) \in \mathbb{R}_+^2$ can be partitioned into the mutually exclusive and collectively exhaustive regions depicted visually in Figure A.1 and furthermore: - In region $A \equiv \{(p_h,p_l)\in \mathbb{R}_+^2 : \; p_h\geq q_h \text{ and } q_l/p_l \geq q_h/p_h\}$, $l$ or $o$ is chosen. - In region $B\equiv\{(p_h,p_l) \in \mathbb{R}_+^2 : \; p_l\geq q_h \text{ and } p_h \leq p_l\}$, $h$ or $o$ is chosen. - In region $C\equiv\{(p_h,p_l) \in \mathbb{R}_+^2: \; p_h \leq q_h \text{ and } q_l/p_l\leq q_h/p_h \}$, $l$ or $h$ is chosen. - In region $D\equiv\{(p_h,p_l) \in \mathbb{R}_+^2: \; p_h>q_h \text{ and } q_l/p_lp_l \}$, $o$ is chosen. - In region $E\equiv\{(p_h,p_l) \in \mathbb{R}_+^2: \; p_hp_l$ and $p_h>q_h$, hence $\Delta_q=q_h$ and $\Delta_p=p_h$. By A3(ii), $V_h=g(q_h;\gamma) \cdot q_h - g(p_h;\gamma) \cdot p_h < 0 = V_o$, and therefore $h$ will never be chosen. This condition can be restated as, $q_h/p_h 0 \\ \Leftrightarrow \qquad \qquad \qquad & \dfrac{q_l}{p_l} &> \dfrac{g(p_h;\gamma)}{g(q_h;\gamma)} > \dfrac{q_h}{p_h}. \qquad \qquad \qquad \qquad \qquad \end{array} $$ *Part II.* If $p_h g(p_h;\gamma) \cdot p_h$, and hence, $$ \begin{aligned} V_l &= g(q_h;\gamma) \cdot q_l - g(p_l;\gamma) \cdot p_l \\ &< g(q_h;\gamma) \cdot q_h - g(p_l;\gamma) \cdot p_l \\ &< g(q_h;\gamma) \cdot q_h - g(p_h;\gamma) \cdot p_h = V_h \end{aligned} $$ and so $l$ is never chosen. Furthermore, a necessary condition for $o$ to be chosen is $V_o>V_h$, which holds whenever, $$ \begin{array}{lrl} & 0 & > g(q_h;\gamma) \cdot q_h - g(p_l;\gamma)\cdot p_h \\ \Leftrightarrow \qquad \qquad \qquad \qquad \qquad & 0 & > g(q_h;\gamma) \cdot q_h - g(p_l;\gamma)\cdot p_l \\ \Leftrightarrow \qquad \qquad \qquad \qquad \qquad & g(p_l;\gamma) \cdot p_l & > g(q_h;\gamma)\cdot q_h \\ \Leftrightarrow \qquad \qquad \qquad \qquad \qquad & p_l & > q_h \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{(by A3(ii)).} \\ \end{array} $$ *Part III.* Suppose $p_h>p_l$ and $p_h 0 = V_o$, and therefore $o$ will never be chosen. Rearranging gives $q_h/p_h>g(p_h;\gamma)/g(q_h;\gamma)$ and therefore, a necessary condition for $l$ to be chosen is $V_h \; \dfrac{q_h}{p_h}. \qquad \qquad \qquad \end{array} $$ *Part IV.* From Part I, $p_h>p_l$ and $p_h>q_h$ implies that $h$ is never chosen and furthermore, that a necessary condition for $l$ to be chosen is that $q_l/p_l>q_h/p_h$. By definition of $D$ all prices must satisfy $q_l/p_l0$ and hence $j^*(p_h,p_l)=l$. Then, $$ \begin{aligned} f_A\left(\dfrac{p_h q_l}{q_h} \right) & = g(q_h) q_l - g(p_h) \dfrac{p_h q_l}{q_h} = g(q_h) \dfrac{q_l p_h}{q_h} \left[\dfrac{q_h}{p_h}-\dfrac{g(p_h)}{g(q_h)} \right] < 0 \end{aligned} $$ as the term in brackets is less than zero (see the proof of Result 1 Part I), and hence $j^*(p_h,p_l)=o$. Then, by the intermediate value theorem, there exists a unique $\bar{p}>0$ such that $f_A(\bar{p})=0$. As $f_A(p_l)$ is strictly decreasing, then for all $p_l\leq\bar{p}$, $j^*(p_h,p_l) =l$, and for all $p_l\geq\bar{p}$, $j^*(p_h,p_l)=o$. Define $A_i=\{(p_h,p_l) \in A : p_l = \bar{p}\}$, $A_l=\{(p_h,p_l) \in A : p_l < \bar{p}\}$, and $A_o=\{(p_h,p_l) \in A : p_l > \bar{p}\}$, then $\{A_l,A_o,A_i\}$ is a unique partition of $A$. To show that $\{A_l,A_o,A_i\}$ is distinct for any $\gamma$, note that $f_A(\bar{p})=0$, implies $\bar{p}(\gamma) = q_l g(q_h;\gamma)/g(p_h;\gamma)$, which is continuous and strictly decreasing in $\gamma$ as $q_h0$ and hence $j^*(p_h,p_l)=h$. $f_B(p_l)<0$ (by A3(ii)) and hence $j^*(p_h,p_l)=o$. Then, by the intermediate value theorem, there exists a unique $\hat{p}$ such that $f_B(\hat{p})=0$. As $f_B(p_h)$ is strictly decreasing, then for all $p_h\leq\hat{p}$, $j^*(p_h,p_l)=h$, and for all $p_h\geq\hat{p}$, $j^*(p_h,p_l)=o$. Define $B_i=\{(p_h,p_l) \in B : p_h = \hat{p}\}$, $B_h=\{(p_h,p_l) \in B : p_h < \hat{p}\}$, and $B_o=\{(p_h,p_l) \in B : p_h > \hat{p}\}$, then $\{B_h,B_o,B_i\}$ is a partition of $B$. Furthermore, $f_B(\hat{p})=0$, implies that $\hat{p}(\gamma) = q_l g(q_h;\gamma)/g(p_l;\gamma)$, which is continuous and strictly decreasing in $\gamma$ as $q_hg(p_h)/g(q_h)$ (see proof of Result 1, Part III) and in particular for $p_h=p_l (q_h/q_l)$, which implies, $q_l/p_l > g(\frac{p_l q_h}{q_l})/g(q_h)$. Therefore, $$ \begin{aligned} f_C\left(\dfrac{p_l q_h}{q_l}\right) & = g(q_h)(q_h-q_l) - g\left(\dfrac{p_l q_h}{q_l}\right) \left[\dfrac{p_l q_h}{q_l} - p_l \right] & = g(q_h)(q_h-q_l) \dfrac{p_l}{q_l} \left[\dfrac{q_l}{p_l} - \dfrac{g(\frac{p_l q_h}{q_l})}{g(q_h)} \right] > 0 \end{aligned} $$ and hence $j^*(p_h,p_l)=h$. Then, by the intermediate value theorem, there exists a unique $\tilde{p}>0$ such that $f_C(\tilde{p})=0$. As $f_C(p_h)$ is strictly decreasing, then for all $p_h\leq \tilde{p}$, $j^*(p_h,p_l)=h$, and for all $p_h\geq \tilde{p}$, $j^*(p_h,p_l)=l$. Define $C_i=\{(p_h,p_l) \in C : p_h = \tilde{p}\}$, $C_h=\{(p_h,p_l) \in C : p_h < \tilde{p}\}$, and $C_l=\{(p_h,p_l) \in C : p_h > \tilde{p}\}$, then $\{C_h,C_l,C_i\}$ is a unique partition of $C$. It remains to show that $\tilde{p}(\gamma)$ is strictly increasing in $\gamma$. Recall that by definition, $f_C(\tilde{p}) = g(q_h;\gamma) (q_h-q_l) - g(\tilde{p};\gamma) (\tilde{p}-p_l) = 0$. Taking the total derivative with respect to $\gamma$ gives, $$ \begin{aligned} & g_\gamma (q_h;\gamma)(q_h-q_l) - \left[ \left( g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) \frac{\partial \tilde{p}(\gamma)}{\partial \gamma} + g_\gamma(\tilde{p}(\gamma);\gamma) \right) (\tilde{p}-p_l) + \frac{\partial \tilde{p}(\gamma)}{\partial \gamma} g(\tilde{p}(\gamma);\gamma) \right] = 0 \\ \Rightarrow \qquad & \dfrac{\partial \tilde{p}(\gamma)}{\partial \gamma} = \dfrac{g_\gamma(q_h;\gamma)(q_h-q_l) - g_\gamma(\tilde{p}(\gamma);\gamma)(\tilde{p}(\gamma)-p_l)}{g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) (\tilde{p}(\gamma)-p_l) + g(\tilde{p}(\gamma);\gamma)} \end{aligned} $$ To sign the denominator, consider two cases. If $g$ is (weakly) increasing in the $\Delta$ (i.e. $g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) \geq 0$), then expression in the denominator is strictly positive. If instead $g$ is (weakly) decreasing in the range (i.e.