Assignment mechanisms: common preferences and information acquisition
aa r X i v : . [ ec on . T H ] J a n Assignment mechanisms: commonpreferences and information acquisition
Georgy Artemov ∗ January 3, 2020
I study costly information acquisition in a two-sided matching problem, such asmatching applicants to schools. An applicant’s utility is a sum of common andidiosyncratic components. The idiosyncratic component is unknown to the appli-cant but can be learned at a cost. When applicants are assigned using an ordinalstrategy-proof mechanism, too few acquire information, generating a significantwelfare loss. Affirmative action and other realistic policies may lead to a Paretoimprovement. As incentives to acquire information differ across mechanisms, ig-noring such incentives may lead to incorrect welfare assessments, for example, incomparing a popular Immediate Assignment and an ordinal strategy-proof mech-anism.
JEL classification:
D47, D82
Keywords: School choice, information acquisition, Deferred Acceptance, Imme-diate Acceptance ∗ Department of Economics, The University of Melbourne, 111 Barry St., Carlton, VIC 3010, Australia.Tel: +61 (3) 83447029; Email [email protected]
The author acknowledges support from theAustralian Research Council (DP160101350). The paper has been previously circulated as “School ChoiceMechanisms, Information Acquisition and School Rankings.” The matching literature typically assumes that applicants know their preferences. The mainquestion then is how to elicit those preferences for school assignments or empirical studies (seePathak, 2016 and reference therein). However, a truthful uninformed applicant submits thepublic ranking, which contains no useful information on their preferences.Strategy-proof mechanisms, such as Deferred Acceptance (DA) or Top-Trading Cycles,are widely advocated (Abdulkadiro˘glu and S¨onmez, 2003; Abdulkadiro˘glu et. al., 2005a,b;Ergin and S¨onmez, 2006). I show that those mechanisms, which coincide with RandomSerial Dictatorship (RD) in my environment, often provide inadequate incentives to acquireinformation. This finding raises the possibility that welfare gains from adopting strategy-proofmechanisms may be routinely overstated unless their adoption is accompanied by policies thatpromote information acquisition.Consider, for example, a school district contemplating a switch from the manipulable Im-mediate Acceptance (IA) mechanism to DA. To evaluate its effects, the district estimates ap-plicants’ preferences by either running a survey, as in De Haan et. al. (2015) or Kapor et. al.(2018), or using empirical methods that rely on submitted Rank-Ordered Lists (ROLs), asin Agarwal and Somaini (2018), Calsamglia et. al. (2018), He (2017) or Hwang (2017). If thedistrict uses preferences estimated under IA to evaluate the performance of both mechanisms,it ignores the mechanisms’ informational incentives. I establish that, under general conditions,IA better incentivizes information acquisition. Hence, after the adoption of DA, the districtmay find that applicants rely on test scores and other readily observable performance mea-sures, rather than school visits. They become “uninformed” and submit more homogeneousROLs than the estimated preferences would suggest, hurting other applicants. This finding does not advocate a return to IA; it advocates the adoption of policies thatencourage information acquisition alongside DA. Lovenheim and Walsh (2018) show empiri-cally that policies may significantly affect the intensity of information search. I explain that Hastings et. al. (2007) argue that cost is an important barrier to acquiring information. An indirect evidenceis provided by Lovenheim and Walsh (2018), who show that an increase in the value of information leads tohigher search intensity. Hastings and Weinstein (2007, 2008); Hoxby and Turner (2015); Kessel and Olme(2018) observe that applicants change their choices when additional information is provided to them,implying that they are not fully informed at the time of the application. A large number of papersreport differential information possessed by applicants in different settings or domains (Dur et. al., 2018;Hastings et. al., 2016; Kapor et. al., 2018; Luflade, 2018). An extensive follow-up literature, starting with Abdulkadiro˘glu et. al. (2011) and Miralles (2008), arguethat DA does not sufficiently account for cardinal preferences. Uninformed applicants may partially be the cause of a small number of over-demanded schools in ROLssubmitted to DA, reported by Abdulkadiro˘glu et. al. (2011). The school ranking is readily available inBoston; for example, it is published by Boston Globe. Any information beyond the public ranking is costly to obtain because it is applicant-specific. I assume that the costs differ among applicants. Such differences may be due to theopportunity cost of time or informational needs. For example, an applicant solely interestedin school religious affiliation should have little difficulty finding this information. An applicantprimarily interested in good discipline is likely to spend much more time learning that, askingmultiple students or their parents.In summary, applicants are ex-ante identical except for their cost of information acquisition.Once they pay the cost, they learn their utilities for all schools. Under these assumptions, boththe equilibria and the social optimum are characterized by fractions of informed applicants andcan be unambiguously compared. Under a mild condition, the equilibrium fraction of informedapplicants is below the social optimum; if there are multiple equilibria, a low-informationequilibrium is Pareto dominated by a high-information equilibrium.Furthermore, the gains from ensuring optimal information acquisition in a strategy-proofmechanism can be substantial. There are environments where this gain is larger than the gainfrom replacing random assignment by a strategy-proof mechanism. I propose three policiesthat lead to Pareto improvement. The assumption that uninformed applicants have the same preferences is not very restrictive. First, if unin-formed applicants have some information beyond the public ranking, but this information is insignificantrelative to the ranking, then uninformed applicants still follow the ranking. Although their informationacquisition decisions may vary, such effects will average out, and similar results would obtain. Second,the results would be the same in the model where a positive mass of applicants can acquire idiosyncraticinformation at zero cost. In the school choice context, parents make the decision about information acquisition, but children enjoythe benefits of going to a better-fitting schools (Dinkelman and Martinez A., 2014). Hence, the gains fromoptimal information acquisition may be larger than I report.
3. Releasing applicants’ priorities at schools before applicants acquire information leadsto Pareto improvement. However, too many medium- and, in some cases, high-priorityapplicants remain uninformed.
2. A school district can prioritize those who decide to acquire information, if it can observesuch a decision, by, for example, collating names of open day visitors. Such a policy hasbeen used in New York City school assignments (Nathanson et. al., 2013). Prioritizedapplicants acquire information, positively affecting non-prioritized applicants.Affirmative action can be viewed as an instance of this policy, in which prioritization isdirected at a specific, target group of applicants. As explained, the policy can benefitboth target and non-target groups. This is important because affirmative action – orany priority redistribution – is often perceived as a zero-sum game in school assignment(Cantillon, 2017).3. A school district can directly subsidize information acquisition, for example, by runninginformation sessions in disadvantaged neighborhoods or translating documents to otherlanguages. Those services are funded by taxing all applicants. Importantly, I do notassume perfectly transferable utility. I only assume that taxes lead to some increase inservice/subsidy provided to a specific group of applicants. Then there exists a level oftaxation that leads to a Pareto improvement.Information acquisition has first been considered by Bade (2015) in the context of houseallocation problems. Bade shows that serial dictatorship provides best informational incentivesamong all non-bossy strategy-proof mechanisms. I show that those incentives are still shortof socially optimal in most cases and propose policies to improve information acquisition. InBade’s setting, the state space is finite, and an applicant learns its partition. I assume thatan applicant learns the true state of a continuum state space.Two other studies, contemporaneous to the present paper, differ from it by both the focusand the setting.Harless and Manjunath (2018) also focus on non-bossy strategy-proof mechanisms. Unlikethis paper, they assume that learning is costless; however, applicants can sample only one ofmultiple uncertain objects. That is, they focus on what to learn rather than whether to learn.Thus, the results of Harless and Manjunath (2018) cannot be directly compared to the resultsof this paper. The timing of the release of applicants’ scores – either before or after they submit their ROLs – has been atopic of recent research (Chen and Pereyra, 2018; Lien et. al., 2016, 2017; Wu and Zhong, 2014). In mostmarkets, scores are released before applicants submit their ROLs; the authors argue that the delayed releaseof scores can achieve ex-ante fairness. However, the change in timing also changes incentives to acquireinformation and any gains from the late release of scores may be canceled out by applicant’s changes intheir information acquisition behavior. Similar interventions have been studied in empirical papers cited earlier, e.g. Hastings et. al. (2007);Hoxby and Turner (2015) γ SD . In Section4, I derive a mild condition when γ SD is below the social optimum. In Section 5, I illustratethe results obtained in Sections 3 and 4. I provide an example with three equilibrium levels ofinformation acquisition. For two levels, my condition holds, and they can be Pareto ranked;both are below the socially optimal level. For the third equilibrium, my condition does nothold, and the equilibrium information acquisition is excessive. In the same section, I show, byexample, that stronger results could only be obtained by imposing additional conditions onthe distribution of utilities that students derive from attending schools. Having establishedin the previous sections that too few applicants acquire information, in Section 6, I discusspolicies that could promote learning. Another important policy issue is whether the switchfrom the popular, but criticized IA to the desirable SD could discourage learning. I addressthis question in Section 7, and establish that, generally, the answer is affirmative. Since mytheoretical results are restricted to the environment consisting of three schools, I stress-test myresults in Section 8. I provide simulations for up to six schools and include the distributionsof utilities, under which we expect over-acquisition of information to occur, based on thetheoretical results. Simulations show that over-acquisition does occasionally occur, but therequired conditions are extreme. I discuss possible extension to the model in Section 9 and5onclude in Section 10. Example 1 illustrates externalities resulting from one applicant’s decision to acquire informa-tion. When only a few applicants are informed, the externality is necessarily positive. As theapplicant deciding to acquire information does not take the positive externality into account,she may opt against acquiring information even when it is socially optimal to do so. Whennearly everyone is informed, the effect is more nuanced: there is a positive externality on theuninformed, but negative externality on the informed. The paper will derive a condition whenthe resulting externality is positive.
Example . Consider a problem in which four applicants, , , and are allocated to threeschools, A, B and C . Schools A and B have one seat each, and school C has two seats. Eachapplicant finds every school acceptable.For each applicant i ∈ { , , , } , i ’s utilities are: u i ( A ) = 1 , u i ( C ) = 0 and u i ( B ) = 1 . u i ( B ) = − . i is A ≻ i B ≻ i C and ordinal preferences of informed applicant i can either be B ≻ i A ≻ i C or A ≻ i C ≻ i B , with equal probability.The applicants are allocated using RD mechanism: they are randomly assigned priorities,1, 2, 3 and 4, then each selects the most desirable school that is still available, in the order oftheir priority. Since RD is a strategy-proof mechanism, I assume each applicant i reports herpreferences truthfully when she submits her ROL R i .If no applicant is informed, then their probabilities of being assigned to a school are identicaland given in Table 1. Schools A B C
Applicants , , , R , , , = ABC .Suppose now that acquires information, learns that B is her top choice, and submits BAC . All other applicants submit
ABC . The probabilities of assignments change to thosein Table 2. Note that would be assigned to school B if her lottery priority is either 1 or 2and would be assigned to C if her lottery priority is 3 or 4. All other applicants do not faceany competition for A from applicant ; their probability of an assignment to A increases to1/3. Indeed, i ∈ { , , } is allocated to A if his priority is 1 (probability is 1/4), or if hispriority is 2 and ’s priority is 1 (probability is 1/12). All other probabilities can be similarlycalculated. 6chools A B C
Applicants , , R = BAC and R , , = ABC .Note that i ∈ { , , } are better off after acquires information: although the total proba-bility of being allocated to either A or B remains the same, the probability of being allocatedto the more preferred school A increases. Thus, the decision of to acquire informationprovides a positive externality to other applicants. I call this positive externality a sorting externality because it leads to a more efficient sorting of applicants to schools, thus increasingtheir expected utility.Suppose next that , and are informed; learns B ≻ A ≻ C and both and learn A ≻ , C ≻ , B . remains uninformed. Then the assignment probabilities are given in Table3. Note that all applicants are better off compared to Table 2.Schools A B C
Applicants , R = BAC , R , = ACB and R = ABC .Finally, suppose that the last applicant, , acquires information and learns B ≻ A ≻ C .Then the probabilities become as in Table 4. Note that applicant is worse off when acquires information, as now has a lower chance of being assigned to school B ; creates a displacement externality. Schools A B C
Applicants , , R , = BAC and R , = ABC .It is intuitive, and I will verify it formally below, that uninformed applicants are alwaysbetter off when other applicants acquire information and reduce the demand for popularschools. The displacement externality is negative, but could only arise when sufficientlymany applicants are informed and prefer B to A . I derive a condition that guarantees nodisplacement externality and that is satisfied in most environments. Then the equilibriuminformation acquisition under RD is below the social optimum.7 Setup
The paper follows the continuum matching framework introduced by Azevedo and Leshno(2016), who show that assignments in large finite economies converge to the assignment inthe continuum economy.A mass 1 of applicants is allocated to three schools, A , B , and C . The quota is q A ≤ / A and q B ≤ / B . The quota at school C is 1. That is, it can accept allapplicants not assigned to schools A and B and can be thought of as an outside option.For notational convenience, all applicants have the same ordinal ranking of two schools, A and C , with normalization u i ( A ) = 1 and u i ( C ) = 0 for every applicant i (see Section 9 fordetails). Applicant i ’s utility of attending school B is a draw u i ( B ) ∈ R from a distribution F ( x ). The distribution is known by and common for all applicants and has mean u B ∈ [0 , x = 0 and x = 1; hence applicants are indifferent between any twoschools with probability zero. I also assume that F (1) ≥ q A / ( q A + q B ); the requirement that F (1) is not “too small” implies that B is not too often preferred to A (recall that u i ( A ) = 1).This condition guarantees that school A is always at least as demanded as B . Applicant i doesnot know u i ( B ), but can learn it at cost c ( i ); only this cost differentiates applicants ex-ante.I assume that c ( i ) is a strictly increasing continuous function with c (0) ≥ In the main body of the paper I will consider a “symmetric case.” It consists of twoadditional assumptions: (i) The capacities of schools A and B are equal: q A = q B = q and (ii)The distribution F ( x ) is symmetric around the mean, with u B = 1 /
2. The latter implies that F ( x − u ) = 1 − F ( x + u ). In the symmetric case, the results depend only on two parametersof distribution F , which deserve their own notations:Φ = F (0) = 1 − F (1) , (1) I = Z −∞ ( − x ) d F ( x ) = Z ∞ ( x − d F ( x ) . (2)The main results of the paper are valid for a non-symmetric case, under more stringentconditions. They are presented in Appendix A.2.The timing is as follows. First, applicants learn their disutility from acquiring informa-tion, c ( i ), which can be assumed to be drawn from an arbitrary atomless distribution. Afterobserving the cost, some applicants choose to learn their private value of attending school B , u i ( B ). Then all applicants submit their ROLs to the centralized clearinghouse withoutobserving anything about other applicants and are assigned to schools. That is, applicant’s This assumption can be relaxed at the expense of some technical complexity. The assumption that applicants do not observe the decisions of the others simplify the notation, but can beeasily dispensed with. Indeed, applicants base their decisions on aggregates, which are known in equilibriumand are not affected by individual deviations. s i = ( e i , ˆ R i ( u i ( B ))), where e i ∈ { , } and ˆ R i is a permutation of A, B and C ,which may depend on the realization of u i ( B ). As a seat in C is guaranteed to any applicant,any ROL is truncated just below C for notational simplicity.An applicant’s priority is determined by a (uniformly distributed) random number r i ∈ [0 , r i < r j , than applicant i has a higher priority than j at every school. In themain part, I assume that applicants do not know their priorities when they decide to acquireinformation; other cases are handled in Sections 6.1 and 9.1.Since each applicant i has the same random priority at all school (given by r i ), DA isequivalent to Random Serial Dictatorship (RD) mechanism. In this section, I find equilibria in the game where applicants acquire information and then areassigned using RD. I first establish, in subsection 3.1, that it is sufficient to focus on applicants’truthful reporting. As applicants report truthfully, I find the total demand for school seatsand choose school admission cutoffs r A , r B , and r C = 1 to equate the demand for and supplyof seats (subsection 3.2). The cutoffs determine the set of schools which are feasible to anapplicant: { X ∈ { A, B, C } : r i ≤ r X } . Applicants are assigned to their highest-rankedfeasible school. Given the cutoffs, I find the expected utility of informed and uninformedapplicants (subsection 3.3); the difference between the two determines the threshold value ofthe applicant’s learning cost. That threshold value determines the equilibrium, as higher-costapplicants stay uninformed and the rest learn (subsection 3.4).
