Ordinal Bayesian Incentive Compatibility of Truth-telling in Stable Mechanisms in Monotonic Matching Markets

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1 Ordinal Bayesian Incentive Compatibility o Truth-telling in Stable Mechanisms in Monotonic Matching Markets Lars Ehlers y Jordi Massó z February Still preliminary Abstract: We study strategic behavior o agents participating in centralized stable mechanisms in two-sided many-to-one matching markets when each agent has incomplete inormation about the preerences o the other agents. It is known that in any stable mechanism there is a strong link between ordinal Bayesian Nash equilibria under incomplete inormation and Nash equilibria under complete inormation: given a common belie, a strategy is an ordinal Bayesian Nash equilibrium in a stable mechanism i and only i at each pro le in the support o the belie the strategy prescribes a Nash equilibrium in the stable L. Ehlers acknowledges nancial support rom the SSHRC (Canada). The work o J. Massó is partially supported by the Spanish Ministry o Economy and Competitiveness, through the Grant ECO P and the Severo Ochoa Programme or Centres o Excellence in R&D (SEV ), and rom the Generalitat de Catalunya, through Grant SGR Part o this research was done while J. Massó was a visiting ellow o CIREQ, at the Université de Montréal. y Département de Sciences Économiques and CIREQ, Université de Montréal. Montréal, Québec H3C 3J7, Canada; lars.ehlers@umontreal.ca z Departament d Economia i d Història Econòmica, Universitat Autònoma de Barcelona, and Barcelona GSE Cerdanyola del Vallès (Barcelona), Spain; jordi.masso@uab.es 1

2 mechanism under complete inormation at the pro le. In this paper we show that in the workers-optimal stable mechanism this link does not hold or truthtelling i rms preerences over amilies o subsets o workers are monotonic responsive extensions o their rankings over individual workers. Firms have monotonic responsive extensions i having all positions lled with acceptable workers is always strictly preerred to having some positions vacant. However, we show that in the rms-optimal stable mechanism the link holds or truthtelling: given a common belie, truth-telling is a monotonic ordinal Bayesian Nash equilibrium in the rms optimal stable mechanism under incomplete inormation i and only i truth-telling is a monotonic Nash equilibrium under complete inormation at each pro le in the support o the common belie. JEL classi cation number: C78; D81; J44. Keywords: Many-to-one matching market; Stability; Incomplete inormation; Monotonic responsive extensions. 1 Introduction Both empirical and theoretical studies o two-sided many-to-one matching markets have been useul in applications. Many such markets have developed centralized market clearing mechanisms (in response to various ailures o the decentralized market) to match the agents rom the two sides: the institutions ( rms, colleges, hospitals, schools, etc.) and the individuals (workers, students, medical interns, children, etc.). 1 The National Resident Matching Program is the most well-studied example o this kind o two-sided matching markets. Each year around 20,000 medical students look or a our-years position in American hospital programs to undertake their medical 1 Roth and Sotomayor (1990) gives a masterul overview o two-sided matching markets. 2

3 internships. 2 In many countries, each year thousands o students seek or positions in colleges, 3 six years old children have to be assigned to public schools or 8 th graders high school students to high schools, 4 as well as civil servants to similar jobs in public positions scattered in di erent cities across a country. All o these entry-level many-to-one matching markets share two speci c eatures. First, workers enter the market by cohorts (oten once per year) and, second, agents are matched through a centralized procedure: an institution (clearinghouse) collects, or each participant, a ranked list o potential partners and proposes, ater processing the pro le o submitted ranked lists, a nal matching between rms and workers. Yet, and in order to survive, the proposed matching has to be stable (relative to the true preerence pro le) in the sense that all agents have to be matched to acceptable partners and no unmatched pair o a rm and a worker preer each other rather than the proposed partners. Stability constitutes a minimal requirement that a matching has to ul ll i the assignment is voluntary rather than compulsory. The literature has considered stability o a matching to be its main characteristic in order to survive. 5 Indeed, many o the successul mechanisms are stable despite the act that there exists no stable mechanism which makes truth-telling a dominant strategy or all agents (Roth, 1982). Thereore, an agent s (submitted) ranked lists o potential partners are not necessarily his true ones and the implemented matching may not be 2 See Roth (1984a, 2002) and Roth and Peranson (1999) or a careul description and analysis o this market. Roth (1991), Kesten (2005), Ünver (2005), and Ehlers (2008) describe and analyze the equivalent UK markets. 3 Romero-Medina (1998) studies the case o Spain and Chen and Kesten (2017) the case o China. 4 Chen and Sönmez (2006), Ergin and Sönmez (2006), and Abdulkadiro¼glu, Che, and Yosuda (2011) study the case o public schools in Boston, Abdulkadiro¼glu and Sönmez (2003) studies the cases o public schools in Boston, Lee County (Florida), Minneapolis and Seattle, Abdulkadiro¼glu, Pathak, and Roth (2005, 2009) study the case o public high schools in New York City, and Calsamiglia and Guell (2014) and Calsamiglia, Fu and Guell (2016) study the case o public schools in Barcelona. 5 See, or instance, Roth (1984a) and Niederle and Roth (2003). 3

4 stable or the true pro le. As a consequence, the literature has studied intensively Nash equilibria (NE) o direct preerence revelation games induced by di erent stable mechanisms or a given true preerence pro le. 6 Nevertheless all this strategic analysis presupposes that the true pro le o preerences is both certain and common knowledge among all agents; the very de nition o Nash equilibrium under complete inormation requires it. Indeed, participants in these markets perceive the outcome o the mechanism as being uncertain because the submitted preerences o the other participants are unknown. To model this uncertainty and to overcome the limitation o the complete inormation set up, we ollow the Bayesian approach by assuming that participants share a common belie (or common prior); namely, nature selects a preerence pro le according to a commonly known probability distribution on the set o pro les. Each agent then becomes aware o his preerence ordering (his type) and submits to the mechanism a preerence ordering according to his strategy (which speci es a preerence ordering or each type). Then, given a mechanism, a strategy pro le (a list o individual strategies) translates the uncertainty that each agent aces about preerence pro les into probability distributions on matchings and hence, on the set o potential partners. Thus, each type, when evaluating the consequences o submitting alternative ranked lists has to evaluate probability distributions (conditional on his type) on the set o potential partners. 7 Since matching markets require to report ranked lists and not their speci c utility representations, we stick to the ordinal setting and assume that probability distributions are evaluated according to the rst-order stochastic dominance criterion. Then, a strategy pro le is an ordinal Bayesian Nash equilibrium (OBNE) i, or every von Neumann Morgenstern (vnm)-utility unction that represents an agent s preerence ordering (his type), the submitted ranked list maximizes his expected utility in the 6 See Dubins and Freedman (1981), Roth (1982, 1984b, 1985), Gale and Sotomayor (1985), Shin and Suh (1996), Sönmez (1997), Ma (1995, 2002), and Alcalde (1996). 7 Roth (1989) is the rst to study Bayesian Nash equilibria (or a xed utility representation o agents preerences) in one-to-one matching markets under incomplete inormation. 4

