1 Introduction

Strategy-proof direct revelation mechanisms are hallmark achievements of mechanism design. Participants do not need to form beliefs about how other participants are going to act or what other participants' preferences (types) might be. Participants should simply report their preferences truthfully because they can do no better regardless of what the other participants do. That is, truthful reporting is a weakly dominant strategy. We focus on strategy-proof student-school (or analogous) matching mechanisms such as deferred acceptance (DA, Gale and Shapley, Reference Gale and Shapley1962), top trading cycles (TTC, Shapley and Scarf, Reference Shapley and Scarf1974, originally proposed by Gale, see also Abdulkadiroğlu and Sönmez, Reference Abdulkadiroğlu and Sönmez2003) or (random) serial dictatorship ((R)SD). In these mechanisms, students report their ordinal preferences by ranking the schools, while schools behave non-strategically by following their priorities over students. Strategy-proofness implies that we would expect students to submit rankings that coincide with their ordinal preferences.

There is growing evidence, however, that students frequently submit rankings that do not follow their ordinal preferences. This has been documented both in the lab (Chen and Sönmez, Reference Chen and Sönmez2006; Pais and Pintér, Reference Pais and Pintér2008; Li, Reference Li2017) and in the field (Gross et al., Reference Gross, DeArmond and Denice2015; Rees-Jones, Reference Rees-Jones2017; Rees-Jones and Skowronek, Reference Rees-Jones and Skowronek2018; Chen and Pereyra, Reference Chen and Pereyra2019; Hassidim et al., Reference Hassidim, Romm and Shorrer2021; Artemov et al., Reference Artemov, Che and He2023; Shorrer and Sovago, Reference Shorrer and Sóvágó2023).Footnote ¹ These non-truthful rankings present a serious challenge since they have the potential to undermine the desired properties of the resulting allocation. Moreover, they can introduce bias or noise to a policymaker's inference about the population distribution of preferences over schools, which can lead to a misallocation of resources across schools.

Hassidim et al. (Reference Hassidim, Marciano, Romm and Shorrer2017) catalogue various explanations for non-truthful reporting in strategy-proof mechanisms, such as game-form misperception,Footnote ² underestimation of the likelihood of obtaining favored schools, and “behavioral” preferences. It is important to understand what factors really lie behind non-truthful reporting in order to help us think about the directions in which to modify these mechanisms or their presentation in order to increase the incidence of truthful reporting.

This paper aims to contribute to this research program. In particular, we examine a prominent behavioral hypothesis that non-truthful reporting might be driven by “disappointment aversion” (Meisner and von Wangenheim, Reference Meisner and von Wangenheim2023). This hypothesis is based on the expectation-based loss aversion (EBLA) models of Köszegi and Rabin (Reference Köszegi and Rabin2007, Reference Köszegi and Rabin2009). In this setting, each student cares not only about the particular school they are assigned to (“consumption utility”), but also about how their realized assignment compares to a reference point (“gain-loss utility”). EBLA can explain why a student ranks a less-preferred school (in terms of consumption utility), where they have a higher probability of obtaining a seat, above a more preferred school, where they have a lower probability of obtaining a seat (e.g., “district school bias,” Chen and Sönmez, Reference Chen and Sönmez2006). By ranking the more preferred school higher, the student sets themselves up to expect to obtain a seat in that school with a higher probability than if they ranked this school lower. But because the student is still not very likely to obtain a seat in this school, they will more likely become disappointed in that their realized utility will fall short of their expected utility. Ranking the less-preferred school first is a way of endogenously reducing the probability and size of this disappointment. Formally, as shown by Meisner and von Wangenheim (Reference Meisner and von Wangenheim2023) using the choice-acclimating personal equilibrium of Köszegi and Rabin (Reference Köszegi and Rabin2007), a necessary, but not a sufficient, condition for optimality of a non-straightforward ranking is that students' preferences exhibit “loss dominance” (see Sect. 2).Footnote ³ Indeed, Dreyfuss et al. (Reference Dreyfuss, Heffetz and Rabin2022) and Dreyfuss et al. (Reference Dreyfuss, Glicksohn, Heffetz and Romm2023) document that EBLA explains observed data better than standard theory.

However, previous experiments have not been designed to distinguish the effect of disappointment aversion from other potential explanations of non-straightforward ranking. Our main contribution is in measuring the effect of disappointment aversion using a design that eliminates the impact of these alternative explanations, which we achieve by simplifying the underlying economic environment. We believe that such a simplification is necessary for proper identification, since more complex environments are susceptible to confounds, in particular game-form misperception. At the same time, our simple design allows us to maximize the effect of disappointment aversion on non-straightforward reporting within our setting. Moreover, we argue that the size of this effect can be used as an (approximate) upper bound for the size of the effect of disappointment aversion in more complex matching settings as well.