$g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) \leq 0$), then by A3(ii), $$ \dfrac{\partial [\tilde{p}(\gamma) \cdot g(\tilde{p}(\gamma);\gamma)]}{\partial \tilde{p}(\gamma)} = g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) \tilde{p}(\gamma) + g(\tilde{p}(\gamma);\gamma) > 0 $$ and hence, $g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) (\tilde{p}(\gamma)-p_l) + g(\tilde{p}(\gamma);\gamma) \geq g_{\tilde{p}}(\tilde{p}(\gamma);\gamma) \tilde{p}(\gamma) + g(\tilde{p}(\gamma);\gamma) > 0$. Therefore the denominator is always strictly positive. To sign the numerator, note that $f_C(\tilde{p}(\gamma))=0$ $$ \Rightarrow \qquad \dfrac{g(q_h;\gamma)}{g(\tilde{p}(\gamma);\gamma)} = \dfrac{\tilde{p}(\gamma) - p_l}{q_h-q_l} \tag{$^*$} $$ Furthermore, as $q_h>\tilde{p}(\gamma)$, $g(q_h;\gamma)/g(\tilde{p}(\gamma);\gamma)$ is strictly increasing in $\gamma$, and hence $$ \dfrac{g_\gamma(q_h;\gamma) \cdot g(\tilde{p}(\gamma);\gamma) - g_\gamma(\tilde{p}(\gamma);\gamma) \cdot g(q_h;\gamma)}{g(\tilde{p}(\gamma);\gamma)^2}>0 $$ Combining the above with $(^*)$ gives, $$ \dfrac{g_\gamma(q_h;\gamma)}{g_\gamma(\tilde{p}(\gamma);\gamma)} > \dfrac{g(q_h;\gamma)}{g(\tilde{p}(\gamma);\gamma)} = \dfrac{\tilde{p}(\gamma) - p_l}{q_h-q_l} $$ which can be re-written as $g(q_h;\gamma) (q_h-q_l) - g_\gamma(\tilde{p}(\gamma);\gamma) (\tilde{p}(\gamma)-p_l) >0$ and therefore, $\partial \tilde{p}(\gamma)/\partial \gamma > 0$. Hence, for $\gamma^\prime \neq \gamma$, it follows that $\tilde{p}(\gamma^\prime) \neq \tilde{p}(\gamma)$ and hence $\{C_h,C_l,C_i\}\neq \{C_h^\prime,C_l^\prime,C_i^\prime\}$. $\blacksquare$ It follows from Result 2 that for any $\gamma\neq\gamma^\prime$, the choice correspondence $j^*(p_h,p_l;\gamma) \neq j^*(p_h,p_l;\gamma^\prime)$ for all $(p_h,p_l)\in\mathbb{R}_+^2$. Therefore, there exists either no $\gamma$ such that $j^*(p_h,p_l;\gamma)=j(p_h,p_l)$ for all $(p_h,p_l)\in\mathbb{R}_+^2$ or there exists a unique $\gamma$ such that $j^*(p_h,p_l;\gamma)=j(p_h,p_l)$ for all $(p_h,p_l)\in\mathbb{R}_+^2$. $\blacksquare$ *Proof of Proposition 2*\ Let $f$ denote the joint distribution of error difference terms. he probability of choosing $h$ or $l$ is: $$ \begin{aligned} P(h \text{ or } l) &= 1 - P(o) \\ &= 1 - \Pr(\widehat{\varepsilon}_{ol}>U_l-U_o)\Pr(\widehat{\varepsilon}_{oh}>U_h-U_o) \\ &= 1- \int_{\widehat{\varepsilon}_{ol}=U_l-U_o}^\infty \int_{\widehat{\varepsilon}_{oh}=U_h-U_o}^\infty f(\widehat{\varepsilon}_{ol},\widehat{\varepsilon}_{oh}) d \widehat{\varepsilon}_{oh} d \widehat{\varepsilon}_{ol} \end{aligned} $$ As $f(\widehat{\varepsilon}_{lo},\widehat{\varepsilon}_{lh}) > 0$ for all $(\widehat{\varepsilon}_{lo},\widehat{\varepsilon}_{lh})$, the probability of choosing $h$ or $l$ is strictly increasing (decreasing) if and only if the bounds of both integrals contract (expand). First, consider how an increase in $p_h$ affects the bounds of the inner integral. By A1, $U_l$ and $U_o$ are independent of $p_h$, while $U_h$ is strictly decreasing in $p_h$. Therefore, the bound $U_h-U_o$ is strictly decreasing in $p_h$ which leads to an strict increase in $P(o)$ and hence a strict decrease in $P(h \text{ or } l)$. Finally, consider how an increase in $p_l$ affects the bounds of the outer integral. By A1, $U_h$ and $U_o$ are independent of $p_l$, while $U_l$ is strictly decreasing in $p_l$. Therefore, the bound $U_l-U_o$ is strictly decreasing in $p_l$ which leads to an strict increase in $P(o)$ and hence a strict decrease in $P(h \text{ or } l)$. $\blacksquare$ *Proof of Proposition 3*\ Let $j,k,k^\prime \in \{h,l,o\}$ denote the three distinct options in choice set $\mathcal{C}(p_h,p_l)$. The probability of choosing $j$ is: $$ \begin{aligned} \qquad \text{(A.4)} P(j) &= \Pr(\widehat{\varepsilon}_{jk}>V_k-V_j,\widehat{\varepsilon}_{jk^\prime}>V_{k^\prime}-V_j) \nonumber \\ &= \int_{\widehat{\varepsilon}_{jk}=V_k-V_j}^\infty \int_{\widehat{\varepsilon}_{jk^\prime}=V_{k^\prime}-V_j}^\infty f(\widehat{\varepsilon}_{jk},\widehat{\varepsilon}_{jk^\prime}) d \widehat{\varepsilon}_{jk} d \widehat{\varepsilon}_{jk^\prime} \end{aligned} $$ As $f(\widehat{\varepsilon}_{jk},\widehat{\varepsilon}_{jk^\prime})>0$ for all $(\widehat{\varepsilon}_{jk},\widehat{\varepsilon}_{jk^\prime})$, the probability of choosing $j$ is strictly increasing (decreasing) if and only if the bounds of both integrals expand (contract). *Part (a): $h$-decoy region*. To see how the integral bounds respond to changes in $p_h$ in the $h$-decoy region, note that for any prices $(p_h,p_l) \in A$, $V_o-V_l = g(p_h;\gamma) p_l- g(q_h;\gamma) q_l$, $V_h-V_o = g(q_h;\gamma) q_h - g(p_h;\gamma) p_h$ and $V_h-V_l = g(q_h;\gamma)(q_h-q_l)-g(p_h;\gamma)(p_h-p_l)$. By the properties of $g$, it follows that $V_h-V_l$ and $V_h-V_o$ are strictly decreasing in $p_h$ for all $\gamma$, while $V_o-V_l$ is strictly increasing in $p_h$ if $\gamma>0$, strictly decreasing in $p_h$ if $\gamma<0$, and unaffected by $p_h$ if $\gamma=0$. Given these properties and the choice probability formula (A.4): - The first integral bound for $P(h)$ is $V_l-V_h$ which is strictly increasing in $p_h$ for all $\gamma$. The second is $V_o-V_h$ which is also strictly increasing in $p_h$ for all $\gamma$. Thus, $P(h)$ is strictly decreasing in $p_h$ for all $\gamma$. - The first integral bound for $P(l)$ is $V_h-V_l$ which is strictly decreasing in $p_h$ for all $\gamma$. The second is $V_o-V_l$ which is strictly increasing in $p_h$ if $\gamma>0$, strictly decreasing in $p_h$ if $\gamma<0$, and unaffected by $p_h$ if $\gamma=0$. Thus, $P(l)$ is strictly increasing in $p_h$ if $\gamma \leq 0$ and ambiguous if $\gamma>0$. - The first integral bound for $P(o)$ is $V_h-V_o$ which is strictly decreasing in $p_h$ for all $\gamma$. The second is $V_l - V_o$ which is strictly decreasing in $p_h$ if $\gamma>0$, strictly increasing in $p_h$ if $\gamma<0$, and unaffected by $p_h$ if $\gamma=0$. Thus, $P(o)$ is strictly increasing in $p_h$ if $\gamma\geq0$ and ambiguous if $\gamma<0$. *Part (b): $l$-decoy region*. For any prices $(p_h,p_l) \in B$, $V_l-V_o = g(q_h;\gamma) q_l - g(p_l;\gamma) p_l$, $V_l-V_h = g(p_l;\gamma) (p_h-p_l) - g(q_h;\gamma)(q_h-q_l)$ and $V_o-V_h = g(p_l;\gamma)p_h-g(q_h;\gamma)q_h$. By the properties of $g$, it follows that $V_l-V_o$ and $V_l-V_h$ are strictly decreasing in $p_l$ for all $\gamma$, while $V_o-V_h$ is strictly increasing in $p_l$ if $\gamma>0$, strictly decreasing in $p_l$ if $\gamma<0$, and unaffected by $p_l$ if $\gamma=0$. Given these properties and the choice probability formula (A.4): - The first integral bound for $P(h)$ is $V_l-V_h$ which is strictly decreasing in $p_l$ for all $\gamma$. The second is $V_o-V_h$ which is strictly increasing in $p_l$ if $\gamma>0$, strictly decreasing in $p_l$ if $\gamma<0$, and unaffected by $p_l$ if $\gamma=0$. Thus, $P(h)$ is strictly increasing in $p_l$ if $\gamma\leq0$ and ambiguous if $\gamma>0$. - The first integral bound for $P(l)$ is $V_h-V_l$ which is strictly increasing in $p_l$ for all $\gamma$. The second is $V_o-V_l$ which is also strictly increasing in $p_l$ if $\gamma>0$. Thus, $P(l)$ is strictly decreasing in $p_l$ for all $\gamma$. - The first integral bound for $P(o)$ is $V_l-V_o$ which is strictly decreasing in $p_l$ for all $\gamma$. The second is $V_h - V_o$ which is strictly decreasing in $p_l$ if $\gamma>0$, strictly increasing in $p_l$ if $\gamma>0$, and unaffected by $p_l$ if $\gamma=0$. Thus, $P(o)$ is strictly increasing if $\gamma\geq0$ and ambiguous if $\gamma<0$. *Part (c): $o$-decoy region*. Fix the price difference $p_h-p_l \equiv c>0$. Then for any prices $(p_h,p_l) \in C$, $V_l-V_o = g(q_h;\gamma) q_l - g(p_h;\gamma) p_l$, $V_h-V_o = g(q_h;\gamma) q_h - g(p_h;\gamma) p_h$ and $V_h-V_l = g(q_h;\gamma)(q_h-q_l)-g(p_h;\gamma)c$. By the properties of $g$, it follows that $V_l-V_o$ and $V_h-V_o$ are strictly decreasing in $p_h$ for all $\gamma$, while $V_h-V_l$ is strictly decreasing in $p_h$ if $\gamma>0$, strictly increasing in $p_h$ if $\gamma<0$, and unaffected by $p_h$ if $\gamma=0$. Given these properties and the choice probability formula (A.4): - The first integral bound for $P(h)$ is $V_o-V_h$ which is strictly increasing in $p_h$ for all $\gamma$. The second is $V_l-V_h$ which is strictly increasing in $p_h$ if $\gamma>0$, strictly