Consider an applicant who takes cutoffs r A and r B as given. Suppose that applicant i picksstrategy s i = ( e i , XY Z ), where XY Z is an arbitrary permutation of { A, B, C } . Then theexpected utility of applicant i is Eu i ( e i , XY Z ) = Pr( r i ≤ r X ) Eu i ( X ) + Prob( r X < r i ≤ r Y ) Eu i ( Y )+ Pr(max { r X , r Y } < r i ≤ r Z ) Eu i ( Z ) − e i c ( i )= r X Eu i ( X ) + max { r Y − r X , } Eu i ( Y ) ( ∵ r X , r Y ∼ U [0 , { r Z − max { r X , r Y } , } Eu i ( Z ) − e i c ( i ) (3) The tie-breaking rule that gives applicants the same priorities at all schools is called “single tie-breaking.”It has efficiency benefits over other rules (Pathak and Sethuraman, 2011). Note that the lower r i , the more schools are feasible to i ; this definition of r i will later allow me to interpret r X as the probability of being admitted to school X . laim 1. Let Eu i ( X ) > Eu i ( Y ) > Eu i ( Z ). Then Eu i ( e i , XY Z ) ≥ Eu i ( e i , R ) , (4)where R is an arbitrary permutation of the set { X, Y, Z } .Furthermore, if r A > r B > r C = 1 and equation (4) holds with equality, then R generate the same probability distribution over school assignments as XY Z .Given Claim 1, I assume that every applicant reports either the truthful ROL – the ROLthat corresponds to the applicant’s expected utilities – or an ROL that generates the sameprobability distribution over school assignments as the truthful ROL.
Denote the fraction of informed applicants by γ . As applicants report truthful ROLs, γ determines the demand for school seats; cutoffs r A and r B equate it to the supply of seats. Claim 2.
For any γ ∈ [0 , r A ≤ r B ; r A = r B only if γ = 1 and F (0) = 1 / B rejected applicant i ( r i > r B ), school A would also reject i ( r i > r A ). Thus, only an applicant who lists A above B in her ROL may be assigned to A .The mass of applicants who list A above B is the sum of (i) (1 − γ ) uninformed applicants;and (ii) γ F (1) informed applicants who learn that u i ( B ) <
1. Thus, the supply of is equal tothe demand for seats in A when r A solves (5): q = Prob( r i ≤ r A ) ((1 − γ ) + γ F (1)) = r A (1 − γ Φ) , (5)where Φ is defined earlier in equation (1) as Φ = F (0) = 1 − F (1).To calculate the cutoff for school B , note that applicants assigned to B are:(i) Informed applicants with R = BAC ; among them, those with r ≤ r B are assigned to B ; and(ii) Applicants with R = ABC who are rejected from A ; among them, those with r A < r ≤ r B are assigned to B .Thus, the following equation determines the cutoff r B : q = r B γ (1 − F (1)) + ( r B − r A ) [(1 − γ ) + γ ( F (1) − F (0))]= r B (1 − γ Φ) − q + r A γ Φ (6)Algebraic manipulations of equations (5) and (6) lead to the following claim.10 laim 3.
The cutoffs in RD game are r A = q − γ Φ , (7) r B = q − γ Φ(1 − γ Φ) , (8)where Φ is defined by equation (1). In this section, I focus on the gain from information acquisition and ignore its cost. As allapplicants are ex-ante identical except for their costs, I suppress index i of an applicant inall expressions for expected utilities. I do not simplify equations for the symmetric case untilClaim 4 to highlight the intuition.Let U (1 , R ( · ) | r A , r B ) denote the expected utility that informed applicant i who submitstrue ROL R i ( u i ( B )) derives from her school assignment, given cutoffs r A and r B . The ROLmay depend on the value u i ( B ). Similarly, let U (0 , ABC | r A , r B ) denote the expected utilityof uninformed applicant i who submits ABC .The expected utility of an uninformed applicant is U (0 , ABC | r A , r B )= Pr( r ≤ r A ) u A + Pr( r A < r ≤ r B ) E ( u i ( B )) + Pr( r > r B ) u C = r A + ( r B − r A ) u B , (9)where r is applicant’s priority. The last equality takes into account u A = 1 and u C = 0.An informed applicant learns R ∈ { ABC, BAC, ACB } with probabilities ( F (1) − F (0)),(1 − F (1)), and F (0) respectively. Her expected utility is U (1 , R ( · ) | r A , r B ) (10)=( F (1) − F (0)) [ r A u A + ( r B − r A ) E ( u i ( B ) | ≤ u i ( B ) ≤
1) + (1 − r B ) u C ]+(1 − F (1)) r B [ E ( u i ( B ) | u i ( B ) >
1) + (1 − r B ) u C ]+ F (0) [ r A u A + (1 − r A ) u C ] . The utility gain from acquiring information is∆ U ( r A , r B ) = U (1 , R ( · ) | r A , r B ) − U (0 , ABC | r A , r B )= r A (1 − F (1))( E ( u i ( B ) | u i ( B ) > −
1) + ( r B − r A ) F (0)(0 − E ( u i ( B ) | u i ( B ) < . (11)11he derivations can be found in Appendix A.1, proof of Claim 4.Formula (11) has a natural interpretation. Suppose i is informed. She changes her ROL onlyif u i ( B ) [0 , − F (1)). Her assignment changes when shesubmits R i = BAC instead of R i = ABC and r i ≤ r A , which happens with probability r A .The gain from the change in the assignment is E ( u i ( B ) | u i ( B ) > −
1. Jointly, it gives thefirst term in equation (11). Similarly, if u i ( B ) <
0, the second term consists of the expectedgain of an assignment to C instead of B , the probability that r A < r i ≤ r B , and the probabilitythat R i = ACB .In symmetric case, equation (11) simplifies to (12).
Claim 4.
An applicant’s gain from acquiring information is given by∆ U ( r B ) = r B I , (12)where I is defined by (2). Consider a function i ∗ , defined as follows: i ∗ (∆ U ) = , if ∆ U < c (0) c − (∆ U ) , if ∆ U ∈ [ c (0) , c (1)] , , if ∆ U > c (1) . (13)This function is well-defined because c ( i ) is strictly increasing and it is continuous because c ( i ) is continuous. For any i ∗ ∈ (0 , U is equal to the cost c ( i ∗ ). Definition 1.
Let Γ be a collection of γ ∈ [0 ,
1] such that: γ = i ∗ (cid:18) q − γ Φ(1 − γ Φ) I (cid:19) Theorem 1.
The set Γ is non-empty. Theorem 2.
For each γ ∗ ∈ Γ , a strategy profile { (ˆ e i , ˆ R i ) } i ∈ N defined as: ˆ e i = if i > γ ∗ if i ≤ γ ∗ , (14)12 nd ˆ R i = BAC if ˆ e i = 1 and u i ( B ) ≥ ACB if ˆ e i = 1 and u i ( B ) ≤ ABC otherwise, (15) is a Nash equilibrium.Furthermore, if a strategy profile { ( e ′ i , R ′ i ) } i ∈ N is a Nash equilibrium, then (i) the fractionof informed applicants, γ ′ , is in Γ and (ii) the assignment is the same as given by the strategyprofile (14)-(15) corresponding to γ ′ , except, possibly, for applicant i = γ ′ . In certain cases, equilibria described in Theorem 2 can be Pareto ranked.
Theorem 3.
Suppose that Γ has at least two elements, γ L and γ H , such that γ L < γ H . If γ H Φ ≤ , then the equilibrium associated with γ H Pareto dominates the equilibrium associatedwith γ L . Note that the condition 3 γ H Φ ≤ ≤ /
2. Hence,the condition holds if γ H ≤ /
3. Second, if γ H = 1, then the condition holds when Φ ≤ / ABC is correct at least1/3 of the time. Example 2 shows that if this condition is not satisfied, then the Paretoranking may be reversed.
I assume that the mechanism – RD – is fixed and the school district directs some applicants toacquire information. Applicants are then free to decide which ROL to submit and, accordingto Claim 1, submit the truthful ROL. In this section, I find the socially optimal fraction ofinformed applicants and show that there is under-acquisition of information in equilibrium.Since applicants with higher index i have higher costs of information acquisition c ( i ) butotherwise are ex-ante identical, the district problem can be expressed as selecting γ ∈ [0 , i ≤ γ are informed and i > γ are not. The value γ SO maximizes SW ( γ ) = (1 − γ ) U (0 , ABC | γ ) + γU (1 , ˆ R | γ ) − Z γ c ( i ) di, (16)where I suppress r A , r B in expression U ( e i , ABC | r A ( γ ) , r B ( γ )). The expression (16) can beinterpreted as the ex-ante – before i knows her cost realization c ( i ) – expected utility of A more general formulation, where the district selects a function e : [0 , → { , } so that applicant i acquires information if e ( i ) = 1 and does not if e ( i ) = 0, can straightforwardly be reduced to the case ofselecting a single value γ ∈ [0 , i . Indeed, if i > γ , which happens with probability (1 − γ ), i is uninformed andobtains U (0 , ABC | γ ). If i ≤ γ , which happens with probability γ , i is informed, obtains U (1 , ˆ R | γ ) and, in expectation, pays the cost R γ c ( i ) di . Thus, γ SO maximizes ex-ante expectedutility of applicant i (recall that all applicants are ex-ante identical).Differentiating with respect to γ , I obtain SW ′ ( γ ) = (cid:18) γ ∆ U ( γ ) + U (0 , ABC | γ ) − Z γ c ( i ) di (cid:19) ′ = [∆ U ( γ ) − c ( γ )] + γ ∆ U ′ ( γ ) + U ′ (0 , ABC | γ ) (17)Expression (17) leads to the following theorem: Theorem 4.
For any γ RD ∈ Γ , if γ RD Φ ≤ / and γ RD < , then γ RD < γ SO . The main observation for the proof of this theorem is that at γ RD ∈ Γ, the first term,∆ U ( γ ) − c ( γ ) is equal to zero. The sign depends on the second term, which is positive if γ RD Φ ≤ / Before proceeding to the examples, it is useful to combine equations (12) and (8) into theequilibrium condition that determines γ RD :∆ U ( γ RD ) = q − γ RD Φ(1 − γ RD Φ) I = c ( γ RD ) . (18) Example . This example illustrates Theorems 3 and 4. There are three equilibria. For thefirst two, γ RD , Φ ≤ / γ RD Φ > / q = 1 / c ( i ) = 0 .
35 + ai , where a is a very small constant; that is, all applicantsface almost identical cost of acquiring information. Let u i ( B ) = 2 with probability 1/2 and u i ( B ) = − Eu i ( B ) = 1 / / I = 1 /
2. The equation (18) becomes∆ U ( γ ) = 13 2 − γ/ − γ/
12 = 0 .
35 + aγ.
It has two solutions: γ ≈ .
20 and γ ≈ .
94. Note that ∆ U (0) < .
35, so { } ∈ Γ. Also notethat ∆ U (1) < .