We simplify the matching environment in two ways. First, we restrict the environment to only two schools, one high-value and one low-value, with no possibility of remaining unmatched. Second, we reduce the mechanism to an equivalent contingent choice task. In doing so, we rely on the two-step framing of any strategy-proof matching mechanism proposed by Katuščák and Kittsteiner (Reference Katuščák and Kittsteiner2024). In the first step, based on the rankings submitted by other students and the non-strategic priorities of the schools, the mechanism determines which schools are obtainable for the applicant. This step does not use the ranking submitted by the applicant in any way. The set of obtainable schools is therefore exogenous to the applicant. In the second step, the ranking submitted by the applicant is used to pick the highest-ranked school from the set of obtainable schools. We simplify the decision situation by putting the applicant directly into the second step. In that step, the applicant faces one of three contingencies: either only one school is obtainable, or only the other, or both (none of them being obtainable is impossible). The probabilities of the three contingencies are given explicitly, without referring to notions of competing applicants, school priorities or acceptance thresholds. Without knowing which contingency is realized, the applicant submits one of two possible rankings of the two schools. If the top-ranked school happens to be obtainable, the applicant is assigned to that school. Otherwise they are assigned to the other school.

Our identification of the impact of disappointment aversion is based on a comparative statics prediction derived from the EBLA model of Köszegi and Rabin (Reference Köszegi and Rabin2007) that the incidence of non-straightforward ranking increases with the difference between the probability of only the low-value school being obtainable and the probability of only the high-value school being obtainable, provided that this difference is positive. Based on this prediction, we use a between-subjects design with two treatments. In Symmetric, the two probabilities are similar, predicting no non-straightforward ranking due to disappointment aversion except for the most extremely loss-dominant students. In Asymmetric, the probability of only the low-value school being obtainable is much higher than the probability of only the high-value school being obtainable, predicting non-straightforward ranking due to disappointment aversion for all but the least loss-dominant students. Assuming that the incidence of noise and errors in ranking behavior is orthogonal to this probability variation, the difference in the rate of non-straightforward ranking between Asymmetric and Symmetric identifies the causal impact of disappointment aversion on non-straightforward ranking. Moreover, the treatment effect captures nearly the maximum extent of non-straightforward ranking that is possible due to disappointment aversion in any matching setting. It hence provides an approximate upper bound for the effect of disappointment aversion in both simple and complex matching settings.

To motivate the two-school simplification, note that in an environment with more than two schools, the data pattern that indicates disappointment aversion (a less-preferred school by consumption utility that is more obtainable is ranked above a more-preferred school that is less obtainable) can also be accounted for by the applicant “misperceiving” the mechanism. As a leading example, consider the “immediate acceptance” misperception under which the applicant believes they face the immediate acceptance, or Boston, mechanism. This can be justified by real-world experience from job, marital or dating markets that are all decentralized in nature and in which matching alternatives disappear over time unless pursued. Under this misperception, “chasing” a more-favored but lower-probability school instead of a less-favored but higher-probability school might be risky since the latter might fill up and leave the applicant at an even less-preferred school if the pursuit of the former fails. As a result, it might be optimal to rank these two schools in order of obtainability rather than consumption value, leading to non-straightforward ranking. By restricting the environment to only two schools, with no possibility of remaining unmatched, we eliminate the risk of “chasing” the more-preferred school since one is at worst guaranteed a seat in the other school.

The contingent choice task reduction simplifies the decision environment. It is aimed at further mitigation of mechanism misperception and other potential confounds of disappointment aversion. For example, by clearly communicating probabilities of various contingencies, we overcome the possible issue of an applicant believing that a low-priority or a highly-competitive school is impossible to obtain and hence, under an otherwise correct perception of strategy-proofness, that this school could without any loss be ranked anywhere or omitted from the ranking altogether (Chen and Pereyra, Reference Chen and Pereyra2019). Also, by not using school priorities as sources of variation in the likelihood of obtainability of various schools, we avoid the possibility of an applicant having “reciprocal preferences,” i.e., interpreting school priorities as proxies for how much each school likes them and modifying their own preferences over schools accordingly due to an update about an unobserved component of match quality or due to reciprocity (Opitz and Schwaiger, Reference Opitz and Schwaiger2023a, Reference Opitz and Schwaigerb). Similarly, since the availability of seats is based on exogenous probabilities, we also avoid the possibility that subjects rank “less-competitive” schools above “more-competitive” ones so as to avoid a possible hit to “ego utility” that might result from a rejection from a more-competitive school (Köszegi, Reference Köszegi2006; Moscariello, Reference Moscariello2023).