decreasing in $p_h$ if $\gamma<0$, and unaffected by $p_h$ if $\gamma=0$. Thus, $P(h)$ is strictly decreasing in $p_h$ if $\gamma\geq0$ and ambiguous if $\gamma<0$. - The first integral bound for $P(l)$ is $V_o-V_l$ which is strictly increasing in $p_h$ for all $\gamma$. The second is $V_h-V_l$ which is strictly decreasing in $p_h$ if $\gamma>0$, strictly increasing in $p_h$ if $\gamma>0$, and unaffected by $p_h$ if $\gamma=0$. Thus, $P(l)$ is strictly decreasing if $\gamma\leq0$ and ambiguous if $\gamma>0$. - The first integral bound for $P(o)$ is $V_h-V_o$ which is strictly decreasing in $p_h$ for all $\gamma$. The second is $V_l - V_o$ which is also strictly decreasing in $p_h$ for all $\gamma$. Thus, $P(o)$ is strictly increasing in $p_h$ for all $\gamma$. $\blacksquare$ ***Proposition 4 (Equivalence for Binary Choice):** Suppose the choice set is $\{(q_h,p_h),(q_l,p_l)\}$ where $q_h$ and $q_l$ are fixed such that $q_h>q_l$ and that A3 holds. Let $j_0^*\equiv \arg\max_{j\in\{h,l\}} U_j$ and $j_\gamma^* \equiv \arg\max_{j\in\{h,l\}} V_j$. Then, $j_\gamma^*=j_0^*$ for all $(p_h,p_l) \in \mathbb{R}_+^2$ and for all $\gamma$.* *Proof*. Given choice set $\mathcal{C}=\{(q_h,p_h),(q_l,p_l)\}$, the quality and price ranges are $\Delta_q=q_h-q_l$ and $\Delta_p=|p_h-p_l|$. First, consider the case where $p_l > p_h$. Both $V_h - V_l = g(q_h-q_l;\gamma) \cdot (q_h-q_l) + g(p_l-p_h;\gamma) \cdot (p_l-p_h) > 0$ and $U_h - U_l = (q_h-q_l) + (p_l-p_h) > 0$, and therefore $j_\gamma^*=j_0^* = h$ for all $\gamma$. For $p_h \geq p_l$, $h$ will be preferred whenever $V_h = g(q_h-q_l;\gamma) \cdot q_h-q_l > \; g(p_h-p_l;\gamma) \cdot (p_h-p_l) = V_l$. This holds if and only if $g(\Delta_q;\gamma) \cdot \Delta_q > \; g(\Delta_p;\gamma) \cdot \Delta_p$. By A3(ii), this holds if and only $\Delta_q > \Delta_p$ which is equivalent to $U_h > U_l$. Likewise, $V_h \leq V_l$ $\Leftrightarrow$ $U_h \leq U_l$, and therefore for all $\gamma$, $j_\gamma^*=j_0^* = h \text{ if } U_h > U_l$ and $j_\gamma^*=j_0^* = l \; \text{ if } U_h < U_l$ , with indifference at $U_h = U_l$. Thus, $j_\gamma^*=j_0^* \text{ for all } (p_h,p_l) \in \mathbb{R}^2_+$ and $\gamma$. $\blacksquare$ ## B Additional Experimental Design Details ### B.1 Products **Table B.1: Products Offered in the Experiment** | | *high* variety | *low* variety | | |:----------------------------------|:---------------|:--------------|:----| | 2-3 *Quantity* | | | | | Lindor Chocolate Truffles | 300 pieces | 100 pieces | | | Cinema Card | 10 passes | 5 passes | | | Starbucks Coffee | 40 cups | 20 cups | | | Gillete Razors | 16 blades | 8 blades | | | Insignia TV | 24 inch | 19 inch | | | Uber Credit | \$80 | \$40 | | | | | | | | *Duration* | | | | | Blue Apron Subscription | 2 weeks | 1 week | | | Yoga Subscription | 2 month | 1 month | | | Wine of the Month Subscription | 2 month | 1 month | | | Cheese of the Month Subscription | 2 month | 1 month | | | Beer of the Month Subscription | 2 month | 1 month | | | Flowers of the Month Subscription | 2 month | 1 month | | | | | | | | *Functionality* | | | | | Roku Streaming Device | Roku Ultra TV | Roku Stick | | | Amazon Tablet | Fire 8 | Fire 7 | | | | | | | | *Brand* | | | | | Bluetooth Speaker | Bose Soundlink | AmazonBasics | | | Water Purifier + 6 Filters | Brita | AmazonBasics | | | Hardcase Luggage | Samsonite | AmazonBasics | | | Rechargable Batteries + Charger | Energizer | AmazonBasics | | | Laptop Backpack | Northface | AmazonBasics | | | Sunglasses | Ray-Ban | Sungait | | | | | | | ### B.2 Screenshots of the Experimental Tasks ![Figure: Example screenshot of the first-stage valuation task. Participants can enter any positive amount and could reveal product details by clicking on the bold text “here”.](assets/example-valuation-task1-ffba64.png) **Example screenshot of the first-stage valuation task. Participants can enter any positive amount and could reveal product details by clicking on the bold text “here”.** ![Figure: Example screenshot of the second-stage choice task. Participants decide whether to purchase a given product by responding “yes” or “no”. Prices start at $0 and increase in increments of $5 up to a maximum of…](assets/example-valuation-task2-95829e.png) **Example screenshot of the second-stage choice task. Participants decide whether to purchase a given product by responding “yes” or “no”. Prices start at \$0 and increase in increments of \$5 up to a maximum of \$100.** ![Figure: Example Screenshots of the incentive animation. The ball highlighted in green is the number chosen by the participant. In the second stage choice task, participants can chose one number (i.e. ball to turn…](assets/incentive1-4819de.png)\ ![Figure: Example Screenshots of the incentive animation. The ball highlighted in green is the number chosen by the participant. In the second stage choice task, participants can chose one number (i.e. ball to turn…](assets/incentive2-a2bbbb.png) **Example Screenshots of the incentive animation. The ball highlighted in green is the number chosen by the participant. In the second stage choice task, participants can chose one number (i.e. ball to turn green). During the second stage valuation task incentive, nine numbers can be chosen. The yellow ball is the randomly chosen number. A new randomly number is drawn every 0.01 seconds and highlighted in yellow. The animation stops when the participant hits the space key (labeled "NEXT"). If the animation stops on the participant’s chosen number, (19 in this example), the participant wins a bonus cash prize of \$80 and their previous choice is implemented for real.** ### B.3 Additional Price Generation Details This section provides the precise method used for drawing prices in the experiment. In the $h$-decoy region (defined by the set $A$ in Section A), prices were draw from zone $m=1,2,3,4,5,6$ by first drawing some $p_h>q_h$ (subject to the restrictions outlined in the main text), and then $p_l$ was drawn at random from the interval bounded by the indifference boundaries for $\bar{\gamma}_m$ and $\bar{\gamma}_{m+1}$; more precisely, zone $m$ in $h$-decoy region was defined as $z_m^A = \{(p_h,p_l) \in A : p_l \in (q_l (q_h/p_h)^{\bar{\gamma}_m}$,$q_l (q_h/p_h)^{\bar{\gamma}_{m+1}} \}$. In the $l$-decoy region (defined by the set $B$ in Section A), prices from zone $m$ were obtained by first drawing some $p_l>q_h$, and then randomly drawing $p_h$ from the interval generated by indifference boundaries $\bar{\gamma}_m$ and $\bar{\gamma}_{m+1}$; specifically, zone $m$ in the $l$-decoy region was defined as $z_m^B = \{(p_h,p_l) \in B : p_h \in (q_h (q_h/p_l)^{\bar{\gamma}_m},q_h (q_h/p_l)^{\bar{\gamma}_{m+1}})\})$. In the $o$-decoy region (defined by the set $C$ in Section A), prices were draw from zone $m$ by first drawing some $p_l\underline{p}_h^*$ and $p_h<\overline{p}_h^*$, where $\underline{p}_h^*$ is the solution to $(q_h/\underline{p}_h^*)^{\bar{\gamma}_m} = (\underline{p}_h^*-p_l)/(q_h-q_l)$ and $\overline{p}_h^*$ is the solution to $(q_h/\overline{p}_h^*)^{\bar{\gamma}_{m+1}} = (\overline{p}_h^*-p_l)/(q_h-q_l)$; hence, $z_m^C = \{(p_h,p_l) \in C : p_h \in (\underline{p}_h^*,\overline{p}_h^*)\}$. ### B.4 Instructions to Participants *Stage 1: Valuation