Designing Mechanisms to Make Welfare- Improving Strategies Focal∗

Daniel E. Fragiadakis Peter Troyan Texas A&M University University of Virginia

Abstract Many institutions use matching algorithms to make assignments. Examples include the allocation of doctors, students and military cadets to hospitals, schools and branches, respectively. Most of the market design literature either imposes strong incentive constraints (such as strategyproofness) or builds mechanisms that, while more efficient than strongly incentive compatible alternatives, require agents to play potentially complex equilibria for the efficiency gains to be realized in practice. Understanding that the effectiveness of welfare-improving strategies relies on the ease with which real-world participants can find them, we carefully design an algorithm that we hypothesize will make such strategies focal. Using a lab experiment, we show that agents do indeed play the intended focal strategies, and doing so yields higher overall welfare than a common alternative mechanism that gives agents dominant strategies of stating their preferences truthfully. Furthermore, we test a mechanism from the field that is similar to ours and find that while both yield comparable levels of average welfare, our algorithm performs significantly better on an individual level. Thus, we show that, if done carefully, this type of behavioral market design is a worthy endeavor that can most promisingly be pursued by a collaboration between institutions, matching theorists, and behavioral and experimental economists. JEL Classification: C78, C91, C92, D61, D63, Keywords: dictatorship, indifference, matching, welfare, efficiency, experiments

∗We are deeply indebted to our graduate school advisors Fuhito Kojima, Muriel Niederle and Al Roth for count- less conversations and encouragement while advising us on this project. We would like to thank Sandro Ambuehl, Blake Barton, Douglas Bernheim, Itay Fainmesser, Clayton Featherstone, Simon Gachter,¨ Guillaume Haeringer, Jeffrey Naecker, Krishna Rao and Charles Sprenger for helpful comments and suggestions. We would also like to thank Liam Anderson and Kimberley Williams from the Stanford Graduate School of Business. Generous support from the B.F. Ha- ley and E.S. Shaw Fellowship (Troyan) and the Leonard W. Ely and Shirley R. Ely Fellowship (Fragiadakis), both through the Stanford Institute for Economic Policy Research, is gratefully acknowledged. Emails: [email protected] and [email protected].

1 1 Introduction

Matching algorithms are used to solve a number of allocation problems. For example, many doctors, students and military cadets have participated in markets where they submit preferences to centralized clearinghouses that in turn use them to make residency, school and branch assignments, respectively (Roth [1984], Abdulkadiroglu˘ and Sonmez¨ [2003], Sonmez¨ and Switzer [2013]). The demand for such mechanisms has spurred an incredible amount of theoretical work on the design of algorithms with desirable properties. Since these mechanisms are designed for practi- cal use, it is important to understand how the design of the mechanism translates into strategic actions from the agents. Most commonly, some type of (Bayesian) Nash equilibrium solution con- cept is used, and the welfare properties of the mechanism are evaluated using these equilibria. A plethora of experiments document departures from equilibrium in strategic environments, however, providing evidence for (and inspiring) models in Behavioral Game Theory (see Camerer [2003]), henceforth BGT. As a result, a mechanism’s efficient equilibria may not be reached in practice. A popular way to address such strategic concerns is to design mechanisms that are impervious to strategizing; that is, to design mechanisms that make it a dominant strategy for individuals to simply report their preferences truthfully. Such mechanisms are called strategyproof. In fact, a substantial proportion of the matching theory literature imposes strategyproofness as a constraint in the design of algorithms. Such work has, in some sense, incorporated BGT into its mechanism design, as imposing strategyproofness gives predicted outcomes that are robust to assumptions on agent beliefs (and thus satisfy the so-called Wilson Docrtine (Wilson 1987) in a very strong sense). The clear benefit of strategyproofness is that even subjects who may not understand the mechanism very clearly are likely to be capable of simply stating their true preferences. The cost of strategyproofness, however, is that strategyproof mechanism may actually be dominated by non- strategyproof (or “manipulable”) mechanisms if the players are sophisticated and are able to reach a (Bayesian) Nash equilibrium. As mentioned, previous results in BGT indicate that optimal equilibrium best-responding from all players is likely to be unrealistic. However, the results from BGT also suggest that, while subjects may depart from equilibrium play, they still do often behave systematically and/or strategically (see Fragiadakis et al. [2013b]). Indeed, a number of non-random models (Cognitive Hi- erarchy, Level-k, Quantal Response Equilibria) have been proposed in BGT that organize non- equilibrium experimental data quite well.1 These observations in BGT suggest a prevalence of behavioral subjects, i.e., agents that may not always reach equilibria, but that nevertheless are capable of implementing rules of thumb or forming and best responding to beliefs. Thus, a fruitful approach for matching theory and market design more generally may be to move away from the efficiency-constraining features of strategyproofness and the complex strategizing required of perfectly optimizing agents, and instead move towards a more “behavioral market design (BMD)” paradigm, i.e., the construction of mechanisms that are optimal for such behavioral subjects . This

1See Camerer et al. [2004] for a description of Cognitive Hierarchy. Nagel [1995] and Stahl II and Wilson [1994] develop the Level-k model. For Quantal Response Equilibria, see McKelvey and Palfrey [1995] for a deﬁnition pertaining to normal form games and McKelvey and Palfrey [1998] for their application to extensive form games.

2 paper takes a step in the direction of BMD as we i) develop an algorithm aimed at focalizing welfare-improving strategies and ii) test it experimentally (alongside two other algorithms); the evidence we obtain shows that BMD can be a successful avenue in practice. We begin with a simple, yet very common, environment: the assignment problem. In an assignment problem, a group of objects must be allocated to a group of individuals without the use of monetary transfers. A number of real-world markets are basic enough to be modeled as assignment problems. Universities, for example, must decide how to allocate housing, courses, and trips in study-abroad programs.2 Firms often allocate tasks and shifts to interchangeable workers without using monetary transfers. Outside of ofﬁcial contexts, assignment problems are also prevalent in many simple situations common to everyday life (e.g., assigning rooms to housemates or profes- sors to ofﬁces), and so providing better mechanisms for assignment problems can raise welfare in a wide variety of informal settings as well, provided the mechanisms are simple enough to easily understand and implement.3 A common approach to solving assignment problems that is probably familiar to most read- ers is the “Random Serial Dictatorship (RSD)”: each agent submits a strict ordering of goods, the agents are ordered randomly, and each is assigned her favorite good in this order. Though strategyproof, RSD’s main drawback is that it may produce assignments whereby it is possible to substantially improve the welfare of some agents at only a small cost to others.4 These socially sub- optimal outcomes arise from the callousness5 of RSD: when assigning a good to an individual, RSD fails to consider the preferences of succeeding agents. This can be especially costly if individuals are indifferent over subsets of goods. For instance, consider two agents, 1 and 2, and two goods, A and B. Suppose agent 1 values A and B equally and is randomly selected to pick before agent 2 who strongly prefers A to B. RSD forces 1 to submit a strict preference ordering and break his indifference. This would lead to a Pareto inferior outcome if 1 decides to report A as his preferred object. A solution to this problem is to simply allow 1 to report his indifference: 1 can be “guaranteed” A or B, then ultimately be assigned B when 2 reports a strict preference for A. Such a “Random Serial Dictatorship with Indifference (Random-SDI)” is proposed by Featherstone and Roth (work in progress) for assigning MBA students to educational trips abroad at Harvard Business School.6 While this solution is successful in the extreme case of true indifference, if 1 has even a slight preference for A over B, purely rational behavior would lead him to report A as his preferred outcome (and receive it), even though the alternative assignment results in higher welfare from a utilitarian social planner’s perspective. The former observation suggests the following approach: incentivize agent 1 to report A and B in the same indifference class–called a bin for shorthand–when he is nearly indifferent between

2See Chen and Sonmez¨ [2002], Chen and Sonmez¨ [2004], Sonmez¨ and Unver¨ [2005],Sonmez¨ and Unver¨ [2010], Budish and Cantillon [2011], Featherstone [2011] and Featherstone and Roth (work in progress). 3Bogomolnaia [2001] and Budish [2011] provide additional examples of assignment problems such as leads to sales- people, players to sports teams, shared capital, airline landing slots. 4This is undesirable according to certain welfare criteria such as those of Rawls [1972] and Harsanyi [1975]. Feather- stone [2011] notes that some institutions may evaluate welfare in similar ways. 5This terminology was introduced by Budish and Cantillon [2011]. 6See Bogomolnaia and Moulin [2004], W. Irving [1994], Jaramillo and Manjunath [2012] and Katta and Sethuraman [2006] for additional discussions of mechanisms that involve the expression of indifference.