35, hence { } / ∈ Γ. That is, there are three equilibria: with 0%, 20% and 94%of applicants acquiring information. Ex-ante utility in each of these cases are 0.5, ≈ . ≈ . γ SO ≈ . . × . × . >
1. In that equilibrium,too many applicants are informed relative to the social optimum.
Example . In this example, I show that if condition γ RD Φ ≤ / F ( x ) such that too many applicants are informed in RD, γ RD > γ SO .Let Φ = (1 + ǫ ) / ǫ > c ( i ) = 1 − ǫ − ǫ I i. Then γ = 1 solves equation (18) since2 − ǫ ) / − (1 + ǫ ) / ǫ − ǫ (2 − ǫ ) . Thus, γ RD = 1. At γ = 1, equation (17) can be written as SW ′ (1) = ∆ U ′ (1) + U ′ (0 , ABC |
1) = r ′ B (1) I + r ′ A (1) + r ′ B (1)2 . (19)Note that, for any ǫ , r ′ B (1) < r ′ A (1) > F ( x ) except through Φ. Thus, for any ǫ , pick F ( x ) so that I > − r ′ A (1) + r ′ B (1)2 r ′ B (1) . An instance of such a distribution is F ( x ) with mass points at x < − x , where x < r ′ A (1)+ r ′ B (1))2 r ′ B (1)(1+ ǫ ) , and uniform on [0 , F ( x ) = x < x ;(1 + ǫ ) / x ∈ [ x , ǫ ) / x (1 − ǫ ) /
3) for x ∈ [0 , − (1 + ǫ ) / x ∈ [1 , − x ];1 for x > − x . In that case, SW ′ ( γ RD ) = SW ′ (1) <
0, thus too many applicants are informed.15
Policy implications
In this section I propose three policies that lead to Pareto improvement by changing incentivesto acquire information. Throughout the section I assume that γ RD < I consider the same model as in Section 3, with the exception that applicants know theirpriorities, r i , before they acquire information; thus, they make their information acquisitiondecisions based on both the priority and the cost. I continue to assume that lottery prioritiesare not correlated with the cost of information acquisition.The policy leads to a Pareto improvement, but under-acquisition of information still oc-curs. Furthermore, it leads to cost distortion: low-priority low-cost applicants do not acquireinformation, while high-priority high-cost applicants do.I first show that revealing priorities leads to Pareto improvement.
Theorem 5.
The utilities of applicants weakly increase when priorities are revealed beforeinformation acquisition. There is a positive mass of applicants for whom the increase is strict.
To see why Pareto improvement is possible, first note that if there are no equilibrium effects– school cutoffs r A and r B remain the same – then each applicant is weakly better off havingmore information. High-priority applicants are more certain that their information would beusable, so they have higher incentives to acquire information. The cutoffs change, and bothschools become more accessible.Note that the total mass of applicants acquiring information may decrease when prioritiesare revealed because low-priority applicants have no incentives to acquire information; it isalso not socially optimal for them to do so.Even though the policy leads to a Pareto-improvement, too few applicants are informed.The argument is presented below and closely follows the one in the main model. Denotecutoffs in schools A and B by r aaa and r bbb , respectively. Denote the fraction of applicants withpriority sufficient to be admitted to school A by γ aaa and to school B by γ bbb ; γγγ = ( γ aaa , γ bbb ).I define the social welfare function as the sum of all utilities, taking into account thatuninformed applicants with priority r i ≤ r aaa have utility 1 and with priority r aaa < r i ≤ r bbb haveutility 1/2; and applicants with r i > r bbb have utility 0: SW ( γγγ ) = r aaa ( γγγ ) (cid:18) γ aaa I − Z γ aaa c ( x ) dx (cid:19) + ( r bbb − r aaa )( γγγ ) (cid:18)
12 + γ bbb I − Z γ bbb c ( x ) dx (cid:19) (20) The policies of information revelation and non-revelation can be seen as two extremes of a more generalpolicy, where the designer reveals priorities only to some applicants. However, in this environment, it isalways (weakly) optimal to reveal priority information to all applicants. γ ∗ solve c ( γ ∗ ) = I and r aaa = q − γ ∗ Φ , r bbb = q − γ ∗ Φ(1 − γ ∗ Φ) . Note that r aaa and r bbb are defined by the same equations as r A and r B ; however, the value of γ ∗ is higher than γ RD . Denote C = 2 γ ∗ Φ (cid:18) Z γ ∗ ( I − c ( x )) dx (cid:19) Theorem 6.
Under the policy of revealing priorities before information acquisition,1. Among applicants with r i < r aaa , too few are informed if C < and too many if C > ;2. Too few applicants with priorities r aaa < r i ≤ r bbb are informed;3. Applicants with priorities r i > r bbb are uninformed and it is socially optimal. Note that, unlike Theorem 4, Theorem 6 unconditionally claims under-acquisition of in-formation by a group of applicants with “medium” priorities. For applicants with “good”priorities, additional condition
C <
C <
1, first note that it holds for sufficiently small γ ∗ for anydistribution F ( x ). Thus, in the field, if the district observes that few applicants are informed,then a policy intervention is likely required. Second, recall that for any γ < γ ∗ , c ( γ ) < I .Thus, the more homogenous the costs are across applicants, the more likely that too few areinformed. Finally, let us suppose that Φ ≤ /
3. For both uniform and normal distributions
I ≤ / C < c ( x ). Thus, condition C <
In this section, I allow the priority to depend on information acquisition decision. If the schooldistrict is the primary provider of information, it may be able to observe the decision directly.Alternatively, it can elicit the intention to acquire information; ex-ante, revealing such anintention is incentive-compatible for applicants. It is essential, however, that the district canelicit the intention before applicants acquire information. Otherwise, an applicant who findsthat school B is worse than C is better off pretending to be uninformed.I assume that the district cannot force applicants to acquire information; however, it can re-allocate seats between informed and uninformed applicants. The re-allocation changes schoolcutoffs and can be done in such a way that more applicants become informed.17lternatively, this scheme can be viewed as mimicking affirmative action. Suppose thatthe district identifies a target group of uninformed applicants who would acquire informationif more seats at school B are available to them. Being uninformed, this group is under-represented in B . The district accounts for the equilibrium response of this group and allocatesmore seats in B to them. The group becomes informed, positively affecting other applicants.I consider the same model as in Section 3, except that the district allocates q A , q B seats touninformed and q A , q B to informed applicants, at schools A and B respectively. Let r A , r B , r A and r B be the equilibrium cutoffs for uninformed and informed applicants. Anticipating thecutoffs, applicants decide whether to acquire information (and face cutoffs r A and r B ) or not(and face cutoffs r A and r B ). The district selects seat allocation so that r B > r RDB . Informedapplicants value seats at school B more than uninformed ones; thus the value of informationincreases. Uninformed applicants are compensated for the loss of seats in B by seats in A . Theorem 7.
Let Φ < / . Suppose that cost function c ( i ) is continuous, strictly increasingand convex. There exist γ p ∈ ( γ RD , , seat allocation q A , q B , q A , q B and an equilibrium suchthat applicants i ≤ γ p are informed, applicants i > γ p are not, and all applicants submittruthful preferences. In this equilibrium, applicants i > γ p have the same expected utility asunder RD and applicants i ≤ γ p are better off. Note that the theorem does not guarantee that γ p = γ SO . Indeed, if γ RD is close to 1, thenthere may be too few uninformed applicants for the redistribution to achieve social optimum.This policy does not introduce a cost distortion, unlike the policy of revealing priorities ex-ante: the lowest-cost applicants acquire information. In this policy, information acquisition is subsidized by imposing a flat tax on all applicants.Specifically, τ units of utility are collected from everyone. Then at least κ > τ is used to subsidize information acquisition of the target group of applicants with indices i ∈ [ γ RD , γ τ ], where γ τ is a policy variable. The parameter κ may capture the district’sinability to perfectly identify the target group (e.g., the district may have to subsidize everyinformed applicant); wider economic distortions of taxation; or inefficiencies in the utilitytransfer. Note that κ may be larger than 1 because the centralized provision of informationmay be more efficient than an individual search, for example when information is providedin other languages for non-native speakers. For simplicity, I assume that (1 − κ ) τ is lost anddoes not enter anyone’s utility; any other assumption would only strengthen my result.The applicants in the target group are given a subsidy c ( i ) − c ( γ RD ), so that their cost18ecomes c ( γ RD ). The tax needed for this scheme is τ ( γ τ ) = 1 κ Z γ τ γ RD (cid:0) c ( x ) − c ( γ RD ) (cid:1) dx, (21)and the utility of applicant i ∈ [ γ RD , γ τ ] in the tax-subsidy scheme describe above is U τi = U ( e ( i ) , ˆ R i | γ τ ) − τ ( γ τ ) − e ( i ) min { c ( i ) , c ( γ RD ) } . Note that U i corresponds to the case where there is no subsidy and γ τ = γ RD . Note alsothat, once I allow for utility transfers, the normalization u A = 1, u C = 0 is no longer withoutthe loss of generality. Theorem 8.
Let Φ < / . For each γ > γ RD , let τ ( γ ) be the total tax collected from allapplicants, given by formula (21). Then for any κ > , there exists γ τ > γ RD such that:(i) every applicant i ∈ [ γ RD , γ τ ] is subsidized so that c ( i ) = c ( γ RD ) and(ii) every applicant is better off: for every i ∈ [0 , , U τi > U i . The proof is based on two observations: (i) the worst-off applicant is uninformed and (ii)the utility of uninformed applicant is increasing with γ . In this section, I show that, when the costs are not “too low,” another popular assignmentmechanism, Immediate Acceptance (IA), provides higher incentives to acquire informationthan does RD. IA is the mechanism that has been widely used until recently, most prominentlyin Boston (Pathak and S¨onmez, 2008), but that has been criticized for its lack of strategy-proofness and for producing unfair (unstable) assignments.I focus exclusively on comparing informational incentives between two mechanisms anddo not analyze welfare. The reason is that the problems with IA, well-documented in theliterature, cannot be addressed in the environment of the present paper. As a side note,uninformed applicants are worse off in IA compared to RD for a range of parameters.Throughout this section, I impose an additional assumption that cumulative distributionfunction F ( x ) is continuous on [0 , e i ∈ { , } ) and (ii)what ROL R i ( · ) to submit ( R i ( · ) ∈ { ABC, AC, BAC, BC, C } ). As C can accommodate allapplicants, schools listed below C are irrelevant and excluded. Denote applicant’s strategyby S i = ( e i , R i ( · )). R i ( · ) may depend on the acquired information about B . Applicants donot observe information decisions and ROLs of others. Each applicant is assigned a randompriority r i , unknown to applicants at the time they submit their ROLs.19he district collects applicants’ ROLs and assigns applicants according to IA algorithm,which, in the given environment, is described as follows. For applicant i , • If the quota in i ’s top-ranked school is not exhausted by applicants who rank thatschool as their own top choice and whose priority is better than r i , i is assigned to hertop-ranked school; otherwise i is rejected. • If i is rejected from her top-ranked school, then that school is removed from R i ; thequotas in all schools are reduced by the mass of applicants assigned to them in thecurrent round; and the process is repeated. • The algorithm stops when all applicants are assigned.The outcome of the IA algorithm can be expressed as a sequence of cutoffs ρ kA , ρ kB , cor-responding to the k th round of the algorithm. The cutoffs are defined as follows: ρ kA =max { r i | i is admitted to A in round k } , ρ kB = max { r i | i is admitted to A in round k } . If noapplicant is admitted to school X in round l , ρ lX = 0. For notational simplicity, I dropsuperscript for round one, so that ρ A = ρ A , ρ B = ρ B .There always exists an equilibrium where schools A and B are full after round 1, which caninformally be described as follows: • For high informational costs: the fraction of informed applicants is γ ∈ [0 , / ABC (with probability at least 2/3) and
BAC . • For low costs: the fraction of informed applicants is γ ∈ [2 / , u ∈ [ u B , u A )such that informed applicants submit ABC when u i ( B ) < ¯ u and BAC otherwise; andall uninformed applicants submit truthful ROL
ABC .The formal statement of the result is Lemma 1 in Appendix. In the proof of Lemma 1, it isalso established that when γ IA ≤ /
3, the cutoffs ρ A and ρ B do not change as more applicantsbecome informed because uninformed applicants change their randomization strategy accord-ingly. Hence, uninformed applicants’ utility does not change when γ IA ≤ /
3. As IA and RDresult in the same assignment at γ = 0, and uninformed applicants’ utility is increasing inRD, the following corollary to Lemma 1 obtains. Corollary 1.
If the cost function c ( x ) is such that c (0) < q Z ∞ ( x − d F ( x ) (22)and c (2 / ≥ q Z ∞ . ( x − . d F ( x ) , (23)then uninformed applicants have higher utility in the equilibrium of RD than in the equilibriumof IA in which schools A and B are full after round 1.20 turn to the informational incentives next. Theorem 9.
Suppose that cost function c ( x ) is such that inequalities c (0) < q Z ∞ . ( x − . d F ( x ) (24) and c (8 / > − / − (8 / Z ∞ ( x − d F ( x ) (25) hold. Then the equilibrium level of information acquisition under IA is higher than equilibriumlevel of information acquisition under RD, γ IA > γ RD . Inequality (24) implies that some applicants are informed under IA. Inequality (25) impliesthat less than 8/9 applicants are informed under RD. The theorem applies to any equilibriumof IA, not only the one described in Lemma 1. When other equilibra exist, they lead to higherinformation acquisition.