Task Instructions* The next few screens will provide detailed instructions about what this survey entails. It is very important that you understand how it works, so please read all instructions carefully. Not only will this allow us to collect more precise data, it will also help you to maximize your payment for participating. In this survey you will view many goods, one at a time. Your task is to enter the amount (in \$) that makes you indifferent between purchasing the good and not purchasing the good. There is no deception or “tricks” in this study; we are interested in your preferences for a variety of goods and your payment will be maximized when you respond truthfully and accurately. At your in-person session at LEEDR, you will have the opportunity to win a bonus prize of \$80, in addition to your fixed payment of \$20. On average, one participant in every five will win this prize and if you are a winner, one of your decisions from this survey may be drawn at random to be implemented for real (around a 20% chance). When a trial is implemented you may end up purchasing a product based on the value you entered. If your value is above the randomly drawn market price, you will receive the product and have that market price deduced from your total payment. Since you don’t know which decision will be selected, you should treat each one as if it was real. The next few screens will explain to you why it is always in your best interest to enter the true maximum value that you are willing to pay for each product. Why is it in your interest to enter your true maximum value for the product? You might think that your best strategy is to enter an amount less than the item is worth to you. This is INCORRECT. The price you pay (i.e. the market price) is determined by a random number generator, and NOT by the price you enter. Thus, if you enter a price less than your true value, you would not be able to affect the price that you pay, but might end up losing the opportunity to buy the item at a “good” price. As an example, suppose the “product” is simply a \$10 bill. If you enter your true maximum willingness-to-pay (i.e. \$10), you will receive the good whenever the randomly drawn market price is less than \$10, and pay that price. For example, if the randomly drawn market price is \$4, you will buy a \$10 bill for only \$4. If the randomly drawn market price is above \$10 you will not buy the \$10 bill. It follows that by bidding your maximum buying price, you make a “profit” since you always end up paying less than the item is worth to you. You should never enter more than your maximum willingness-to-pay, as you may end up paying more than \$10 for a \$10 bill. What happens if you enter a price less than your true value? For example, suppose you say you are only willing to pay \$1 for a \$10 bill and the randomly drawn market price is \$5. As the amount you entered is less than the randomly drawn market price, you would miss out on the opportunity to purchase the \$10 bill for only \$5! Your maximum buying price for the products that you will see next will not always be as obvious as it is for a \$10 bill. However, the logic is the exact same—you will maximize your payout for participating when you enter your true valuation for the product as precisely as possible. Example 1: Let’s go through a hypothetical example. Suppose that you stated that your maximum buying price for a golden egg was \$50. If the randomly drawn market price is \$14, then what outcome would occur? - \$0 will be deducted from your payment and you will receive no golden eggs. - \$14 will be deducted from your balance and you will receive a golden egg. - \$64 will be deducted from your balance and you will receive a golden egg. - \$50 will be deducted from your balance and you will receive a golden egg. - \$14 will be deducted from your balance and you will receive no golden eggs. That’s correct! As the randomly drawn price of the golden egg is below your maximum buying price, you will purchase it for the market price of \$14. Example 2: Let’s go through another hypothetical example. Suppose that you stated that your maximum buying price for a silver egg was \$25. If the randomly drawn price was \$32, then what outcome would occur? - \$0 will be deducted from your payment and you will receive no silver eggs. - \$32 will be deducted from your balance and you will receive a silver egg. - \$25 will be deducted from your balance and you will receive a no silver eggs. - \$25 will be deducted from your balance and you will receive a silver egg. - \$32 will be deducted from your balance and you will receive no silver eggs. That’s correct! As the randomly drawn price of the silver egg is above your maximum buying price, you will not purchase it and \$0 will be deducted from your payment. Thank you for your attention so far. When you are ready to begin the survey, click “Next”. *Stage 2: Choice Task Instructions* Welcome to the LEEDR Lab and thank you for agreeing to take part in this study. Your base payment for participating is \$20 and you will have several opportunities to win an additional \$80. Around one in every five participants will win this bonus payment. There are two parts to today’s session—phase 1 and phase 2—and each should take around 20 minutes to complete. There is no deception or “tricks” involved; we are interested in your preferences and your payment will be maximized when you respond truthfully and accurately. The next few screens will provide detailed instructions about phase 1. It is very important that you understand how it works and how your payment will be calculated, so please read all instructions carefully. Not only will this allow us to collect more precise data, it will also help to maximize your payment for participating. If anything is unclear at any point, raise your hand and a lab assistant will be right over to help clarify. In phase 1, three options will be displayed onscreen and your job is simply to choose the one you like the most. Each screen will feature two kinds of the same product, with prices displayed on an attached tag, along with the option of not buying either. You will see many choices like this, one after another, and each time you will be asked to choose your preferred option using the keys labeled “A”, “B” and “C” on the keyboard in front of you. In total, you will make 18 choices in phase 1, with each differing in product assortment and prices. Each choice that you make today might be implemented for real, so it is important that you treat each decision as if you are actually purchasing one of the options. After each of the 18 choices that you make, you will be asked to choose a number between 1 and 144. You will then draw a number at random and if the number you chose comes up, then the following will occur: - You will win a bonus payment of \$80. - You will purchase whichever option you just chose at the posted price. If you do purchase a product, the buying price will be deducted from your final payment (i.e. \$20 plus bonuses). Purchased products will be delivered electronically where possible or shipped to a location of your choosing. Note that the price you see will never exceed your payment, so