5 direct preerence revelation game induced by the common belie and the mechanism. 8 In two earlier papers we have studied strategic incentives (i.e., OBNE) induced by centralized stable matching mechanisms under incomplete inormation. Speci cally Theorem 1 in Ehlers and Massó (2015) shows that, or many-to-one matching markets, there is a strong link between NE under complete inormation and OBNE under incomplete inormation. Namely, given a common belie, a strategy pro le is an OBNE under incomplete inormation in a stable mechanism i and only i, or any pro le in the support o the common belie, the submitted pro le is a NE under complete inormation at the true pro le in the direct preerence revelation game induced by the stable mechanism. This result implies the ormer Theorem 1 in Ehlers and Massó (2007) which says that, or one-to-one matching markets, truth-telling is an OBNE in the Bayesian direct revelation game induced by a common belie and a stable mechanism i and only i the support o the common belie is contained in the set o pro les with a unique stable matching. However, the proos o the two results use responsive extensions o rms preerences that are not monotonic. And we think that, in many applications (or instance, those described at the beginning o this Introduction), monotonicity is the norm: rms consider that having all positions lled with acceptable workers is always strictly preerred to having some positions vacant. In particular, in settings where, although workers may have di erent individual qualities or a rm, those di erences are not su ciently important to compensate the loss o two acceptable workers with only one worker. In this paper we rst de ne monotonic NE and monotonic OBNE by requiring that when rms compare sets o workers (or NE) or probability distributions by means o the rst order stochastic dominance criterion (or OBNE), they only use 8 This notion was introduced by d Aspremont and Peleg (1988) who call it ordinal Bayesian incentive-compatibility. Majumdar and Sen (2004) use it to relax strategy-prooness in the Gibbard- Satterthwaite Theorem. Majumdar (2003), Pais (2005, 2008), and Ehlers and Massó (2007, 2015) have already used, among others, this ordinal equilibrium notion in two-sided matching markets. 5

6 monotonic responsive extensions. Secondly, we restrict attention only to truth-telling. In particular, in Theorem 1 here, we show that truth-telling is a monotonic OBNE in a stable mechanism under incomplete inormation only i the support o the common belie is contained in the set o pro les with a unique stable matching. We then present Example 1 showing that the link between complete and incomplete inormation o Theorem 1 in Ehlers and Massó (2015) is broken when rms evaluate subsets o workers only with monotonic responsive extensions o their rankings on individual workers. Speci cally, in the many-to-one matching market o Example 1, truth-telling is a monotonic OBNE in the workers-optimal stable mechanism, and yet there is a preerence pro le in the support o the common belie at which truth-telling is not a monotonic NE in any stable mechanism (in particular, in the workers-optimal stable mechanism) under complete inormation at this pro le. Hence, to hold, Theorem 1 in Ehlers and Massó (2015) requires that rms ought to have non-monotonic responsive extensions. Nevertheless, we show that in Example 1 truth-telling is not a monotonic OBNE in the rms-optimal stable mechanism. This act raises the question o whether or not the link between incomplete and complete inormation holds or this stable mechanism, even when rms responsive preerences are restricted to be monotonic. Theorem 2 answers this question positively: given a common belie, truth-telling is a monotonic OBNE in the rms optimal stable mechanism under incomplete inormation i and only i the support o the common belie is contained in the set o all pro les where truth-telling is a monotonic NE in the rms optimal stable mechanism under complete inormation. The paper is organized as ollows. Section 2 describes the many-to-one matching market with responsive preerences. Section 3 introduces incomplete inormation and the notion o OBNE (and its monotonic variant), states and proves our two results (Theorems 2 and 3) and presents Example 1. 6

7 2 Many-To-One Matching Markets 2.1 Agents, Quotas, and Preerences The agents o a many-to-one matching market consist o two disjoint sets, the set o rms F and the set o workers W. A generic rm will be denoted by, a generic worker by w, and a generic agent by v 2 V F [ W. Both the set o workers W and the set o rms F are assumed to be nite. While workers can only work or at most one rm, rms may hire di erent numbers o workers. For each rm, there is a maximum number q 1 o workers that may hire, s quota. Let q = (q ) 2F be the vector o quotas. Each worker w has a strict preerence ordering P w over F [ ;g, where ; means the prospect o not being hired by any rm. Each rm has a strict preerence ordering P over W [ ;g, where ; stands or leaving a position un lled. 9 A pro le P = (P v ) v2v is a list o preerence orderings, one or each agent. To emphasize the preerence orderings o a subset o agents S V we oten denote a pro le P by (P S ; P S ). Let P v be the set o all preerence orderings o agent v. Let P = v2v P v be the set o all pro les and let P v denote the set v 0 2V nvgp v 0. Since agent v might have to compare potentially the same partner, we denote by R v the weak preerence ordering corresponding to P v ; namely, or v 0 ; v 00 2 V [ ;g, v 0 R v v 00 means either v 0 = v 00 or v 0 P v v 00. Momentarily x a worker w and his preerence ordering P w. Given v 2 F [ ;g, let B(v; P w ) be the weak upper contour set o P w at v; i.e., B(v; P w ) = v 0 2 F [ ;g j v 0 R w vg. Let A(P w ) be the set o acceptable rms or w according to P w ; i.e., A(P w ) = 2 F j P w ;g: Given a subset S F [ ;g, let P w js denote the restriction o P w to S. Similarly, given P 2 P, v 2 W [ ;g, 9 Observe that rm s preerence ordering is only de ned on the set W [ ;g in spite o the act that rm will be matched to a subset in 2 W. Hence, should be able to compare also subsets o workers with cardinality larger than one. For this reason, we will later extend any preerence ordering P on W [ ;g to a responsive preerence P on 2W (to be de ned in Subsection 2.4). Although each preerence P on W [;g admits many responsive extensions on 2 W ; the set o stable matchings is invariant to the particular chosen responsive extensions. 7