Example . If inequality (25) does not hold, more applicants may acquire information underRD than under IA. Let c ( x ) = 3 q/
4. Suppose that u i ( B ) can take one of four values: -1 and 2with probabilities 1/3, ǫ and 1 − ǫ with probabilities 1/6. Then Φ = 1 / R ∞ ( x − d F ( x ) =1 /
3. For any ¯ u ∈ [ u B , − ǫ ), F (¯ u ) = 1 / R ∞ ¯ u ( x − ¯ u ) d F ( x ) = 2 / − ǫ ) / − ¯ u/ / − ǫ/ − ¯ u/ γ RD is determined from the equation3 q c ( γ RD ) = q − γ RD / − γ RD /
13 = 3 q − γ RD (3 − γ RD ) which is solved when γ RD = 1: all applicants are informed under RD .The equations (A.27) and (A.28) in the Appendix that describe equilibrium under IA canbe simplified to ¯ u = γ IA − γ IA and 34 = 13 γ IA (cid:18) − γ IA − γ IA − ǫ (cid:19) (26)Note that at γ IA = 1, the RHS of equation (26) is equal to (2 − ǫ ) / < / ǫ >
0. Thus, γ IA < γ RD . In fact, γ IA increases as ǫ decreases; hence γ IA < (25 − √ / ≈ . This probability distribution does not satisfy the assumption on F ( x ) maintained through the section.However, the choice of ǫ will ensure that the results are still valid in this specification.
21n turn, the bound on γ IA implies that ¯ u < ( √ − / ≈ .
94. Thus, for any ǫ < . u < ǫ holds and γ IA found above is the fraction of applicants acquiringinformation in IA. This fraction is lower in IA than in RD. My model has three schools and the utility of only the middle-ranked school is uncertain.Using simulations, I show that the results are robust to adding more schools and uncertainty.Specifically, information is under-acquired except in the case where middle-ranked schoolshave much higher uncertainty than the rest. That is, my model where only middle schoolis uncertain is most unlikely to yield low information acquisition. As the model predicts,under-acquisition disappears only at very high levels of γ RD and is more likely to occur whenthe distribution has fat tails. In other words, in nearly all environments a policy promotinginformation acquisition would be beneficial. On average, the magnitude of the benefits iscomparable to moving a quarter of a population to a higher-ranked school, but may be muchlarger in some cases.The simulations are conducted as follows. I assume that the applicant who pays the costlearns utilities from all schools, so she does not face a problem of which schools to learn about.The cost function is c ( i ) = c × i . Let S be the set of schools, with |S| = N and σ s be thestandard deviation of the idiosyncratic component of a utility that an applicant derives fromattending school s ∈ S . For each profile of school standard deviations ( σ s ) s ∈S , and for eachlevel of information acquisition γ RD ∈ { . , . , . . . , . , } , I find the cost coefficient c ∗ such that γ RD is the equilibrium level of information acquisition. I then find the sociallyoptimal level of information acquisition, γ SO , that corresponds to cost function c ∗ × i . As γ RD varies from 0.01 to 1, I have 100 simulations for different cost functions and for each profileof standard deviations.I vary the number of schools between 3 to 6. The distribution of applicants’ utilities ateach school s ∈ S , F s ( x ), follows either normal or uniform distribution, with the standarddeviation being an integer between 0 and 9.First, I ask how often a policy that encourages information acquisition is not needed. Ireport all cases when γ RD ≥ γ SO in Table 5. On average, it happens in less than 2% ofall simulations, mostly when the standard deviations of top- and bottom-ranked schools aresmall and the standard deviations of some of the middle-ranked schools are large (note thathe column ∆ σ ≤ σ = max s ∈ S \{ s ,s N } { σ s } − min s ∈{ s ,s N } { σ s } , where s and s N are the top- and bottom-ranked schools. The majority of cases of over-acquisition ofinformation happen when ∆ σ is equal to 8 or 9. Furthermore, most instances of over-acquistion22ccur when γ RD is close to 1 and more frequent under normal than under uniform distribution.Table 5: Over-acquisition of information, by the number of schools and the difference of stan-dard deviations between top and bottom schools ↓ ∆ σ ≤ ≤ T h r ee Obs. 67 95 114 87 9 48 71 57% Total .17 2.55 4.07 5.62 .03 1.35 2.65 3.84 γ RD Mean .97 .96 .95 .94 .99 .94 .97 .96 γ RD Min .91 .89 .87 .85 .97 .98 .91 .90 F o u r Obs. 965 1501 2007 1590 395 632 961 838% Total 0.23 2.24 3.78 5.16 .10 1.01 1.92 2.85 γ RD Mean .96 .95 .95 .94 .96 .95 .95 .95 γ RD Min .79 .76 .73 .70 .85 .81 .78 .78 F i v e Obs. 12738 18617 24850 20170 5174 8623 11961 10409% Total 0.29 2.09 3.33 4.44 0.13 1.04 1.72 2.43 γ RD Mean .96 .96 .95 .95 .97 .96 .96 .95 γ RD Min .76 .72 .69 .68 .84 .80 .77 .77 S i x Obs. 177129 229737 299629 247350 66287 99721 142223 128050% Total 0.39 2.17 3.24 4.20 0.16 1.02 1.65 2.30 γ RD Mean .97 .96 .96 .95 .97 .97 .96 .96 γ RD Min .76 .72 .69 .68 .83 .80 .77 .77
Notes:
Reported are the cases of over-acquisition of information for a different number of schools. ∆ σ = max s ∈ S \{ s ,s N } { σ s } − min s ∈{ s ,s N } { σ s } , where s and s N are top- and bottom-ranked schools; % Total is the percentage of observations with over-acquisition of information from all observation with a given ∆ σ . Having established that the policy is beneficial in most cases, I turn to evaluating its ben-efits. I report the difference between socially optimal and equilibrium levels of informationacquisition, ∆ γ = γ RD − γ SO , and welfare gains, ∆ SW = SW ( γ SO ) − SW ( γ RD ). Table 6shows the results. The average difference may be seen as moderate, but it is obscured by high-and low-cost cases. The low-cost case is where γ SO = 1 but γ RD <
1. A further cost decreasedoes not change γ SO , but increases γ RD , shrinking the difference. The high-cost case is whenboth γ RD and γ SO are small in absolute terms, but their ratio is large. To mitigate the effectsof low and high costs, I also report the mean for a restricted sample γ RD ∈ [0 . , . γ and ∆ SW when variances of all schoolsare equal. The means are similar to the full sample, suggesting that unusual configurations23f standard deviations are not the main driver of the results.In summary, simulations suggest that the results reported in the theoretical three-schoolmodel strengthen in more complex markets.Table 6: The differences in the fraction of informed applicants (∆ γ ) and in social welfare(∆ SW ) at socially optimal and equilibrium levels Normal Distribution Uniform Distribution −
100 10 −
100 10 −
100 10 −
100 10 −
500 10 −
800 10 − − ∆ γ = γ SO − γ RD Mean .22 .20 .18 .17 .25 .23 .21 .20p90 .49 .43 .38 .34 .53 .46 .41 .38p95 .56 .49 .44 .39 .59 .53 .47 .43max .76 .75 .75 .77 .82 .82 .81 .82
Mean, γ RD ∈ [ . ,. .45 .39 .35 .31 .46 .42 .38 .35 Mean, samevariance .24 .23 .22 .21 .27 .26 .24 .24∆ SW = SW ( γ SO ) − SW ( γ RD )Mean .24 .22 .21 .19 .17 .16 .15 .14p90 .64 .59 .53 .47 .42 .39 .35 .30p95 .76 .72 .66 .60 .50 .49 .44 .39max 1.05 1.09 1.12 1.12 .65 .73 .78 .79 Mean, γ RD ∈ [ . ,. .44 .38 .34 .32 .31 .28 .25 .23 Mean, samevariance .25 .24 .23 .23 .15 .14 .13 .13
Notes:
Observation is a combination of a profile of standard deviations and a cost. For each school s , σ s is an integer between 0and 9, making 10 | S | profiles, where | S | is the number of schools. For each profile, I simulate the market using 100 different costfunctions c ( x ) = cx , chosen so that γ RD ∈ { . , . . . . } . Thus, there are 10 | S | +2 observations. Excluded are observationswhere applicants cannot gain from information acquisition: the standard deviation profiles of all zeros for a normal distributionand profiles where, for every school s ∈ S , σ s ∈ { , } and if σ s = 1, then σ s − = σ s +1 = 0 ( σ − and σ | S | +1 are taken to bezero) for uniform distribution. pX is a corresponding X’s percentile. Last two columns in each sub-tables report means for arestricted sample: when γ RD ∈ (0 . , .
7] (40% of the sample) and when σ s = σ s ′ for all schools s, s ′ ∈ S (900 observations perdistribution). I have assumed that only a lottery determines applicants’ priorities. In a typical school choicecontext, “in-zone” applicants are prioritized over the others. I argue below that the resultsand intuition carry over to this environment.Let aaa, bbb, nnn be groups of applicants in zones of schools A , B , and of neither school, respectively.That is, if i ∈ ggg and j / ∈ ggg , where ggg ∈ { aaa, bbb } , then i has a higher priority at ggg than j . Iassume that an applicant is never rejected from her zone school. The priorities of out-of-zone24pplicants are determined randomly, as in the main model; r A and r B refer to the cutoffs facedby out-of-zone applicants.As different groups of applicants face different cutoffs, they may make different informationacquisition decisions. I denote the fractions of informed applicants in zones of A , B andneither school by γ aaa , γ bbb and γ nnn ; γγγ = ( γ aaa , γ bbb , γ nnn ). For each of these groups, the equation∆ U ggg ( γγγ ) = c ( γ ggg ) , ggg ∈ { aaa, bbb, nnn } determines the fractions of informed applicants.The social welfare is defined, as before, as the sum of utilities of all applicants: SW ( γγγ ) = X ggg ∈{ aaa,bbb,nnn } (cid:20) γ ggg ∆ U ggg ( γγγ ) + U ggg (0 , ABC | γγγ ) − Z γ ggg c ( i ) di (cid:21) It can be shown that applicants in group bbb always under-acquire information and applicantsin groups aaa and nnn under-acquire information if | nnn | ≥ | bbb | , Φ ≤ /
3, and I is sufficiently small.The derivations are similar to those in Section 6.2, which discusses the policy of alteringapplicants’ priorities in response to their information acquisition decisions. C above A I have assumed that relative ranking of schools A and C is certain. My results would onlybe strengthened if their ranking is uncertain. Indeed, such uncertainty would add three moreschool rankings: CAB , CBA and
BCA . When an applicant learns
CAB or CBA , then she isnot competing for desirable seats in schools A and B , hence creates a positive externality. Theprobabilities of assignments to schools A , B and C are the same for applicants who submit BCA and
BAC , because an applicant who is rejected from B is never assigned to A . Thus,the intuition provided for ranking BAC extends to
BCA .
10 Conclusion
Strategy-proof mechanisms have been widely advocated as replacements of manipulable mech-anisms such as Immediate Acceptance, but their interactions with a larger environment requiremore research. In this paper, I show that such mechanisms provide low incentives to acquireinformation and their adoption needs to be followed by policies that promote informationacquisition. Some policies that would encourage information acquisition are realistic and ob-served in real life. I also argue that some empirical conclusions regarding welfare comparisonsof mechanisms may be invalid if informational incentives are taken into account.25
I am grateful to Suren Basov, Scott Kominers, and Bobby Pakzad-Hurson for suggestionsand insightful discussions. I thank Ivan Balbuzanov, Estelle Cantillon, Yeon-Koo Che, DavidDelacretaz, Guillaume Haeringer, Yinghua He, Fuhito Kojima, Jacob Leshno, Shengwu Li,Simon Loertscher, Matt Jones, Michael Ostrovsky, Marek Pycia, Alvin Roth, William Thomp-son, Utku Unver, Steven Williams, seminar participants at Buenos Aires, Deakin, HigherSchool of Economics, Hitotsubashi, Monash, Stanford, Tsukuba, Matching in Practice Work-shop (Toulouse), SCW Conference (Boston), a Workshop at Victoria University of Wellington,International Workshop of Game Theory Society (Sao Paulo) for their comments. I gratefullyacknowledge support from the Australian Research Council grant DP160101350 and a FacultyResearch Grant at the University of Melbourne.
References
Abdulkadiro˘glu, A., Y.-K. Che, and
Y. Yasuda (2011): “Resolving Conflicting Pref-erences in School Choice: The “Boston Mechanism” Reconsidered,”
American EconomicReview , 101(1), 399–410.
Abdulkadiro˘glu, A., P. A. Pathak, and
A. E. Roth (2005a): “The New York CityHigh School Match,”
American Economic Review Papers and Proceedings , 95, 364–367.
Abdulkadiro˘glu, A., P. A. Pathak, A. E. Roth, and
T. S¨onmez (2005b): “TheBoston Public School Match,”
American Economic Review Papers and Proceedings , 95,368–372.
Abdulkadiro˘glu, A., and
T. S¨onmez (2003): “School Choice: A Mechanism DesignApproach,”
American Economic Review , 93, 729–747.
Agarwal, N., and
P. Somaini (2018): “Demand Analysis Using Strategic Reports: AnApplication to a School Choice Mechanism,”
Econometrica , 86(2), 391–444.
Azevedo, E. M., and
J. D. Leshno (2016): “A Supply and Demand Framework forTwo-Sided Matching Markets,”
Journal of Political Economy , 124, 1235–1268.
Aziz, H., P. Bir´o, S. Gaspers, R. de Haan, N. Mattei, and
B. Rastegari (2016):“Stable Matching with Uncertain Linear Preferences,” in
International Symposium on Al-gorithmic Game Theory , pp. 195–206. Springer.
Bade, S. (2015): “Serial Dictatorship: The Unique Optimal Allocation Rule when Informa-tion is Endogenous,”
Theoretical Economics , 10(2), 385–410.
Bergemann, D., and
J. V¨alim¨aki (2006): “Information in Mechanism Design,” in
Proceed-ings of the 9th World Congress of the Econometric Society , ed. by R. Blundell, W. Newey, and
T. Persson. Cambridge University Press.26 alsamglia, C., C. Fu, and
M. G¨uell (2018): “Structural Estimation of a Model ofSchool Choices: The Boston Mechanism vs. its Alternatives,” Discussion paper, NationalBureau of Economic Research.