3 them. In Section 3, we show an example where all three agents in a market receive higher expected utilities when individuals are rewarded for expressing indifference compared to running RSD. (Each agent in this example has strict preferences, and hence running Random-SDI is theoret- ically equivalent to running RSD.) The rewards work by modifying RSD in two ways: 1) allowing indifference and 2) ordering an agent before another if the former agent creates bins of larger size (i.e., expresses more indifference) than the latter agent. In this paper, we create such a mechanism which we call the “Sizes Serial Dictatorship with Indifference (Sizes-SDI)”. The way Sizes-SDI exemplifies a behavioral approach to market design is that it provides agents with a clear focal strategy: place multiple goods in the same bin if and only if the goods are similar in value. We call such behavior “cardinal-following”, as it involves using cardinal utilities to make binning decisions. The benefit of cardinal-following behavior is that it can provide an individual (as well as others) with higher expected utilities compared to Random-SDI, as illustrated in Section 3. We anticipate cardinal-following behavior to occur under Sizes-SDI because the algorithm creates a clear cost and benefit to placing multiple goods in the same bin: on the one hand, placing more goods in a bin yields a higher chance of receiving some good in the bin, but on the other hand, grants the algorithm flexibility in deciding which good is assigned from within the bin. Because the mechanism makes these strategies focal, we expect agents to be able to discover and implement them easily, and in so doing, raise the overall welfare of the final assignment. Of course, whether this will be true is ultimately an empirical question, and so we use a lab experiment to test whether Sizes-SDI improves welfare over RSD in practice, and whether such an improvement is due to the behavior we expect, given the design of the mechanism.7 In our main experimental treatment, laboratory subjects are assigned goods according to Sizes- SDI. A significant benefit of using the lab is that it allows us to construct and observe individuals’ underlying monetary payoffs for the set of goods (our proxy for welfare). Using this informa- tion, we can evaluate the welfare of any matching without having to infer true preferences from the submitted reports. In the field, however, the underlying preferences are often unobtainable; only submitted preferences may be available. In this case, accurate welfare analysis of a particular match’s outcome is restricted to algorithms where we have good reason to assume individuals are stating their preferences truthfully, i.e., strategyproof mechanisms. Being that we are explicitly studying the manipulable Sizes-SDI mechanism, we find a laboratory investigation necessary. Once lab subjects observe their values for the various goods, they place the goods into bins which are in turn used to compute assignments under Sizes-SDI. We highlight our main positive result: Sizes-SDI delivers aggregate earnings that are significantly higher than those that (the strategyproof) RSD would produce under truth-telling. In terms of the behavior under Sizes-SDI that generates the welfare gains, we find that the incentives of the mechanism largely work as orig- inally intended: individuals express indifference over goods of similar value, i.e., a substantial proportion of behavior is cardinal-following.

7The question may arise of why we do not simply ask agents to report cardinal utilities directly. While this is a possible approach, it may be difficult for agents to formulate their reports in such mechanisms, and the strategic properties may become quite complex (this is discussed further in the Literature Review below). We are interested in mechanisms that will be more efficient than strategyproof alternatives, while still providing the participants with simple strategies and heuristics that will ensure that these potential welfare gains will be realized in practice. As we show, Sizes-SDI satisfies both of these goals.

4 While our main result indicates that compromising strategyproofness in search of welfare gains can be worthwhile in practice, another question comes to mind when drawing on insights from behavioral and experimental economics more generally: if individuals were simply encouraged to provide positive externalities through expressing indifference, would social preferences lead them to do so “for free”? In light of the vast amount of literature showing that individuals have social preferences (see Kahneman et al. [1986], Fehr and Schmidt [1999] and Charness and Rabin [2002]), some of the welfare gains from Sizes-SDI may have occurred if we had simply allowed (and not incentivized) indifference. As a control treatment, we test Random-SDI to investigate whether simply allowing agents to express weak ordinal preferences (even though their true preferences are strict) will induce any reports of indifference. Interestingly, Random-SDI performs significantly better than RSD on the aggregate level, contradicting the theoretical prediction of truth-telling. The welfare gains come from some individuals expressing indifference when informed that it can greatly improve the welfare of others while only imposing small costs on themselves. Though effective, Random- SDI’s welfare improvements over RSD are limited: Random-SDI does not perform nearly as well as Sizes-SDI. Thus, while some of the gains from our manipulable algorithm can be obtained by simply allowing indifference, even more improvements are possible when actually incentivizing it. The positive results obtained from Sizes-SDI suggest that incentivizing indifference in serial dictatorships can help shape the design of future algorithms. At the same time, Sizes-SDI is only one potential way to relax strategyproofness in order to achieve efficiency gains. Indeed, orga- nizations facing assignment problems in the field have actually experienced the shortcomings of RSD and have attempted to modify it to achieve welfare gains. For instance, in its first year of using a matching algorithm to assign its students to educational trips abroad, the Stanford Gradu- ate School of Business (GSB) used RSD. Unsatisfied with its results, a group of students proposed a (manipulable) serial dictatorship with indifference that orders agents endogenously based on their submitted reports, though in a markedly different manner than Sizes-SDI. Rather than using bin sizes to order agents, the Stanford GSB uses bin contents: agents are ordered earlier for highly ranking goods that other agents do not rank highly.8 Given its incentives, we henceforth refer to the Stanford GSB’s algorithm as the “Contents Serial Dictatorship with Indifference (Contents-SDI)”. While both Contents-SDI and Sizes-SDI are endogenously ordered serial dictatorships with indifference, it is unclear what strategies are focal under Contents-SDI, and hence, what the resulting welfare implications are of the mechanism. Accordingly, we test Contents-SDI in the lab, alongside Sizes-SDI. In terms of aggregate welfare, we find Contents-SDI and Sizes-SDI fare similarly, providing further confirmation that manipulable algorithms can yield welfare gains in practice. However, earnings are much less evenly distributed in Contents-SDI. Furthermore, when moving from RSD to Contents-SDI, roughly 30 percent of subjects are made worse off; when moving RSD to Sizes- SDI, however, all but 5 percent of subjects are made better off. Thus, Sizes-SDI performs better than Contents-SDI on an individual level. 8We discuss this in more detail in Section 4.

5 In analyzing the behavior of Contents-SDI, we find that while some behavior is cardinal-following, even more is “ordinal-violating”, i.e., involves a good B being placed in a more preferred bin than a good A, even though B yields a strictly lower payoff than A. In this example, we say that good B is “augmented”. The extent to which ordinal-violating reports are focal in Contents-SDI may not be surprising given that the algorithm orders individuals earlier who highly rank unpopu- lar goods. (It is common knowledge that individuals have the same true ordinal rankings over goods.) More often than not, however, when individuals augment undesirable goods and receive earlier positions in the agent ordering, they receive the augmented goods; thus, ordinal-violations are generally harmful in practice. So, while moving away from strategyproof mechanisms may be desirable, exactly how this is done should be carefully considered, as agents may not be perfectly rational and may instead be tempted by natural focal points or heuristics that are built in to the mechanism. In summary, this paper raises awareness of a gap in the existing matching literature. Namely, most prior work has focused on designing 1) strategyproof algorithms that make reporting simple, but may result in lower efficiency or 2) manipulable algorithms with strong efficiency properties in equilibrium, but that may require complex strategizing on the part of the agents. There exists, however, substantial room for the design of algorithms that are well-suited for “behavioral” subjects that are not perfectly rational, but nonetheless are able to identify and implement focal strategies and heuristics. Such subjects have been largely identified in numerous Behavioral Game Theory experiments. In this paper, we design a novel manipulable algorithm that provides agents with “easy” non-truthful strategies and document its welfare improvements over more common strategyproof mechanisms under experimental conditions. We also show that another algorithm with similarly good intentions, but with more complicated incentives, also gives rise to similar welfare improvements in the aggregate, but the more complex incentives result in more players making errors and the overall improvements being distributed more unevenly across subjects. Thus, we argue that developing new mechanisms with the intent of catering to behavioral subjects can be a promising venture, but should be done carefully. Lastly, we hope the present study spurs an inter- est in a more behavioral approach to market design on the part of institutions, market designers, and behavioral and experimental economists alike. The remainder of the paper is organized as follows. Section 2 provides review of relevant literature. Section 3 illustrates a motivating example. Section 4 defines the environment. Section 5 describes the experimental design. Section 6 discusses the experimental results. Section 7 con- cludes.

2 Related Literature

Matching algorithms are used by a variety of institutions. For instance, medical residents in many countries are assigned to hospitals using mechanisms that translate doctor and hospital preferences into matchings (Roth [1984], Roth and Peranson [1999], Kamada and Kojima [2013]). Educa- tional institutions, in the United States and abroad, often use preferences of students and schools to determine which students attend which schools (Abdulkadiroglu˘ and Sonmez¨ [2003], Abdulka-

6 diroglu˘ et al. [2011], Abdulkadiroglu˘ et al. [2005], Erdil and Ergin [2008]). Yet another example of a large institution using matching algorithms is the United States military that considers cadets’ preferences when making branch assignments (Sonmez¨ [2013], Sonmez¨ and Switzer [2013]). The demand for matching algorithms in the field has spurred a large body of research designing mechanisms with desirable properties. Much of the literature either (i) imposes strong incentive constraints (such as strategyproofness or “approximate strategyproofness”) that may hinder efficiency or (ii) assumes agents play a (possibly complex) Bayesian Nash equilibrium.9 In regards to (i), a vast literature that has focused on designing mechanisms that are strategyproof.10 Strategyproof mechanisms advance the so-called Wilson Doctrine (Wilson [1987]), which argues that to be successful, market design should not be sensitive to specific assumptions on agent beliefs. Strategyproof algorithms satisfy the Wilson Doctrine in its strongest sense, since truthful reporting is optimal for any beliefs agents may have. Pathak and Sonmez¨ [2008] argue that this is particularly important in school choice settings, and interpret strategyproofness as a way to “level the playing field” between nave and sophisticated participants; strategyproofness may help distribute welfare more evenly. As far as case (ii), many papers identify the extent to which relaxing incentive properties can raise efficiency. Hylland and Zeckhauser [1979] design an algorithm that is symmetric and (ex- ante) Pareto optimal.11 Miralles [2008], Abdulkadiroglu˘ et al. [2011], and Troyan [2012] use a similar efficiency notion in a school choice context to argue that the equilibrium outcomes of the commonly used “Boston” mechanism (also known as immediate acceptance) may ex-ante Pareto dominate those of the strategyproof deferred acceptance mechanism that is often recommended by economists. However, these results only hold if agents play (potentially complex) equilibrium strategies that depend heavily on their beliefs about other agents (indeed, the cited papers themselves cannot solve for the optimal strategies, and prove their results using strategies that are defined only implicitly). There is some experimental evidence to support the conclusion that it may be difficult for subjects to reach equilibria in manipulable mechanisms. Featherstone and Niederle [2008] find that subjects cannot reach non-truthful equilibria under the Boston mechanism. In another related study, Chen and Sonmez¨ [2006] find that experimental subjects earn less under Boston compared to DA. However, not all manipulable mechanisms are created equal. Indeed, a central contribution of the Behavioral Game Theory (BGT) literature is to show that while individuals may tend to depart from equilibria in strategic settings, they often do so in systematic ways (see Camerer [2003]). Thus, if mechanisms are designed in ways that clarify and focalize the intended manipulations