you will always leave with a positive amount of money. All purchases are final, so please consider each decision carefully. Finally, it is important to remember that each decision is independent; your choice and the outcome of one decision have no bearing on future options or outcomes. Everything “resets” after each decision, so try not to think about your previous decisions when you are faced with a new choice. Thank you for your attention so far. To summarize: - Respond each time as if your choice will be implemented for real. - Treat each decision as if it were the only choice you make today. - You will leave with a positive amount of money (and possibly a product). When you are ready to begin, press “NEXT” to proceed. *Stage 2: Valuation Task Instructions* In phase 2 you will view several goods, one at a time. Each time your task is simply to decide whether or not you would like to purchase the displayed good at the posted price. Prices will start at zero (i.e. free!) and begin to increase. Once the price gets sufficiently high, you can switch your preference from “buy” to “don’t buy”. Remember, there is no deception or “tricks” in this study; we are interested in your preferences for a variety of goods and your payment will be maximized when you respond truthfully and accurately. If anything is unclear, raise your hand and a lab assistant will be right over to help clarify. You may end up purchasing a product based on the decisions that you’re about to make, so it important that you treat each one as if it were real. At the end of phase 2, you will have another opportunity to win a bonus prize of \$80. You will be able to select 9 numbers (i.e. green balls) between 1 and 144, and if any of your numbers are drawn, the following will occur: - You will win a bonus payment of \$80. - A decision from phase 2 or your online survey will drawn at random and implemented for real. If a decision from phase 2 is chosen to be implemented, the price will be determined by a random draw. Crucially, since you don’t know what price will be selected, treat each buying decision for a given good as if it were the only one you make. The prices that you will see next have been pre-selected and, therefore, your answers will have no impact on the prices you will see. Put differently, it is always in your best interest to respond truthfully because it will maximize your payment for participating. If you end up purchasing a product, the buying price will be deducted from your final payment (i.e. \$20 plus bonuses). All products will be delivered electronically where possible or shipped to a location of your choosing. Note that the price you see will never exceed your payment, so you will always leave with a positive amount of money. All purchases are final, so please consider each decision carefully. Thank you for your attention so far. To summarize: - Respond each time as if your choice will be implemented for real. - Treat each decision as if it were the only choice you make today. - You will leave with a positive amount of money (and possibly a product). When you are ready to begin, press “NEXT” to proceed. ## C Additional Analysis and Robustness ### C.1 Heterogeneous-Agent Structural Estimates This section takes a parametric approach to estimating individual heterogeneity in the range-weighting parameter $\gamma$. With 18 observations per participant, it is possible to estimate the model separately for each individual in the sample—thus estimating a $(\gamma_i,\lambda_i)$ for each $i$.[^18] The grey bars in Figure C.1 plot the distribution of the point estimates for both $\gamma_i$ and $\lambda_i$. The majority of the $\gamma_i$ point estimates are consistent with relative thinking: 75.3% of point estimates are negative. However, the results also reveal substantial heterogeneity in the magnitude of both $\gamma$ and $\lambda$. ![Figure C.1: Distribution of individual-level estimates. Each (γ_(i), λ_(i)) is estimated from equation 1 using 18 choice observations per person. The attribute weights are normalized to sum to 2. The solid black…](assets/gamma-dist-21720e.png) ![Figure C.1: Distribution of individual-level estimates. Each (γ_(i), λ_(i)) is estimated from equation 1 using 18 choice observations per person. The attribute weights are normalized to sum to 2. The solid black…](assets/lambda-dist-fd0562.png) **Figure C.1: Distribution of individual-level estimates. Each ($\gamma_i,\lambda_i$) is estimated from equation (1) using 18 choice observations per person. The attribute weights are normalized to sum to 2. The solid black lines depict the implied distributions of $\gamma_i$ and $\lambda_i$ based on the parametric model estimates in column (3) of Table C.1.** Individual-level estimates based on small samples are inherently noisy, complicating inference. To obtain more precise estimates of the distributions of $\gamma$ and $\lambda$, I develop a parametric model of heterogeneity. Figure C.1 suggests that both parameters follow an approximately log-normal distribution. Therefore, let $$ \begin{aligned} \log(\gamma_i+\kappa) &= \text{\textbf{x}}_i \boldsymbol \beta_\gamma + \epsilon_\gamma \qquad \text{(C.1)} \\ \log(\lambda_i) &= \text{\textbf{x}}_i \boldsymbol \beta_\lambda + \epsilon_\lambda \qquad \text{(C.2)} \end{aligned} $$ where $(\epsilon_\lambda,\epsilon_\gamma)$ are independent normally distributed errors with mean zero, **x**$_i$ is a vector of individual-specific demographic controls, and $\kappa$ is a shift parameter that allows $\gamma_i$ to take negative values while preserving the log-normal structure. Gender, age, age-squared, educational attainment, income bracket, and marital status are all included in **x**$_i$. Rather than estimating two parameters per individual (300 in total), this approach estimates individual-specific means as a function of demographics, $\mu_{\gamma,i} \equiv \text{\textbf{x}}_i \boldsymbol \beta_\gamma$ and $\mu_{\lambda,i} \equiv \text{\textbf{x}}_i \boldsymbol \beta_\lambda$, along with variances indexed by $\sigma_\gamma$ and $\sigma_\lambda$. I estimate the distribution parameters, $\theta=(\boldsymbol \beta_\gamma,\boldsymbol \beta_\lambda,\sigma_\gamma,\sigma_\lambda,\kappa)$ using simulated maximum likelihood (Train 2009). Table C.1 presents the heterogeneous-agent model results. The main statistics of interest are the expected value and standard deviation of $\gamma_i$ and $\lambda_i$. Column (1) presents the estimates of the surplus-maximizing benchmark, $\gamma_i=0$ for all $i$. The scale-parameter estimates imply large and variable choice noise relative to the average surpluses on offer. Column (2) includes demographics as mean shifters as described by equation (C.2). The parameter estimates are quantitatively similar and there is no significant improvement in model fit based on a log-likelihood ratio test. Column (3) allows for range-dependent attribute weighting as indexed by $\gamma_i$, but restricts the model to a constant mean: $\mu_{\gamma,i}=\mu_{\gamma}$ and $\mu_{\lambda,i}=\mu_{\lambda}$ for all $i$. The estimated expected value of $\gamma$ is $-0.31$, which is close to the homogeneous model estimate. The standard deviation estimate of 0.40 indicates substantial