Cantillon, E. (2017): “Broadening the Market Design Approach to School Choice,”
OxfordReview of Economic Policy , 33(4), 613–634.
Chade, H., G. Lewis, and
L. Smith (2014): “Student Portfolios and the College Admis-sions Problem,”
Review of Economic Studies , 81(3), 971–1002.
Chakraborty, A., A. Citanna, and
M. Ostrovsky (2010): “Two-sided matching withinterdependent values,”
Journal of Economic Theory , 145(1), 85–105.
Chen, L., and
J. S. Pereyra (2018): “Self-selection in School Choice,” Discussion paper.
Chen, Y., and
Y. He (2018a): “Information Acquisition and Provision in School Choice: ATheoretical Characterization,” Discussion paper.(2018b): “Information Acquisition and Provision in School Choice: An ExperimentalStudy,” Discussion paper.
Das, S., and
Z. Li (2014): “The Role of Common and Private Signals in Two-sided Matchingwith Interviews,” in
Web and Internet Economics , ed. by T.-Y. Liu, Q. Qi, and
Y. Ye, pp.492–497. Springer International Publishing.
De Haan, M., P. A. Gautier, H. Oosterbeek, and
B. van der Klaauw (2015):“The Performance of School Assignment Mechanisms in Practice,” Discussion paper, TheInstitute for the Study of Labor (IZA).
Dinkelman, T., and
C. Martinez A. (2014): “Investing in Schooling In Chile: The Roleof Information about Financial Aid for Higher Education,”
The Review of Economics andStatistics , 96(2), 244–257.
Drummond, J., and
C. Boutilier (2013): “Elicitation and Approximately Stable Matchingwith Partial Preferences.,” in
IJCAI , pp. 97–105.(2014): “Preference Elicitation and Interview Minimization in Stable Matchings.,”in
AAAI , pp. 645–653.
Dur, U., R. G. Hammond, and
T. Morrill (2018): “Identifying the Harm of ManipulableSchool-choice Mechanisms,”
American Economic Journal: Economic Policy , 10(1), 187–213.
Ehlers, L., and
J. Mass´o (2015): “Matching Markets under (In)complete Information,”
Journal of Economic Theory , 157, 295–314.
Ergin, H., and
T. S¨onmez (2006): “Games of School Choice under the Boston Mechanism,”
Journal of Public Economics , 90, 215–237. 27 arless, P., and
V. Manjunath (2018): “Learning Matters: Reappraising Object Alloca-tion Rules when Agents Strategically Investigate,”
International Economic Review , 59(2),557–592.
Hastings, J. S., C. A. Neilson, A. Ramirez, and
S. D. Zimmerman (2016):“(Un)informed College and Major Choice: Evidence from Linked Survey and Adminis-trative Data,”
Economics of Education Review , 51, 136–151.
Hastings, J. S., R. Van Weelden, and
J. Weinstein (2007): “Preferences, Information,and Parental Choice Behavior in Public School Choice,” Discussion paper, National Bureauof Economic Research.
Hastings, J. S., and
J. M. Weinstein (2007): “No Child Left Behind: Estimating theImpact on Choices and Student Outcomes,” Working Paper 13009, National Bureau ofEconomics Research.
Hastings, J. S., and
J. M. Weinstein (2008): “Information, School Choice, and Aca-demic Achievement: Evidence from two Experiments,”
The Quarterly journal of economics ,123(4), 1373–1414.
He, Y. (2017): “Gaming the Boston School Choice Mechanism in Beijing,” Discussion paper,Rice University.
Hoxby, C. M., and
S. Turner (2015): “What High-achieving Low-income Students Knowabout College,”
American Economic Review , 105(5), 514–517.
Hwang, S. (2017): “How Does Heterogeneity In Beliefs Affect Students In the Boston Mech-anism?,” Discussion paper.
Kadam, S. V. (2015): “Interviewing in Matching Markets,” Discussion paper.
Kapor, A., C. A. Neilson, and
S. D. Zimmerman (2018): “Heterogeneous Beliefs andSchool Choice Mechanisms,” Discussion paper, National Bureau of Economic Research.
Kessel, D., and
E. Olme (2018): “Are Parents Uninformed? The Impact of School Per-formance Information on School Choice Behavior and Student Assignment,” Discussionpaper.
Lee, R. S., and
M. Schwarz (2017): “Interviewing in Two-sided Matching Markets,”
RAND Journal of Economics , 48(3), 835–855.
Lien, J. W., J. Zheng, and
X. Zhong (2016): “Preference Submission Timing in SchoolChoice Matching: Testing Fairness and Efficiency in the Laboratory,”
Experimental Eco-nomics , 19(1), 116–150.(2017): “Ex-ante Fairness in the Boston and Serial Dictatorship Mechanisms underPre-exam and Post-exam Preference Submission,”
Games and Economic Behavior , 101,98–120. 28 ien, Y. (2009): “Costly Interviews and Non-Assortative Matchings,” Discussion paper,California Institute of Technology.
Liu, Q., G. J. Mailath, A. Postlewaite, and
L. Samuelson (2014): “Stable Matchingwith Incomplete Information,”
Econometrica , 82(2), 541–587.
Lovenheim, M. F., and
P. Walsh (2018): “Does Choice Increase Information? Evidencefrom Online School Search Behavior,”
Economics of Education Review , 62, 91–103.
Luflade, M. (2018): “The Value of Information in Centralized School Choice Systems,”Discussion paper, Duke University.
Miralles, A. (2008): “School Choice: The Case for the Boston Mechanism,” Discussionpaper.
Nathanson, L., S. Corcoran, and
C. Baker-Smith (2013): “High School Choice in NewYork City: A Report on the School Choices and Placements of Low-Achieving Students,”Discussion paper, Research Alliance for New York City Schools.
Pathak, P. (2016): “What Really Matters in Designing School Choice Mechanisms,” in
Advances in Economics and Econometrics, 11th World Congress of the Econometric Society ,ed. by L. Samuelson. Cambridge University Press, Cambridge, UK.
Pathak, P., and
J. Sethuraman (2011): “Lotteries in Student Assignment: An Equiva-lence Result,”
Theortical Economics , 6, 1–17.
Pathak, P. A., and
T. S¨onmez (2008): “Leveling the Playing Field: Sincere and Sophis-ticated Players in the Boston Mechanism,”
American Economic Review , 98(4), 1636–1652.
Rastegari, B., A. Condon, N. Immorlica, and
K. Leyton-Brown (2013): “Two-sided Matching with Partial Information,” in
Proceedings of the fourteenth ACM conferenceon Electronic commerce , pp. 733–750. ACM.
Wu, B., and
X. Zhong (2014): “Matching Mechanisms and Matching Quality: Evidencefrom a Top University in China,”
Games and Economic Behavior , 84, 196–215.29 ppendix to “Assignment mechanisms: commonpreferences and information acquisition”
A.1 Omitted proofs
Proof of Claim 1.
First, note that all the permutations of schools { X, Y, Z } form a groupwith respect to the operation of composition of permutations. I call this operation “groupmultiplication” and denote it by ∗ . A well-known property of this group (indeed, of any S n ) isthat any permutation can be written as a product of transpositions (a transposition is a per-mutation that exchanges the places of just two elements, leaving all other elements intact). A.1
Now, group S has three transpositions,: (1 , , (2 ,
3) and (1 , i, k ) denotes the trans-position that exchanges places of elements i and k . However, only two of transpositions areindependent, because (1 ,
3) = (1 , ∗ (2 , ,
2) and (2 , Eu i ( X ) > Eu i ( Y ) then Eu i ( e i , XY Z ) − Eu i ( e i , Y XZ )= r X Eu i ( X ) + max { r Y − r X , } Eu i ( Y ) − r Y Eu i ( Y ) − max { r X − r Y , } Eu i ( X ) = min { r X , r Y } ( Eu i ( X ) − Eu i ( Y )) ≥ , (A.1)with strict inequality if min { r X , r Y } >
0. Note that this result does not depend on the utilityof Z .(ii) If Eu i ( Y ) > Eu i ( Z ) then Eu i ( e i , XY Z ) > Eu i ( e i , XZY )= max { r Y − r X , } Eu i ( Y ) + max { r Z − max { r X , r Y } , } Eu i ( Z ) − max { r Z − r X , } Eu i ( Z ) − max { r Y − max { r X , r Z } , } Eu i ( Y )= max { min { r Y , r Z } − r X , } ( Eu i ( Y ) − Eu i ( Z )) ≥ , with strict inequality if min { r Y , r Z } > r X . Note that this result does not depend on the utilityof X .Thus, the first part of Claim 1 is established.To prove the second part of Claim 1, first note that min { r X , r Y } >
0, hence inequalityin (A.1) is strict. That is, X must be listed as the top school and I only need to compare XY Z and
XZY . If
XY Z generates a different probability distribution than
XZY , then u i ( e i , XY Z ) > u i ( e i , XZY ), which contradicts the statement of the theorem. A.2
Proof of Claim 2.
The proof is by contradiction. Suppose that r A ≥ r B ; thus, anyone rejectedfrom A is assigned to C . The mass γ (1 − F (1)) lists school B as the top choice.Consider first the case where γ (1 − F (1)) ≥ q . Then equating supply (LHS of equations(A.2) and (A.3)) and demand (RHS of equations (A.2) and (A.3)) gives us q = r B γ (1 − F (1)) (A.2) q = r A [(1 − γ ) + γ F (1)] + ( r A − r B ) γ (1 − F (1)) = r A − q (A.3) A.1
For the general theory of permutation groups see, for example, Dixon and Martimer (1996).
A.2
Note that if r A < r B < r C , then an applicant whose true preferences are BAC gets the same probabilitydistribution over possible assignments when she submits
BAC , her true preferences, and
BCA . This isbecause if she is rejected from B , then r i > r B > r A and she will also be rejected from school A . A–1n turn, this implies the following conditions on the cut-offs: r A = 2 q, r B = qγ (1 − F (1))Recall that F (1) ≥ / γ ≤
1, thus γ (1 −F (1)) ≥
2, hence r A ≤ r B , with strict inequalityif γ < F (1) < / γ (1 − F (1)) < q . In this case every applicant to B is accepted,including an applicant with r i = 1, so r B = 1. Yet, as we assumed r A ≥ r B , it means r A = 1as well. Thus, schools A and B accept all students, but the total quota of these two schoolsis 2 q ≤ / < Proof of Claim 3.
First, I re-write equations (5) and (6) for the symmetric case as: q = r A ((1 − γ ) + γ F (1)) = r A (1 − γ Φ) (A.4) q = r B γ Φ + ( r B − r A ) [1 − γ (1 − F (1)) − γ Φ)]= r B (1 − γ Φ) − r A (1 − γ Φ) (A.5)Using equation (A.4), I can re-write equation (A.5) as r B (1 − γ Φ) = q (cid:18) − γ Φ1 − γ Φ (cid:19) = q − γ Φ1 − γ Φ . Proof of Claim 4.
The difference between equation (6) and equation (5) can be written, takeninto account u A = 1 , u C = 0, as∆ U ( r A , r B ) = r A ( F (1) − F (0)) + ( r B − r A ) Z xd F ( x ) + r B Z ∞ xd F ( x )+ r A F (0)= r A (1 − F (1))( E ( u i ( B ) | u i ( B ) > − r B − r A ) F (0)(0 − E ( u i ( B ) | u i ( B ) < Z ∞ ( x − d F ( x ) = Z −∞ ( − x ) F ( x ) . Proof of Theorem 1.
Define mapping φ : [0 , [0 ,
1] as γ = i ∗ (∆ U ( γ )) ≡ φ ( γ ). Thismapping is continuous and into itself. Therefore, by Brower’s fixed point theorem, it has afixed point. Proof of Theorem 2.
Suppose ( e ′ i , R ′ i ) = (ˆ e i , ˆ R i ) is a strategy of applicant i . According toClaim 1, Eu i ( e ′ i , ˆ R i ) ≥ Eu i ( e ′ i , R ′ i ). A–2ote that fraction γ ∗ is informed. By construction of i ∗ (∆ U ), and because γ ∗ ∈ Γ, ∆ U ( γ ∗ ) = c ( γ ∗ ). Thus, if i ≤ γ ∗ , then Eu i (1 , ˆ R i ) ≥ Eu i (0 , ˆ R i ) and if i < γ ∗ , then Eu i (0 , ˆ R i ) >Eu i (1 , ˆ R i ). Thus, a deviation to ( e ′ i , R ′ i ) does not increase the utility of i and (ˆ e i , ˆ R i ) is a Nashequilibrium.Next, I show that any Nash equilibrium leads to the same choice of information acquisitionand the same probability distribution over allocations, given the realized preferences of theapplicants. Suppose that { ( e ′ i , R ′ i ) } i ∈ N = { (ˆ e i , ˆ R i ) } i ∈ N is a Nash equilibrium.According to Claim 1, strategy profile { ( e ′ i , ˆ R i ) } i ∈ N , with the same information choice butpossibly different submitted ROL, is also a Nash equilibrium and generates the same probabil-ity distribution as { ( e ′ i , R ′ i ) } i ∈ N . Indeed, any player i has a profitable deviation from ( e ′ i , R ′ i )to ( e ′ i , ˆ R i ) unless Eu i ( e ′ i , R ′ i ) = Eu i ( e ′ i , ˆ R i ). Then, according to Claim 1, the equality impliesthe same probability distribution of i ’s assignment under both ( e ′ i , R ′ i ) and ( e ′ i , ˆ R i ).Next, I show that the information acquisition choice must be the same under { ( e ′ i , R ′ i ) } i ∈ N and { (ˆ e i , ˆ R i ) } i ∈ N . Let γ ′ be the fraction of informed applicants in equilibrium { e ′ i } i ∈ N . Sup-pose that γ ′ / ∈ Γ; then c ( γ ′ ) = ∆ U ( γ ′ ). In this case, some applicants can profitably choose adifferent level of information acquisition, as shown below.First, consider the case c ( γ ′ ) < ∆ U ( γ ′ ). As c ( x ) is a continuous function, there exists i ′ > γ ′ such that c ( i ′ ) < ∆ U ( γ ′ ). Since c ( i ) < ∆ U ( γ ′ ) for all i ≤ i ′ , then any i such that e i = 0 has aprofitable deviation to (1 , ˆ R i ). Hence, all i ≤ i ′ are informed in Nash equilibrium { ( e ′ i , ˆ R i ) } i ∈ N .Thus, the fraction of informed applicants is at least i ′ . As i ′ > γ ′ , it contradicts the initialassumption that γ ′ is informed.Second, if c ( γ ′ ) > ∆ U ( γ ′ ), there is i ′ < γ ′ such that c ( i ′ ) > ∆ U ( γ ′ ). As i > i ′ areuninformed, at least (1 − i ′ ) > (1 − γ ′ ) applicants are uninformed, which contradicts theassumption that γ ′ is informed.Therefore, I have shown that if { ( e ′ i , ˆ R i ) } i ∈ N is a Nash equilibrium, then γ ′ ∈ Γ. By asimilar argument, applicants with index i < γ ′ do and with i > γ ′ do not acquire information.That is, { ( e ′ i , ˆ R i ) } i ∈ N is one of the Nash equilibria considered earlier. As has been shown, itgenerates the same probability as one of the equilibria { (ˆ e i , ˆ R i ) } i ∈ N . Proof of Theorem 3.