9The necessary tradeoff between incentives and efficiency is well-known in many areas of economics. In the specific context of the assignment problem we consider, the impossibility of strategyproofness and efficiency has been studied in many previous papers. For instance, Zhou [1990] shows the incompatibility of symmetry, strategyproofness and Pareto optimality. Bogomolnaia [2001] and Featherstone [2011] show that strategyproofness makes it impossible to design a mechanism that is “ordinally efficient” or “rank efficient”, respectively. 10See, for example, Abdulkadiroglu˘ and Sonmez¨ [2003], Papai´ [2000], Pathak and Sonmez¨ [2008], Pathak and Sonmez¨ [2012], Hatfield and Kojima [2009], Kamada and Kojima [2013], Fragiadakis and Troyan [2015] andFragiadakis et al. [2013a]. 11Symmetry means that if two agents submit the same report, they receive the same probability shares over each good in the market. Pareto optimality means that, given submitted preferences, the probabilities that agents receive various goods are efficient, i.e., for any trade in shares that strictly increased some individual’s expected utility, another individual’s expected utility must be strictly reduced.

7 (those that will increase efficiency), subjects may invoke these strategies and improve welfare. The broad contribution of this paper is the design of a novel algorithm that we show to be well-suited for these sorts of “behavioral” subjects that have intermediate levels of strategic sophistication. We decide to build our mechanism off of the Random Serial Dictatorship (RSD); the reasons for this decision are twofold. First, even if never having submitted preferences to a centralized mechanism, individuals are likely to have encountered serial dictatorships before, i.e., when they get on a bus and pick their favorite available seat or when they pick their favorite available meal on an airplane. Second, the prevalence of dictatorships in the field is likely to be partially a result of their simplicity. Given that market design in general is founded upon generating algorithms that are “user-friendly” (a feature which should only be more important in a behavioral approach to market design), we think RSD is a an appropriate stepping stone on the path towards greater efficiency. In the remaining sections, we illustrate how the inefficiencies of RSD can be reduced by not just allowing indifference (as has been done in prior work), but actually incentivizing indifference (a practice that, to our knowledge, we are the first to study). When the incentives for reporting indifference produce clear and focal strategies, we show via experiments that participants will find these intended strategies, and, in doing so, the overall efficiency of the market is increased.

3 Motivating Example

3.1 The Random Serial Dictatorship (RSD)

The Random Serial Dictatorship (RSD) is a common way to allocate scarce resources to individuals. First, agents submit strict ordinal rankings over goods. Second, individuals are ordered randomly. Third, each individual is assigned her favorite (available) good, one at a time, according to the ordering. RSD has the advantage of being strategyproof, i.e., each agent has a dominant strategy of stating her preferences truthfully. Unfortunately, RSD is callous, i.e., it assigns an agent to a good without considering how this affects succeeding individuals. This can be especially costly if individuals have weak preferences since RSD does not allow reports of indifference. To illustrate the welfare effects of forcing agents to strictly rank goods over which they are indifferent, we consider a market of three agents, Alex, Beth and Claire, and three goods, g1, g2 and g3. Each agent is to receive a different good. The agents’ preferences over goods are shown in

Table 1. All agents can maximize their expected utilities by reporting g1 ﬁrst, then g2, followed by g3, i.e. by reporting X = g1 g2 g3. (The expectation is taken over the possible agent orderings. The utilities and reports are ﬁxed in the expectation.) Agents are free to make other reports as well.

The report Y = g2 g1 g3, for instance, is also payoff-maximizing for Alex, as he is indifferent between g1 and g2. The X and Y columns in Table 1 show the agents’ expected utilities under RSD when Alex reports X and Y, respectively, given that Beth and Claire each state X in both columns. Although Alex receives the same expected utility by stating X or Y, Beth and Claire are worse off when Alex reports X versus when he states Y. Fortunately, we can modify RSD to generate the higher set of utilities simply by allowing Alex to report his indifference between g1 and g2.

8 goods XYZ g1 g2 g3 g1 g2 g3 g2 g1 g3 {g1, g2} g3 Alex 1 1 0 0.67 0.67 0.67 Beth 1 0.2 0 0.4 0.53 0.53 Claire 1 0.2 0 0.4 0.53 0.53 utilities ←−−−−−RSD−−−−−→ Random-SDI

TABLE 1.—Fixing the reports of Beth and Claire at X, the columns show the expected utilities for all agents for different reports made by Alex. Columns X and Y show the agents’ expected utilities under RSD when Alex reports X and Y, respectively; we see that RSD yields different expected utilities depending on how Alex breaks his indifference. All agents are weakly better off in RSD when Alex reports Y versus X. If he reports Z under RSDI, we reach expected utilities (in column Z) that are equivalent to those obtained in column Y under RSD.

3.2 The Random Serial Dictatorship with Indifference (Random-SDI)

To remedy the problem illustrated in Section 3.1, suppose we make a single modiﬁcation to RSD: allow agents to submit weak ordinal rankings. Thus, allow Alex to express an indifference between g1 and g2 and a strict preference of each over g3; we denote this report as Z = {g1, g2} g3, where {...} represents an indifference class. We call this new mechanism the Random Serial Dic- tatorship with Indifference (Random-SDI), which, like RSD, is strategyproof. If Alex reports his true preferences, Z, while Beth and Claire both state X, the rightmost column of Table 1 shows the expected utilities generated from Random-SDI, which are equivalent to the higher set of RSD utilities in Table 1.12 Random-SDI achieves the higher column of RSD expected utilities because, whenever Alex is

ﬁrst in line, it “guarantees” him the receipt of g1 or g2. Then, upon reaching the second agent in line, Random-SDI assigns Alex g2 in order to give the second agent her ﬁrst choice, g1. Alex would also receive g2 if he appeared second in the ordering. The last agent in the ordering always receives g3. Thus, Alex has a 2/3 chance of getting g2 (utility of 1) and a 1/3 chance of getting g3 (utility of 0), which yields the expected utility of .67 shown in Table 1. (Section 4.2.3 describes Random-SDI in more detail.) While moving from RSD to Random-SDI can improve welfare, we note that the improvement is knife-edged: if Alex’s utility for g2 is slightly perturbed to, say, 0.98, Alex’s (optimal) truthful report is now X, and reporting Z makes Alex worse off. From a utilitarian social planner’s perspective, it would be welfare-improving if Alex were somehow subsidized for making the report Z instead of X.

3.3 Incentivizing indifference

Keeping with our last example of Alex, Beth and Claire, we may wish to ﬁnd a way to incentivize

Alex to report Z, even when he slightly prefers g1 to g2. One approach would be to order individuals at random who make identical reports, but order those who report “more” indifference earlier than those who reports “less”.13 For instance, if Alex, Beth and Claire report Z, X and X, respec-

12Random-SDI is currently used at the Harvard Business School to assign its MBA agents to educational goods abroad (Featherstone and Roth (work in progress)). 13There are many ways to deﬁne what it means for one report to express “more” indifference than another.

9 goods X average Z g1 g2 g3 g1 g2 g3 utilities {g1, g2} g3 Alex 1 0.98 0 0.66 ) ( 0.65 Beth 1 0.2 0 0.4 .49 .57 0.53 Claire 1 0.2 0 0.4 0.53 utilities randomly ordering agents, i.e., Random-SDI

TABLE 2.—In the X and Z columns, Beth and Claire each state X. Average utility is .49 under the Random- SDI if Alex makes his truthful report X (shown in column X). If Alex were to state Z, Random-SDI would yield an average utility of .57 (shown in column Z). tively, order Alex first with probability 1, followed by a random ordering of Beth and Claire. Alex receives g2 with certainty, leaving g1 and g3 to Beth and Claire, each with a chance of 0.5. Table 3 shows that every agent is better off under these reports compared to if they were all to report their preferences truthfully (column Z versus column X). There are many ways to define what it means for one report to express “more” indifference than another, based on the sizes of the bins or their contents. In the next section, we define mechanisms to do this formally.

goods X average Z g1 g2 g3 g1 g2 g3 utilities {g1, g2} g3 Alex 1 0.98 0 0.66 ) ( 0.98 Beth 1 0.2 0 0.4 .49 .66 0.5 Claire 1 0.2 0 0.4 0.5 utilities rewarding agents who express indifference

TABLE 3.—Suppose we order agents earlier who express more indifference. In the X and Z columns, Beth and Claire each state X, while in the X column, Alex reports X and in the Z column, Alex reports Z. Considering Alex’s expected utilities, he has an incentive to deviate to report Z over X, as he is better off receiving g2 with probability 1 than receiving each good with a 1/3 chance; Beth and Claire are better off as well when Alex reports Z versus X.