8 and S W [ ;g, we de ne B(v; P ), A(P ), and P js. Given P and w; w 0 2 W, let P w$w0 stand or the ordering where w and w 0 switch positions in P. 10 For example, i P : w 1 w 2 w 3 w 4 ;, then P w 1$w 3 : w 3 w 2 w 1 w 4 ;. 11 A many-to-one matching market (also known as a college admissions problem) is a quadruple (F; W; q; P ). Because F, W and q will oten remain xed, a problem will simply be a pro le P 2 P. I q = 1 or all 2 F, (F; W; q; P ) is called a one-to-one matching market (also known as a marriage market). 2.2 Stable Matchings The assignment problem consists o matching workers with rms keeping the bilateral nature o their relationship, complying with rms capacities given by their quotas, and allowing or the possibility that both workers and rms may remain unmatched. Formally, given a college admissions problem (F; W; q; P ), a matching is a mapping rom the set V to the set o all subsets o V such that: (m1) or all w 2 W, either j (w)j = 1 and (w) F or else (w) = ;; (m2) or all 2 F, () W and j()j q ; and (m3) (w) = g i and only i w 2 (). Abusing notation, we will oten write (w) = instead o (w) = g. I (w) = ; we say that w is unmatched at and i j()j < q we say that has q j()j un lled positions at ; is unmatched at when it has q un lled positions at. Let M denote the set o all matchings. Not all matchings are equally likely. Stability o a matching is considered to be its main characteristic in order to survive. A matching is stable i no agent is matched to 10 Formally, (i) P w$w0 jw [ ;gnw; w 0 g = P jw [ ;gnw; w 0 g, (ii) wp w 0 i w 0 P w$w0 w 0, (iii) or all v 2 W [ ;gnw; w 0 g, vp w i vp w$w0 w 0, and (iv) or all v 2 W [ ;gnw; w 0 g, vp w 0 i vp w$w0 w. 11 We will use the convention that P : w 1 w 2 w 3 w 4 ; means w 1 P w 2 P w 3 P w 4 P ;. 8

9 an unacceptable partner (individual rationality) and no unmatched worker- rm pair mutually preers each other to (one o) their current assignments (pair-wise stability). That is, given a many-to-one matching market P 2 P, a matching 2 M is stable (at P ) i (s1) or all w 2 W, (w)r w ;; (s2) or all 2 F and all w 2 (), wp ;; and (s3) there is no pair (w; ) 2 W F such that w =2 (), P w (w), and either wp w 0 or some w 0 2 () or wp ; i j()j < q. Notice that this de nition declares a matching to be stable i it is not blocked (in the sense o the core) by either individual agents or unmatched pairs. Gale and Shapley (1962) establishes that all college admissions problems have a non-empty set o stable matchings and Roth and Sotomayor (Theorem 3.3, 1990) states that larger coalitions do not have additional (weak) blocking power because the set o stable matchings coincides with the core. We denote by C(P ) the non-empty set o stable matchings at P (or the core o P ). 2.3 Stable Matching Mechanisms Whether or not a matching is stable depends on the preerence orderings o agents, and since they are private inormation, agents have to be asked about them. A mechanism requires each agent v to report some preerence ordering P v and associates a matching with any reported pro le P. Namely, a (direct revelation) mechanism is a unction ' : P! M mapping each preerence pro le P 2 P to a matching ' [P ] 2 M. Then ' [P ] (v) is the match o agent v at preerence pro le P under mechanism '. Note that, or all w 2 W, '[P ](w) 2 F [ ;g and, or all 2 F, '[P ]() 2 2 W. A mechanism ' is stable i or all P 2 P, ' [P ] 2 C (P ). The most popular stable mechanisms are the two associated to the two deerredacceptance algorithms (DA-algorithms) (Gale and Shapley, 1962): the rms-proposing 9

10 DA-algorithm, as a mechanism denoted by DA F, and the workers-proposing DAalgorithm, as a mechanism denoted by DA W. Fix a pro le P 2 P. At any step o the algorithm in which workers make o ers, each worker w proposes to the most-preerred rm among the set o rms that have not already rejected w during previous steps, while a rm accepts the q -most preerred workers among the set o current o ers plus the set o workers provisionally matched to in the previous step (i any). The algorithm stops at the step when either all o ers are accepted or workers have no more acceptable rms to whom they want to make an o er; the provisional matching becomes then de nite and is denoted by DA W [P ]: At any step o the algorithm in which rms make o ers, each rm proposes itsel to set o q -most preerred workers who have not already rejected during the previous steps, while each worker w accepts the o er o the best rm among the set o current o ers plus the one made by the rm provisionally matched in the previous step (i any). The algorithm stops at the step at which all o ers are accepted; the provisional matching becomes then de nite and is denoted by DA F [P ]: Responsive Extensions The notion o a mechanism in which rms (like workers) only submit rankings on individual agents ts with most o the mechanisms used in real lie centralized matching markets. But a mechanism matches each rm to a set o workers, taking into account only s preerence ordering P over individual workers. Thus, to study rms incentives in direct revelation mechanisms, preerence orderings o rms over individual workers have to be extended to preerence orderings over subsets o workers. But 12 It is well-known that or each pro le P 2 P, (i) DA W [P ] matches each worker w with w s best partner, among all partners that w is matched to across all stable matchings, and (ii) DA F [P ] matches each rm with s best set o partners according to any responsive extension o P (see Subsection 2.4 below or the de nition o responsiveness), among all partners that is matched to across all stable matchings. This is why they are named workers-optimal and rms-optimal stable matchings, respectively. 10

11 a rm may have di erent rankings over subsets o workers respecting its quota q and the ranking P over individual workers. For instance, let W = w 1 ; w 2 ; w 3 ; w 4 g be the set o workers and let P be such that P : w 1 w 2 w 3 w 4 ; and q = 2. While it is reasonable to assume that, under the absence o very strong complementarities among workers, the set w 1 ; w 2 g is preerred by to the set w 3 ; w 4 g or to the set w 1 ; w 3 g, rm s preerence between the sets w 1 ; w 4 g and w 2 ; w 3 g is ambiguous since P does not convey this inormation. Following the literature, 13 we will only require these extensions to be responsive in the sense that replacing a worker in a set (or an un lled position) by a better worker (or an acceptable worker) makes a set more preerred; or example, in all responsive extensions w 1 ; w 2 g is preerred to w 1 g, to w 3 ; w 4 g and to w 1 ; w 3 g but or some responsive extensions w 1 ; w 4 g is preerred to w 2 ; w 3 g while or other responsive extensions w 2 ; w 3 g is preerred to w 1 ; w 4 g. For later purposes we also introduce the notion o monotonic responsive extensions under which having all positions lled with acceptable workers is always strictly preerred to having some positions vacant. De nition 1 (Responsive Extensions) (a) The preerence extension P over 2W is responsive to the preerence ordering P over W [ ;g i or all S 2 2 W, all w 2 S, and all w 0 =2 S: (r1) jsj > q implies ;P S: (r2) S [ w 0 gp S i and only i jsj < q and w 0 P ;. (r3) S [ w 0 gp Snwg i and only i w0 P w. (b) We say that a responsive extension P o P is monotonic i or all S; S W such that js 0 j < jsj q and S A(P ), we have SP S0. Observe that responsive extensions are not necessarily monotonic. For instance, let W = w 1 ; w 2 ; w 3 g be the set o workers and consider a rm with q = 2 and 13 See or instance, Roth and Sotomayor (1990). 11