There are three groups of applicants: [0 , γ L ] who are informed in bothequilibria; ( γ L , γ H ], who are informed in γ H , but remain uninformed in γ L equilibrium; and( γ H , γ H than under γ L because their utilitypositively depend on r A ( γ ) and r B ( γ ), which are, in turn, increasing: see formulae (9) and(10) which can be re-written for symmetric distribution as: U (0 , ABC | γ ) = ( r A ( γ ) + r B ( γ )) / U (1 , ˆ R | γ ) = r B Z ∞ ( − x ) d F ( x ) + r A ( γ ) + r B ( γ )2 . Thus, I need to evaluate the effect on the group ( γ L , γ H ]; these applicants are informed inonly one of the equilibria.Since for any i < γ H , U (1 , ˆ R | γ H ) − c ( i ) ≥ U (1 , ˆ R | γ H ) − c ( γ H ) and applicant γ H chooses toacquire information, then U (1 , ˆ R | γ H ) − c ( i ) ≥ U (1 , ˆ R | γ H ) − c ( γ H ) ≥ U (0 , ABC | γ H ) > U (0 , ABC | γ L ) . A–3hus, every applicant i ∈ ( γ L , γ H ] is better off in the equilibrium associated with γ H . Proof of Theorem 4.
I first establish the following claim.
Claim 5.
Cutoff r A ( γ ) is strictly increasing for γ ∈ [0 , r B ( γ ) is strictly increasingfor γ ∈ [0 , / (3Φ)) and strictly decreasing for γ ∈ (1 / (3Φ) , Proof of Claim 5.
From equation (7), r ′ A ( γ ) = q Φ(1 − γ Φ) . Hence, as F (1) < r ′ A ( γ ) > γ ∈ [0 , r ′ B ( γ ) = q Φ 1 − γ Φ(1 − γ Φ) Hence r ′ B ( γ ) > γ Φ < γ RD solves ∆ U ( γ RD ) = c ( γ RD ).Thus, the derivative of the social welfare function is SW ′ ( γ RD ) = γ RD ∆ U ′ ( γ RD ) + U ′ (0 , ABC | γ RD ) . Recall that U (0 , ABC | γ ) = ( r A + r B ) / U ( γ ) = r B × Constant . According to Claim5, r ′ A ( γ ) > r ′ B ( γ ) ≥ γ ∈ [0 , / (3Φ)]. Thus, U ′ (0 , ABC | γ ) > U ′ ( γ ) ≥ γ ∈ [0 , / (3Φ)]. Proof of Theorem 5.
Consider first the policy of revealing priorities. Let r aaa and r bbb be cutoffsfor schools A and B and denote applicants with r i ≤ r aaa by aaa and applicants with r aaa < r i ≤ r bbb by bbb . Denote the fractions of informed applicants in aaa by γ aaa and in bbb by γ bbb . Note that applicantsnot belonging to these two groups are assigned to C regardless of what ROL they submit, soall of them are uninformed (except, possibly, a zero-cost applicant). Denote γγγ = ( γ aaa , γ bbb , U aaa ( γγγ ) = c ( γ aaa ) and ∆ U bbb ( γγγ ) = c ( γ bbb ), where∆ U ggg is the net gain of acquiring information for groups ggg ∈ { aaa, bbb } , given by the formula∆ U ggg ( γγγ ) = I (A.6)Note that the formula is the same for both groups, so the cost cutoff and, because cost andpriority are not correlated, the equilibrium level of information acquisition are the same forboth groups. To see why formula (A.6) is correct, consider group aaa . Applicants in that groupare guaranteed a place in A . If they acquire information and find that B is better than A , theyare assigned to B with probability 1; their gain is Φ ( E ( u i ( B ) | u i ( B ) > −
1) = I . Similarly,applicants in group bbb are guaranteed a seat in B and would only go to C instead of B if theyfind that C is better for them than B . The gain is Φ (1 / − E ( u i ( B ) | u i ( B ) < I .A–4utoffs r aaa and r bbb can be calculated by equating supply of and demand for seats in schools A and B (as in equations (5) and (6)): r aaa = q − γ aaa Φ = q − γ ∗ Φ , (A.7) r bbb = q − γ aaa Φ − γ bbb Φ(1 − γ aaa Φ)(1 − γ bbb Φ) = q − γ ∗ Φ(1 − γ ∗ Φ) , (A.8)Equations (A.7), (A.8) are identical to equations that determine r A and r B (equations (7), (8)),except for the equilibrium level of information acquisition γ ∗ . Recall that γ RD is determinedby equation r B I = c ( i ); as r B < c ( i ) is strictly increasing, it implies γ ∗ > γ RD , r aaa > r A and r bbb > r B . Then, for any applicant, the assignment under r aaa , r bbb is weakly better than under r A , r B .Next, I show that there is a positive mass of applicants who are better off. Consider arbitraryapplicant i with priority r i ∈ ( r A , r aaa ) who is uninformed when priorities are not revealed hencesubmits R i = ABC . If i remains uninformed when priorities are revealed (and submits R i ),then i is assigned to B in the former case and A in the latter. Hence, uninformed i is betteroff under the latter policy; informed i must then also be better off. Hence, there is a positivemass of applicants who are better off. Proof of Theorem 6.
Differentiating SW ( γγγ ) defined in (20) with respect to γ aaa and γ bbb , andtaking into account that r aaa does not depend on γ bbb , I get ∂SW ( γγγ ) ∂γ aaa = 12 ∂∂γ aaa ( r aaa + r bbb )( γγγ ) + ∂r aaa ( γγγ ) ∂γ aaa (cid:18) ( γ aaa − γ bbb ) I + Z γ bbb γ aaa c ( x ) dx (cid:19) + r aaa ( γγγ ) ( I − c ( γ aaa )) + ∂r bbb ( γγγ ) ∂γ aaa Z γ bbb ( I − c ( x )) dx∂SW ( γγγ ) ∂γ bbb = ∂r bbb ( γγγ ) ∂γ bbb (cid:18)
12 + Z γ bbb ( I − c ( x )) dx (cid:19) + ( r bbb − r aaa )( γγγ ) ( I − c ( γ bbb ))Recall that in equilibrium γ aaa = γ bbb = γ ∗ and I = c ( γ ∗ ); thus, in equilibrium the derivativescollapse to: ∂SW ( γγγ ∗ ) ∂γ aaa = 12 ∂∂γ aaa ( r aaa + r bbb )( γγγ ∗ ) + ∂r bbb ( γ ∗ ) ∂γ aaa Z γ ∗ ( I − c ( x )) dx (A.9) ∂SW ( γγγ ∗ ) ∂γ bbb = ∂r bbb ( γγγ ∗ ) ∂γ bbb (cid:18)
12 + Z γ ∗ ( I − c ( x )) dx (cid:19) . (A.10)As I > c ( i ) for any i < γ ∗ , Z γ ∗ ( I − c ( x )) dx > . Thus, the signs of the derivatives depend on ∂∂γ aaa ( r aaa + r bbb ) ( γγγ ∗ ), ∂r bbb ( γγγ ∗ ) ∂γ aaa and ∂r bbb ( γγγ ∗ ) ∂γ bbb , which areA–5iven below: ∂∂γ aaa ( r aaa + r bbb ) ( γγγ ) = q Φ(1 − γ bbb Φ)(1 − γ aaa Φ) (1 − γ bbb Φ) = q Φ(1 − γ ∗ Φ)(1 − γ ∗ Φ) > ∂r bbb ( γγγ ) ∂γ aaa = − qγ bbb Φ (1 − γ aaa Φ) (1 − γ bbb Φ) = − qγ ∗ Φ (1 − γ ∗ Φ) < ∂r bbb ( γγγ ) ∂γ bbb = q Φ(1 − γ aaa Φ)(1 − γ aaa Φ)(1 − γ bbb Φ) = q Φ(1 − γ ∗ Φ)(1 − γ ∗ Φ) > ∂SW ( γγγ ∗ ) ∂γ aaa > C <
1, while the derivative ∂SW ( γγγ ∗ ) ∂γ bbb > γ ∗ and F ( x ). Proof of Theorem 7.
The proof is based on the following claim establishing (A.12)–(A.15).
Claim 6.
Suppose γ RD ∈ (0 , • i ∈ [0 , γ RD ] acquire information, • i ∈ ( γ RD ,
1] do not acquire information, and • all applicants report their preferences R i truthfully.Let γ p ∈ [ γ RD , i ∈ ( γ p ,
1] are assigned to schools A and B with quotas ( q A , q B ) and applicants i ∈ [0 , γ p ] are assigned to schools A and B with quotas( q − q A , q − q B ), where( q A , q B ) = ((cid:0) (1 − γ p )( r RDA + r RDB ) / , (cid:1) if (1 − γ p )( r RDA + r RDB ) ≤ q (cid:0) q, (1 − γ p )( r RDA + r RDB ) − q (cid:1) if (1 − γ p )( r RDA + r RDB ) > q. (A.11)Then the cutoffs of schools A and B , ( r A , r B , r A , r B ), are such that r RDA ≥ r A (A.12) r B ≥ r RDB (A.13) r A + r B = r RDA + r RDB (A.14) r A + r B ≥ r RDA + r RDB (A.15)
Proof.