4 Environment

4.1 The Model

Our model consist of a set of agents, A = {1, . . . , N}, and a set of goods, G = {g1,..., gM}.

Goods have capacities C = {c1,..., cM}, where ci corresponds to the capacity of good gi. There are ∑M 14 enough spaces to accommodate all agents, so we also assume i=1 ci = N. The mechanisms we study in this paper involve agents submitting ordered partitions of the set j of goods, G. We call each member of a partition a “bin”. We let Bi denote the set of goods agent 1 M i places in bin j. Thus, to represent the submitted report of agent i, we write Bi = Bi ... Bi SM k x y where k=1 Bi = G and Bi ∩ Bi = φ for all x 6= y. Other than this, we impose only one additional x x+1 restriction: Bi = φ =⇒ Bi = φ for all x < M. This says that bins cannot be “skipped” when 14 ∑M The theoretical results we show also hold when we let i=1 ci > N. Additionally, we have no reason to believe our experimental results would depend on the equality being strict or weak.

10 reporting weak ordinal preferences. In other words, if i places some goods in her fourth and sixth 4 6 5 bins (min{|Bi |, |Bi |} ≥ 1), then she must place at least one good in her ﬁfth bin (|Bi | > 0). We let B denote the set of an agent’s allowable reports and BN denote the collective set of all agents’ allowable reports.

4.2 The Mechanisms

The algorithms we study all have different ways of translating submitted reports into ﬁnal assignments. They can all be thought of as serial dictatorships, however, because they

1. order individuals (randomly or as a function of the submitted reports)

2. assign goods to individuals using the ordering established in 1.

The most straightforward serial dictatorship is the Random Serial Dictatorship (RSD). After individuals submit strict ordinal rankings over goods, RSD begins by ordering individuals at random. Then, RSD assigns individuals their favorite available goods, one-by-one. Visually, RSD can be viewed as (randomly ordered) agents standing in a line, and one at a time, walking up to a table and picking their favorite good off a table. When a agent approaches the table, the agent has a simple strategy: pick her favorite good. In terms of actually submitting her preferences to the RSD algorithm, this corresponds to each agent having a dominant strategy of ranking her goods truthfully,15 i.e. RSD is strategyproof.

4.2.1 The Sizes Serial Dictatorship with Indifference (Sizes-SDI)

Agent Odering Rule of Sizes-SDI: For each pair of agents, i and j, ﬁnd the smallest integer k k k k k such that |Bi | 6= |Bj |. If such a k exists, order i before j if |Bi | > |Bj | and order j before i and if k k |Bj | > |Bi |. If no such k exists, order i before j with probability 0.5.

In words, Sizes-SDI first orders agents according to the sizes of their first bins: agents with more goods in their first bins being ordered earlier. Then, for agents with the same number of goods in their first bins, second bins are considered. Agents with more goods in their second bins are ordered earlier. This “lexicographic” ordering procedure continues onto later bins in a similar fashion. While there are other ordering rules that could have been chosen, we think this procedure is relatively easy to explain to individuals–the Stanford GSB uses a lexicographic ordering procedure as well–and hence, a desirable Behavioral Market Design choice. To see an example of this ordering procedure, we provide an example of a market of 5 agents and 5 goods, each with unit capacity; see the Sizes-SDI panel on the left in Figure 1. (For now, ignore the fact that some of the goods are written in capital letters and ignore the circles and

15If a agent’s truthful preferences involve indifference, she can technically not submit her true preferences in RSD; she must break her indifference in RSD. Any strict report that “monotonically breaks indifference”, however, is a dominant strategy. By “monotonically breaks indifference”, we mean that if gi is stated as strictly preferred to tj, then the truthful preference of the agent must be a weak preference of gi to tj. For example, if true preferences are {g1, g2} > {g3}, there are two ways to monotonically break indifference: {g1} > {g2} > {g3} and {g2} > {g1} > {g3}. The report {g1} > {g3} > {g2} breaks indifference, but not monotonically.

11 boxes.) The entries in the table indicate the goods that each agent places in each bin. For example, 2 consider the i = 3 row and Bi column; the cell at their intersection point contains goods 3 and 5, indicating that agent 3 placed these goods in her second bin. Now let’s see how the agents are ordered. Agents 1 and 2 and are the first two agents in the ordering because they have the first bins of largest size. Furthermore, 1 comes before 2 because 1’s first bin is larger than 2’s. Agent 3 is ordered before agents 4 and 5 because 3’s second bin is largest in size. Finally, 4 is ordered before 5 because 4’s third bin is larger in size than 5’s.

Sizes-SDI Contents-SDI 1 2 3 4 5 1 2 3 4 5 agent Bi Bi Bi Bi Bi a aal agent Bi Bi Bi Bi Bi i = 1 g1, g2, G4 g3, g5 i = 1 g1, g2, G4 g3, g5

i = 2 G1, g2 g3, g5 g4 i = 5 G3 g1 g2 g4 g5

i = 3 G2 g3, g4 g1 g5 i = 2 G1, g2 g3, g5 g4 i = 4 g2 g1 G3 ,g4 g5 i = 3 G2 g3, g4 g1 g5

i = 5 g3 g1 g2 g4 G5 i = 4 g2 g1 g3,g4 G5

FIGURE 1.—The left and right panels–corresponding to Sizes-SDI and Contents-SDI, respectively–show the agent orderings and assignments made from a single set of reports. Each “agent” column signiﬁes the agent ordering (where higher agents are ordered earlier). Each cell indicates the composition of an agent’s bin. 2 For instance, in each panel, the cell intersected by the i = 3 row and Bi column indicates that agent 3 places goods 3 and 4 in her second bin. The good that an individual receives is indicated by a capital letter. For example, under both mechanisms, individuals 1, 2 and 3 receive goods 4, 1 and 2, respectively. Boxes and circles are drawn around individuals and goods that are matched differently across the two mechanisms. The agent ordering under Contents-SDI is produced using Popularity 1 from Figure 2. (It can be veriﬁed that the assignments would not change under Popularity 2 from Figure 2.)

To complete the definition of Sizes-SDI, we must describe how it assigns goods to agents after the agents have been ordered. In a manner analogous to how RSD assigns goods to individuals, Sizes-SDI sequentially provides each agent with an assigned bin. Specifically, agent i0s assigned bin will be her most preferred bin such that there exists an allocation of goods to agents where (1) agent i receives a good from her assigned bin, and (2) all agents preceding i in the ordering receive goods from their respective assigned bins. Once all agents are assigned bins, each agent is assigned a good from her assigned bin. In the event that multiple such allocations possible, Sizes-SDI picks one at random. This completes the definition of Sizes-SDI. To see an example of this procedure, see the Sizes-SDI panel in Figure 1. Sizes-SDI begins with 1 agent 1, who is assigned B1. Next, Sizes-SDI attempts to assign agent 2 her most preferred bin, 1 1 B2. This is possible, given the composition B2 and the constraint that 1 must receive a good from 1 1 1 B1. Once the algorithm moves onto agent 3, it assigns B3. Under the assignment of bins Bi to i = 1, 2, 3, any feasible matching must grant of goods 4, 1 and 2 to agents 1, 2 and 3, respectively. This is shown with capital letters in the Sizes-SDI panel in Figure 1. When Sizes-SDI arrives at 3 agent 4, the most preferred bin she can be assigned is B4, which ultimately gives her good 3. Then, 5 agent 5 receives good 5 from B5. Lastly, we note that in this particular example, the bin assignments that were made gave rise to a unique way to assign all agents to goods from their assigned bins.

12 4.2.2 The Contents Serial Dictatorship with Indifference (Contents-SDI)

Agent Odering Rule of Contents-SDI: a Step 1) For each good, count the number of agents who place it in their ﬁrst bins. In other words, 1 compute each good g’s “demand” as {i : g ∈ Bi for some agent i} . a Step 2) Order the goods strictly from most to least demanded, breaking ties randomly. Call the resulting ordering good “popularity”. We distinguish between “demand” and “popularity” of goods since two goods may have the same demand but must have different popularities. a k k Step 3) For each pair of agents, i and j, ﬁnd the smallest integer k such that Bi 6= Bj . If such a k k k k k k k exists, let t be the least popular good in (Bi ∪ Bj ) − (Bi ∩ Bj ). If t ∈ Bi , order i before j. If t ∈ Bj , order j before i. If no such k exists, order i before j with probability 0.5.

For a fixed set of submitted reports Contents-SDI and Sizes-SDI can yield different agent orderings, as shown in Figure 1. To see how the agent ordering for the Contents-SDI panel is created, we follow the three-step procedure above. First, we calculate each good’s demand (see Figure 2). Then the popularities of the goods are chosen according to the demands. Since goods 3 and 4 have the same demands, there are two popularities that can be used (see Figure 2). Under Popularity 1, good 4 is the least popular good that appears in an agent’s first bin under the reports from Figure 1. Agent 1 is the only individual with good 4 in his first bin, and hence, is ordered first. Then, agent 5 is ordered second, since she is the only agent with good 3 in her first bin. (Note that even though agent 2 has more goods in her first bin, agent 5 is ordered before agent 2.) Then, agent 2 is ordered before agents 3 and 4 because agent 2’s first bin is a superset of the first bins of agents 3 and 4. Finally, agent 3 is ordered before agent 4 because, according to their second bins, agent 3’s least popular good is less popular than agent 4’s least popular good. Having ordered the agents, Contents-SDI proceeds to make bin assignments and then chooses a matching at random from the set of feasible allocations that satisfy the bin assignments (much like Sizes-SDI). The way Contents-SDI makes bin assignments is nearly identical to the manner of Sizes-SDI. For details, see Appendix A.3.