12 P : w 1 w 2 w 3 ;. Responsiveness does not rule out that worker w 1 may be more desirable than workers w 2 and w 3 together. For instance, the ordering w 1 ; w 2 gp w 1 ; w 3 gp w 1 gp w 2 ; w 3 gp w 2 gp w 3 gp ; is responsive to P but not monotonic since w 1 gp w 2; w 3 g and w 2 ; w 3 g A(P ). For some applications monotonicity is meaningul; or instance, in matching problems where leaving positions un lled is very costly. Given a responsive extension P o P, let R denote its corresponding weak preerence ordering on 2 W. Moreover, given S 2 2 W, let B(S; P ) be the weak upper contour set o P at S; i.e., B(S; P ) = S0 2 2 W j S 0 R Sg. Given P 2 P, we denote by resp(p ) and mresp(p ) the sets o responsive and monotonic responsive extensions o P, respectively. 2.5 Nash Equilibrium under Complete Inormation Clearly any mechanism ' : P! M and any true pro le P 2 P de ne a direct (ordinal) preerence revelation game under complete inormation or which we can de ne the natural (ordinal) notions o Nash equilibrium and monotonic Nash equilibrium. De nition 2 (Nash Equilibrium) (a) Let P 2 P be a pro le. A pro le P 0 is a Nash equilibrium (NE) under complete inormation P in the direct preerence revelation game induced by the mechanism ' i or all w 2 W, '[P 0 ](w)r w '[ ^P w ; P 0 w](w) or all ^P w 2 P w, and or all 2 F and all P 2 resp(p ), '[P 0 ]()R '[ ^P ; P 0 ]() or all ^P 2 P. (b) Let P 2 P be a pro le. A pro le P 0 is a monotonic Nash equilibrium (monotonic NE) under complete inormation P in the direct preerence revelation game induced by the mechanism ' i i or all w 2 W, '[P 0 ](w)r w '[ ^P w ; P 0 w](w) or all ^P w 2 P w, and or all 2 F and all P 2 mresp(p ), '[P 0 ]()R '[ ^P ; P 0 ]() or all ^P 2 P. 12

13 A large literature on matching studies Nash equilibrium under complete inormation in direct preerence revelation games induced by stable mechanisms; in particular, or the mechanisms DA W and DA F. However, or many applications the assumption that the true pro le is common knowledge is extremely unrealistic. In the next section we present an ordinal way o studying Bayesian Nash Equilibria under incomplete inormation, which has already been used (see Ehlers and Massó (2007, 2015), or instance). 3 Incomplete Inormation 3.1 Preliminaries and OBNE We depart rom the very strong assumption that the true pro le is common knowledge and consider the Bayesian direct preerence revelation game induced by a mechanism and a belie about the true pro le, which is shared among all agents. A common belie is a probability distribution P ~ over P. Given a pro le P and the common belie P ~, Pr P ~ = P g is the probability that P ~ assigns to the event that the true pro le is P. 14 Given v 2 V, let P ~ v denote the marginal distribution o P ~ over P v. Observe that, ollowing the Bayesian approach, the common belie P ~ describes agents uncertainty about the true pro le beore agents learn their types. Now, given a common belie ~P and a preerence ordering P v (agent v s type), let P ~ v j Pv denote the probability distribution which P ~ induces over P v conditional on P v. It describes agent v s uncertainty about the preerences o the other agents, given that his preerence ordering is P v. This ormulation does not require symmetry nor independence o belies; conditional belies might be very correlated i agents use similar sources to orm them (i.e., rankings, grades, recommendation letters, etc.). An agent with incomplete inormation about the others preerence orderings 14 Strictly speaking P ~ cannot be set equal to P because P ~ is not a random variable but a probability distribution on P. However, or convenience we use this notation as i P ~ were a random variable. 13

14 (more importantly, about their submitted lists) will perceive the outcome o a mechanism as being uncertain. A random matching ~ is a probability distribution over the set o matchings M. Given a matching and the random matching ~, Pr~ = g is the probability that ~ assigns to matching. But the uncertainty important or agent v is not over matchings but over v s set o potential partners. Let ~(w) denote the probability distribution which ~ induces over worker w s set o potential partners F [ ;g and let ~() denote the probability distribution which ~ induces over rm s set o potential partners 2 W. Namely, or w 2 W and all v 2 F [ ;g; X Pr~(w) = vg = Pr~ = g 2Mj(w)=vg and or 2 F and all S 2 2 W, X Pr~() = Sg = Pr~ = g: 2Mj()=Sg A mechanism ' and a common belie P ~ de ne a direct (ordinal) preerence revelation game under incomplete inormation as ollows. Beore submitting a list to the mechanism, agents learn their types. Thus, a strategy o agent v is a unction s v : P v! P v speciying or each type o agent v, P v, a list that v submits to the mechanism, s v (P v ). 15 A strategy pro le is a list s = (s v ) v2v o strategies speciying or each true pro le P a submitted pro le s(p ). Given a mechanism ' : P! M and a common belie P ~ over P, a strategy pro le s : P! P induces a random matching '[s( P ~ )] in the ollowing way: or each 2 M, X Pr P ~ = P g P 2Pj'[s(P )]=g is the probability o matching. But ater learning their types, agents also update their belies. Hence, the relevant random matching or agent v, given his type P v and 15 Ehlers and Massó (2015) contains the analysis when agents may use mixed strategies, and shows that the addition o the randomness coming rom mixed strategies does not change the results. Hence, we restrict here our analysis to pure strategies. 14

15 a strategy pro le s, is '[s v (P v ); s v ( P ~ v j Pv )] (where s v ( P ~ v j Pv ) is the probability distribution over P v which s v and P ~ induce conditional on P v ). But again, the relevant uncertainty that agent v aces is given by '[s v (P v ) ; s v ( P ~ v j Pv )] (v), the probability distribution which the random matching '[s v (P v ) ; s v ( P ~ v j Pv )] induces over v s set o potential partners. Observe that this uncertainty is held by each agent v, given his type P v, beore the mechanism proposes a matching. Then, ater collecting the preerence pro le s(p ), the mechanism ' proposes a unique matching '[s(p )], the one that agents have to carry out. Our stability condition on the mechanism applies to this proposed matching (i.e., it is the standard notion o stability under complete inormation). This is why we do not need an ex ante notion o stability relative to the incomplete inormation environment when the partners are still uncertain and no particular matching has been proposed yet. To evaluate the consequences o their declared preerence orderings agents will compare their induced probability distributions on the set o their potential partners according to the rst-order stochastic dominance criterion. De nition 3 (First-Order Stochastic Dominance) (o1) A random matching ~ rst-order stochastically P w dominates a random matching ~ 0, denoted by ~ (w) m Pw ~ 0 (w), i or all v 2 F [ ;g ; X X Pr~ (w) = v 0 g Pr~ 0 (w) = v 0 g: v 0 2F [;gjv 0 R wvg v 0 2F [;gjv 0 R wvg (o2) A random matching ~ rst-order stochastically P dominates a random matching ~ 0, denoted by ~ () m P ~ 0 (), i or all P 2 resp(p ) and all S 2 2 W, 16 X X Pr~ () = S 0 g Pr~ 0 () = S 0 g: (1) S 0 22 W js 0 R Sg S 0 22 W js 0 R Sg 16 Observe that this de nition requires that ~ rst-order stochastically dominates ~ 0 according to all responsive extensions o P. Note that this requirement is meaningul since the clearinghouse observes rms rankings over individual workers only and not which responsive extension they use to compare sets o workers. 15