heterogeneity relative to the mean. For example, the estimates imply that surplus maximization lies 0.8 standard deviations above the mean and the limiting case of relative thinking ($\gamma=-1$) is 1.7 standard deviations below the mean. Allowing $\gamma_i$ and $\lambda_i$ to follow a mixture distribution does not yield a significant improvement in fit ($p>0.05$). This supports individual-level results showing there is a single, relative-thinking type that differs in the intensity but not the direction of range weighting. **Table C.1: Heterogeneous Model Estimates** | | \(1\) | \(2\) | \(3\) | \(4\) | |:---|:---|:---|:---|:---| | *Range Weighting: $\gamma$* | | | | | | Mean Expected Value | 0 | 0 | $-$0.31 | $-$0.31 | | | – | – | \[$-$0.39,$-$0.22\] | \[$-$0.40,$-$0.21\] | | Mean Standard Deviation | 0 | 0 | 0.40 | 0.37 | | | – | – | \[0.30,0.51\] | \[0.26,0.47\] | | *Scale: $\lambda$* | | | | | | Mean Expected Value | 12.98 | 13.03 | 9.07 | 9.05 | | | \[11.31,15.09\] | \[11.21,15.30\] | \[7.90,10.31\] | \[7.95,11.68\] | | Mean Standard Deviation | 8.18 | 7.49 | 5.46 | 4.69 | | | \[5.46,11.50\] | \[4.40,10.01\] | \[3.67,7.56\] | \[2.75,8.87\] | | Ranging Weighting | No | No | Yes | Yes | | Demographics | No | Yes | No | Yes | | \# Obs. | 2,700 | 2,700 | 2,700 | 2,700 | | \# Ind. | 150 | 150 | 150 | 150 | | $SLL(\hat{\theta})$ | $-$2,175 | $-$2,168 | $-$2,070 | $-$2,057 | | Comparison Model | – | \(1\) | \(1\) | \(3\) | | $LR$ Statistic | – | 13.86 | 209.9 | 26.8 | | $p$-value | – | 0.127 | $<$0.001 | 0.061 | Individual-cluster-robust bootstrapped 95% confidence intervals in brackets. The attribute weighting function is defined by $g(\Delta_x;\gamma_i)= \left(\Delta_x\right)^{\gamma_i}$ for attribute $x \in \{q,p\}$. $\lambda_i$ is the scale parameter of the type-1 extreme value error for individual $i$. Heterogeneity is modeled by $\log(\gamma_i+\kappa) \sim N(\text{\textbf{x}}_i \boldsymbol \beta_\gamma,\sigma_\gamma)$ and $\log(\lambda_i) \sim LN(\text{\textbf{x}}_i \boldsymbol \beta_\lambda \,\sigma_\lambda)$. Demographic variables included in **x**$_i$ are gender, age, age-squared, educational attainment, income bracket, and martial status. Column (4) adds the full set of demographic variables to the estimation, hence estimating equations (C.1) and (C.2). Including observable characteristics does not alter the expected-value estimates, but does yield smaller standard deviations. There is a modest improvement in model fit based on a likelihood-ratio test, though it is not statistically significant at the conventional level. The solid black lines in Figure C.1 illustrate the implied population distributions of $\gamma$ and $\lambda$ using the preferred specification in column (3) of Table C.1. Overall, the estimated distributions closely mirror the individual estimates. ### C.2 Additional Tests of Price Signaling In this section, I describe additional analyses to assess whether price signaling can explain the experimental results. First, I consider whether the results in Section 5.1 are sensitive to the definition of the price-signal index, $s_{ig}$. Rather than defining an index, I restrict the analysis to situations in which participants observed opposing price signals between their first and second encounters with a product. For example, do participants who first observe low ($o$-decoy) prices make choices consistent with relative thinking when they later experience higher ($h$-decoy) prices? This approach does not rely on any assumptions about how prices signaled quality but comes at the expense of statistical power. Table C.2 presents the results. In panel (a), I restrict the sample to participants who first experienced $o$-decoy prices and then made a choice at $h$-decoy prices. Consistent with relative thinking, higher prices of the decoy option led to an increase in the probability of choosing $l$ and a decrease in the probability of buying $o$. Panel (b) restricts the analysis to participants who saw $o$-decoy prices followed by $l$-decoy prices for a good. The estimates are again consistent with relative thinking: higher decoy prices increased the probability of buying $h$ and decreased the probability of not buying. Finally, panel (c) analyzes the choice behavior of participants who first saw $h$- or $l$-decoy prices and then made a choice at $o$-decoy prices. The results confirm that, despite observing conflicting price signals, parallel price increases caused demand to shift from $l$ to $h$. **Table C.2: Choice Probability Estimates with Conflicting Price Signals**
(1) (2) (3)
Dependent Variable: $\qquad \quad$ Buy high Buy low Don’t Buy
Panel (a): $\boldsymbol h$-decoy region
Previous price region: $o$-decoy
Price of $h$ $-$0.19 0.55 $-$0.38
(0.11) (0.18) (0.21)
Panel (b): $\boldsymbol l$-decoy region
Previous price region: $o$-decoy
Price of $l$ 0.40 0.13 $-$0.56
(0.23) (0.09) (0.23)
Panel (c): $\boldsymbol o$-decoy region
Previous price region: $h$- or $l$-decoy
Price of $h$ 0.96 $-$1.21 0.20
(0.30) (0.34) (0.16)
Average marginal effects derived from logistic regression estimates. The regressions either control for the price of $l$ (panel a), the price of $h$ (panel b), or the price difference between $h$ and $l$ (panel c). All numbers reported in percentages. Panel (a) contains 153 observations across 103 participants, panel (b) contains 185 observations across 113 participants, and panel (c) contains 316 observations from 138 participants. Individual cluster-robust standard errors in parentheses. Next, I explore whether the structural estimates are robust to controlling for price signaling quality. As described in Section 5.1, relative thinking and price signaling make different predictions when participants make more than one choice per good. The panel structure of the data therefore allows me to separately identify these two forces. Formally, I estimate the equations 2 and 3. In the estimation, I set initial quality equal to the first-stage reservation values. Column (1) of Table C.3 reproduces the baseline estimates for $\alpha_h=\alpha_l=0$—see column (2) of Table C.6. Column (2) allows price to signal quality, but restricts the model to no range weighting ($\gamma=0$). The estimates are consistent with prices signaling quality, with a larger impact for the high-quality option relative to the low-quality option. Column (3) allows for both range weighting and price signaling quality. The results show that positive price signals had a small, statistically insignificant impact on perceived quality. Furthermore, the attribute-weighting parameter $\gamma$ is robust to the inclusion of price signaling. The range-weighting specification in column (1) provides the best fit to the data as measured by BIC. A likelihood-ratio test confirms that adding price signaling yields no improvement in the model fit relative to the range-weighting only specification ($p=0.21$). **Table C.3: Price as a Signal of Quality Estimates** | | \(1\) | \(2\) | \(3\) | |:---|:---|:---|:---| | $\widehat{\gamma}$ | $-$0.31 | | $-$0.24 | | | \[$-$0.40,$-$0.21\] | | \[$-$0.36,$-$0.11\] | | $\widehat{\alpha}_h$ | | 0.21 | 0.09 | | | | \[0.14,0.29\] | \[$-$0.04,0.19\] | | $\widehat{\alpha}_l$ | | 0.12 | 0.06 | | | | \[0.03,0.21\] | \[$-$0.04,0.15\] | | $\widehat{\lambda}$ | 11.78 | 12.89 | 11.89 | | | \[10.52,13.33\] | \[11.65,14.37\] | \[10.58,13.29\] | | $BIC$ | 4748.3 | 4771.2 | 4761.0 | | Individuals | 150 | 150 | 150 | | Observations $\qquad$ | 2,700 | 2,700 | 2,700 | Individual-cluster-robust bootstrapped 95% confidence intervals in brackets. Estimation allows for price signaling quality and range-dependent attribute weighting as defined in equations (2) and (3). Column (1) restricts the estimation to $\alpha_h=\alpha_l=0$ and column (2) restricts the estimation to $\gamma=0$. ### C.3 Modeling Lagged Price Dependency To model the impact of the previous price ranges, I define the effective price range as $$ \tilde{\Delta}_{p,it} = \sum_{k=0}^K \phi_{ik} \Delta_{p,it-k} $$ where $\phi_{i0}=1-\sum_{k=0}^K \phi_{ik}$ and $K$ is the number of lags included in the specification. Modelling price range spillovers amounts to simply replacing $\Delta_{p,t}$ by $\tilde{\Delta}_{p,t}$ in determining the attribute weights. In terms of estimation, more structure is required to estimate the parameters $\boldsymbol \phi_{i}=(\phi_{i1},...,\phi_{iK})$. To model heterogeneity across individuals let, $$ \left[ \begin{array}{c} \log(\gamma_i+\kappa) \\ \log(\lambda_i) \\ \boldsymbol \phi_i \end{array}\right] \sim N \left( \left[ \begin{array}{c} \mu_\gamma \\ \mu_\lambda \\ \boldsymbol \mu_\phi \end{array}\right] ,\left[ \begin{array}{ccc} \sigma_\gamma & \cdot & \cdot \\ 0 & \sigma_\lambda & \cdot \\ 0 & 0 & \boldsymbol \Sigma_\phi \end{array} \right] \right) $$ where $\boldsymbol \mu_\phi=(\mu_1,...,\mu_K)$ and $\boldsymbol \Sigma_\phi=\text{diag}(\sigma_1,...,\sigma_K)$. Note that $K=0$ is equivalent to the baseline model with no lagged price dependency. Given this specification, the parameters $\theta=(\mu_\gamma,\mu_\lambda,\boldsymbol \mu_\phi,\sigma_\gamma,\sigma_\lambda,\boldsymbol \Sigma_\phi,\kappa)$, can be estimated using simulated maximum likelihood. Table C.4 reports the results for $K \in \{0,1,2,3\}$. **Table C.4: Lagged-Price-Range Estimates** | | | | | | |:---|:---|:---|:---|:---| | | \(1\) | \(2\) | \(3\) | \(4\) | | | $K=0$ | $K=1$ | $K=2$ | $K=3$ | | Mean $\gamma_i$ | $-$0.31 | $-$0.32 | $-$0.31 | $-$0.28 | | | \[$-$0.39,$-$0.22\] | \[$-$0.41,$-$0.23\] | \[$-$0.39,$-$0.10\] | \[$-$0.38,$-$0.08\] | | SD $\gamma_i$ | 0.40 | 0.44 | 0.46 | 0.50 | | | \[0.30,0.51\] | \[0.34,0.57\] | \[0.33,0.73\] | \[0.36,0.71\] | | Mean $\lambda_i$ | 9.07 | 9.20 | 9.21 | 9.07 | | | \[7.90,10.31\] | \[7.87,10.68\] | \[7.84,10.77\] | \[7.74,10.78\] | | SD $\lambda_i$ | 5.46 | 5.95 | 6.13 | 5.95 | | | \[3.67,7.56\] | \[4.07,8.19\] | \[4.09,9.01\] | \[3.91,8.86\] | | Mean $\phi_{1i}$ | | 0.07 | 0.06 | 0.05 | | | | \[0.01,0.18\] | \[0.01,0.21\] | \[$-$0.01,0.20\] | | SD $\phi_{1i}$ | | 0.10 | 0.05 | 0.07 | | | | \[0.00,0.22\] | \[0.00,0.27\] | \[0.00,0.26\] | | Mean $\phi_{2i}$ | | | 0.02 | 0.02 | | | | | \[$-$0.04,0.17\] | \[$-$0.03,0.14\] | | SD $\phi_{2i}$ | | | 0.07 | 0.00 | | | | | \[0.00,0.24\] | \[0.00,0.24\] | | Mean $\phi_{3i}$ | | | | 0.03 | | | | | | \[$-$0.03,0.10\] | | SD $\phi_{3i}$ | | | | 0.01 | | | | | | \[0.00,0.06\] | | $SLL(\hat{\theta})$ | $-$2,175 | $-$1,954 | $-$1,830 | $-$1,706 | | Individuals | 150 | 150 | 150 | 150 | | Observations | 2,700 | 2,550 | 2,400 | 2,250 | Individual-cluster-robust bootstrapped 95% confidence intervals in brackets. The attribute weighting function is $g(\Delta_x;\gamma_i)= \left(\Delta_x\right)^{\gamma_i}$ for attribute $x \in \{q,p\}$. The effective price range is $\tilde{\Delta}_{p,it} = \sum_{k=0}^K \phi_{ik} \Delta_{p,it-k}$ where $\phi_{i0}=1-\sum_{k=0}^K \phi_{ik}$ and $K$ is the number of lags included in the specification. Attribute weights are normalized to sum to two. $\lambda_i$ is the scale parameter of an additive type-1 extreme value error for individual $i$. Heterogeneity is modeled by $\log(\gamma_i+\kappa) \sim N(\mu_\gamma,\sigma_\gamma)$, $\log(\lambda_i) \sim LN(\mu_\lambda \,\sigma_\lambda)$, and $\boldsymbol \phi_i \sim N(\mu_\phi,\boldsymbol \Sigma_\phi)$. ### C.4 Reservation Values The experimental design elicited two reservation value proxies of quality for both the high- and low-quality variants of the nine products that each participant saw during the choice task. In Section 4, I used the average of these reservation values as a proxy for quality in the estimation. In this section, I explore whether there are systematic differences between these two values and estimate the optimal weighting of these two measures. Table C.5 presents the average first- and second-stage reservation values by product. The values are systematically higher in the first stage relative to the second stage. This different could be driven by differences in methodology. The first elicitation used an open-ended free response with no upper bound. To the extend that participants may have perceived a relatively high upper bound on prices, this could have decreased price sensitivity and therefore led to inflated reservation values relative to the underlying quality. It is also possible that participants misunderstood the incentive mechanism, which could cause them to report inflated reservation values (Cason and Plott 2014). Conversely, I elicited the second-stage reservation values using an ascending price. Research has shown the whether prices are increasing or decreasing leads to framing effects (Andersen et al. 2006). In this case, participants might have stated lower reservation values due to the ascending format. **Table C.5: First- vs. Second-Stage Quality Elicitions (in \$)**
First Stage Second Stage RRP$^*$
Variant high (s.d.) low (s.d.) high (s.d.) low (s.d.) high low
Quantity
Chocolate Truffles 43.3 (22.0) 18.1 (8.9) 27.5 (18.6) 15.4 (8.0) 75.0 35.0
Cinema Passes 56.6 (23.4) 30.6 (12) 45.4 (25.5) 25.7 (14.4) 95.0 50.0
Starbucks Coffee 53.8 (22.6) 28.8 (11.6) 36.1 (21.6) 22.5 (14.2) 100.0 50.0
Gillete Razors 24.1 (14.2) 13.4 (7.7) 20.3 (9.7) 11.8 (5.8) 52.9 21.0
Insignia TV 57.9 (22.3) 39.9 (19.1) 39.7 (26.7) 28.7 (18.5) 80.0 60.0
Uber Credit 61.4 (17.3) 30.0 (8.2) 50.1 (22.6) 26.9 (11) 80.0 40.0
Duration
Meal Kit 49.3 (22.3) 28.3 (12.7) 38.2 (20.3) 23.8 (14.8) 120.0 60.0
Yoga Subscription 52.4 (23.6) 29.1 (13.1) 37.5 (20.4) 22.3 (10.7) 100.0 50.0
Wine of the Month 42.9 (20.9) 23.7 (11.1) 35.8 (20.2) 21.8 (10.9) 79.9 40.0
Cheese of the Month 33.6 (18.5) 18.0 (9.5) 28.8 (18.2) 17.5 (9.8) 97.9 49.0
Beer of the Month 38.8 (16.8) 21.5 (9.0) 31.3 (15.6) 19.7 (10.7) 85.9 43.0
Flowers of the Month 42.9 (19.4) 24.4 (10.6) 36.0 (19.4) 21.2 (10.8) 97.9 49.0
Functionality
Roku 50.6 (22.1) 26.6 (13.7) 34.0 (19.3) 19.2 (11.3) 85.3 50.0
Amazon Tablet 56.2 (19.4) 38.7 (16.0) 44.6 (19.5) 32.2 (16.7) 80.0 50.0
Brand
Bluetooth Speaker 40.4 (19.8) 21.3 (9.6) 27.7 (18.5) 15.9 (9.8) 80.0 20.0
Water Purifier 33.9 (15.7) 23.9 (12.8) 27.9 (11.2) 18.7 (9.1) 60.0 48.0
Hardcase Luggage 59.4 (19.5) 42.0 (17.1) 41.1 (20.8) 27.6 (14.5) 75.0 50.0
Rechargeable Batteries 20.9 (8.3) 15.2 (6.3) 17.6 (9.8) 13.4 (6.9) 40.3 37.0
Laptop Backpack 47.2 (22.5) 25.1 (14.4) 35.8 (17.6) 20.9 (11.4) 78.9 30.0
Sunglasses 44.3 (22.6) 18.1 (11.3) 29.3 (16.1) 15.0 (7.4) 95.0 15.0
