Playing on a Level Field: Sincere and Sophisticated Players in the Boston Mechanism with a Coarse Priority Structure
PPlaying on a Level Field:Sincere and Sophisticated Players in the BostonMechanism with a Coarse Priority Structure
Moshe Babaioff Yannai A. Gonczarowski Assaf Romm ∗ June 9, 2020
Abstract
Who gains and who loses from a manipulable school-choice mechanism? Study-ing the outcomes of sincere and sophisticated students under the manipulableBoston Mechanism as compared with the strategy-proof Deferred Acceptance, weprovide robust “anything-goes” theorems for large random markets with coarse pri-ority structures. I.e., there are many sincere and sophisticated students who preferthe Boston Mechanism to Deferred Acceptance, and vice versa. Some populationsmay even benefit from being sincere (if also perceived as such). Our findings rec-oncile qualitative differences between previous theory and known empirical results.We conclude by studying market forces that can influence the choice between thesemechanisms.
School districts all over the world are increasingly realizing the benefits of letting parentschoose the educational environment that best fits their children. The move from a zoningpolicy to a choice-oriented process, together with the scarcity of seats in some of the mosthighly demanded schools, calls for a regulated procedure or an algorithm for assigningseats. The Boston mechanism (henceforth BM), also known as Immediate Acceptance,is one of the most popular mechanisms for seat allocation. The appeal of BM lies in itssimplicity in terms of both intuition and implementation. The mechanism, which also ∗ First draft: February 2018. Babaioff: Microsoft Research, email : [email protected]: Microsoft Research, email : [email protected]; the research was conducted whileGonczarowski was affiliated also with the Hebrew University of Jerusalem. Romm: Stanford Univer-sity and the Hebrew University of Jerusalem, email : [email protected]; the research was conductedwhile Romm was co-affiliated with Microsoft Research. This paper greatly benefited from discus-sions with Scott Duke Kominers, Tal Lancewicki, D´eborah Marciano, Alvin Roth, and Ran Shorrer.Yannai Gonczarowski was supported by the Adams Fellowship Program of the Israel Academy of Sciencesand Humanities; his work was supported by Israel Science Foundation (ISF) grant 1435/14 administeredby the Israeli Academy of Sciences, by United States-Israel Binational Science Foundation (BSF) grant2014389, and by the European Research Council (ERC) under the auspices of the European Union’sHorizon 2020 research and innovation program (grant 740282). The work of Assaf Romm is supportedby the Maurice Falk Institute, by Israel Science Foundation (ISF) grant 1780/16, by United States-IsraelBinational Science Foundation (BSF) grant 2016015, and by Koret Young Israeli Scholars Program. a r X i v : . [ c s . G T ] J un akes the parents’ preferences into account, first maximizes the number of students whoget their first choice, then maximizes the number of students who get their second choice,and so on. When performing this step-by-step maximization, the mechanism selectswho gets admitted to overdemanded schools according to some (possibly school-specific)priority ordering. This process is so simple that in some small municipalities it is carriedout manually using spreadsheet software.Despite its seemingly straightforward description, the drawback of BM is that it issusceptible to strategic manipulation. Students (or parents) who carefully consider theworkings of the mechanism can submit a rank-order list (ROL) that does not representtheir true preferences, but may help them to be admitted to one of their (true) topschools. This raises two closely related concerns: first, do students benefit or are theyharmed from using a manipulable mechanism and, second, do students who are moresophisticated and more informed get an unfair advantage in public-school admissions?A good way to measure gain and loss in this context is to compare the outcome of BMwith that of the student-proposing Deferred Acceptance algorithm (Gale and Shapley,1962; henceforth DA), which is strategy-proof and therefore does not give an advantageto sophisticated students (Dubins and Freedman, 1981; Roth, 1982). We say that astudent receives a positive absolute gain (from BM) if her expected utility under (anequilibrium of) BM is higher than her utility under (the dominant-strategy equilibriumof) DA. We say that a student receives a relative gain with respect to another student(or to a counterfactual version of the same student) if the absolute gain of the former ishigher. Conveniently, a comparison of the relative gains between two types of the samestudent who differ only in their level of sophistication reduces to a comparison of theirexpected utilities under BM, since their outcome under DA is the same.The first to approach these questions were Pathak and S¨onmez (2008). They showpositive absolute gain for sophisticated students, by demonstrating that a sophisticatedstudent weakly prefers her outcome under the Pareto-dominant equilibrium of BM toher outcome under DA. They then prove positive relative gains for sophisticated typescompared with sincere types of the same player. That is, holding everything else fixedand focusing on the Pareto-dominant equilibria of BM, a player is weakly better off beingsophisticated rather than sincere.It should be noted that both of these results require that schools have strict priorityorderings over students—an assumption that is often unrealistic. In most cases, schoolshave a coarse priority structure and ties are resolved using a random tie-breaking rule.For example, children who have siblings attending a specific school may be given higherpriority, but among children with such siblings there is no strict order, nor is there oneamong children who do not have siblings at that school. Allowing for a coarse prioritystructure, Abdulkadiro˘glu et al. (2011) first prove strictly positive absolute gains whenall players are sophisticated, a result that is driven by the potential of BM to betterexpress cardinal utility levels through preference reports. Abdulkadiro˘glu et al. thenshow that in the presence of both sophisticated and sincere students, sincere students maypotentially experience positive absolute gains as well, as they benefit from their strategicpeers’ demand shading under BM, and gain higher probability of being admitted to The comparison of BM to DA is appealing also because BM was replaced by DA in several schoolchoice systems that were redesigned by economists, such as in Boston (Abdulkadiro˘glu et al., 2005) andNew York City (Abdulkadiro˘glu et al., 2009). For sincere students the comparison is ambiguous. Miralles (2009) states a similar result in a model with no sincere students. That being said, in theirmodel it is always better to be sophisticated (by assumption, since sophistication levelsare random and unobserved), and thus their model always predicts nonnegative relativegains for sophisticated types compared with sincere types. Their results crucially rely onall players having common ordinal preferences, and on specific symmetry assumptions: intheir model, cardinal utility levels are drawn i.i.d. from a random distribution, players aresincere or sophisticated with equal probability, and the results apply only to symmetricequilibria. As noted by Troyan (2012), the welfare comparisons of Abdulkadiro˘glu et al.are also sensitive to the assumption that there is only a single priority class, an assumptionthat mandates symmetric tie-breaking.This paper complements, extends, and hopefully clarifies existing results on the ab-solute and relative gains of sincere and sophisticated students. Regarding relative gains,we observe that under weak priorities, being sincere can sometimes be an advantagerather than a liability. In fact, it is a likely situation for many students even in largerandom markets. The intuition is that a sincere student commits to list her top schoolsfirst, even if they are overdemanded, and in doing so she crowds out the competitionmore than a sophisticated