16 All mechanisms used in centralized matching markets are ordinal. In other words the only inormation available or a clearinghouse are the agents ordinal preerences over potential partners. In such an environment a strategy pro le is an ordinal Bayesian Nash equilibrium whenever, or any agent s true ordinal preerence, submitting the rank list speci ed by his strategy maximizes his expected utility or every vnm-utility representation o his true preerence. This requires that an agent s strategy only depends on the ordinal ranking induced by his vnm-utility unction (i any). Moreover, ordinal strategies are well-de ned i an agent only observes his ordinal ranking and may have (still) little inormation about his utilities o his potential partners. Below we de ne the notions o ordinal Bayesian Nash equilibrium De nition 3 (Ordinal Bayesian Nash Equilibrium) Let P ~ be a common belie. Then a strategy pro le s is an ordinal Bayesian Nash equilibrium (OBNE) in the mechanism ' under incomplete inormation P ~ i and only i or all v 2 V and all P v 2 P v such that Pr P ~ v = P v g > 0, '[s v (P v ); s v ( ~ P v j Pv )](v) m Pv '[P 0 v; s v ( ~ P v j Pv )](v) or all P 0 v 2 P v : 17 (2) Similarly, a strategy pro le s is a monotonic ordinal Bayesian Nash equilibrium (monotonic OBNE) in the mechanism ' under incomplete inormation ~ P i and only i condition (2) holds or all w 2 W but, or all 2 F, (2) is replaced by where m mon P '[s (P ); s ( ~ P j P )]() m mon P '[P 0 ; s ( ~ P j P )]() or all P 0 2 P ; means that condition (1) in (o2) holds only or all P 2 mresp(p ). Theorem 1 in Ehlers and Massó (2015) characterize OBNE in any stable mechanism under incomplete inormation ~ P by means o the NE in the stable mechanism under complete inormation. Theorem 1 (Ehlers and Massó, 2015, Theorem 1) Let P ~ be a common belie, s be a strategy pro le, and ' be a stable mechanism. Then, s is an OBNE in the stable mechanism ' under incomplete inormation P ~ i and only i or any pro le P 16

17 in the support o ~ P, s(p ) is a Nash equilibrium under complete inormation P in the direct preerence revelation game induced by '. 3.2 Truth-telling with Monotonicity We will ocus on truth-telling. Ehlers and Massó (2007, Corollary 5) shows that truth-telling is an OBNE in a stable mechanism under incomplete inormation P ~ i and only i the support o P ~ is contained in the set o pro les with a singleton core. This is an immediate consequence o Theorem 1. Here, we rst show that or truth-telling to be a monotonic OBNE in a stable mechanism it is still necessary that the common belie puts positive probability only on pro les with singleton core. Thereore, even in monotonic matching markets the core is unique or any realized (or observed) pro le i truth-telling is an equilibrium. Note, however, that here we cannot use Theorem 1 because we only allow or monotonic responsive extensions. Theorem 2 Let P ~ be a common belie and assume that truth-telling is a monotonic OBNE in a stable mechanism under incomplete inormation P ~. Then, the support o ~P is contained in the set o all pro les with a singleton core. Proo. Let ' be a stable mechanism. To obtain a contradiction, suppose that there is some P 2 P such that both Pr P ~ = P g > 0 and jc(p )j 2. By stability o ', '[P ] 2 C(P ). Let 2 C(P )n'[p ]g. I there is some w 2 W such that (w)p w '[P ](w), then similarly to Ehlers and Massó (2007) we can show that truthtelling cannot be a monotonic OBNE. Thus, '[P ](w)r w (w) or all w 2 W: since DA W [P ] is the workers-optimal stable matching, '[P ] = DA W [P ]: (3) Obviously, (3) needs to hold or all pro les belonging to the support o P ~. Since 6= '[P ] and '[P ] = DA W [P ], there exists some 2 F such that () 6= '[P ](). Then, by Roth and Sotomayor (1990, Theorems 5.12 and 5.13), j()j = 17

18 j'[p ]()j = q. By Roth and Sotomayor (1989, Theorem 4), we have wp w 0 or all w 2 () and all w 0 2 '[P ]()n(). Thus, ()P '[P ]() (4) or all monotonic responsive extensions P o P. Let w 2 () be such that wr w or all w 2 (). Let P 0 2 P be such that (i) A(P 0 ) = B( w; P ) and (ii) P 0 ja(p 0 ) = P ja(p 0 ). By construction o P 0, 2 C(P 0 ; P ). Hence, '[P 0 ; P ]() = () or i '[P 0 ; P ]() 6= (), then, again by Roth and Sotomayor (1989, Theorem 4), or all w 2 '[P 0 ; P ]() and all w 0 2 ()n'[p 0 ; P ](), wp 0 w0. Thus, by P 0 ja(p 0 ) = P ja(p 0 ) and (4), '[P 0 ; P ]()P '[P ]() (5) or all monotonic responsive extensions P o P. Let P be the monotonic responsive extension o P such that or all W W, W 0 P () i and only i (i) jw 0 j = q (note that j()j = q ) and (ii) there is a one-to-one mapping h : W 0! () such that w 0 R h(w 0 ) or all w 0 2 W 0 with strict preerence holding or some worker in W 0. In other words, W 0 is strictly preerred to () under P only i W 0 is unambiguously strictly preerred to () or all monotonic responsive extensions o P. Let P 0 2 P be such that both Pr ~ P = (P ; P 0 )g > 0 and '[P ; P 0 ]() 2 B((); P ). We want to show that '[P 0 ; P 0 ]() 2 B((); P ): Let ^ = '[P ; P 0 ] and 0 = '[P 0 ; P 0 ]. By the de nition o P and P 0, we have that ^()R (), ^() A(P 0 ) and j^()j = q. Thus, by A(P 0 ) = B( w; P ) and P 0 ja(p 0 ) = P ja(p 0 ), ^ 2 C(P 0 ; P 0 ). Since 0 ; ^ 2 C(P 0 ; P 0 ), Theorem 4 o Roth and Sotomayor (1989) tells us that one o the ollowing two cases needs to hold. Case 1: w 0 R ^w or all w () and ^w 2 ^()n 0 (). Thus, 0 ()R ^() and since ^()R (), 0 ()R (). Hence, '[P 0 ; P 0 ]() 2 B((); P ). 18