Good g5 g4 g3 g1 g2 Demand 0 1 1 2 4 Popularity 1 1 2 3 4 5 Popularity 2 1 3 2 4 5

FIGURE 2.—According to the reports from Figure 1, we compute the following demand for each good. For example, since there are 4 individuals who place good 2 in their ﬁrst bins, the demand of good 2 is 4. The goods are sorted by their Demand. Given that goods 3 and 4 have the same demands, there are two popularities that can be used (Popularity 1 and Popularity 2). Less popular goods are assigned lower popularities. Boxes and circles are drawn around Popularities that differ between the two possibilities.

13 4.2.3 The Random Serial Dictatorship with Indifference (Random-SDI)

Agent Ordering Rule of Random-SDI: Random (hence the name). The way that Random-SDI proceeds once an agent ordering is realized is procedurally identical and hence mathematically equivalent to that of Sizes-SDI (previously described Section 4.2.1 and Figure 1.)

4.3 The Preferences

An advantage of RSD is that it is “ex-post” Pareto optimal: once RSD makes a final assignment, any reallocation of goods that makes one agent strictly better off necessarily makes another strictly worse off.16 In certain environments, however, this property “loses its bite”. For instance, Ab- dulkadiroglu˘ et al. [2011] make the observation that, in the extreme case of common ordinal preferences, every single assignment is ex-post Pareto efficient. Furthermore, in this setting, since all agents will make identical reports, RSD will assign goods at random. Put differently, when ordinal preferences are the same across individuals, the only way to improve allocations beyond a random assignment is to implement a mechanism that provides agents with a channel through which they can express their cardinal preferences, i.e., the intensities with which individuals value goods. To cleanly study the extent that we can effectively construct such “cardinal-utility” channels, we isolate cardinal-utility from ordinal preferences by endowing agents with the same ordinal–but different cardinal–preferences. Incidentally, such an environment appears in several theoretical papers as well.17 Abdulka- diroglu˘ et al. [2011] set up a Bayesian framework where they show that each type of agent is weakly better off (compared to RSD) under any symmetric equilibrium of the game induced by the “Boston” algorithm. In Appendix A.1, we show that this theorem can be adapted to show analogous welfare improvements of Sizes-SDI and Contents-SDI over RSD. Furthermore, Abdulka- diroglu˘ et al. [2011] argue that this approximation to highly correlated preferences is justifiable in a school choice market where some schools are much more demanded than others. Data suggest that the MBA students at the Stanford GSB have highly correlated ordinal preferences as well. In its first academic year (2007-08) of using an algorithm to generate trip assignments, the Stanford GSB implemented RSD. According to the assignments made and the submitted preferences, 53% of students received their 1st choice trips and over 10% received their 8th choices or worse. Unsat- isfied with the resulting match, some MBA students devised Contents-SDI, which has been used ever since. In each of the five academic years spanning 2008-09 to 2012-13, the number of students receiving trips from their 1st bins was 90% or more. Furthermore, no students received trips in their 8th bins or worse in any of these years. At first glance, it appears that the Stanford GSB achieved far superior outcomes under Contents-SDI in comparison to RSD. However, because Contents-SDI is not a strategyproof mechanism, we cannot reliably evaluate its outcomes using the stated reports at hand; we have no guarantee that these reports are truthful. In order to have complete observ- ability of not just the submitted, but of the underlying true values that subjects have over goods, we turn to a lab investigation of Sizes-SDI, Contents-SDI and Random-SDI.

16This efﬁciency result assumes that individuals have strict preferences (see the example in Section 1). In the presence of indifference, Random-SDI can be implemented to guarantee ex-post Pareto optimality. 17See Abdulkadiroglu˘ et al. [2011] Troyan [2012] and Miralles [2008]

14 5 Experiment

5.1 Assignment Games

In our experiment, participants play six-person “Assignment Games”. At the outset of the game, a player is shown her Payoff Function, which indicates a value in Experimental Currency Units (ECU)18 for each good. Players’ Payoff Functions are drawn independently and uniformly from a set of ten possible Payoff Functions shown in Figure 3. Players simultaneously place goods into bins to provide weak ordinal rankings over goods. Depending on the treatment, subjects are matched to goods according to either Contents-SDI, Sizes-SDI or Random-SDI.

g1 g2 g3 g4 g5 g6 d g1 g2 g3 g4 g5 g6 Payoff Function4,5 99 97 95 93 13 3 Payoff Function2,3 99 97 17 7 5 3 Payoff Function3,5 99 97 95 15 13 3 Payoff Function1,5 99 19 17 15 13 3 Payoff Function3.4 99 97 95 15 5 3 Payoff Function1,4 99 19 17 15 5 3 Payoff Function2,5 99 97 17 15 13 3 Payoff Function1,3 99 19 17 7 5 3 Payoff Function2,4 99 97 17 15 5 3 Payoff Function1,2 99 19 9 7 5 3

FIGURE 3.—Each of the ten pairs of numbers that can be generated from the set X = {1, 2, 3, 4, 5} are used to create each Payoff Function (PF). Denote a pair as a, b where a < b. PFa,b(g1) = 99 for all a, b,. For all x ∈ X, PFa,b(gx+1) = PFa,b(gx) − Y, where Y = 80 or 10, if x = a or b, respectively. Otherwise, Y = 2.

5.2 Procedures

Our experiment was run in ztree19 in Spring 2013 at a large university in the United States. It consisted of 72 subjects, mostly undergraduates. Each student participated in exactly one of three treatments. The three treatments were Random, Sizes and Contents, where Random-SDI, Sizes- SDI and Contents-SDI, were the respective algorithms used to create matchings in the Assignment Games that subjects played. Each of the three treatments had 24 students. Within a treatment, the 24 subjects were further partitioned into groups of 6 students. All groups of 6 were ﬁxed through- out the entire experiment and strategically independent. Once subjects were seated, the experi- menter presented a PowerPoint presentation of the instructions,20 which generally took about 15 minutes. Afterwards, subjects played 20 Assignment Games, one per period. At the start of each period, Payoff Functions were redrawn and subjects had 60 seconds to place goods into bins.21 After each period, subjects were given feedback (for about 30 seconds) of all previous periods: they saw their own Payoff Functions, decisions and assignments.22 At the end of the 20 periods, each subject received the value of a good she was assigned in a randomly chosen period. Students received show up payments of $5, earned $11.47 on average from their decisions, and received $5 for completing the study. The experiment lasted about 75 minutes.

18Four ECU are equivalent to one US Dollar. 19Fischbacher [2007] 20The instructions for each treatment are shown in Appendix A.4. 21Time limits were generally obeyed, sometimes being broken by just a few seconds. 22While many real-world matching markets are one-shot, agents often have time to learn about algorithms before making their reports. This feedback was an attempt to simulate learning in the lab. The general process of simulating learning from the ﬁeld through repetition and feedback in the lab has been used in many experimental studies.

15 6 Results

6.1 Analysis of Welfare

We begin our earnings analysis on a “market” level. In other words, for a particular Assignment Game–which depicts a matching market of 6 subjects–we compute three forms of earnings:

1 earningsEmpirical: the sum of subjects’ expected earnings, computed using the algorithm in the given Assignment Game and the subjects’ submitted reports

2 earningsRSD: the sum of subjects’ expected earnings, computed using the Random Serial Dic- tatorship, assuming truthful reports

3 earningsSoc.Plan: the highest total earnings that a Social Planner could reach over all possible assignments (given the subjects’ drawn Payoff Functions in the Assignment Game)

Using 1 , 2 and 3 , we can compute an Assignment Game’s “Scaled Earnings”.

earningsEmpirical − earningsRSD Scaled Earnings = (1) earningsSoc.Plan − earningsRSD

Scaled Earnings are valuable for two reasons. First, when considering a single mechanism we tested experimentally, say Sizes-SDI, Scaled Earnings quickly allows us to see i) whether the mechanism in question outperforms RSD (it does if and only if Scaled Earnings are positive) and ii) how close the mechanism comes to achieving the Social Planner’s welfare (it comes closer as Scaled Earnings approach 123). Second, when making pairwise comparisons of the mechanisms we tested, Scaled Earnings control for the possibility that subjects in one treatment, on average, may have drawn better payoff functions than subjects in the other treatment. Put differently, if one mechanism yields higher earnings than another, we do not want it to simply be because the former happened to endow subjects with better payoff functions. In our experiment, subjects receive feedback after each Assignment Game. Hence, we cannot safely assume that the sequence of 20 Assignment Games played by a group are statistically independent observations. Nonetheless, because each group was completely strategically isolated from all the others (even within the same treatment), we can use each group’s average Scaled Earn- ings over the 20 periods (shown in Table 4) to build a set of statistically independent observations that we use to obtain our two results regarding aggregate welfare (Results 1 and 2).

Result 1. The Scaled Earnings in the Contents, Sizes and Random treatments are each signiﬁcantly greater in comparison to the RSD prediction. (Two-tailed Mann-Whitney tests at 5% signiﬁcance, or “TTMW5”.)

Result 1 states that in all three of our tested algorithms, subjects earn signiﬁcantly more money on average than they would if we were to simply run RSD (assuming that individuals state their

23By construction, Scaled Earnings are never greater than 1.

16 true preferences under RSD). Thus, indifference need not be incentivized in order to achieve welfare gains. Nonetheless, incentives can yield further gains over simply allowing it (Result 2).