To the extent that our identification is based on comparing rates of non-straightforward ranking in two different treatments, it might appear that our simplifications are not necessary since the impact of potential confounds would be differenced out. However, in a more complex environment, confounds such as game form misperception would interact with changes in obtainability of various schools in a way that would predict behavioral change in the same direction as disappointment aversion. This underlines why it is important to simplify the mechanism in order to eliminate or at least mitigate these confounds.Footnote ⁴

We find an economically small and statistically marginally significant treatment effect in the direction predicted by disappointment aversion. The rate of non-straightforward ranking is $3.8\%$ in Symmetric and $7.5\%$ in Asymmetric.Footnote ⁵ The treatment effect of 3.72 percentage points is statistically marginally significant. The $95\%$ confidence interval for the treatment effect is $(-0.72,8.16)$ percentage points. In comparison, rates of non-straightforward ranking estimated in the experimental literature range from 14 to 75% (Table A.1).Footnote ⁶ Since the treatment effect provides an approximate upper bound on the extent of non-straightforward ranking due to disappointment aversion in any matching setting, we conclude that disappointment aversion in fact accounts for a small amount (approximately 3.72 percentage points) of the non-straightforward ranking typically observed in school-choice matching experiments (from 14 to 75%).

The rest of the paper is organized as follows. Section 2 introduces a theoretical framework that formally illustrates the comparative statics that the experimental design is based on. Section 3 describes details of the experimental design. Section 4 presents our empirical findings. Section 5 discusses the findings. Finally, Section 6 concludes.

2 Theory

Consider the following binary contingent choice situation: a student applicant wants to obtain a seat at one of two schools, A or B. The consumption utility, or valuation, of school A is larger than that of school B, i.e. $v_A>v_B$ . The applicant knows the probabilities $p_A>0$ , $p_B>0$ , and $p_{\mathrm{AB}}=1-p_A-p_B<1$ , which are, respectively, the probability of the contingency that only school A has an obtainable seat, only school B has an obtainable seat, or both schools have an obtainable seat. The applicant is asked to rank the two schools, either A over B, denoted by AB (the straightforward ranking) or B over A, denoted by BA (the non-straightforward ranking). If the top-ranked school is obtainable, the applicant obtains a seat in that school, otherwise they obtain a seat in the other school. Note that it is impossible for the applicant to end up with a seat in neither school. Therefore if the applicant maximizes their expected consumption utility and $p_{\mathrm{AB}}>0$ , ranking AB is strictly dominant, regardless of $p_A$ and $p_B$ , and hence the mechanism is strategy-proof.

If the applicant's utility exhibits reference dependence, then following Kahneman and Tversky (Reference Kahneman and Tversky1979), we assume that the utility of the applicant consists of the consumption utility from the actual assignment (either $v_A$ or $v_B$ ) and a gain-loss component, which enters the utility function additively with a weight of $\eta >0$ . Moreover, it is assumed that, relative to positive surprises, negative surprises get a weight $\lambda >1$ capturing the applicant's individual degree of loss aversion. Hence, if the applicant obtains a seat at school $s\in \{A,B\}$ and expects to obtain a seat at school $r\in \{A,B\},$ their utility is given by

(1) $\begin{array}{c}u(s\mid r)=v_s+\left(\begin{array}{cc}\eta (v_s-v_r)& \text{if}{v}_s\ge v_r,\\ \lambda \eta (v_s-v_r)& \text{if}{v}_s<v_r.\end{array}\right)\end{array}$

Under reference-dependent preferences, it is key to model what the reference point is and how it is affected by the decision maker's choice. We follow Köszegi and Rabin (Reference Köszegi and Rabin2007) in assuming that the reference point is stochastic and given by the lottery over the two schools that results from the chosen ranking (a so-called choice-acclimating personal equilibrium). If the applicant ranks AB, the probability of obtaining a seat at school A is $q_A=p_A+p_{\mathrm{AB}}$ and the probability of obtaining a seat at school B is $q_B=p_B$ . Analogously, if the applicant ranks BA, the probability of obtaining a seat at school B is $q_B=p_B+p_{\mathrm{AB}}$ and the probability of obtaining a seat at school A is $q_A=p_A$ . Therefore the expected utility of ending up in school s is