19 Case 2: ^wr w 0 or all ^w 2 ^() and w ()n^(). I ^() 6= 0 (), then ^()P 0 (): (6) But then 0 () A(P 0 ) = B( w; P ) A(P ) and by P 0 ja(p 0 ) = P ja(p ), we have 0 2 C(P ; P 0 ). Note that (P ; P 0 ) is a pro le belonging to the support o ~ P and thus, by (3), ^ = '[P ; P 0 ] = DA W [P ; P 0 ]. Now (6) contradicts the acts that DA W chooses the stable matching in C(P ; P 0 ) which is worst or the rms and 0 2 C(P ; P 0 ). Thus, ^() = 0 () and 0 ()R ^(). Since ^()R (), 0 ()R (). Hence, '[P 0 ; P 0 ]() 2 B((); P ). Cases 1 and 2 show that i P 0 2 P is such that Pr P ~ = (P ; P 0 )g > 0 and '[P ; P 0 ]() 2 B((); P ), then '[P 0 ; P 0 ]() 2 B((); P ). Since Pr ~ P j P = P g > 0, '[P ]() =2 B((); P ), and '[P 0 ; P ]() 2 B((); P ), it ollows that Pr'[P 0 ; ~ P j P ]() 2 B((); P )g > Pr'[P ; ~ P j P ]() 2 B((); P )g; which means truth-telling is not a monotonic OBNE in the stable mechanism '. It is natural to ask when the link in Theorem 1 breaks in many-to-one matching markets. We will provide an example o a market in which our ormer main result (stated here as Theorem 1) is not true. More precisely, we will see that Theorem 1 depends on rms having responsive extensions which are not monotonic. In markets where workers are close substitutes, we may restrict the set o responsive extension rms are allowed to have. More precisely, in environments where it is (very) costly to leave some positions un lled, responsive extensions are necessarily monotonic. For instance, in medical markets or a certain speciality hospitals may not be able to provide ull medical service in case a position is let vacant. In such environments it is natural to assume that rms have only monotonic responsive preerences and ocus on monotonic OBNE 19

20 Recall that Roth (1985) exhibits an example o a pro le with a singleton core where a rm pro tably manipulates any stable mechanism and truth-telling is not a NE under complete inormation. Below we exhibit an example o a many-to-one matching market where (i) truth-telling is a monotonic OBNE or the stable mechanism DA W (and any pro le in the support has a singleton core) and (ii) there is a pro le belonging to the support o the common belie at which truth-telling is not a monotonic NE under complete inormation. Thereore, the link in Theorem 1 is broken in one direction 18 when all rms have only monotonic responsive extensions and one rm has a capacity o at least two. Example 1 Consider a many-to-one matching market with three rms F = 1 ; 2 ; 3 g and our workers W = w 1 ; w 2 ; w 3 ; w 4 g. Firm 1 has capacity q 1 = 2 and rms 2 and 3 have capacity q 2 = q 3 = 1. Consider the common belie P ~ with Pr P ~ = P g = p and Pr P ~ = P g = 1 p, where p < 1=2, and P and P are the ollowing pro les: P 1 P 2 P 3 P w1 P w2 P w3 P w4 P1 P2 P3 Pw1 Pw2 Pw3 Pw4 w 1 w 1 w w 1 w 2 w w 2 w 2 w w 2 w 3 w 3 w w 3 : w 4 w 4 w 4 w 4 Note that P 1 = P 1. It is straightorward to veriy that both pro les have a singleton core and C(P ) = g and C( P ) = g, where A and A : w 3 ; w 4 g w 2 g w 1 g w 1 ; w 3 g w 2 g w 4 g Let ' be a stable mechanism. Thus, by stability o ', '[P ] = and '[ P ] =. First we will show that or the pro le P truth-telling is not a monotonic Nash equilibrium 18 It is clear that the other direction is true or any arbitrary game.o incomplete inormation. 20

21 under complete inormation P in the direct preerence revelation game induced by '. Let P P 1 be such that P 0 1 : w 1 w 2 w 4 ;w 3. Then C(P 0 1 ; P 1 ) = 0 g where w 1 ; w 4 g w 2 g w 3 g A : Hence, by stability o ', '[P 0 1 ; P 1 ] = 0. Obviously, or all responsive (and hence, monotonic) extensions P 1 o P we have w 1 ; w 4 gp 1 w 3 ; w 4 g, which is equivalent to '[P 0 1 ; P 1 ]( 1 )P 1 '[P ]( 1 ). Thereore, truth-telling is not a monotonic Nash equilibrium in any stable mechanism ' under complete inormation P (and pro le P belongs to the support o the common belie ~ P ). On the other hand we will show that truth-telling is a monotonic OBNE in the stable mechanism DA W under incomplete inormation ~ P. Note that or all v 2 V n 1 g, i v observes his preerence relation, then v knows whether P was realized or P was realized. Since at both o P and P the core is a singleton and rms 2 and 3 have quota one, it ollows that v cannot gain by a deviation. Next we consider rm 1. All arguments except or the last one apply to any stable mechanism '. Observe that P 1 = P 1 and the random matching '[P 1 ; ~ P 1 j P1 ] assigns to 1 the set w 3 ; w 4 g with probability p and the set w 1 ; w 3 g with probability 1 p. Let P 1 be a monotonic responsive extension o P 1 and P 00 1 that 2 P 1. We show '[P 1 ; ~ P 1 j P1 ]( 1 ) m P 1 '[P 00 1 ; ~ P 1 j P1 ]( 1 ): (7) We distinguish two cases. First, suppose that j'[p 00 1 ; P 1 ]( 1 )j = 1 or j'[p 00 1 ; P 1 ]( 1 )j = 1. Now i (7) does not hold, then by monotonicity o P 1 and the act that when submitting P 1, 1 is assigned the set w 3 ; w 4 g with probability p and the set w 1 ; w 3 g with probability 1 p (where 1 p > 1=2), we must have that '[P 00 1 ; P 1 ]( 1 )P 1 w 1 ; w 3 g or '[P 00 1 ; P 1 ]( 1 )P 1 w 1 ; w 3 g. Obviously, rom the de nition o P 1, the last is impossible. Thus, '[P 00 1 ; P 1 ]( 1 )P 1 w 1 ; w 3 g and, by responsiveness o P 1, we must have '[P 00 1 ; P 1 ]( 1 ) = w 1 ; w 2 g. But then, without loss o generality, we would have DA F [P 00 1 ; P 1 ]( 1 ) = w 1 ; w 2 g (because DA F chooses the most preerred sta- 21