student would. That is, other students who are sophisticatedinevitably take the sincere student’s commitment into account and in doing so they ra-tionally move toward ranking other schools higher. The question of whether a student isbetter off being sincere or sophisticated is thus similar to the decision faced by a Stack-elberg leader in a meta-game in which she can either commit to one specific strategy (besincere) or not commit to any strategy and play according to a Nash equilibrium profile(be sophisticated).Our first main result, Theorem 3.1, demonstrates that the first-mover advantage ofsincere students is prominent even in large random markets. Our model employs a ran-dom market-generating process that produces markets with heterogeneous preferencesand varying demands by mixtures of sophisticated and sincere students. We show thatthere is in expectation a constant fraction of students who strictly prefer to be sincere. Nevertheless, the sincerity advantage under BM is sensitive both to market structure andto the extent that the players’ sophistication is common knowledge. Roughly put, theoverall effect of being sincere as compared with being sophisticated is comprised bothof the negative effect of not responding to excess demand (identified by Pathak andS¨onmez, 2008), which manifests in both strict-priority and weak-priority environments,and of the positive effect of crowding out the competition, which is completely absent instrict-priority environments. For example, if two or more of each student’s top schoolsare likely to be overdemanded, sincerity will allow a student to crowd out the competitionin her top school. However, a sincere student who does not get her first choice pays animplicit cost of not having ranked her second choice first, which could potentially haverepresented a better trade-off between utility and admission probability. This nonrespon-siveness to the excess demand of the top-choice school may or may not overshadow thebenefit of the crowding-out effect. Apesteguia and Ballester (2012) further demonstrate that positive absolute gains are possible forsincere students even prior to knowing their utility levels from being assigned to different schools. We study markets that are “large” in the sense that there are at the same time many schools andmany students (as in Immorlica and Mahdian, 2005; Kojima and Pathak, 2009). A different model usedin the literature is one with many students (a continuum) but a fixed number of schools (Azevedo andLeshno, 2016). This paper does not attempt to study the latter model. Perhaps less surprisingly there is also a constant fraction of students who strictly prefer to besophisticated.
The mechanism-design approach to school choice begins with Abdulkadiro˘glu and S¨onmez(2003). Since then much of the academic literature has focused on the application ofDA to various environments. BM itself was brought to the attention of economists byAbdulkadiro˘glu et al. (2005), who redesigned Boston Public Schools’ existing mechanismand replaced it with DA.Abdulkadiro˘glu et al. (2005) noted that BM was prone to strategic manipulation,and pointed to anecdotal evidence suggesting that indeed some parents in Boston actedstrategically. Following that, strategic behavior was demonstrated in both experimentallabs (e.g., Chen and S¨onmez, 2006, and many follow-up designs) and real-world envi-ronments (see, e.g., Calsamiglia and G¨uell, 2018). Ergin and S¨onmez (2006) show thatwhen priorities are strict and all students are sophisticated, moving from BM to DA isweakly beneficial for all students. As mentioned, Pathak and S¨onmez (2008) generalizethis statement, claiming that the strategic manipulability of BM also has fairness im-plications, as it gives an advantage to sophisticated students. An experimental findingalong the same lines was recently presented by Basteck and Mantovani (2018). Similar (ifinconclusive) theoretical results for an environment with boundedly sophisticated (level- k ) players are described by Zhang (2016). Dur et al. (2018a) study strategic behavior ofstudents and its implications in the field. Akbarpour and van Dijk (2018) note that act-ing sincerely is not necessarily related to being strategically unsophisticated or sufferingfrom lack of information; it can also be the result of having better outside options.Strategic sophistication (or lack thereof) and its effect on designing matching marketshas recently been at the focus of a number of works that study preference misrepresen-tation under the strategy-proof DA. Experimental evidence for preference misrepresenta-tion have existed for more than a decade, starting with the pioneering work of (Chen andS¨onmez, 2006). More recently, Rees-Jones (2018) and Rees-Jones and Skowronek (2018)studied this phenomenon in the American market for new medical residents, and a fewother authors pointed at field evidence for preference misrepresentation in college admis-sion markets (Artemov et al., 2017; Hassidim et al., 2020; Shorrer and S´ov´ag´o, 2018).Some explanations for this behavior are suggested by Hassidim et al. (2017), and moreformal treatments of some of them are provided by Ashlagi and Gonczarowski (2018) andDreyfuss et al. (2019).Finally, this paper deals with the effects of weak priorities on the workings of BM. Itthus contributes to a recent line of works that deal with weak priorities and tie-breaking See also Kojima (2008) for similar results under substitutable priority structures.
Schools, Students, and Preferences
We adopt much of the notation used by Pathakand S¨onmez (2008). There is a finite set of schools , S = { s , . . . , s m } and a finite set of students , I = { i , . . . , i n } . Each school s j has a capacity q s j . The (vNM) utility of a stu-dent i ∈ I from being assigned to school s ∈ S is u i ( s ), the utility from being unmatchedis (normalized to) zero, and we assume that students are risk-neutral. Throughout thepaper, we assume that the priority structure of the schools is coarse to the extent thatall students belong in the same priority class in all schools. While this is a simplifyingassumption, it is not far from the properties of many real-life matching markets, and theresulting phenomena efficiently convey our main messages. The Boston Mechanism
Each student must report an ROL to BM, and then themechanism is run as follows:0. For each school s , a strict ordering of students τ s is drawn uniformly at randomfrom the set of all permutations over I . This ordering remains unknown to thestudents.1. Each student applies to the school that she ranked as her first choice. A schoolwhose capacity is at least the number of students who applied permanently admitsall of them. A school s whose capacity is less than the number of students whoapplied permanently fills its capacity with a subset of these applicants, who arechosen according to τ s .2. Each student who was not admitted in the first round applies to her second choice.A school whose remaining capacity (taking into account the slots taken by all of thestudents admitted in the first round) is at least the number of students who appliedin this round permanently admits all of them. A school s whose remaining capacityis less than the number of students who applied in this round permanently fills itsremaining capacity with a subset of these applicants, who are chosen according to τ s . ... k . Each student who was not admitted in the previous rounds applies to her k thchoice. A school whose remaining capacity (taking into account the slots takenby all of the students admitted in all previous rounds) is at least the number ofstudents who applied in this round permanently admits all of them. A school s whose remaining capacity is less than the number of students who applied in thisround permanently fills its remaining capacity with a subset of these applicants,who are chosen according to τ s . This is the multiple tie-breaking rule (MTB). However, all our results remain qualitatively the samefor other random tie-breaking rules, including the widely used single tie-breaking rule (STB) under whichall schools use the same random ordering of students.