(2) $\begin{array}{c}{E}_r\left[u(s\mid r)\right]=v_s+q_{r\ne s}\left(\begin{array}{cc}\eta (v_s-v_r)& \text{if}{v}_s\ge v_r,\\ \lambda \eta (v_s-v_r)& \text{if}{v}_s<v_r.\end{array}\right)\end{array}$

As a result, the overall expected utility given the applicant's ranking choice is given by

(3) $\begin{array}{cc}{E}_s\left\{E_r\left[u\left(s\right|r)\right]\right\}=& q_A[v_A+q_B\eta (v_A-v_B)]+q_B[v_B-q_A\lambda \eta (v_A-v_B)]\\ =& \underset{\text{consumption utility component}}{\underbrace{q_A{v}_A+q_B{v}_B}}-\underset{\text{gain-loss component}}{\underbrace{q_A{q}_B\Lambda (v_A-v_B)}},\end{array}$

where $\Lambda \equiv \eta (\lambda -1)$ is referred to as the loss dominance parameter. Intuitively, for the gain-loss component, there is a positive payoff surprise $v_A-v_B$ if the applicant expects a seat in school B but ends up in A and there is a negative payoff surprise $-(v_A-v_B)$ if the applicant expects to obtain a seat in school A but ends up in B. The compound probability of each surprise is $q_A{q}_B$ . Surprises are weighted by $\eta$ relative to the consumption utility and negative surprises get an additional weight of $\lambda$ , leading to the definition of $\Lambda$ .

Note that having classical preferences corresponds to either $\eta =0$ (no weight on gain-loss utility) or $\lambda =1$ (no loss aversion). These two are isomorphic parametrizations. In fact, any pair $(\eta ,\lambda )$ that results in the same value of $\Lambda$ constitutes an isomorphic representation of the very same EBLA preferences. In the jargon of empirical research, within the context of EBLA preferences, $\eta$ and $\lambda$ are not separately identified. This is because, under EBLA, the chosen lottery determines the reference point, so it is impossible to exogenously manipulate the reference point, which is necessary to separately identify $\eta$ and $\lambda$ . As a result, we carry on the analysis by using the composite preference parameter $\Lambda$ only.

Ranking AB gives the expected utility

(4) $\begin{array}{c}EU\left(AB\right)=(p_A+p_{\mathrm{AB}})v_A+p_B{v}_B-(p_A+p_{\mathrm{AB}})p_B\Lambda (v_A-v_B),\end{array}$

whereas ranking BA gives the expected utility

(5) $\begin{array}{c}EU\left(BA\right)=p_A{v}_A+(p_B+p_{\mathrm{AB}})v_B-p_A(p_B+p_{\mathrm{AB}})\Lambda (v_A-v_B).\end{array}$

This means that the expected utility difference between the two rankings is

(6) $\begin{array}{cc}EU\left(AB\right)-EU\left(BA\right)=& \stackrel{\text{diff. consumption utility component}}{\overbrace{p_{\mathrm{AB}}(v_A-v_B)}}-\stackrel{\text{diff. gain-loss component}}{\overbrace{\Lambda (p_B-p_A)p_{\mathrm{AB}}(v_A-v_B)}}\\ =& p_{\mathrm{AB}}(v_A-v_B)[1-\Lambda (p_B-p_A)].\end{array}$

This reveals that ranking BA is optimal if and only if $\Lambda (p_B-p_A)>1$ . A necessary condition is that $\Lambda >1$ , i.e. that the applicant exhibits “loss dominance.”

The next section describes our experimental design based on this theoretical foundation.

3 Experimental design

We frame the choice situation as one where subjects try to obtain an employment position in one of two institutes instead of a seat in one of two schools. In comparison to a school matching framing, we deem this framing to be more fitting to a situation in which one obtains a monetary reward for a match. We run two between-subjects treatments in which each subject makes a single choice of ranking the two institutes and faces two possible outcomes: obtaining a position in institute A or obtaining a position in institute B. In the actual instructions (see part A of the Supplementary Material), the two institutes are labeled using neutral names “Circle” and “Square”, with the two labels being randomly matched at the subject level to the labels A and B that we use in the paper. The two treatments differ in the probability distribution over the three contingencies. In Symmetric, the contingency probabilities are chosen such that disappointment aversion predicts that nearly all subjects will provide a straightforward ranking. In Asymmetric, the contingency probabilities are instead chosen such that almost everyone who is loss-dominant is predicted to provide a non-straightforward ranking due to disappointment aversion. As a result, we would expect to see a significantly higher rate of non-straightforward ranking in Asymmetric compared to Symmetric if disappointment aversion is an important factor behind non-straightforward rankings. On the other hand, if disappointment aversion is not an important factor behind non-straightforward rankings, we would expect to see little to no difference in the rate of non-straightforward ranking between the two treatments.