Result 2. The Scaled Earnings in the Contents and Sizes treatments are not signiﬁcantly different from one another, yet both are signiﬁcantly greater than the scaled earnings in the Random treatment. (TTMW5.)

Treatments Group Contents Sizes Random Highest 0.62 0.60 0.33 0.49 0.57 0.32 0.43 0.54 0.29 Lowest 0.34 0.35 0.27 Mean 0.47 0.51 0.30

TABLE 4.—The table shows each group’s Scaled Earnings, averaged over the 20 periods in the experiment. In each column, groups are organized according to their average Scaled Earnings: highest at the top to lowest at the bottom. The ﬁnal row in the table shows the mean of these averages. Recalling that Scaled Earnings of 0 correspond to RSD earnings, the data indicate that all three mechanisms outperform RSD; see Result 1. The data also indicate that Sizes-SDI and Contents-SDI do markedly better than Random-SDI; see Result 2.

While Results 1 and 2 document average welfare improvements, institutions may hesitate to shift away from RSD if welfare gains are enjoyed rather unequally across individuals, especially if some individuals are actually made worse off. Accordingly, we turn to an individual earnings analysis and compute two earnings measures for each individual:

1 earningsEmpirical: the subjects’ experimental earnings, averaged over all 20 periods

2 earningsRSD: the subjects’ earnings under the Random Serial Dictatorship, assuming truthful reports, averaged over all 20 periods

Using 1 and 2 , we can compute an individual’s “Differenced Earnings”.

Differenced Earnings = earningsEmpirical − earningsRSD (2)

For each treatment, we plot individuals’ Differenced Earnings (averaged over the 20 periods in the experiment) in Figure 4. To best see how they are distributed, individuals are ordered in the plot, from lowest to highest Differenced Earnings, by treatment. We see that the Differenced Earn- ings vary much more in the Contents treatment in comparison to Sizes and Random. Furthermore, though it is slightly hard to tell in the plot, we can compute the percentage of individuals in each treatment with negative Differenced Earnings. These numbers are 4%, 8% and 29% for the Sizes, Random and Contents treatments, respectively. We use these percentages to obtain Result 3.

Result 3. Relative to RSD, a significantly larger proportion of individuals fare worse under Contents-SDI than Sizes-SDI; all other comparisons are insignificant. (Two-tailed Fisher’s Exact tests at 5% significance.)

17 Taken altogether, Figure 4 and Results 1 through 3 show that relaxing strategyproofness can yield overall welfare improvements, but not all approaches will be equally successful, even when the focus is narrowed to a speciﬁc class of algorithms (endogenously ordered serial dictatorships). We ﬁnd that the mechanism we design, Sizes-SDI, outperforms Contents-SDI and Random-SDI. In the discussion that follows, we consider subject behavior in each of the three treatments and relate it to the previously stated welfare results.

FIGURE 4.—The ﬁgure shows each individual’s Differenced Earnings, averaged over the 20 periods in the experiment. Within each treatment, subjects are ordered according to their Differenced Earn- ings. We see that the Differenced Earnings are fairly evenly distributed in the Random treatment; this is expected, as Random-SDI is strategyproof. Dif- ferenced Earnings in the Sizes treatment are less evenly distributed, but roughly stochastically dominate those from the Random treatment, indicating that moving from Random-SDI to Sizes-SDI would be an approximate Pareto improvement. Mov- ing to Contents-SDI, on the other hand, makes some subjects substantially better off (top right corner of graph), but at the expense of making some subjects substantially worse off (bottom left corner of graph).AAAA AAAAA AAAAAAAAA AAAAAAAAAAAAA AAAAA AAA A

6.2 Analysis of Behavior

In an Assignment Game, each individual has 4,683 different reports she can make. Certain reports are much more frequently observed than others. To present the behavioral ﬁndings, Sec- tions 6.2.3 through 6.2.2 present reports that are particularly prevalent in Random-SDI, Sizes-SDI and Contents-SDI, respectively. Section 6.2.4 considers all remaining reports.

6.2.1 Cardinal-Following Reports (most common in Sizes-SDI)

The motivation for BMD is a faith in individuals being willing and able to state their preferences in welfare-improving ways when presented with manipulable algorithms that are “user-friendly”. Accordingly, we built our mechanism, Sizes-SDI, with transparent incentives aimed to induce a simple rule of thumb of placing goods of similar value into the same bins. Using this rule of thumb, we can deﬁne two such reports for each of the payoff functions in our experiment (Deﬁnition 6.1). Figure 5 shows these two reports for each of the Payoff Functions that subjects may receive in our experiment.

i Deﬁnition 6.1. Let vj denote agent i’s utility for good j. Consider an agent i. For a and b such that i i i i i i va − va+1 = 80, vb − vb+1 = 10 and vw − vw+1 = 2 for w ∈ {1, 2, 3, 4, 5}\{a, b},

18 1 the Cardinal-Following 1 (CF1) report is {g1, ..., ga} > {ga+1, ..., gb} > {gb+1, ..., g6}, and

2 the Cardinal-Following 2 (CF2) report is {g1, ..., ga} > {ga+1, ..., g6}.

The reason we initially believed Sizes-SDI would be successful in improving welfare was that its incentives would induce CF1 and CF2 reports. This hypothesis was reasonably conﬁrmed; in the Sizes, Contents and Random treatments, the percentages of reports that are CF1 or CF2 are 52, 30 and 28, respectively. We use these ﬁgures to obtain Result 4.

Result 4. CF1 and CF2 reports are signiﬁcantly more common in the Sizes treatment compared to the Random and Contents treatments. (Two-tailed Fisher’s Exact tests at 1% signiﬁcance.)

In light of Random-SDI being strategyproof, it may not be surprising that cardinal-following behavior is less common in the Random treatment than the Sizes treatment. What’s more interesting is that cardinal-following behavior is more prevalent in Sizes-SDI than Contents-SDI, demonstrating that our mechanism design was more successful in generating the intended behavior than the more “ad-hoc” mechanism used by the Stanford GSB to assign its MBA students to foreign trips.

FIGURE 5.—These are the Payoff Functions from Figure 3, color-coded to illustrate the CF1 and CF2 reports. Under CF1, goods in black, gray and white cells are placed into bins 1, 2 and 3, respectively. Under CF2, goods in black and gray cells are placed into bin 1, while goods in white cells are placed into bin 2.

6.2.2 Ordinal-Violating Reports (most common in Contents-SDI)

The name given to Contents-SDI was inspired by the way in which the algorithm orders individuals based on their reports: it considers the contents of bins and orders individuals earlier who highly rank goods that others deem undesirable. It turns out that these incentives produce an overwhelming proportion of ordinal-violating reports (see Deﬁnition 6.2).

Deﬁnition 6.2. If a subject values a good gi more than a good gj but reports gj in a strictly more-preferred bin than gi, this report is ordinal-violating; otherwise, the report is ordinal-following.

In the Contents, Random and Sizes treatments, the percentages of reports that are ordinal- violating are 28, 6 and 3, respectively. We use these ﬁgures to obtain Result 5.

Result 5. Ordinal-violating reports are significantly more common in the Contents treatment compared to the Random and Sizes treatments. (Two-tailed Fisher’s Exact tests at 1% significance.) Furthermore, 81% of the ordinal-violating reports in the Contents treatment “involve bin 1” (see Definition 6.3).

19 Deﬁnition 6.3. An ordinal-violating report involves bin 1 if there exists gi and gj such that gi is strictly preferred to gj but gj is placed in bin 1 while gi is not. (See Example 1.)

Example 1. Reports 1 and 2 are both ordinal-violating, but only Report 1 involves bin 1.

Report 1 = {good 1, good 4} {good 2} ... Report 2 = {good 1, good 2} {good 4} {good 3} ...

Result 6. Of the ordinal-violations that involve bin 1 (in the Contents treatment), roughly 63% of these reports result in the receipt of the least preferred good in bin 1.

Given Result 6, it may come as no surprise that–in the Contents treatment–the average ordinal- violating report yields average earnings of 32.14 ECU while the average ordinal-following report yields earnings of 51.52 ECU.24 Given the previous ﬁnding that a relatively large proportion of individuals fare worse under Contents-SDI than RSD (Result 3), we may suspect that these individuals are the ones submitting the most ordinal-violating reports. In Figure 6, we plot each individual (represented by a marker) in the Contents treatment according to two statistics; the hor- izontal axis shows the number of ordinal violating reports submitted and the vertical axis shows Differenced Earnings, averaged over all 20 periods. We see a clear trend that illustrates individuals making more ordinal-violating reports are generally earning less money.

FIGURE 6.—The ﬁgure shows each individual’s Differenced Earnings, averaged over the 20 periods in the experiment. Subjects with negative Differenced Earnings are marked as solid black square markers; 6 out of 7 of these individuals are committing 11 or more ordinal- violations while all subjects with positive Scaled Earnings commit 11 or fewer ordinal- violations.AAAA AAAAA AAAAAAAAA AAAAAAAAAAAAA AAAAA AAA A

Given that ordinal-violations are much more prevalent in the Contents treatment than the Sizes treatment, it’s puzzling how average Scaled Earnings in these two treatments are not signiﬁcantly different from one another, since ordinal-violations are so harmful. Solving this puzzle involves considering the “market” implications of ordinal-violations. In other words, while our previous results show that ordinal-violations hurt the individuals committing them, they may in fact be beneﬁtting others in the market. To investigate this, we partition the set of Assignment Games (in the Contents treatment) into two subsets. Assignment Games were no subject submitted an ordinal-violating report were placed into one subset. The remaining Assignment Games were placed into the other subset. Then, we compare the average Scaled Earnings from each subset. For the “no ordinal-violating reports” subset, the average Scaled Earnings are 0.47, compared to 0.50

24The samples are not independent, so we do not compare them using a Mann-Whitney test. Nonetheless, the difference is meaningful, translating to $4.85.