22 ble matching rom the rms point o view). 19 Since we have C(P ) = g and DA F [P 1 ; P 1 ]( 1 ) = w 3 ; w 4 g, this would imply that in the corresponding one-toone matching problem DA F is group manipulable by the two copies o 1 (with each copy gaining strictly), a contradiction to the result o Dubins and Freedman (1981). 20 Second, suppose that j'[p 00 1 ; P 1 ]( 1 )j = 2 and j'[p 00 1 ; P 1 ]( 1 )j = 2. Then by de- nition o P 1 and j'[p 00 1 ; P 1 ]( 1 )j = 2, we must have '[P 00 1 ; P 1 ]( 1 ) = w 1 ; w 3 g. Thus, by stability o ', w 1 ; w 3 g A(P 00 1 ). I or all 00 2 C(P 00 1 ; P 1 ), 00 (w 4 ) = ;, then w 4 =2 A(P 00 1 ) and by de nition o P 1 and w 3 2 A(P 00 1 ), DA W [P 00 1 ; P 1 ]( 1 ) = w 3 g. Then 1 does not ll all its positions at the workers-optimal matching and by Roth and Sotomayor (1990), 1 is matched to the same set o workers at all stable matchings. Hence, '[P 00 1 ; P 1 ]( 1 ) = w 3 g and (7) holds (because when submitting P 00 1, rm 1 is matched with probability p to w 3 g and with probability 1 p to w 1 ; w 3 g). I or some 00 2 C(P 00 1 ; P 1 ), 00 (w 4 ) 6= ;, then by de nition o P 1, 00 (w 4 ) = 1 ; otherwise the pair (w 2 ; 2 ) would block 00 at (P 00 1 ; P 1 ) i 00 (w 4 ) = 2 and the pair (w 1 ; 3 ) would block 00 at (P 00 1 ; P 1 ) i 00 (w 4 ) = 3. Thus, by w 3 ; w 4 g A(P 00 1 ), 2 C(P 00 1 ; P 1 ) and DA W [P 00 1 ; P 1 ] =. Hence, (7) holds or the stable mechanism DA W. Note that Example 1 does not contradict Theorem 1. When considering the nonmonotonic responsive extension P 1 such that w 1 gp 1 w 3 ; w 4 g, then rm 1 gains by submitting the list ^P 1 where worker w 1 is the unique acceptable worker (i.e. A( ^P 1 ) = w 1 g). Then we have both '[ ^P 1 ; P 1 ]( 1 ) = w 1 g and '[ ^P 1 ; P 1 ]( 1 ) = w 1 g, which means that truth-telling is not an OBNE in any stable mechanism ' under incomplete inormation ~ P. 19 I DA F [P 00 1 ; P 1 ]( 1 ) 6= w 1 ; w 2 g, then choose P such that A(P ) = w 1 ; w 2 g. obtain DA F [P ; P 1 ]( 1 ) = w 1 ; w 2 g. Then we 20 Their result says that in a marriage market no group o rms can pro tably manipulate DA F at the true pro le under complete inormation (with strict preerence holding or all rms belonging to the group). 22

23 Furthermore, in Example 1 truth-telling is not a monotonic OBNE in DA F under incomplete inormation ~ P : To see that, consider the preerence P 00 1 Example 1. We have 0 DA F [P 00 1 ; P 1 ] = Thus, w 1 ; w 4 g w 2 w 3 1 A and DA F [P 00 1 ; P 1 ] = : w 1 w 2 w 4 w 3 ; in w 1 ; w 3 g w 2 w 4 1 A : PrDA F [P 00 1 ; ~ P 1 j P1 ]( 1 ) 2 B(w 1 ; w 4 g; P 1 )g = 1 PrDA F [P 1 ; ~ P 1 j P1 ]( 1 ) 2 B(w 1 ; w 4 g; P 1 )g = 1 p or all (monotonic) responsive extensions P 1 o P 1 (i.e. this is an unambiguous deviation or rm 1 because it does not depend on the choice o the responsive extension o P ). This means that truth-telling is not a monotonic OBNE in DA F under incomplete inormation ~ P. One would think that a naïve way to show that in DA F truth-telling is a monotonic OBNE (but not or some pro le in the support) by enlarging the above example and adding the pro le P w 3$w 4 1 (where the roles o w 3 and w 4 are exchanged in P 1 ) to the support o ~ P 1 j P1. By symmetry, then we would have DA F [P 00 1 ; P w 3$w 4 1 ]( 1 ) = w 3 ; w 4 g and DA F [P 1 ; P w 3$w 4 1 ]( 1 ) = w 1 ; w 3 g, meaning that under pro le P w 3$w 4 1 rm 1 strictly preers reporting P 1 to reporting P Observe that since DA F [P 1 ; P w 3$w 4 1 ] is individually rational at (P 1 ; P w 3$w 4 1 ) and it matches each worker w to w s best rm, DA F [P 1 ; P w 3$w 4 1 ] 2 C(P 1 ; P w 3$w 4 ): However, this would then give an immediate 1 contradiction to Theorem 2: DA F [P 00 1 ; P w 3$w 4 1 ]; DA F [P 1 ; P w 3$w 4 1 ] 2 C(P 1 ; P w 3$w 4 1 ) implies that the core is not a singleton set under pro le (P 1 ; P w 3$w 4 1 ) which belongs to the support o ~ P. 3.3 The Link or Truth-telling in the DA F In this subsection we establish that i agents truth-tell in the DA F stable mechanism, the strong link o Theorem 1 between complete and incomplete inormation carries over to monotonic matching markets. 23

24 Theorem 3 Let P ~ be a common belie. Then, truth-telling is a monotonic OBNE in DA F under incomplete inormation P ~ i and only i the support o P ~ is contained in the set o all pro les where truth-telling is a monotonic NE in DA F under complete inormation Proo o Theorem 3 Let P ~ be a common belie. (() Suppose that or all pro les in the support o P ~, truth-telling is a monotonic NE in DA F under complete inormation. Let P be such that Pr P ~ = P g > 0. Then, or all w 2 W and all Pw 0 2 P w, DA F [P ](w)r w DA F [Pw; 0 P w ](w) and or all 2 F and all P 0 2 P ; DA F [P ]()R DA F [P 0 ; P ]() or all P 2 mresp(p ). Hence, or all w 2 W and all P w 2 P w such that Pr P ~ w = P w g > 0, we have that or all Pw 0 2 P w ; DA F [P w ; P ~ w j Pw ](w) m Pw DA F [Pw; 0 P ~ w j Pw )](w); and or all 2 F and all P 2 P such that Pr P ~ = P g > 0, we have that or all P 0 2 P ; DA F [P ; P ~ j P ]() m P DA F [P; 0 P ~ j P )](): That is, P is a monotonic OBNE in DA F under P ~, the desired conclusion. ()) Let P be such that Pr P ~ = P g > 0 and assume that P is a monotonic OBNE in DA F under incomplete inormation P ~. By Theorem 2, jc(p )j = 1. To obtain a contradiction suppose that in addition P is not a monotonic NE in DA F under complete inormation P. We rst show in Lemma 1 below that then some rm with quota o at least two has a pro table deviation such that (i) the rm is only matched to acceptable workers, (ii) the rm is matched to q workers and (iii) or some monotonic responsive extension the rm strictly preers the assigned workers to the the set received under truth-telling. 24