The Deferred-Acceptance Mechanism
Each student must report an ROL to DA,and then the mechanism is run as follows:0. For each school s , a strict ordering of students τ s is drawn uniformly at randomfrom the set of all permutations over I . This ordering remains unknown to thestudents.1. Each student applies to the school that she ranked as her first choice. A schoolwhose capacity is at least the number of students who applied tentatively admitsall of them. A school s whose capacity is less than the number of students whoapplied tentatively admits a subset of these applicants who fill its capacity and whoare chosen according to τ s , and permanently rejects all other applicants.... k . Each student applies to her favorite school among those that have not rejected heryet. (Thus a student who was tentatively admitted to a school in the previousround reapplies to the same school in this round.) A school whose capacity is atleast the number of students who apply in this round tentatively admits all of them.A school s whose capacity is less than the number of students who apply in thisround tentatively admits a subset of these applicants who fill its capacity and whoare chosen according to τ s , and permanently rejects all other applicants....The mechanism terminates when a round with no rejections occurs, following whichall tentative admissions from this round become permanent. The resulting outcome isthe student-optimal stable outcome. Sincerity and Sophistication
Students are either sincere or sophisticated.
Sincere students truthfully report an ROL according to their utilities, while sophisticated studentscan submit any ROL regardless of their utilities. When analyzing the Boston mechanism,we look at a Nash equilibrium of the preference-reporting game among the sophisticatedstudents. (We assume truthful reporting under DA, as this constitutes a dominant-strategy equilibrium.)
Utilities
While some of the examples in this paper use specific given utilities for eachstudent, our main results apply to the uniform ( n ; u , . . . , u k ) model , which we now de-scribe. In this model, which is defined by fixed utilities u > · · · > u k and a size n ,there are n students and n schools, where each of the schools has a capacity of exactly1. For each student, we draw ordinal preferences uniformly at random, and set the util-ity for this student from being matched to a school that she ranks in place (cid:96) to be u (cid:96) .Formally, for each student i ∈ I we independently and uniformly draw k distinct schools s π , . . . , s π k , and set u i ( s π ) = u , u i ( s π ) = u , etc., where u (cid:96) = 0 for all (cid:96) > k . Theshort-list assumption is mostly for technical convenience, and most results and proofsare easily adapted to the case of unbounded lists. Specifically, we do not rely on having See footnote 8. The only uncertaintythat players face is due to the random tie-breaking rule used in both mechanisms.
Our first main result speaks to the indeterminacy of the relative gains of sophisticatedstudents as compared with sincere students under BM. As mentioned, under DA bothtypes get the same utility. Therefore, we only need to verify how many students preferto be sophisticated and how many prefer to be sincere under BM.We show that in large random markets there are many students who prefer beingsophisticated, and many who prefer being sincere. This is thus an “anything goes” kindof theorem that stands in sharp contrast to the case of strict priorities, where, as Pathakand S¨onmez (2008) show, each student weakly prefers being sophisticated. Formally, weprove that in large random markets, for a large family of cardinal utility levels, and forany nontrivial proportion of sincere and sophisticated students, the expected number ofstudents who prefer being sincere is linear in n , as is the expected number of studentswho prefer being sophisticated. Theorem 3.1 (Relative Gains are Often Positive for Some and Negative for Others) . Let k = 2 and let u > u > · u > . For any < p < , there exists τ > such thatfor any large enough n , when each student is sincere with probability p independently ofother students, in the uniform ( n ; u , u ) model both of the following hold:1. There exists a set of sincere students of expected size at least τ n , such that eachstudent in this set strictly prefers any equilibrium had she been sophisticated toany equilibrium (in which she is sincere). Furthermore, each student in this setmaintains this strict preference regardless of whether or not any other studentsbecome sophisticated and/or sincere.2. For any equilibrium (in which only the sophisticated students strategize), there existsa set of sophisticated students of expected size at least τ n , such that each student inthis set strictly prefers any equilibrium had she been sincere to the given equilibrium.Furthermore, each student in this set maintains this strict preference regardlessof whether or not any other students in this set become sincere and regardless ofwhether or not any sincere students become sophisticated. Remark 3.2.
For p = 0 only the second part of the theorem holds. For p = 1 only thefirst part of the theorem holds. Remark 3.3.
While we use a fixed cardinal utility form, the exact values are qualitativelyinconsequential. What is important is that u and u are not too far apart, as otherwiseall students would always rank their top school first. In Appendix B we do analyze a more general model with incomplete information on sophisticationlevels. The expectation is taken over preferences and sophistication levels. The expectation is taken over preferences and sophistication levels (for any given mapping fromrealized preferences and sophistication levels to equilibria). emark 3.4. It is straightforward to generalize this result and all other theorems andlemmas (except for the first part of Theorem 4.2, as mentioned above) to arbitrary valuesof k and even to preferences of unbounded length. One easy way to achieve this is toreplace the current restrictions on the cardinal form with restrictions that mandate asignificant decrease in students’ utility when they receive their third alternative or worse.Before presenting the proof, we provide some intuition using two examples. We firstillustrate the basic idea using an example in which we restrict our focus to symmetricequilibria. Example 3.5 (Sincerity Can be an Advantage: Symmetric Equilibrium) . Let S = { s , s } and let I = { i , i , i } . Each s ∈ S has capacity q s = 1. Every i ∈ I has u i ( s ) = 3 and u i ( s ) = 2. Suppose that i and i are sophisticated.If i is also sophisticated, the only symmetric equilibrium is for each student to report s (cid:31) s with probability 0 . s (cid:31) s with probability 0 .