20 for the other subset. Thus, it seems that while ordinal-violating reports result in worse outcomes for those committing them, the effect on others is not very detrimental. This may not be surprising given that more ordinal-violations lead to lower payoffs which are obtained from receiving worse goods. Given that individuals have common ordinal preferences, one individual receiving a worse good makes another agent able to receive a better one.

6.2.3 Truthful Reports (most common in Random-SDI)

Of the three algorithms we test experimentally, only Random-SDI is strategyproof. Thus, in theory, much more truth-telling would occur in Random-SDI than our other tested mechanisms. We ﬁnd this to be the case in practice: in the Random, Contents and Sizes treatments, truthful reports were made 21%, 2% and less than 1% of the time, respectively. We use these ﬁgures to obtain Result 7.

Result 7. Truthful reports are signiﬁcantly more common in the Random treatment compared to the Sizes and Contents treatments. (Two-tailed Fisher’s Exact tests at 1% signiﬁcance.)

Given that subjects can maximize their expected earnings by stating their preferences truthfully, but that over three quarters of reports are non-truthful in the Random treatment, altruistic attitudes may have inﬂuenced behavior. The extent to which this supposed altruism affected decisions was, however, limited: less than 14% of reports in Random-SDI involved placing a good in the ﬁrst bin that was valued less than 93 ECU.

6.2.4 Organizing All Reports

Sections 6.2.3 through 6.2.2 presented forms of behavior that were particularly characteristic of one of the mechanisms we tested. There remain signiﬁcant proportions of behavior, however, that do not fall into these classes of reports. Thus, we now move to a discussion of all reports expressed in the three experimental treatments. To organize the data, we make use of the mechanics of Sizes- 25 SDI and Contents-SDI. More speciﬁcally, we use a strict weak ordering,

25It satisﬁes irreﬂexivity, asymmetry, transitivity and transitivity of incomparability.

21 had instead made the No-Binning report.26 Table 5 shows that behavior is highly variant across treatments, suggesting that individuals respond strongly to the incentives provided in these different mechanisms. Thus, market designers have the potential to strongly inﬂuence the reports that agents make by altering the underlying mechanics of the algorithms they construct; the challenge is to make the welfare-improving strategies as focal as possible.

Deﬁnition 6.4. Let the truthful report x = {g1} > {g2} > {g3} > {g4} > {g5} > {g6} be also referred to as the No-Binning (NB) report.

Deﬁnition 6.5. Let the report x = {g1, g2, g3, g4, g5, g6} be referred to as the Full-Binning (FB) report.

CF2 FB CF1

SC SC SC < < < y y y

SC SC SC < < < = Ordinal-Violating= No-Binning (NB) = CF1 = CF2 = Full-Binning (FB) y y NB y CF1 y CF2 y total Contents-SDI 133 9 70 120 27 24 96 1 480 Sizes-SDI 12 2 33 139 31 109 152 2 480 Random-SDI 29 102 186 126 6 10 21 0 480 y = Ordinal-Following

TABLE 5.—Each report (“y”) is categorized across all three treatments. For each treatment, the three most common forms of behavior are shown in bold. Cells indicating cardinal-following strategies are in gray. In general, we see much more “binning” (i.e., placing several goods into the same bin) in the Sizes treatment than the Random treatment; this is expected, given the incentives to express indifference in the Sizes treatment. In the Contents treatment, we see ordinal-violations are the most common type of report made, which, again, is not startling given the rewards for highly ranking trips that others express as undesirable.

6.3 Summary of Results

In terms of aggregate welfare, all three of the algorithms we test show improvements over RSD. This is true even for Random-SDI, where theoretical predictions yield no welfare gains over RSD for agents who purely seek to maximize their own payoffs. Thus, simply allowing indifference can yield welfare improvements, even under true preferences that are strict. Nonetheless, the aggregate earnings are signiﬁcantly higher under Sizes-SDI and Contents-SDI than Random-SDI, showing that even more welfare can be obtained by actually incentivizing agents to express indifference. On the individual level, Contents-SDI performs worse than our other tested algorithms, demonstrating that ordering individuals according to the sizes of their bins may be better than considering their contents.

26Her movement is “weak” because her actual report, as well as No-Binning and CF1 may all involve less binning than her counterparts according to

22 In terms of behavior, Sizes-SDI does a reasonable job in evoking the behavior we intended to induce when designing the mechanism: more than half of agents’ reports in the Sizes-SDI treatment are cardinal-following. Thus, these reports seem to be focal when subjects are faced with Sizes-SDI. While cardinal-following behavior is also present under Contents-SDI, an even larger number of reports are ordinal-violating. Thus, Contents-SDI does not focalize the same type of reports than Sizes-SDI. We ﬁnd that ordinal-violating reports tend to be unproﬁtable and that some subjects make much more of these reports than others. This drives the inequality in earnings that we observe in the Contents treatment.

7 Conclusion

The broad contribution of this paper is to raise awareness of an area in the market design literature that has thus far been given limited attention: the design of mechanisms that are meant for individuals with intermediate levels of strategic sophistication (or “Behavioral Market Design”). While a large body of research has designed manipulable algorithms with equilibria having strong efficiency properties, reaching these equilibria may involve strategies that are too complex for subjects to implement in practice. Many previous papers have eliminated this concern entirely by simply requiring that the algorithms they design be strategyproof, i.e., that they grant each individual with a dominant strategy of stating her preferences truthfully. In light of the previous theoretical results showing that strategyproofness can constrain efficiency, we argue that imposing strategyproofness may be “going to far”. In other words, even if subjects are unable to reach equilibria, perhaps their strategic sophistication is high enough to identify and implement “easy” strategies if they are made focal through careful mechanism design. Indeed, a plethora of Behav- ioral Game Theory experiments show that subjects who depart from equilibria may nonetheless behave systematically, strategically, and follow heuristics. The more narrow contribution of this paper is to take a concrete step in the direction of Be- havioral Market Design and attempt to construct an algorithm with incentives that make welfare- improving strategies focal. Specifically, we create a serial dictatorship with indifference that orders individuals earlier who form bins of larger size. The mechanics of our algorithm create a tradeoff to placing multiple goods in a bin: while placing more goods increases the overall probability of receiving a good from the bin, some goods in the bin may be received with lower chances. This tradeoff, we hypothesizes, induces a sensible rule of thumb of placing goods into the same bin if and only if these goods are similar in value. We experimentally test our mechanism in the lab and find that more often than not, a report obeys this rule of thumb strategy; in turn, subjects earn more under our mechanism in comparison to a common strategyproof alternative. Hence, we show that manipulable mechanisms work, but should be designed carefully, as we find that another closely related manipulable focalizes a different type of preference manipulation that harms some individuals. To conclude, we think there is substantial room for advancements in “Behav- ioral Market Design”, a research area that we hope to see progress through a joint effort on the part of institutions, market designers, and behavioral and experimental economists alike.

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26 A Appendix

A.1 Equilibria with Weak Welfare Improvements

Suppose it is common knowledge that each agent draws von-Neumann Morgenstern utility values W V V M from the ﬁnite set ⊂ using the same distribution, f ,where = {(v1,..., vM) ∈ [0, 1] |v1 > v2... > vM, v1 = 1 and vM = 0} and vi denotes the utility from good gi. We can follow the reason- ing from Abdulkadiroglu˘ et al. [2011] and Troyan [2012] and prove Theorem 1 below.

Theorem 1. Each type of agent is weakly better off under any symmetric Bayesian Nash Equilibrium of the game induced by Contents-SDI or Sizes-SDI compared to RSDI.

Proof. The proof follows a similar argument to Abdulkadiroglu˘ et al. [2011] (also used in Troyan

[2012]). Note that, from the ex-ante perspective, each agent is assigned good gi or better with c1+···+ci ci probability N under RSDI. Thus, each agent receives ti with probability N . Now, consider any symmetric equilibrium of the Contents-SDI or Sizes-SDI mechanism in which every agent ∗ ∗ plays the same strategy σ . Let σ ∈ {σ (v)}v∈V be any mixed strategy used in an equilibrium of

Contents-SDI or Sizes-SDI. Let Pgm (σ) be the probability that a agent is assigned to good gm given that all other agents follow strategy σ∗. Note that in equilibrium, the following equation must hold for each good gm:‘

∗ N ∑ Pgm (σ (v)) f (v) = cm v∈V ∗ Now, ﬁx any type of agent v˜ ∈ V. Suppose this agent plays the strategy σ˜ ≡ ∑v∈V σ (v) f (v). In other words, σ˜ randomizes over the equilibrium strategies σ∗(v) of each type v ∈ V, where the weight placed on strategy σ∗(v) is the relative weight of type v in the population, f (v). By playing this strategy, the probability of this agent being assigned to good gm is

∗ cm Pgm (σ˜ ) = ∑ Ptm (σ (v)) f (v) = , v∈V N which equals the chance of receiving gm under RSDI. Since σ˜ need not be an equilibrium strategy, we must have, for every type of agent v˜,

M M M ∗ cm ∑ Pgm (σ (v˜))v˜m ≥ ∑ Pgm (σ˜ )v˜m = ∑ v˜m m=1 m=1 m=1 N where the left hand side is the expected utility in the Contents-SDI or Sizes-SDI equilibrium and the right hand side is the expected utility under RSDI.