25 Lemma 1 Assume P is such that jc(p )j = 1: I P is not a monotonic NE in DA F under complete inormation P, then there exists a rm 2 F such that q 2 and or some ^P we have (i) DA F [ ^P ; P ]() A(P ), (ii) jda F [ ^P ; P ]()j = q = jda F [P ]()j, and (iii) DA F [ ^P ; P ()P DA F [P ]() or some monotonic responsive extension P P. o Proo o Lemma 1. Let P 2 P be such that jc(p )j = 1. Then, using a similar argument than the one used in the su ciency proo o Theorem 1 in Ehlers and Massó (2007), it can be seen that no worker has a pro table deviation rom P in DA F. The same argument applies to any rm with quota one. Thus, i P is not a monotonic NE in DA F under complete inormation P, then or some 2 F with q 2 and ^P, we have DA F [ ^P ; P ]()P DA F [P ]() (8) or some P 2 mresp(p ). Because P is responsive, we have DA F [ ^P ; P ]() \ A(P )R DA F [ ^P ; P ](). Thus, by (8) and monotonicity o P, jda F [ ^P ; P ]() \ A(P )j jda F [P ]()j: (9) But then we may suppose w.l.o.g. that DA F [ ^P ; P ]() A(P ). To see that, assume DA F [ ^P ; P ]() * A(P ): Then, choose any preerence P 0 2 P such that A(P 0 ) = DA F [ ^P ; P ]() \ A(P ). Now consider the matching market where the set o workers ^W = DA F [ ^P ; P ]()na(p ) is not present with pro le (P 0 ; P ^W [g ). Then, it is easy to see that DA F [P 0 ; P ^W [g ]() = A(P 0 ). Now letting workers in ^W reenter the market, rms should weakly gain meaning that DA F [P 0 ; P ]() = A(P 0 ).21 This means that has a pro table deviation rom P in DA F where is 21 See Theorem 2.25 in Roth and Sotomayor (1990). 25

26 only matched to acceptable workers. Thus, DA F [ ^P ; P ]() A(P ) holds. But the inclusion is (i) in the Lemma. Next we show that jda F [ ^P ; P ]()j = jda F [P ]()j. To obtain a contradiction, assume otherwise. By (9) and (i), we have jda F [ ^P ; P ]()j > jda F [P ]()j. Then at DA F [P ] rm has some positions un lled and the ranking o P over A(P ) is irrelevant or the stability o DA F [P ] under P. Let k = jda F [ ^P ; P ]()j. In particular, DA F [P ] is stable under P i rm s quota is reduced rom q to k. W.l.o.g. we may suppose that A( ^P ) = DA F [ ^P ; P ](). Now again in any matching which is stable under ( ^P ; P ) rm is matched to DA F [ ^P ; P ](). Again rm s quota may be reduced to k. But now consider P 0 such that A(P 0 ) = A(P ) and DA F [ ^P ; P ]() are the rst k most preerred workers under P 0. But then both DA F [ ^P ; P ] and DA F [P ] must be stable under (P 0 ; P ), which is a contradiction because jda F [ ^P ; P ]()j 6= jda F [P ]()j and any rm is matched to the same number o workers at all stable matchings under the pro le (P 0 ; P ]). We have shown jda F [ ^P ; P ]()j = jda F [P ]()j. Furthermore, i jda F [P ]()j < q, then, by (i), both matchings DA F [P ] and DA F [ ^P ; P ] are stable under P: By (8), DA F [ ^P ; P ]() 6= DA F [P ]() which means that jc(p )j 6= 1, a contradiction. Hence, jda F [P ]()j = q = jda F [ ^P ; P ]()j (which is (ii)). Now (iii) just ollows rom (8). We now proceed with the proo o Theorem 3. Let 2 F; with q 2, and ^P be the rm and its preerences identi ed in Lemma 1, or which (i), (ii) and (iii) hold. Letting ^ = DA F [ ^P ; P ] and = DA F [P ], order the workers in ^() and () according to P : let ^() = ^w 1 ; : : : ; ^w q g and () = w 1 ; : : : ; w q g. Furthermore, because o (i)-(iii) and we are using DA F, we may assume w.l.o.g. that (a) A( ^P ) = B(w q ; P ) [ B( ^w q ; P ), (b) ^P j^() = P j^(), (c) ^P jb( ^w q ; ^P ) = P jb( ^w q ; ^P ), and 26

27 (d) ^P ja( ^P )nb( ^w q ; ^P ) = P ja( ^P )nb( ^w q ; ^P ). Note that since ^() B( ^w q ; ^P ); (c) implies (b), which in turn implies ^() B( ^w q ; P ). We distinguish two cases: ^w q P w q and w q R ^w q. Case 1: ^w q P w q. Let W consist o the q workers which are P -least preerred in B(w q ; P )nw q g. Then choose the monotonic responsive extension P o P such that W 0 R W i W 0 contains q workers in B(w q ; P )nw q g. Let P 00 and P ja(p ) = ^P ja(p 00 be such that A(P 00 ) = B(w q ; P )nw q g ). Then ^ 2 C(P 00 ; P ) and by construction, DA F [P 00 ; P ]() contains q workers in A(P 00 ), but DA F [P ]() contains at most q 1 workers in A(P 00). DA F [P 00 ; P ]() R W P DA F [P ; P ](): (10) To obtain a contradiction with the act that P is a monotonic OBNE in DA F under incomplete inormation ~ P, we consider any subpro le P 0 such that Pr ~ P = (P ; P 0 )g > 0 and restrict the attention to the upper contour set o P at W : We want to show that i DA F [P ; P 0 ]() R W, then DA F [P 00 ; P 0 ]() R W. By the de nitions o W and P ; DA F [P ; P 0 ]() R W implies that DA F [P ; P 0 ]() contains q workers in A(P 00 ). I DA F [P 00 ; P ]() contains at most q 1 workers in A(P 00 ), then the ranking does not matter and this matching is also stable (letting A(P 000 ) = A(P ) and P ja(p ) = P ja(p )) under (P ; P 0 ). Furthermore, DA F [P ; P ] is stable under (P ; P 0 ) and DAF [P ; P 0 ] = q. But then would be matched to di erent numbers o workers under di erent matchings belonging to C(P 000 ; P 0 ), a contradiction. Thus, DA F [P 00 ; P 0 ]() contains q workers in A(P 00 ). Now by our choice o W and P, we obtain or any pro le P 0 in the support o ~P j P, DA F [P ; P 0 ]() R W implies DA F [P 00 ; P 0 ]() R W. Then, together with (10) and Pr ~ P = (P ; P )g > 0; this implication means that truth-telling is not an OBNE in DA F under ~ P because or the monotonic responsive extension P o P ; PrDA F [P 00 ; ~ P j P ]() 2 B( W ; P )g > PrDA F [P ; ~ P j P ]() 2 B( W ; P )g: 27