2. This leaves i (aswell as i and i ) with an expected utility of / .If i is sincere (and therefore reports s (cid:31) s with probability 1), then the onlyequilibrium in which i and i play symmetrically is for each of them to report s (cid:31) s with probability 0 . s (cid:31) s with probability 0 . / < / ). This leaves s with an expected utility of / > / .The mechanics behind Example 3.5 are as follows: in a symmetric equilibrium amongall students, since each student plays a mixed strategy, her expected utilities from eachof the two pure strategies s (cid:31) s and s (cid:31) s that she plays with positive probability arethe same. By becoming sincere, i in essence becomes a Stackelberg leader who commitsto the strategy s (cid:31) s (instead of mixing it with s (cid:31) s ), and by doing so, she “crowdsout” the other (sophisticated) students in the school s , breaking the equality betweenthe expected utilities of the other students from playing s (cid:31) s and s (cid:31) s , therebyshifting their mixed strategy in the new symmetric equilibrium toward playing s (cid:31) s with higher probability and playing s (cid:31) s with lower probability. Since other studentsplay s (cid:31) s with lower probability, the utility of i from playing s (cid:31) s given the newstrategies for all other students is higher than her utility from playing s (cid:31) s given theold strategies for all other students, which was her utility in the old equilibrium (sinceshe played s (cid:31) s with positive probability in that equilibrium). Using this fundamentalidea, we can generalize Example 3.5 to include more students and/or other utilities.While the market in Example 3.5 may seem to be very carefully crafted, e.g., in termsof alignment, Lemma 3.7 below shows that the phenomenon identified in that exampleis in fact generic, and occurs often in a large uniform market. Before proceeding tothat lemma, we present a somewhat more complicated example where we do not restrictourselves merely to symmetric equilibria. Example 3.6 (Sincerity Can be an Advantage) . Let S = { s , s , . . . , s } and I = { i , i , . . . , i } . Each s ∈ S has capacity q s = 1. Every i ∈ I has utility 4 from be-ing matched to her first choice, utility 3 from being matched to her second choice, andutility 0 otherwise. The preferences of the students are as follows:1. (cid:31) i : s , s (i.e., i prefers s first and s second),2. (cid:31) i : s , s , 9. (cid:31) i : s , s ,4. (cid:31) i : s , s ,5. (cid:31) i : s , s .Suppose that i and i are sophisticated. Regardless of whether i , i , and i are sophis-ticated or not, we note that they will rank truthfully (since the second choice of each isguaranteed). Thus, there are three possible equilibria: (1) i ranks truthfully, i ranks s at the top. In this case, the utility of i is / = 2and the utility of i is / .(2) i ranks truthfully, i ranks s at the top. In this case, the utility of i is / = 2and the utility of i is / .(3) i and i rank truthfully with probability / and rank their second choice at thetop with probability / . In this case, the utility of each of these students is / .We note that Equilibrium (1) is strictly preferred by i to the other two equilibria, andis obtained if i is sincere. Similarly, Equilibrium (2) is strictly preferred by i to theother two equilibria, and is obtained if i is sincere. Thus, regardless of the equilibrium,at least one of the two students, i or i , strictly prefers to become sincere.Our next result shows that in a large random market and under a broad range ofcardinal utilities, Example 3.6 repeats linearly many times. Lemma 3.7 (When Sophistication is Prevalent, Sincerity is Often an Advantage) . Let k = 2 and let u > u > · u > . There exists a constant τ > such that for anylarge enough n , when all students are sophisticated, in the uniform ( n ; u , u ) model thefollowing holds: For any equilibrium (in which all of the students strategize), there existsa set of students of expected size at least τ n , such that each student in this set strictlyprefers any equilibrium had she been sincere to the given equilibrium. Furthermore, eachstudent in this set maintains this strict preference regardless of whether or not any otherstudents in this set become sincere. Lemma 3.7 analyzes a random market where all students are sophisticated. By con-trast, in a large uniform market where all students are sincere, it is obviously weaklybeneficial to become sophisticated. As Lemma 3.8 shows, this is also strictly beneficialfor linearly many students.
Lemma 3.8 (When Sincerity is Prevalent, it is Often a Disadvantage) .
1. If all students are sincere, then each student weakly prefers to become sophisticated. This example also demonstrates that a sincere student may be matched to different schools acrossdifferent equilibria (here this can be observed for i and for i ). Furthermore, this holds even whenrestricting to pure-strategy Pareto-dominant equilibria, and even when looking only at utilities and notat the actual school to which the student is assigned. This phenomenon is in contrast to the case of strictpriorities, where Pathak and S¨onmez (2008, Proposition 2) show that each sincere student is matched tothe same school across all equilibria. The expectation is taken over preferences (for any given mapping from realized preferences to equi-libria). In such equilibria all students except herself strategize, while she is truthtelling. . Let k = 2 and let u > u > · u > . There exists a constant τ > such thatfor any large enough n , when all students are sincere, in the uniform ( n ; u , u ) model there exists a set of students of expected size at least τ n , such that eachstudent in this set strictly prefers any equilibrium had she been sophisticated tothe outcome (where all students are sincere). Furthermore, each student in this setmaintains this strict preference (relative to equilibrium outcomes in which she issincere) regardless of whether or not any other students become sophisticated. We are now ready to complete the proof of Theorem 3.1.
Proof of Theorem 3.1.
By a calculation similar to that in the proof of Lemma 3.8, theset of sincere students z satisfying the conditions in that proof is of expected size atleast p · τ · n , where τ is as defined there, and so the first statement holds for τ = p · τ (recall that under the conditions in that proof, y, x, w will rank truthfully even ifthey are sophisticated, since their second choice is guaranteed, and so we multiply τ bythe p probability of z being sincere). By a calculation similar to that in the proof ofLemma 3.7/Example 3.6, the second statement holds for τ = (1 − p ) · τ , where τ is asdefined in Lemma 3.7 (once again, x, w, v will rank truthfully regardless of whether ornot they are sophisticated, since their second choice is guaranteed, and so we multiplyby the (1 − p ) probability of both z and y being sophisticated). The theorem then holdswith τ = min { τ , τ } . Remark 3.9.
The fact that the proof uses only ROLs of length 2 immediately impliesthat the result also holds when BM is replaced by any First-Choice Maximal mechanism(Dur et al., 2018b), and in particular with the Corrected Boston Mechanism (Miralles,2009), the Modified Boston Mechanism (Dur, 2019), and the Adaptive Boston Mechanism(Mennle and Seuken, 2017). The result also holds under the Secure Boston Mechanism(Dur et al., 2019) since the probability that a student is ranked first by the tie-breakingrule is only / n , and the proof remains almost exactly the same. Remark 3.10.
It is possible to extend Theorem 3.1 to a model with finitely many popu-lations and with incomplete information on the sophistication levels of individual students(as long as the probability of being sophisticated is not too low). In this extension, eachpopulation contains a constant share of the students, and belonging to a population im-plies some population-specific probability of being either sincere or sophisticated. Fordetails see Appendix B.