While the proof of Theorem 1 is non-constructive, as calculating equilibrium strategies of Sizes-SDI and Contents-SDI turns out to be rather difﬁcult, we are able to show how “cardinal- following” strategies can constitute an equilibrium in a Bayesian game induced by Sizes-SDI (see Appendix A.2).

27 A.2 Equilibria with Strict Welfare Improvements

Consider a Bayesian game with three individuals and three goods g1, g2, g3 (each with unit capacity). It is common knowledge that ach individual has an i.i.d. chance of p of being a “high” type and 1 − p of being a “low” type, where high types and low types have utilities of u1 = 1, and high ∆ low u3 = 0, but u2 = and u2 = δ, where ui is the utility received through obtaining good gi and 0 < δ < .5 < ∆ < 1. Agents only observe their own types.

1−2δ Proposition 1. Given the previously described parameters of the market, if p ≤ 2−δ , then each player playing the pure strategy {g1, g2} > {g3} if a high type and {g1} > {g2, g3} if a low type constitute a (cardinal-following) Bayesian Nash Equilibria in the game induced by Sizes-SDI.

Proof. We begin by showing that under any realized reports from other agents in the proposed equilibrium, and irrespective of agent i’s type, agent i always has a best response that is either the CF-high report ({g1, g2} > {g3}) or the CF-low report ({g1} > {g2, g3}). Note that there are 4 possible reports that the other agents could make: 2 CF-low, 2 CF-high, or 1 CF-low and 1 CF-high. Below, we show the best response reports for each of these cases.

i’s best response reports by others low type high type 2 CF-low CF-low CF-high 2 CF-high CF-high CF-high 1 CF-low, 1 CF-high CF-high CF-high

Notice from the above table that if i is a high-type, then in all realized states of the world, i is best-responding in the proposed equilibrium. However, if i is a low-type, we need to make sure that i’s best response, in expectation, is to play CF-low, and not deviate to CF-high.

probability i is low (given i is low) expected utility from playing 2 1+δ 2 low (1 − p) CF-low = 3 > δ = CF-high 2 1+δ 2 high p CF-low = 0 < 3 = CF-high 1 1+δ 1 high, 1 low 2p(1 − p) CF-low = 2 < 2 = CF-high

2δ − 1 1 + δ δ EU(CF ) − EU(CF ) = (1 − p)2 + p2 + 2p(1 − p) high low 3 3 2

h1 − 2δ δ i 1 − 2δ 2 + δ 1 − 2δ p(2 + δ) − (1 − 2δ) = 2p + − = 2p − = 3 2 3 6 3 3

1 − 2δ Thus, EU(l) ≥ EU(h) ⇐⇒ p(2 + δ) ≤ (1 − 2δ) ⇐⇒ p ≤ 2 − δ

28 The next question in which we may be interested is whether or not high type and low type agents earn greater expected utility under Sizes-SDI’s previously described cardinal-following equilibrium than under the truth-telling equilibrium of RSD. In the next theorem, we show that agents weakly prefer Sizes-SDI to RSD for precisely the values of p for which Sizes-SDI’s cardinal- following equilibrium holds.

1−2δ Proposition 2. If p ≤ 2−δ , then low and high type agents receive weakly higher expected utility under Sizes-SDI’s cardinal-following equilibrium compared to RSD’s truth-telling equilibrium.

Proof. We begin by noting that for states of the world where all agents are of the same type, both equilibria yield identical expected utilities for all agents. (In the case where all agents are low 1+δ types, agents earn expected utilities of 3 and in the case where all agents are high types, agents 1+∆ earn expected utilities of 3 .) Case 1: i is a high type. Subcase 1: other agents are both low types. Sizes-SDI’s equilibrium yield’s expected utility for i ∆ 1+∆ of , since i receives B with certainty. Compared to receiving 3 under RSD, Sizes-SDI is strictly preferred since

1 + ∆ ∆ > ⇐⇒ 3∆ > 1 + ∆ ⇐⇒ 2∆ > 1 ⇐⇒ ∆ > .5 3 Subcase 2: other agents are 1 low type and 1 high type. Sizes-SDI’s equilibrium yield’s expected 1+∆ utility for i of 2 , since i receives A and B each with probability 1/2. Sizes-SDI is thus strictly 1+∆ preferred to receiving 3 under RSD. Hence, for high types, for no realized state of the world is the expected utility greater under RSD than in Sizes-SDI and in some states, Sizes-SDI’s expected utility is even strictly greater. Case 2: i is a low type. Subcase 1: other agents are both high types. Sizes-SDI’s equilibrium yields expected earnings of 0, 1+δ compared to RSD’s which yields expected earnings of 3 . Thus, RSD is preferred to Sizes-SDI. Subcase 2: other agents are 1 low type and 1 high type. Sizes-SDI’s equilibrium yields expected 1 1 1+δ earnings of 2 , since i receives A and C each with probability 2 . Compared to receiving 3 under RSD, Sizes-SDI is strictly preferred since

1 1 + δ 3 > ⇐⇒ > 1 + δ ⇐⇒ .5 > δ 2 3 2 ! ! 1 + δ 1 1 + δ EU(Sizes ) ≥ EU(RSD) ⇐⇒ p2 0 − + 2p(1 − p) − ≥ 0 SDI 3 2 3

! ! ! ! 1 + δ 2 + 2δ δ − 2 1 − 2δ ⇐⇒ p2 − 1 + p 1 − ≥ 0 ⇐⇒ p2 + p ≥ 0 3 3 3 3

! ! 1 − 2δ 2 − δ 1 − 2δ ⇐⇒ p ≥ p2 ⇐⇒ 1 − 2δ ≥ p(2 − δ) ⇐⇒ p ≤ 3 3 2 − δ

29 A.3 Further Details on the Mechanics of Contents-SDI

Here, we show how Contents-SDI may sometimes yield a different assignment than Sizes-SDI under the same reports and resulting agent orderings. To see how the process differs, we provide an example of a market of 5 agents and 5 goods, each with unit capacity (see Figure 7). Sizes-SDI and Contents-SDI produce the same student ordering under these reports.

Sizes-SDI/RSDI Contents-SDI 1 2 3 4 5 1 2 3 4 5 agent Bi Bi Bi Bi Bi D Bi Bi Bi Bi Bi i = 1 G1 g2, g3 g4 g5 G1 g2, g3 g4 g5

i = 2 g1 G2 g3, g4, g5 g1 g2 g3, g4, G5

i = 3 g2 g1 g3, G4 g5 G2 g1 g3, g4 g5

i = 4 g2 g1 G3 g4, g5 g2 g1 G3 g4, g5

i = 5 g2 g1 g3 g4 G5 g2 g1 g3 G4 g5

FIGURE 7.—The left and right panels–corresponding to Sizes-SDI/RSDI and Contents-SDI, respectively– show the (same) agent ordering and (different) assignments made from a single set of reports. Each “agent” column signiﬁes the agent ordering (where higher agents are ordered earlier). Each cell indicates the com- 2 position of an agent’s bin. For instance, in each panel, the cell intersected by the i = 3 row and Bi column indicates that agent 3 places good 1 in her second bin. The good that an individual receives is indicated by a capital letter. For example, under both mechanisms, individuals 1 and 4 receive goods 1 and 3, respectively. Boxes and circles are drawn around individuals and goods that are matched differently across the two mechanisms.

Like RSDI and Sizes-SDI, the first agent in the ordering of Contents-SDI is always assigned her first bin, thus agent 1 receives good 1. However, unlike RSDI and Sizes-SDI, the next bin assignment 1 made by Contents-SDI may not necessarily be to agent 2. Contents-SDI assigns B2 to agent 2 if it can; 1 if it cannot, it tentatively “skips” agent 2 and tries to assign agent 3 his first bin, B3. It can, and thus agent 3 receives g2 in Contents-SDI. After this bin assignment, Contents-SDI continues in this manner according to the agent ordering and only attempts to assign agents their first bins. As shown in Figure 7, no more agents can be assigned their first bins.27 At this point, Contents-SDI starts at the beginning of the agent ordering and makes bin assignments to the remaining agents: 2, 4 and 5. It thus attempts to assign agent 2 her second bin, but this is not possible, since agent

3 will receive g2. Now, however, instead of skipping agent 2 (as it did before), the algorithm will 3 move to B2 and assign it to agent 2. Thus the algorithm only skips agents when it fails to assign them their ﬁrst bins. The algorithm proceeds by allocating goods g3 and g4 to agents 4 and 5, respectively, leaving agent 2 with good g5. Lastly, we note that the bin assignments made under Contents-SDI in the particular example above give rise to a unique assignment of goods to agents. As in RSDI and Sizes-SDI, if there are multiple good assignments that satisfy all bin assignments, Contents-SDI chooses one of the good assignments at random.

27It is possible for agent 1 to be the only agent guaranteed her ﬁrst bin. For example, this would occur if all agents had ﬁrst bins containing just g1.

30 A.4 Instructions

Instructions were presented using a PowerPoint presentation. We first show the slides shown that were common to all treatments, then we show those that are treatment-specific, i.e., that define the particular mechanism for the given treatment. After going through the slide show as a group, subjects received printouts of the instructions to use for reference as they made their decisions. Subjects also received pens and scratch paper but no calculators.

Instructions common to all treatments

31 32 33 Additional slides for the Comments treatment

34 35 36 Additional slides for the Sizes treatment

37 38 Additional slides for the Random treatment