$^*$RRP = Recommended Retail Price While the quality estimates are subject to noise and systematic differences, they both contain information about participants’ quality perceptions. Building on the intuition in Gillen et al. (2019), I estimate the optimal combination of these measures to obtain a more precise estimate of quality. I define quality as $q_{ijt} = \tau q_{ijt}^I + (1-\tau) q_{ijt}^{II}$, where $q_{ijt}^I$ is the first reservation value and $q_{ijt}^{II}$ is the second. Column (1) of Table C.6 reproduces the baseline structural estimates ($\tau=0.5$). Columns (2) and (3) report the estimates using the first reservation value ($\tau=1$) or the second reservation value ($\tau=0$). The $\gamma$ estimate based on the first reservation values is $-$0.31, which is similar in magnitude to the baseline model. Estimated range weighting is more pronounced when the second reservation values are used as proxies for quality, $\hat{\gamma}=-0.47$. Column (4) estimates an optimal $\tau$ of 0.25, which implies a 3-to-1 weight on the second reservation value relative to the first. This shift weight towards the second reservation value leads to more pronounced range-based attribute weighting: the $\gamma$ estimate of $-0.42$ lies close to the upper bound of the baseline estimate. Finally, I assess whether using the first or second reservation values changes the overall pattern of results. Table C.7 presents the premiums implied by the range weighting estimates using the first or the second reservation values. Overall, while there is some difference in the magnitude of the implied premiums, I find economically meaningful premiums ranging from 11% to 36% across the three regions. **Table C.6: Alternative Quality Specification Estimates** | | | | | | |:---|:---|:---|:---|:---| | | \(1\) | \(2\) | \(3\) | \(4\) | | *Reservation Values:* $\qquad$ | Average | First | Second | Optimal | | $\widehat{\gamma}$ | $-$0.34 | $-$0.31 | $-$0.47 | $-$0.42 | | | \[$-$0.42,$-$0.26\] | \[$-$0.39,$-$0.21\] | \[$-$0.54,$-$0.40\] | \[$-$0.49,$-$0.34\] | | $\widehat{\lambda}$ | 9.19 | 11.78 | 9.61 | 9.19 | | | \[8.18,10.27\] | \[10.45,13.21\] | \[8.48,10.86\] | \[8.19,10.34\] | | $\tau$ | 0.5 | 1 | 0 | 0.25 | | | – | – | – | \[0.15,0.37\] | Individual-cluster-robust bootstrapped 95% confidence intervals in brackets. The attribute weighting function is defined by $g(\Delta_x;\gamma)= \left(\Delta_x\right)^{\gamma}$ for attribute $x \in \{q,p\}$. $\lambda$ is the scale parameter of the type-1 extreme value error. Quality is defined as $q_{ijt} = \tau q_{ijt}^I + (1-\tau) q_{ijt}^{II}$. Columns (1), (2), and (3) provide the estimates based on fixed values of $\tau$. Column (4) estimates the optimal $\tau$. Observations: 2,700. Individuals: 150. **Table C.7: Quantifying Range-Dependent Attribute Weighting: 1st Stage Reservation Values**
(1) (2) (3) (4)
Surplus Max Lower Bound Estimate Upper Bound
Panel A: First-Stage Reservation Values:
Range weighting: $\gamma =$ 0 $-$0.39 $-$0.31 $-$0.21
Quality: $(q_h,q_l)$ = (\$46, \$26)
$\boldsymbol h$-decoy region ($p_h$=\$66)
Willingness-to-pay for $l$ \$26.0 \$29.9 \$29.1 \$28.0
% premium vs. surplus max – 15.2% 11.8% 7.9%
$\boldsymbol l$-decoy region ($p_l$=\$65)
Willingness-to-pay for $h$ \$46.0 \$52.6 \$51.2 \$49.5
% premium vs. surplus max – 14.4% 11.3% 7.5%
$\boldsymbol o$-decoy region ($p_h$=\$30)
Willingness-to-pay for $l$ \$10.0 \$13.1 \$12.5 \$11.7
% premium vs. surplus max – 30.7% 24.8% 17.2%
Panel B: Second-Stage Reservation Values:
Range weighting: $\gamma =$ 0 $-$0.54 $-$0.47 $-$0.40
Quality: $(q_h,q_l)$ = (\$35, \$21)
$\boldsymbol h$-decoy region ($p_h$=\$58)
Willingness-to-pay for $l$ \$21.0 \$27.6 \$26.6 \$25.7
% premium vs. surplus max – 31.4% 26.8% 22.4%
$\boldsymbol l$-decoy region ($p_l$=\$67)
Willingness-to-pay for $h$ \$35.0 \$49.7 \$47.5 \$45.2
% premium vs. surplus max – 21.6% 35.7% 29.7%
$\boldsymbol o$-decoy region ($p_h$=\$26)
Willingness-to-pay for $l$ \$12.0 \$14.1 \$13.8 \$13.6
% premium vs. surplus max – 17.3% 15.2% 13.1%
The price in each region is set to the average realized prices in each decoy region, and each region is defined using the specified reservation values. The reported premiums are measured in percentage terms relative to the surplus maximization amount in column (1). The $\gamma$ values are taken from columns (2) and (3) of Table C.6. Willingness-to-pay is (i) $WTP(\gamma) = q_l (q_h/p_h)^\gamma$ in the $h$-decoy region, (ii) $WTP(\gamma) = q_h (q_h/p_l)^\gamma$ in the $l$-decoy region, and (iii) defined implicitly by $(q_h/p_h)^\gamma = (p_h - WTP(\gamma))/(q_h-q_l)$ in the $o$-decoy region. ## References Andersen, Steffen, Glenn W Harrison, Morten Igel Lau, and E Elisabet Rutström. 2006. “Elicitation Using Multiple Price List Formats.” *Experimental Economics* 9 (4): 383–405. Andersson, Ola, Jim Ingebretsen Carlson, and Erik Wengström. 2021. “Differences Attract: An Experimental Study of Focusing in Economic Choice.” *The Economic Journal*, ahead of print. . 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In contrast, I consider a simple choice context in which both theories make clear testable predictions that are derived assuming only the basic proprieties of the attribute-weighting function. [^2]: This is the limiting case implied under the assumption that $\Delta \cdot g(\Delta;\gamma)$ is strictly increasing in $\Delta$. [^3]: In Appendix A, I show that this holds nonparametrically under weak regularity conditions on the range-weighting function (see Proposition 4). This result relates to the discussion of balanced choices in both KS and BRS; the authors show that if $h$ has a higher quality but also a higher price, then the surplus maximizing choice will prevail. I show that equivalence between surplus maximization and range-weighting extends to all possible price pairs $(p_h,p_l) \in \mathbb{R}_+^2$ in the two-attribute case. [^4]: All proofs can be found in Appendix A. [^5]: The proof of Proposition 1 provides a formal definition of these three regions. [^6]: A key assumption of range-weighting models is that agents narrowly frame each choice. If participants instead view decisions as part of a broader choice set, then reservation values may be biased. For example, if participants treat the Becker et al. (1964) (BDM) mechanism as a lottery and in the absence of an explicit upper bound on price participants assume some large $\bar{p}$, then my quality proxies may not reflect true quality. To the extent that experimental prices were imprecise due to inaccurate quality proxies, my estimates will be biased towards a null effect. [^7]: One advantage of this non-student sample is that it is more diverse compared to the student participant pool. This allows for richer modeling of individual differences in range-dependent attribute weighting. [^8]: The average cash payment upon completion was \$34 and the average value of goods received was \$6 (15 individuals received products with an average value of \$56.05). [^9]: The BDM mechanism was executed as follows. A price, $p^*$, was drawn at random from a uniform distribution over the support $(0,p_{max})$, where $p_{max}$ was set to be the total payment (e.g. \$20 participation fee + the \$80 bonus). If willingness-to-pay was below the randomly drawn price ($WTP