In their paper, Pathak and S¨onmez (2008) compare, for sophisticated students, the out-comes of BM with those of DA (the student-optimal stable mechanism). They showthat sophisticated students weakly prefer the Pareto-optimal equilibrium of BM to thestudent-optimal stable matching. Furthermore, they show that the set of all equilibria ofBM in any given economy coincides with the set of all stable matchings in an “augmented The expectation is taken over preferences. In such equilibria she strategizes, while all other students are truthtelling. When priorities are strict, a unique Pareto-optimal equilibrium indeed exists.
Proposition 4.1 (Sophisticated Students May Prefer DA to BM) . There exists a match-ing market with weak priorities that satisfies both of the following conditions:1. The set of Nash equilibrium outcomes of BM in this matching market is differentfrom the set of stable matchings under the “augmented economy” defined by Pathakand S¨onmez (2008) for this market. Furthermore, the union of the supports of theoutcomes in the former set is different than in the latter set.2. BM has only one equilibrium (which is thus a Pareto-dominant equilibrium) in thismarket. The utility of each sophisticated student in the equilibrium of BM is strictlylower than her utility in the DA outcome. The utility of each sincere student in theequilibrium of BM is strictly higher than her utility in the DA outcome.Proof.
Let S = { s , s } and let I = { i , . . . , i } . Every s ∈ S has q s = 1. Every i ∈ I prefers s first, and receives utility 4 from being matched to s . Students i and i aresincere and do not wish to be matched to s . Students i and i are sophisticated andeach receive utility 3 from being matched to s . Note that the augmented economy inthis case is the same as the original economy.In the unique BM equilibrium, each of the sophisticated students goes to s first, andso gets utility / ; each sincere student therefore gets utility / = 2 in this equilibrium.The union of the supports of all BM equilibrium outcomes is therefore the set of allmatchings where some sincere student i ∈ { i , i } is matched to s and some sophisticatedstudent i ∈ { i , i } is matched to s .In the student-optimal stable mechanism, each sophisticated student goes to s first,and so receives utility / · / · ( / · / · / ) = / > / ; each sincere student thereforegets utility / = 1 < i ∈ I is matched to s and some sophisticated student j ∈ { i , i } with j (cid:54) = i ismatched to s .As in the study of the trade-off between sincerity and sophistication, one may askwhether, and to what extent, the above-demonstrated preference of sophisticated studentsfor DA over BM remains prominent in a large random market. The second main result ofthis paper shows that, generally, both a constant fraction of any population prefers DAto BM and a constant fraction of any population prefers BM to DA, in expectation. Theorem 4.2 (Absolute Gains are Often Positive for Some and Negative for Others) . Let k = 2 . There exists a constant τ > such that for any large enough n , for anyutilities u > u > , and for any fixed assignment of students into sophisticated andsincere types, in the uniform ( n ; u , u ) model both of the following hold:1. There exists a set consisting of an expected fraction of at least τ of the sophisticatedstudents and an expected fraction of at least τ of the sincere students, such thateach student in this set strictly prefers the DA outcome to any equilibrium of BM. The expectation is taken over preferences. . There exists a set consisting of an expected fraction of at least τ of the sophisticatedstudents and an expected fraction of at least τ of the sincere students, such thateach student in this set strictly prefers any equilibrium of BM to the DA outcome.Furthermore, each student in each of the above sets maintains this strict preference evenif the assignment of students into sophisticated and sincere types changes arbitrarily. Remark 4.3.
Remarks 3.9 and 3.10 also hold for Theorem 4.2. Moreover, a strongerversion of Remark 3.10 actually holds, as there is no need to make any restrictions onthe share of sophisticated students in the population.
The results of Section 3 demonstrate a positive effect of the sincerity of a student i inBM: the ability to crowd out others by essentially becoming a Stackelberg leader whocommits to ranking in accordance with her true preference, forcing (other) sophisticatedstudents to respond to this commitment by reducing demand for the school in whichstudent i is interested. We observe that this effect completely disappears when prioritiesare strict. Indeed, since the outcome of BM under strict priorities (given a fixed profileof students’ preferences) is deterministic rather than randomized, if student i is harmedby the competition of student j for a certain school (in the case of strict preferences, thismeans that student j has higher priority at this school), then student i committing toapply to that school does not deter student j from applying to that school, as student j has priority in that school and so will not be crowded out by student i .As demonstrated by Pathak and S¨onmez (2008) in the context of strict priorities,being sincere also has an adverse effect: not responding to excess demand for one’sfavorite school. As it turns out, this effect can still be manifested in markets with weakpriorities. In fact, in some cases the sincerity of a student i may not crowd out anystudents whatsoever, but may nonetheless harm student i as she does not respond toexcess demand for her favorite school. This is precisely what happens in the analysisof Lemma 3.8 for student z . In that example, her sincerity does not crowd out thecompetition, but it does cause her to not respond to excess demand.What happens when both effects of sincerity are present? Which effect dominates:the positive effect of crowding out others or the negative effect of not responding to excessdemand? In Lemma 3.8, the competition that student z faces for school a is completelydecoupled from her competition for school b . As long as this feature is maintained, itis clear that by only tweaking the demand for the second choices of z ’s competition forschool a , we may change whether, and to what extent, a sincere z is able to crowd out Once again, the expectation is taken over preferences. A slightly different way to put things is to say that the players’ decisions of whether to be sincereare strategic substitutes. If many of the other players are sincere, being sincere as well becomes lessappealing, as it crowds out others less efficiently. We do not want to overemphasize this descriptionbecause being sincere or sophisticated in our model is not a strategic choice, but rather the dimensionalong which we compare expected welfare. z from beingassigned to any of the schools. This way, we may easily create variants of this marketwhere the positive effect dominates the negative one or vice versa. A more interestingquestion, therefore, is what happens when the markets for school a and for school b are entangled? In other words, is the lack of symmetry among the different studentsin Lemma 3.8 required for sincerity to be disadvantageous? To be more specific, whathappens in the extreme case where all students are symmetric: which effect of sinceritydominates then? The following example alludes to an answer to this question. Example 5.1 (Negative Effects of Sincerity May Dominate Positive Effects) . Let S = { s , s , s } and let I = { i , i , . . . , i n } . The schools have the following capacities: q s = 1 ,q s = 1 ,q s = n − . Every i ∈ I has u i ( s ) = 9, u i ( s ) = 1, and u i ( s ) = n − . Suppose that i , i , . . . , i n areall sophisticated.If i is also sophisticated, then in a symmetric equilibrium all students get the sameexpected utility, and since the sum of utilities is 9 + 1 + ( n − · n − = 10 .