3.1 Parameter choices

If all subjects follow the theory presented in the previous section, then the contingency probabilities could in the limit be chosen such that all subjects would rank straightforwardly in Symmetric, whereas all subjects with $\Lambda >1$ (loss dominance) would rank non-straightforwardly in Asymmetric. This could be achieved by setting $p_A\ge p_B$ in Symmetric and by letting $p_A\to 0$ , $p_{\mathrm{AB}}\to 0$ and $p_B\to 1$ in Asymmetric. This way, the treatment effect (Asymmetric minus Symmetric) on the rate of non-straightforward ranking would identify the total fraction of loss-dominant subjects. As shown by Masatlioglu and Raymond (Reference Masatlioglu and Raymond2016) for a general choice environment with EBLA preferences and by Meisner and von Wangenheim (Reference Meisner and von Wangenheim2023) specifically for school choice with any number of students and schools, a necessary condition for non-straightforward ranking is $\Lambda >1$ (see (6)). As a result, in any choice environment with EBLA preferences, and in any school choice environment in particular, one cannot obtain a larger theoretical treatment effect of disappointment aversion than with the proposed limit parametrization. That is, we are in principle not losing any ability to identify the effect of disappointment aversion by reducing the school choice problem to the very simple setting that we use. In fact, with the limit parametrization, the treatment effect provides an upper bound for the effect of disappointment aversion on non-straightforward ranking across the entire domain of matching environments.

However, we must move away from these theoretical limits for our experiment. First, in Asymmetric, the limit parametrization comes at the cost of weakening incentives: as $p_{\mathrm{AB}}\to 0$ , flipping the ranking makes almost no difference to the resulting outcome lottery. If (in the spirit of the random utility model) subjects' ranking behavior includes noise whose effect declines with the predicted expected utility difference, any realistic design requires choosing a non-negligible probability $p_{\mathrm{AB}}$ . Next, we aim to make Symmetric and Asymmetric as structurally similar to each other as possible. In order for the rankings to affect the outcome distributions identically, we use the same value of $p_{\mathrm{AB}}$ in both treatments. Also, in order to have three contingencies with positive probability in both treatments, we set $p_A>0$ but small in Asymmetric. Furthermore, to avoid ranking differences between the treatments based solely on which singleton contingency is more likely, we set $p_A<p_B$ but close to each other in Symmetric. Table 1 displays the parameters used in each treatment. We set $v_A$ to €10, $v_B$ to €4 and $p_{\mathrm{AB}}=0.1$ in both treatments. In Asymmetric, we set $p_A=0.05$ and $p_B=0.85$ . In Symmetric, we set $p_A=0.4$ and $p_B=0.5$ .Footnote ⁷

Table 1

Experimental design parametrization

	$v_A$	$v_B$	$p_A$	$p_B$	$p_{\mathrm{AB}}$
Symmetric	€10	€4	0.4	0.5	0.1
Asymmetric	€10	€4	0.05	0.85	0.1

With these parameters, the theory predicts that anyone with $\Lambda >1.25$ in Asymmetric and $\Lambda >10$ in Symmetric will provide a non-straightforward ranking. Since estimates suggest that the fraction of individuals with $\Lambda >10$ is negligible,Footnote ⁸ in terms of predicted ranking, this design comes very close to the theoretical limit discussed above. The only empirically relevant deviation from this limit comes from subjects with $\Lambda \in (1,1.25]$ , who are predicted to rank straightforwardly in Asymmetric. However, this small “slack” relative to the theoretical limit is unlikely to lead to an underestimate of the effect of disappointment aversion in comparison to many existing experimental settings since these settings are also not parametrized to reach that limit. Importantly, our parametrization is tighter than those of other papers that have investigated the impact of EBLA preferences on non-straightforward ranking (Dreyfuss et al., Reference Dreyfuss, Heffetz and Rabin2022, Reference Dreyfuss, Glicksohn, Heffetz and Romm2023). In these settings, non-straightforward ranking is predicted only for $\lambda >3$ , which, under the assumption that $\eta =1$ , is equivalent to $\Lambda >2$ in our setting.