Consider first the case where (1 − γ p )( r RDA + r RDB ) ≤ q .Given that (1 − γ p ) applicants are assigned to q A seats in school A , r A satisfies equation r A (1 − γ p ) = q A := (1 − γ p )( r RDA + r RDB ) /
2. Thus, r A = ( r RDA + r RDB ) /
2. As no applicants areassigned to school B , r B = 0. We thus established (A.14).Cutoff r A equates supply of and demand for seats in school A : q − (1 − γ p )( r RDA + r RDB ) / r A ( γ p − γ RD (1 − F (1))) , (A.16)which can be rewritten, taking into account that, from (5), q = r RDA (1 − γ RD (1 − F (1))) andsubstituting 1 − F (1) = Φ as r RDA (1 − γ RD Φ) − (1 − γ p )( r RDA + r RDB ) / r A ( γ p − γ RD Φ)A–6s r RDB > r
RDA (see Claim 2),(1 − γ RD Φ) r RDA − (1 − γ p ) r RDA = ( γ p − γ RD Φ) r RDA > r A ( γ p − γ RD Φ) , and r RDA > r A follows.Similarly, cutoff r B is determined by equation q = r B γ RD (1 − F (1)) + ( r B − r A )( γ p − γ RD (1 − F (1) + F (0)))which can be simplified to q = r A γ RD Φ + ( r B − r A )( γ p − γ RD Φ) . (A.17)To find r A + r B , I multiply equation (A.16) by 2 and add to equation (A.17), to obtain:3 q − (1 − γ p )( r RDA + r RDB ) = r A γ RD Φ + ( r B + r A )( γ p − γ RD Φ) . (A.18)Similarly, equation (6) can be re-written as q − r RDA γ Φ = ( r RDB − r RDA ) (cid:0) − γ RD Φ (cid:1) , (A.19)Adding equations (5) multiplied by 2 and (A.19), both evaluated at γ RD , I obtain3 q = r RDA γ RD Φ + ( r RDB + r RDA )(1 − γ RD Φ); (A.20)substituting the expression for 3 q into (A.18), I obtain: r RDA γ RD Φ + ( r RDB + r RDA )(1 − γ RD Φ) − (1 − γ p )( r RDA + r RDB )= r A γ RD Φ + ( r B + r A )( γ p − γ RD Φ) , which further can be simplified to r RDA γ RD Φ + ( r RDB + r RDA )( γ p − γ RD Φ) = r A γ RD Φ + ( r B + r A )( γ p − γ RD Φ) . Hence (( r B + r A ) − ( r RDB + r RDA ))( γ p − γ RD Φ) = ( r RDA − r A ) γ RD Φ . (A.21)Recall that r RDA > r A . From (A.21), r A + r B > r RDA + r RDB and, from two preceding inequalities, r B > r RDB .Consider now the case (1 − γ p )( r RDA + r RDB ) > q . Recall that no places in school A areavailable to applicants with i ∈ ( γ p , B or C . Cutoff r B isdetermined by the equation q − q B = γ p (1 − Φ) r B . (A.22)A–7hen q − q B = 3 q − (1 − γ p )( r RDA + r RDB )= r RDA γ RD Φ + ( r RDB + r RDA )( γ p − γ RD Φ) = r RDA γ p + r RDB ( γ p − γ RD Φ) , where the second line follows from equation (A.20).Rewriting γ p (1 − Φ) r B = ( γ p − γ RD Φ) r B + ( γ RD − γ p )Φ r B , I establish r RDA γ p + ( γ p − γ RD )Φ r B = ( γ p − γ RD Φ)( r B − r RDB ) . As γ p ≥ γ RD , the LHS of the equation is positive; γ p − γ RD Φ is also positive, so r B > r RDB .To establish that r B ≥ r RDA + r RDB , I rewrite equation (A.22), again using equation (A.20),as r RDA γ RD Φ + ( r RDB + r RDA )(1 − γ RD Φ) − (1 − γ p )( r RDA + r RDB ) = γ p r B − γ p Φ r B . Thus, r RDA γ RD Φ + ( r RDB + r RDA )( γ p − γ RD Φ) = γ p r B − γ p Φ r B γ p ( r RDB + r RDA ) = γ p r B + r RDB γ RD Φ − γ p Φ r B γ p ( r RDB + r RDA ) ≤ γ p r B + ( r RDB − r B ) γ RD Φ < γ p r B , where the inequalities in the last line follow from γ p ≥ γ RD and r B > r RDB .We are now in a position to prove Theorem 7.Note that Claim 1 applies: there is no profitable deviation from submitting the truthfulROL ˆ R i .Fix x ∈ ( γ RD ,
1) and the choice of quotas ( q A , q B , q A , q B ) corresponding to x , given by equa-tion (A.11) with γ p = x . For these quotas, there are cutoffs ( r A ( x ) , r B ( x )) and ( r A ( x ) , r B ( x ))faced by uninformed and informed applicants, respectively.Consider function G ( x ) = U ((1 , ˆ R ) | x ) − c ( x ) − U RD ((0 , ABC ) | x ) , where U ((1 , ˆ R ) | x ) = U ((1 , ˆ R ) | r A ( x ) , r B ( x )) is the expected utility of targeted informed ap-plicants and U RD ((0 , ABC ) | x ) = U ((0 , ABC ) | r A ( x ) , r B ( x )) is expected utility of uninformedapplicants before the intervention (recall that the index in utility function U is suppressedbecause the expression is the same for all applicants); this function is defined for x ∈ [ γ RD , G ( γ RD ) >
0. Indeed, from Claim 6, U RD ((0 , ABC ) | γ RD ) = ( r RDA ( γ RD ) + r RDB ( γ RD )) / < ( r A ( γ RD ) + r B ( γ RD )) / U ((0 , ABC ) | γ RD ) . Thus, G ( γ RD ) > U ((1 , ˆ R ) | γ RD ) − c ( γ RD ) − U ((0 , ABC ) | γ RD ) . A–8ext, recall, from equation (12), that U ((1 , ˆ R ) | γ RD ) − U ((0 , ABC ) | γ RD ) = r B ( γ RD ) Z −∞ ( − x ) d F ( x ) > r RDB ( γ RD ) Z −∞ ( − x ) d F ( x ) = U RD ((1 , ˆ R ) | γ RD ) − U RD ((0 , ABC ) | γ RD ) . Thus, G ( γ RD ) > U RD ((1 , ˆ R ) | γ RD ) − U RD ((0 , ABC ) | γ RD ) − c ( γ RD ) = 0 , (A.23)where the last equality follows from γ RD being an equilibrium fraction of applicants acquiringinformation in RD.Consider now γ = 1. In that case, q A = q B = q and r A (1) = r RDA (1) , r B (1) = r RDB (1).Thus U ((1 , ˆ R ) |
1) = U RD ((1 , ˆ R ) | γ = 1 is not an equilibrium fraction of applicantsacquiring information, G (1) = U RD ((1 , R ) | − U RD ((0 , ABC ) | − c (1) < . (A.24)As G ( x ) is a linear combination of continuous functions r A ( x ) , r B ( x ) , r RDA ( x ) , r RDB ( x ) , and c ( x ), it is continuous. It is larger than zero at γ RD (equation A.23) and less than zero at 1(equation A.24). Thus, there is x ∗ such that G ( x ∗ ) = 0.Finally, note that for any x ∈ [ γ RD , U RD ((0 , ABC ) | x ) = U ((0 , ABC ) | x ) , A.3 hence U ((1 , R ) | x ∗ ) − c ( x ∗ ) − U ((0 , ABC ) | x ∗ ) = G ( x ∗ ) = 0 . Thus i = x ∗ is indifferent to acquiring information. Hence, there is an equilibrium where i ≤ x ∗ are informed and i > x ∗ are uninformed.The utility of applicants i ≥ x ∗ is the same in RD with original quotas and in RD with q A , q B . It follows directly from Claim 6 for applicants i > x ∗ ; for i = x ∗ , it follows from thefact that i is indifferent to acquiring information.Thus, what remains to be shown is that i < x ∗ are better off. First, applicants i ∈ [ γ RD , x ∗ ),who are uninformed in RD with q A = q B = q , but choose to acquire information under theseat re-allocation scheme, are weakly better off. Indeed, they can always deviate to acquiringno information and get the same utility as in the unmodified RD; given that they choose toacquire information in equilibrium means that this deviation is not profitable. Applicantswith i ≤ γ RD are informed in both regimes. By Claim 6 if only γ RD fraction is informed, then r B ( γ RD ) > r RDB and r A ( γ RD ) + r B ( γ RD ) > r RDB + r RDB ; hence these applicants are better offwith quotas ( q A , q B ) even if the fraction of informed applicants is the same. As the equilibriumfraction of informed applicants with quotas ( q A , q B ) is higher than γ RD , and since r B ( x ) and r A ( x ) + r B ( x ) increase when x increases (see Claim 11), informed applicants are better offwith quotas ( q A , q B ). A.3
Cutoffs r A ( x ) , r B ( x ) are not defined for x = 1. A–9 roof of Theorem 8.
Consider the following maximization problem:max γ ≥ γ RD U (0 , ABC | γ ) − κ Z γγ RD (cid:0) c ( x ) − c ( γ RD ) (cid:1) dx. The derivative of the function above is U ′ (0 , ABC | γ ) − κ (cid:0) c ( γ ) − c ( γ RD ) (cid:1) , which is positive at γ = γ RD , since U ′ (0 , ABC | γ ) >
0. Thus, at γ RD , U (0 , ABC | γ ) − τ isincreasing and τ = 0. Pick some γ τ such that U (0 , ABC | γ τ ) − τ > U (0 , ABC | γ RD ) . The inequality above implies that uninformed applicants i ∈ ( γ τ ,
1] are better off. Forinformed applicant i ∈ [0 , γ RD ], U τi = U (0 , ABC | γ τ ) + ∆ U ( γ τ ) − τ ≥ U (0 , ABC | γ τ ) + ∆ U ( γ RD ) − τ> U (0 , ABC | γ RD ) + ∆ U ( γ RD ) = U i . (A.25)As ∆ U ( γ τ ) > ∆ U ( γ RD ) ≥ c ( i ), these applicants continue to acquire information.Finally, consider applicants i ∈ [ γ RD , γ τ ]. Each of these applicants face the same cost, c ( γ RD ) and, for each of them, ∆ U ( γ τ ) > ∆ U ( γ RD ) = c ( γ RD ). Thus, they are informed. Thenequation (A.25) applies to these applicants.I therefore conclude that U τi > U i for all i ∈ [0 , Lemma 1.
There exists a Bayes-Nash equilibrium ( e i , R i ( · )) i ∈ N of IA game where both schools A and B are full after round one. The equilibria are characterized by three values: the fractionof informed applicants γ ∈ [0 , u ∈ [ u B , u A ), and the probability α ∈ [2 / , • Every applicant i < γ is informed ( e i = 1); every applicant i > γ is not informed( e i = 0); and applicant i = γ may be either. • Every informed applicant i submits R i = BAC or R i = BC if u i ( B ) > ¯ u and submits R i = ABC or R i = AC if u i ( B ) < ¯ u . An applicant i with u i ( B ) = ¯ u submits R i ∈{ BAC, BC, ABC, AC } . • Mass α of uninformed applicants submits ABC or AC and mass (1 − α ) submits BAC or BC .The values γ, ¯ u and α are determined as follows.There are four non-overlapping conditions on costs: (Ia) T < c (0); (Ib) c (0) ≤ T ( u B ) To prove Lemma 1, I first establish two claims. Claim 7. Consider equilibrium s ∗ = ( e i , R i ( · )) i ∈ N . If e j = 1, then for any i < j , e i = 1. If e j = 0, then for any i > j , e i = 0. Proof of Claim 7. Suppose that applicant j acquires information. Since s ∗ is a Nash equilib-rium, then U (1 , R j ( · )) − c ( j ) ≥ U (0 , R ′ ) for any R ′ . Since c ( j ) > c ( i ) for any i < j , then U (1 , R j ( · )) − c ( i ) > U (0 , R ′ ) for any R ′ . Thus, (0 , R ′ ) cannot be a Nash equilibrium strategyof applicant i .The proof for the second part of the statement is identical. Claim 8. Listing C as top choice is strictly dominated. Formally, the following strategies arestrictly dominated: • (1 , R i ( x )) such that the set { x ∈ R | R i ( x ) = C } has a positive measure; and • (0 , C ) Proof of Claim 8. If C is listed as top choice in R i , then applicant i is assigned to C . If A is listed as top choice and r i ≤ q , then i is guaranteed to be assigned to A . Considera strategy ( e i , R ′ i ) such that R ′ i ( x ) = R i ( x ) whenever C is not top ranked in R i ( x ) and R ′ i ( x ) = AC otherwise. Then the assignment is identical except in the cases when i is assignedto C under R i ( x ) and to A under R ′ i ( x ). There is a positive measure of these cases, hence U ( e i , R ′ i ( x )) > U ( e i , R i ( x )).I am now in a position to prove Lemma 1. I distinguish the following cases: (a) when alluninformed applicants submit AC ; (b) when uninformed applicants are indifferent betweensubmitting AC and BC ; and (c) when all uninformed applicants submit BC .Claim 7 establishes that there is a cutoff γ ∈ [0 , 1] that determines information acquisitionchoice. Claim 8 establishes that I need to consider strategies { ABC, AC, BAC, BC } only. AsI am looking for an equilibrium where schools A and B are full after the first round, applicant’sassignment will not change if she submits ABC or AC and BAC or BC . Thus, it is sufficientto consider AC and BC only. A–11ecall that if applicant i lists AC and has priority r i ≤ ρ A , i is assigned to A ; if i lists BC and has priority r i ≤ ρ B , i is assigned to B ; and i is assigned to C in all other cases. Thus,the expected utility of applicant i when she submits ROLs AC and BC can be written as: U ( e i , AC ) = ρ A u A U ( e i , BC ) = ρ B u i ( B ) (A.29)For given ρ A , ρ B , define ¯ u = ρ A ρ B u A . (A.30)As ρ B > 0, ¯ u is well-defined. In equilibrium, applicant i with u i ( B ) > ¯ u submits BC and i with u i ( B ) < ¯ u submits AC .Note that (a) if ¯ u > u B , all uninformed applicants submit AC ; (b) if ¯ u = u B , uninformedapplicants are indifferent; and (c) if ¯ u < u B , all uninformed applicants submit BC . Case (a) : Suppose that ¯ u > u B .If γ (1 − F (¯ u )) ≥ q , cutoffs ρ A and ρ B are determined by the following supply-demandequations: q = ρ A ((1 − γ ) + γ F (¯ u )) = ρ A (1 − γ (1 − F (¯ u ))) (A.31) q = ρ B γ (1 − F (¯ u )) (A.32)If γ (1 − F (¯ u )) < q , then ρ B = 1. I will rule out this case later.Combining (A.30), (A.31) and (A.32), we obtain (A.28): h u A ¯ u + 1 i (1 − F (¯ u )) = γ − , Note that at ¯ u = u B , γ = 2 / u A /u B = 2 and F ( u B ) = 1 / u increases,the LHS of equation (A.28) monotonically