5, eachstudent’s expected utility is . n .If i is sincere (and therefore reports s (cid:31) s (cid:31) s with probability 1), then for largeenough n , the only equilibrium in which i , . . . , i n play symmetrically is for each of themto report s (cid:31) s (cid:31) s with some probability t , and report s (cid:31) s (cid:31) s with probability − t . In Appendix C, we show that t ≥ for large enough n , which implies that as n grows large, the expected utility of i approaches t ( n − < tn ≤ . n , and so for largeenough n the expected utility of i must be strictly smaller than . n , and thus becomingsincere harms i .Example 5.1 demonstrates that even when students share the same preferences andthe same cardinal utility function, the crowding-out effect does not necessarily dominatethe negative implications of being sincere and not responding to excess demand. Sincerityturns out to be a package deal, coupling together the inability to respond to excess demandwith the power of crowding-out others, and in this case the negative effect outweighs thepositive one. While student i , if she is sincere, slightly increases her chances of beingadmitted to school s , she completely forfeits her chances of being admitted to school s by approaching school s in the second round; nevertheless, she does so even though inthis round school s is already very likely to be full. A symmetric equilibrium exists since the game is finite and symmetric (Nash, 1951). Moreover, onecan verify that when n is large enough, the only symmetric equilibrium is for each student to report s (cid:31) s (cid:31) s with some probability t (cid:48) , and report s (cid:31) s (cid:31) s with probability 1 − t (cid:48) . An equilibrium in which i , . . . , i n play symmetrically exists since the game is finite and symmetricwith respect to these students (Nash, 1951). To see why only these two strategies are played for largeenough n , notice that an application to s or s in the second stage has an exponentially small probabilityof success, and therefore when n is large enough, ranking s second gives higher utility as it ensuresadmission to school s (since i will not apply to it before the third round). .2 The Choice of Mechanism:Reduced Competition vs. Reduced Options The results of Section 4 demonstrate that for a sophisticated student i , BM may havea negative effect as it may reduce the options of student i , effectively forcing her toex-ante choose to apply to only one overdemanded school. As we noted in Section 5.1,this negative effect also completely disappears when priorities are strict. Indeed, in anequilibrium of BM, since the outcome is deterministic rather than randomized, student i never has any reason to apply to a school she will not be (deterministically) acceptedto, and therefore she is unharmed by being effectively forced to ex-ante choose to applyto only one overdemanded school: as she will never be rejected by this school, she isunharmed by giving up her “plan B.”The positive effect of using BM is discussed by Pathak and S¨onmez (2008) in thecontext of strict priorities: reduced competition (in particular, for one’s top choice),which can also be seen to manifest in markets with weak priorities. This is demonstratedby the following example, which also sheds light on the interplay between this positiveeffect of reduced competition and the above-discussed adverse effect of reduced options. Example 5.2 (Positive Effects of BM Compared with DA May Dominate NegativeEffects) . Let S = { s , s } and for ease of presentation let the number of students n belarge and divisible by 12. Each s ∈ S has capacity q s = 1. Each student has utility 1for her most-preferred school, and utility 1 − ε , for very small ε , for her second-preferredschool, with one quarter of the students preferring s the most, and the remaining threequarters of the students preferring s the most. Under DA, each student has a probabilityof roughly / n of being admitted to each school, for a total expected utility of roughly / n .Under BM, if a large fraction (say, one half) of the students are sophisticated, then inequilibrium roughly half of the students will apply to each school, for an expected utilityof roughly / n for each student. If, however, only a small fraction (say, one twelfth) ofthe students are sophisticated, then only these students and the quarter of the studentswho truly prefer s will apply to s , resulting in lower competition (at most n / students)for s than for s and an expected utility of strictly more than / n for each sophisticatedstudent (and for each sincere student who prefers s ). Therefore, in this case sophisticatedstudents strictly prefer BM to DA.The question that naturally arises is which of the two effects generally dominates for asophisticated student: reduced competition or reduced options. It seems that the answerdepends on whether the “overall competition” was reduced.To give one example along the lines of Example 5.2, consider a sophisticated student i and call the top of her preference list that consists of overdemanded schools her “overde-manded set.” If all other students have the same overdemanded set of size o (and theirpriorities are randomly selected), then under DA, student i essentially faces competitionfrom all the other students for the o schools in her overdemanded set. Under BM, stu-dent i faces competition from an average of / o of the other students in one school and,therefore, if this competition is evenly spread (weighted by the utilities of student i forthe different schools), then DA and BM give similar utility to student i , whereas if thiscompetition is not evenly spread then BM gives higher utility to student i .In contrast to the example of a shared overdemanded set, suppose that all otherstudents have very attractive outside options, to the extent that each other student j only prefers one of the schools in student i ’s overdemanded set to j ’s outside option. In15uch a case, student i most certainly prefers DA, because BM not only does not reducethe competition student i faces but, in fact, reduces her options. This is precisely whathappens in the analysis of Theorem 4.2 for student z . In that example, BM reduces theoptions of student i without reducing the competition she faces. Market designers often encounter markets that are governed by manipulable mechanisms,with BM being possibly the most prominent example. Common sense dictates that itis better to switch to a strategy-proof mechanism, as it allows the designer to directlyoptimize some target function (e.g., efficiency), subject to certain desirable constraints(e.g., stability), and to preserve incentive compatibility. When a manipulable mechanismis in place, it is difficult to predict what properties the likely outcome will satisfy, andwhether the strategic situation will give an advantage to some populations over others.In the specific case of BM, it intuitively seems that the use of this mechanism favorsplayers who recognize strategic opportunities over players who do not. Indeed, as Pathakand S¨onmez (2008) show, in a strict-priority environment a student weakly prefers beingsophisticated to being sincere, and BM weakly benefits sophisticated players (comparedwith DA). This constitutes a very strong argument against BM, as strategic sophisticationand access to relevant information can be highly correlated with socioeconomic status and,in any case, these are definitely not the criteria based on which the matching should bedetermined. However, as we show in this paper, when considering a reasonable scenarioin which the school district uses a coarse priority structure rather than a strict one, thereare several new effects that may reverse the previous theoretical prediction.The question, therefore, is what should we recommend to practitioners. In Section 5we tried to convey our view that understanding the market structure is crucial for provid-ing sound advice. That being said, it is also fair to assess that in most real-life markets,reacting to excess demand is quite straightforward in terms of strategic behavior, whereascrowding out others is more demanding as it requires a reputation for being sincere andfor not shying away from competition. This suggests that, at the end of the day, whilepolicy-makers who are focused on efficiency and absolute gains should probably adopt BM(which takes into account some students’ cardinal preferences), in most markets policy-makers who are mostly interested in fairness considerations and relative gains should stillfavor DA. That being said, our results imply that this is not as automatic a recommen-dation as may have widely been believed, and should be taken with a grain of salt, asin some specific markets with special structures BM may not only benefit all players,but also provide the same or higher gain for students who are not sophisticated or notinformed about the market. As such students many times come from populations froma weaker socioeconomic background, in such markets turning to BM may help keep oreven increase diversity in highly coveted schools.16
Omitted Proofs
Proof of Theorem 4.2.
Let u > u >
0. We first observe that if there exist three students z, y, x and four schools a, b, c, d such that the following conditions hold:1. (cid:31) z : a, b ,2. (cid:31) y : a, c ,3. (cid:31) x : b, d ,4. a, b, c, d are not preferred first or second by any other student,then regardless of whether each of these three students (and, in fact, regardless of whetherany student) is sophisticated or sincere, z strictly prefers the DA outcome to any equi-librium of BM and x strictly prefers any equilibrium of BM to the DA outcome. Firstnote that regardless of whether z , y , and x are sophisticated or sincere, they will ranktruthfully under any equilibrium of BM even if they are sophisticated. For y and x thisholds since their second choice is guaranteed, and for z this holds as she prefers getting a with probability / to getting b with probability / . Now, under DA z gets utility / · u + / · u (since with probability 0 . z beats y in the competition for a , and withthe remaining probability 0 . . z beats x in thecompetition for b ) while under any equilibrium of BM z gets utility u , which is strictlysmaller. Under DA x gets utility / · u + / · ( / · u + / · u ) (since with probability0 . z beats y in the competition for a and so x is the only applicant for b , and withthe remaining probability 0 . z competes with x for b and so x wins and gets b withprobability 0 .
5, and loses and gets d with probability 0 .
5) while under any equilibrium ofBM x gets utility u , which is strictly greater.For Part 1, let us calculate the probability that for any given z ∈ I , there exist y, x ∈ I such that the above four conditions are satisfied. Let a denote z ’s most-preferred schooland b denote her second-preferred school; this probability is given by some y ∈ I ranks a first(let c denote hersecond-preferred school) (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) − (cid:18) n − n (cid:19) n − (cid:33) · y does notrank b second(i.e., c (cid:54) = b ) (cid:122) (cid:125)(cid:124) (cid:123) n − n − · some x ∈ I ranks b first(let d denote hersecond-preferred school) (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) − (cid:18) n − n (cid:19) n − (cid:33) · x does not rank a / c second(i.e., d / ∈ { a, c } ) (cid:122) (cid:125)(cid:124) (cid:123) n − n − ·· (cid:18) n − n · n − n − (cid:19) n − (cid:124) (cid:123)(cid:122) (cid:125) No other i ∈ I ranks a / b / c / d first or second −−−→ n →∞ (cid:18) − e (cid:19) · e Let τ be any constant slightly smaller than (1 − / e ) · / e ; then we have that for largeenough n , both the expected fraction of sophisticated students z for which such y, x ∈ I exist and the expected fraction of sincere students z for which such y, x ∈ I exist are atleast τ , and so we have identified an appropriate set of students that satisfies Part 1.For Part 2, let us calculate the probability that for any given x ∈ I , there exist z, y ∈ I such that the above four conditions are satisfied. Indeed, let b denote x ’s most-preferred17chool and d denote her second-preferred school; this probability is given by some z ∈ I ranks b second(let a denote hermost-preferred school) (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) − (cid:18) n − n (cid:19) n − (cid:33) · z does notrank d first(i.e., a (cid:54) = d ) (cid:122) (cid:125)(cid:124) (cid:123) n − n − · some y ∈ I ranks a first(let c denote hersecond-preferred school) (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) − (cid:18) n − n (cid:19) n − (cid:33) · y does not rank b / d second(i.e., c / ∈ { b, d } ) (cid:122) (cid:125)(cid:124) (cid:123) n − n − ·· (cid:18) n − n · n − n − (cid:19) n − (cid:124) (cid:123)(cid:122) (cid:125) No other i ∈ I ranks a / b / c / d first or second −−−→ n →∞ (cid:18) − e (cid:19) · e Since this is the same probability calculated above for Part 1, the proof of Part 2 iscompleted similarly to that of Part 1.
B Incomplete Information about Sophistication
In Theorem 3.1 we assumed that each player’s probability of being sincere is p inde-pendently of other students, and that the realizations of these draws (i.e., whether eachstudent is sincere or sophisticated) are common knowledge. Furthermore, we ignored theissue of reputation (that is, the probability of being sincere) constituting a property of populations rather than of individuals , and just assumed that no two students belongto the same population. We have made these assumptions to ease the presentation, yetthe result extends even when we relax them significantly. In this appendix, we present amore robust version of Theorem 3.1 in which each student belongs to a population, eachpopulation is characterized by a probability of being sincere (and this probability is nottoo high), and each player’s realized sophistication level is private information.We consider the following “population model” for generating a market. We assumethat there are M populations. Each student belongs to one population independently atrandom, and her probability of belonging to population ν is w ν ∈ (0 , ν , each student is independently at random determined to be sincere withprobability p ν , and sophisticated otherwise; the realization of this draw is private and isknown to no other student. Theorem B.1.