decrease. Thus, γ is uniquely determined for everyvalue of ¯ u and ¯ u ( γ ) is a monotonically increasing function. At ¯ u = u A , the LHS becomes2(1 − F ( u A )) < 1, hence γ > 1. Thus, for any value of γ ∈ (2 / , u ( γ ) ∈ ( u B , u A ).Next, I find applicant i who is indifferent to acquiring information. Recall that an un-informed applicant submits AC in equilibrium; her expected utility is U (0 , AC ) = ρ A u A (equation A.29). If i is informed, then, with probability F (¯ u ), u i ( B ) < ¯ u and i submits R i = AC ; with probability (1 − F (¯ u )), u i ( B ) > ¯ u and i submits R i = BC . Thus, U (1 , R i ( · )) = (1 − F (¯ u )) ρ B E ( u i ( B ) | u i ( B ) > ¯ u ) + F (¯ u ) ρ A u A , so that i ’a indifference condition can be expressed as(1 − F (¯ u ))( ρ B E ( u ( B ) | u B > ¯ u ) − ρ A u A )= (1 − F (¯ u )) ρ B ( E ( u ( B ) | u B > ¯ u ) − ¯ u )= q h u A ¯ u + 1 i Z ∞ ¯ u ( x − ¯ u ) d F ( x ) = c ( i ) , (A.33)A–12here the first equality following from the definition of ¯ u and in the second equality ρ B isexpressed as ρ B = q (cid:2) u A ¯ u + 1 (cid:3) using equation (A.28).In equation (A.33), the LHS decreases and RHS increases with γ . To see this, recall that i is indifferent to acquiring information; hence γ = i and RHS of equation (A.33) can be writtenas c ( γ ). As γ increases, RHS increases. Recall also that for any value γ ∈ (2 / , u ( γ ) determined by equation (A.28) and as γ increases, ¯ u also increases. Furthermore, as ¯ u increases, LHS decreases.Therefore, equation (A.33) has a solution γ ∈ (2 / , 1] only if c (2 / ≤ q Z ∞ u B ( x − u B ) d F ( x ) (A.34)and if solution exists, then it is unique. If equation (A.34) holds, but either γ (¯ u ) > γ = 1 and ¯ u is determined by equation h u A ¯ u + 1 i (1 − F (¯ u )) = 1 . This case applies when q h u A ¯ u + 1 i Z ∞ ¯ u ( x − ¯ u ) d F ( x ) ≥ c (1) . All applicants are informed; these with u i ( B ) > ¯ u submit BC and these with u i ( B ) < ¯ u submit AC .I return now to the case where γ (1 − F (¯ u )) < q and ρ B = 1. Recall that in Case (a) alluninformed applicants submit AC , so U (0 , AC ) = ρ A ≥ ρ B / / U (0 , BC ). That is, ρ A ≥ / 2. At the same time, using equation (A.31) and conditions γ (1 − F (¯ u )) < q and q ≤ / 3, I obtain ρ A = q − γ (1 − F (¯ u )) < q − q ≤ , which is inconsistent with ρ A ≥ / Case (b) : Suppose ¯ u = u B . Uninformed applicants are indifferent between AC and BC .I assume that α fraction of uninformed applicants submit AC and (1 − α ) fraction apply to BC . As ¯ u = u B , one-half of all informed students submit BC and one-half submit AC . Thus,the cutoffs ρ A and ρ B are given by: q = ρ A ( γ/ − γ ) α ) ,q = ρ B ( γ/ − γ )(1 − α ))Given that uninformed students are indifferent between submitting AC and BC , hence U (0 , AC ) = U (0 , BC ), the following condition must hold: ρ A u A = ρ B u B , (A.35)A–13hich implies, taking into account the formulae for ρ A and ρ B and u A = 2 u B ,3( γ/ − γ )(1 − α )) = 1 . (A.36)It then follows that ρ B = 3 q and ρ A = 1 . q . Furthermore, it must be that α ∈ [0 , 1] and γ ∈ [0 , γ ∈ [0 , / 3] and α ∈ [2 / , i is indifferent to acquiring information when the following equality holds:12 ρ B E ( u i ( B ) | u i ( B ) > u B ) + 12 ρ A u A − c ( i ) = ρ A u A . Using expression for ρ B , equation (A.35) and taking into account that i = γ , I obtain32 q ( E ( u ( B ) | u ( B ) > u B ) − u B ) = 3 q Z ∞ u B ( x − u B ) d F ( x ) = c ( γ ) . As the LHS of this equation is a constant and the RHS is monotonically increasing with γ ,the solution with γ ∈ [0 , / 3] exists if only if c (0) ≤ q Z ∞ u B ( x − u B ) d F ( x ) ≤ c (2 / , and it is unique.If 3 q R ∞ u B ( x − u B ) d F ( x ) < c (0), then γ = 0: no one acquires information. In that case,equation (A.36) becomes 3(1 − α ) = 1, hence α = 2 / 3. If 3 q R ∞ u B ( x − u B ) d F ( x ) > c (1), thenimplied γ is equal to one, which leads to Case (a). Case (c) : Uninformed applicants submit BC and ¯ u < u B . In this case, only informedapplicants submit AC and, given that ¯ u < u B , less than half of them do so. Thus, ρ A > q ,as if 2 q/γ < 1, then ρ A = 2 q/γ ≥ q and if 2 q/γ ≥ ρ A = 1.The cutoff at school B is determined by equation q = ρ B ( γ/ − γ )) = ρ B (1 − γ/ ρ B = q − γ/ ≤ q . Thus, ρ A > ρ B : anyone who is accepted to school B would beaccepted to school A , if applied. School A is more valuable for both uninformed applicantsand for informed applicants whose u i ( B ) ∈ (¯ u, u A ). Those applicants submit BC in Case (c),but have a profitable deviation to AC . There is no equilibrium in this case. Proof of Theorem 9. Before proving Theorem 9, I establish two claims that allow me to con-sider only the equilibrium described in Lemma 1. Claim 9. If there is an equilibrium ( e i , R i ( · )) of IA game such that school B has unfilledseats in round 2, then the fraction of informed applicants, γ IA , is larger than the fraction ofinformed applicants in RD game, γ RD . Proof of Claim 9. As B has unfilled seats in round 2, any applicant to B in round 1 is accepted,so ρ B = 1. In round 2, applicants with r i ≤ ρ B, are accepted. The first-round cutoff for A is ρ A and, as A is full after round 1, ρ A, = 0. Note that only the applicants who are rejectedfrom A apply to B in round two; then, for any such applicant i , r i > ρ A . As a positive massof applicants is accepted to B in round 2, ρ B, > ρ A . The last inequality also implies that, forA–14n uninformed applicant, ABC yields a higher utility than AC , as she has a chance to gainadmission to school B if unsuccessful at school A .Let ¯ u be a solution to equation ρ A + ( ρ B, − ρ A )¯ u = ¯ u (A.37)The LHS is a utility of applicant i whose u i ( B ) = ¯ u and who submits ABC ; the RHS is i ’s utility when she submits BC . Thus, any applicant with u i ( B ) > ¯ u submits BC and anyapplicant with u i ( B ) < ¯ u submits ABC . Equation (A.37) can be re-written as ρ B, − ρ A = 1 − ρ A ¯ u The expected utility of an informed applicant can be calculated as follows. U IA (1 , R ( · )) = Z ∞ ¯ u xd F ( x ) + Z ¯ u ( ρ A + ( ρ B, − ρ A ) x ) d F ( x ) + Z −∞ ρ A d F ( x )= Z ∞ ¯ u xd F ( x ) + ( ρ B, − ρ A ) Z ¯ u xd F ( x ) + ρ A Z ¯ u −∞ d F ( x )As uninformed applicants submit ABC , their utility is U IA (0 , ABC ) = ρ A − ρ B, u B The difference in utilities between informed and uninformed applicants – which depends on γ through ρ A , ρ B, and ¯ u – is∆ U IA ( γ ) = Z ∞ ¯ u xd F ( x ) − ( ρ B, − ρ A ) (cid:18)Z −∞ + Z ∞ ¯ u (cid:19) xd F ( x ) − ρ A Z ∞ ¯ u d F ( x )= Z ∞ ( x − d F ( x ) + ρ A Z u x ¯ u d F ( x ) + ρ A ¯ u Z ∞ d F ( x ) − ρ A Z ∞ ¯ u d F ( x ) ≥ Z ∞ ( x − d F ( x )Recall the inequality for the difference in utility for RD: ∆ U RD ( γ ) ≤ . q R ∞ ( x − d F ( x ).As I assume that q ≤ / 3, ∆ U RD ( γ ) < ∆ U IA ( γ ) for any γ ∈ [0 , γ RD < γ IA . Claim 10. There is no equilibrium under IA where A has unfilled seats after round 1. Proof of Claim 10. Suppose not. Since school A has unfilled seats after round 1, any applicantwho prefers A to B would submit ROL that lists A as the top choice. That is, (1 − γ ) massof uninformed applicants and γ F (1) mass of informed applicants will apply to A in round 1.Thus, the total mass (1 − γ ) + γ F (1) = 1 − γ F (0) ≥ / A , while the quota at A is less or equal to 1/3. This is a contradiction to A being unfilled after round 1.I am now in a position to prove Theorem 9.A–15ecall that γ RD solves the problem c ( γ RD ) = q − γ RD Φ(1 − γ RD Φ) Z ∞ ( x − d F ( x ) (A.38)Suppose that γ RD < / γ IA ≤ γ RD .If γ IA < / 3, then γ IA solves c ( γ IA ) = 3 q Z ∞ u B ( x − u B ) d F ( x ) . (A.39)If 2 / ≤ γ IA < / 9, then γ IA solves c ( γ IA ) = qγ IA (1 − F (¯ u )) Z ∞ ¯ u ( x − ¯ u ) d F ( x ) > q (8 / − / Z ∞ ( x − d F ( x ) = 2 . q Z ∞ ( x − d F ( x ) , (A.40)where the inequality follows from ¯ u < u A .Note that − x (1 − x ) ≤ . 25 for x ∈ [0 , / c ( x ) is an increasing function,it means γ IA > γ RD , a contradiction to my initial assumption. Proof of Corollary 1. Note that equation (22) implies that some applicants are informed inRD (see equation (12)) and equation (23) implies that no more than 2 / γ = 0, IA and RD yield the same utility. As cutoffs are increasing (Claim 5), U (0 , ABC | r A ( γ ) , r B ( γ )) > U (0 , ABC | r A (0) , r B (0)) when γ > 0. When γ ≤ / 3, utility of an uninformed applicantunder IA assignment is unchanged and equal to U (0 , ABC | r A (0) , r B (0)), as cutoffs ρ A and ρ B do not change when γ ∈ [0 , / A.2 General Case In this section, I revisit the conclusion of Section 4 that information acquisition in the equi-librium of RD game is below the socially optimal level for the case where q A ≤ q B and thedistribution is not symmetric. Equation (5) then becomes q A = Prob( r i ≤ r A ) ((1 − γ ) + γ F (1)) = r A ((1 − γ ) + γ F (1)) , (A.41)and equation (6) becomes q B = r B γ (1 − F (1)) + ( r B − r A ) [(1 − γ ) + γ ( F (1) − F (0))] . (A.42)First, note that Theorem 1, which establishes the existence of equilibrium, and 2, whichdescribes the equilibrium of the RD game, still apply, as they rely on the continuity of functionsA–16 A ( x ) and r B ( x ), but not on their particular form. That is, the equilibrium exists and theequilibrium value of γ RD solves ∆ U ( γ RD ) = c ( γ RD ).The derivative of SW ( γ RD ) (given by (17), as that equation does not rely on symmetry) is SW ′ ( γ RD ) = γ RD ∆ U ′ ( γ RD ) + U ′ (0 , ABC | γ RD ) . The expressions for U (0 , ABC | γ ) and ∆ U also do not rely on symmetry and are given byequations 9 and 11: U (0 , ABC | r A , r B ) = ( r A + r B ) / U ( r A , r B ) = r A ( I A − I C ) + r B I A , where I A = R ∞ ( x − d F ( x ), I C = R −∞ ( − x ) d F ( x ), Φ A = 1 − F (1) and Φ C = F (0). Todetermine the sign of SW ′ ( γ RD ), I find r ′ A = q A Φ A (1 − γ Φ A ) r ′ B = q A Φ C (1 − γ Φ A ) + γ Φ A (Φ C − A ))(1 − γ Φ A ) (1 − γ Φ C ) + ( q B − q A )Φ C (1 − γ Φ C ) ( r A + r B ) ′ = q A Φ A (1 − γ Φ C ) + Φ C (1 − − γ (Φ A − Φ C ))Φ A )(1 − γ Φ A ) (1 − γ Φ C ) + ( q B − q A )Φ C (1 − γ Φ C ) ( r B − r A ) ′ = q A Φ C − Φ A − γ Φ A Φ C (1 − γ Φ A )(1 − γ Φ A ) (1 − γ Φ C ) + ( q B − q A )Φ C (1 − γ Φ C ) Algebraic manipulations show the following claim: Claim 11. For any Φ A , Φ C , q A , and q B ≥ q A , the following inequalities hold:1. r ′ A > γ ∈ [0 , r A + r B ) ′ > γ ∈ [0 , r ′ B ≥ γ Φ A ≤ / γ Φ C ≤ / I C and a small I A , the gain from information acquisition can benegative and arbitrarily large. Hence, r ′ A ( I A − I C ) < SW ′ ( γ RD ) < 0: more applicants are informed in RD than socially optimal. Below I provide two necessaryconditions for γ SO > γ RD that impose some discipline on I C ; arguably, these conditions areusually met in practice. Claim 12. Too few applicants are informed, γ SO > γ RD , if:1. Φ C = 0;2. γ RD ( I C − I A ) < / γ RD Φ A ≤ / γ RD Φ C ≤ / F ( x ) is truncated at 0. This casecorresponds to the mechanism where applicants allocated to B can opt to go to C . The seatsA–17acated by these applicants are not re-allocated, possibly because B accepts more applicantsthan its quota anticipating that some will opt to go to C .Case 2 could be violated only when there is a significant uncertainty about schools A and C (so that both I A and I C are large) or when the distribution is significantly skewed toward C . Proof. First, note that when q B > q A , r ′ ( A ) is the same as when q B = q A and all otherderivatives increase. Thus, to save on notation, I assume that q B = q A ; all arguments holdwhen q B > q A .Case 1. If Φ C = 0, then I C = 0. As ( r A + r B ) ′ and r ′ B are both positive, ∆ U ′ ( γ RD ) > U ′ (0 , ABC | γ RD ) > 0. Hence, SW ′ ( γ RD ) > γ SO > γ RD .In Case 2, SW ′ ( γ RD ) = r ′ A (1 / − γ RD ( I C − I A )) + r ′ B (1 / γ RD I A ) > . The first term is positive by assumption and because r ′ A > 0; the second term is positivebecause r ′ B ≥ γ RD Φ A ≤ / γ RD Φ C ≤ / 3. Thus, the Social Welfare is increasing at γ RD and γ RD < γ SO . References Dixon, J. D., and B. Martimer (1996):