Let k = 2 , let u > u > · u > , and set t ∗ = u − u ) u . For anypopulation model where p ν < t ∗ for every ν , there exists τ > such that for any largeenough n , in the uniform ( n ; u , u ) population model both of the following hold:1. There exists a set of students of expected size at least τ n , such that each student i that belongs to any population ν in this set ex ante strictly prefers any equilibriumhad all the students in ν been sophisticated and had that been common knowledge to any equilibrium (in which each student in ν is sincere with probability p ν and As with Theorem 3.1, it is straightforward to generalize this result to arbitrary values of k , and evento preferences of unbounded length. The expectation is taken over preferences and population associations. That is, the strict preference is in expectation over her and others’ sophistication. In such equilibria sophisticated students and all students in ν strategize, while all other students aretruthtelling. his probability is common knowledge ). Furthermore, each student in this setmaintains this strict preference regardless of whether other populations’ probabilitiesof being sincere change.2. For any equilibrium (in which each student that belongs to any population ν issincere with probability p ν and this is common knowledge, and in which only so-phisticated students strategize), there exists a set of students of expected size atleast τ n , such that each student i that belongs to any population ν in this set ex antestrictly prefers any equilibrium had all students in ν been sincere and had that beencommon knowledge to the given equilibrium. Furthermore, each student in this setmaintains this strict preference regardless of whether or not any other students inthis set individually have different probabilities of being sincere and regardless ofwhether or not other populations’ probabilities of being sincere decrease.Proof. For Part 1, we can still use Lemma 3.8. In its proof it is of no importance whetherstudents y , x , and w are sophisticated or not, as each reports truthfully regardless. Since z strictly prefers being sophisticated to being sincere, she also prefers being sophisticatedto being sincere with any positive probability.For Part 2, we slightly modify the argument in Lemma 3.7. It is once again of noimportance whether x , w , and v are sophisticated or not, as each reports truthfullyregardless. As for z and y , we need them to belong to different populations, which wedenote by ν z and ν x respectively (this happens with fixed probability w ν z · w ν y , summedover all possible choices for distinct ν z and ν y ). The reason for having z and y belongto two different populations is that we do not want their sophistication levels to beinterdependent when we change the sincerity probability of the population of one ofthem to 1.In any equilibrium, within any gadget that contains players z , y , x , w , and v , thereare theoretically five possible ways in which the sophisticated types of players z and y can play:(1) The sophisticated type of z ranks truthfully, and the sophisticated type of y ranksher second choice at the top.(2) The sophisticated type of y ranks truthfully, and the sophisticated type of z ranksher second choice at the top.(3) The sophisticated types of z and y play a mixed strategy.(4) The sophisticated types of z and y rank truthfully.(5) The sophisticated types of z and y rank their second choice at the top.Equilibrium (4) is ruled out by the restriction that u > u . Equilibrium (5) is ruledout by our assumption that p ν < t ∗ . Indeed, if the sophisticated types of the two players In such equilibria sophisticated students strategize, while all other students (including those in ν that are not sophisticated) are truthtelling. The expectation is taken over preferences and population associations (for any given mapping fromrealized preferences and population associations, to equilibria). That is, the strict preference is once again in expectation over her and others’ sophistication. In such equilibria sophisticated students, but not students in ν , strategize, while all other students,including all students in ν , are truthtelling. z would havereceived utility u , while by reporting truthfully they could have received utility p ν y · u + (1 − p ν y ) · u = (cid:18) − p ν y (cid:19) · u > (cid:18) − · u − u ) u (cid:19) · u = 12 u . Under Equilibria (1) and (2), either z or y strictly prefers to become sincere. Indeed,whichever one of the two currently reports her second choice first gets utility u , andby becoming sincere she will made the sophisticated type of the other player switch toreporting her own second choice first, and will then get utility (cid:0) − p ν y (cid:1) · u , which isagain higher than u .Finally, we turn to analyzing Equilibrium (3). The mixed equilibrium in this case isfor the sophisticated type of each player i ∈ { z, y } to report truthfully with probability t ∗ − p νi − p νi . This causes each player to face a truthful opponent with probability t ∗ , whichleaves her indifferent between reporting truthfully and reporting her second choice first.In particular, this also means that the sincere type of each player gets the same utilityas her sophisticated type, and so both of these types get utility u . Therefore, as inEquilibria (1) and (2), z and y strictly prefer to become sincere. C Calculation for Example 5.1
In this appendix, we will estimate p , as defined in Example 5.1. Thus, we assume that i is sincere (and so reports s (cid:31) s (cid:31) s ), and focus on analyzing the utility of i . Wefirst claim that for large enough n , it holds that p ≥ / . Indeed, if it were the case that p < / for such n , then the expected utility of u from listing s first would be strictlyhigher than her expected utility from listing s first—a contradiction. Fix m = n − δ = / . Applying a Chernoff bound (see, e.g., Mitzenmacher and Upfal, 2005, p. 67),we have that the number a of students among i , . . . , i n who rank s first satisfiesPr (cid:2) a ≥ (1 + δ ) · mp (cid:3) ≤ exp (cid:18) − δ · pm (cid:19) ≤ exp (cid:18) − δ · m (cid:19) = exp ( − α · m ) , (1)where we have set α = − δ / .We begin by estimating the utility of i from reporting s (cid:31) s (cid:31) s . By Equation (1),this expected utility is at least e − αm · − e − αm ) · (cid:18) Probability that i is admittedto s when m · (1 + δ ) p sophisticated students plus i also apply in addition to i (cid:122) (cid:125)(cid:124) (cid:123)
12 + m · (1 + δ ) p · (cid:18) −
12 + m · (1 + δ ) p (cid:19) · n − n − · n − (cid:19) ≥≥
92 + m · (1 + δ ) p + (cid:18) − m (cid:19) · (cid:18) − m (cid:19) · m − − e − αm ≥≥
92 + m · (1 + δ ) p + (cid:18) − m (cid:19) · m − − e − αm . We now estimate the utility of i from reporting s (cid:31) s (cid:31) s . By Equation (1), this20xpected utility is at most e − αm · − e − αm ) · (cid:18) Probability that i is admittedto s when m · (cid:0) − (1 + δ ) p (cid:1) sophisticated studentsalso apply in addition to i (cid:122) (cid:125)(cid:124) (cid:123)
11 + m · (cid:0) − (1 + δ ) p (cid:1) + (cid:32) −
11 + m · (cid:0) − (1 + δ ) p (cid:1) (cid:33) · n − (cid:33) ≤≤ e − αm + 11 + m · (cid:0) − (1 + δ ) p (cid:1) + 12( m − . As these two expected utilities are equal (since p is the mixing probability in equilib-rium), combining the two estimates above we get92 + m · (1 + δ ) p ≤ e − αm + 11 + m · (cid:0) − (1 + δ ) p (cid:1) + 32 m ( m − . Therefore,9 + 9 m · (cid:0) − (1 + δ ) p (cid:1) ≤ (cid:0) m · (1 + δ ) p (cid:1) · (cid:16) m · (cid:0) − (1 + δ ) p (cid:1)(cid:17) · (cid:18) e − αm + 32 m ( m − (cid:19) + 2 + m · (1 + δ ) p ≤≤ m + 2)( m + 1) · e − αm + 4 + m · (1 + δ ) p for n large enough (and therefore m large enough) such that ( m +2)( m +1) m ( m − ≤ / . Therefore,5 − m + 2)( m + 1) · e − αm + 9 m ≤ mp · (1 + δ ) , and so, for n large enough (and therefore m large enough) such that 10( m + 2)( m + 1) · e αm ≤
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