MATCHING MARKETS UNDER (IN)COMPLETE INFORMATION

We introduce incomplete information to centralized many-to-one matching markets. This is important because in real life markets (i) any agent is uncertain about the other agents' true preferences and (ii) most entry-level matching is many-to-one (and not one-to-one). We show that given a common prior, a strategy profile is an ordinal Bayesian Nash equilibrium under incomplete information in a stable mechanism if and only if, for any true profile in the support of the common prior, the submitted profile is a Nash equilibrium under complete information in the direct preference revelation game induced by the stable mechanism.


Introduction
Centralized many-to-one matching markets operate as follows to match the agents from two sides, the firms (colleges, hospitals, schools, etc.) and the workers (students, medical interns, children, etc.): a centralized clearinghouse collects for each participant a ranked list of potential partners and matches via a mechanism firms and workers on the basis of the submitted ranked lists. In applications many of the successful mechanisms are stable. 12 The literature has considered stability of a matching (in the sense that all agents have to be matched to acceptable partners and no unmatched pair of a firm and a worker prefer each other rather than the proposed partners) to be its main characteristic in order to survive. 3 This is puzzling because there exists no stable mechanism which makes truth-telling a dominant strategy for all agents (Roth, 1982). Therefore, an agent's (submitted) ranked lists of potential partners are not necessarily his true ones and the implemented matching may not be stable for the true profile. The literature has studied intensively Nash equilibria of direct preference revelation games induced by different stable mechanisms under complete information. 4 We use the (ordinally) Bayesian approach in many-to-one matching markets by assuming that nature selects a preference profile according to a commonly known probability distribution on the set of profiles (a common prior). 5 Since matching 1 See Roth (1984a), Roth and Peranson (1999), and Roth (2002) for a careful description and analysis of the American entry-level medical market. Roth (1991), Kesten (2005), Ünver (2005), and Ehlers (2008) describe and analyze the equivalent UK markets. 2 Chen and Sönmez (2006), Ergin and Sönmez (2006), and Abdulkadiroglu, Che, and Yosuda (2011) study the case of public schools in Boston, Abdulkadiroglu and Sönmez (2003) studies the cases of public schools in Boston, Lee County (Florida), Minneapolis, and Seattle, and Abdulkadiroglu, Pathak, and Roth (2005Roth ( , 2009) study the case of public high schools in New York City. 3 See, for instance, Roth (1984a) and Niederle and Roth (2003). 4 See Dubins and Freedman (1981), Roth (1982Roth ( , 1984bRoth ( , 1985, Gale and Sotomayor (1985), Shin and Suh (1996), Sönmez (1997), Ma (1995Ma ( , 2002, and Alcalde (1996). 5 Roth (1989) is the first paper studying strategic incentives generated by stable mechanisms markets require to report ranked lists and not their specific utility representations, we stick to the ordinal setting and assume that probability distributions are evaluated according to the first-order stochastic dominance criterion. Then, a strategy profile is an ordinal Bayesian Nash equilibrium (OBNE) if, for every von Neumann Morgenstern (vNM)-utility function of an agent's preference ordering (his type), the submitted ranked list maximizes his expected utility in the direct preference revelation game induced by the common prior and the mechanism. 6 For direct preference revelation games under incomplete information induced by a stable mechanism, our main result shows a link between Nash equilibria under complete information and OBNE under incomplete information. More precisely, Theorem 1 states that, given a common prior, a strategy profile is an OBNE under incomplete information in a stable mechanism if and only if for any profile in the support of the common prior, the submitted profile is a Nash equilibrium under complete information at the true profile in the direct preference revelation game induced by the stable mechanism.
The paper is organized as follows. Section 2 describes the many-to-one matching market with responsive preferences and introduces incomplete information and the notion of ordinal Bayesian Nash equilibrium. Section 3 states our main result, Theorem 1, and its applications. Section 4 discusses variations of our main result and the Appendix contains all proofs.
under incomplete information. He shows that for particular vNM-utility representations of the ordinal preferences, Bayesian Nash equilibria under incomplete information may not satisfy appealing properties of Nash equilibria under complete information. Chakraborty, Citanna, and Ostrovsky (2010) study two-sided matching markets with interdependent preferences. 6 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-proofness in the Gibbard-Satterthwaite Theorem. Majumdar (2003), Pais (2005), and Ehlers and Massó (2007) have already used this ordinal equilibrium notion in one-to-one matching markets. Each firm f has a strict preference ordering P f over W ∪ {∅}, where ∅ stands for leaving a position unfilled. A profile P = (P v ) v∈V is a list of preference orderings.
Given S ⊆ V , we sometimes write (P S , P −S ) instead of P . Let P v be the set of all preference orderings of agent v. Let P = × v∈V P v be the set of all profiles and let , and P f |S. A many-to-one matching market (or college admissions problem) is a quadruple (F, W, q, P ). Because F , W and q remain fixed, a problem is simply a profile P ∈ P. If q f = 1 for all f ∈ F , (F, W, q, P ) is called a one-to-one matching market.
A matching is a function µ : V → 2 V satisfying the following: (m1) for all w ∈ W , Let M denote the set of all matchings. Given either [wP f ∅ and |µ(f )| < q f ] or [wP f w for some w ∈ µ(f )] (pairwise stability). Let C(P ) denote the set of stable matchings at P (or the core of P ). A (direct) mechanism is a function ϕ : P → M. A mechanism ϕ is stable if for all P ∈ P, ϕ[P ] is stable at P . The most popular stable mechanisms are the deferred-acceptance algorithms (DAalgorithms) (Gale and Shapley, 1962): the firm-proposing DA-algorithm is denoted by DA F and the worker-proposing DA-algorithm is denoted by DA W .
A mechanism matches each firm f to a set of workers, taking into account only f 's preference ordering P f over individual workers. To study firms' incentives, preference orderings of firms over individual workers have to be extended to preference orderings over subsets of workers. The preference extension P * f over 2 W is responsive to P f ∈ P f if for all S ∈ 2 W , all w ∈ S, and all w / Let R * f denote the weak preference associated with P * f and resp(P f ) denote the set of all responsive extensions of P f . Moreover, given S ∈ 2 W , let B(S, P * f ) be the weak upper contour set of P * f at S; i.e., B(S, P * f ) = {S ∈ 2 W | S R * f S}. Any mechanism and any true profile define a direct (ordinal) preference revelation game under complete information for which we can define the natural (ordinal) notion of Nash equilibrium. Given a mechanism ϕ and P, P ∈ P, P is a Nash equilibrium (NE) in the mechanism ϕ under complete information P if (n1) for all w ∈ W , ϕ[P ](w)R w ϕ[P w , P −w ](w) for allP w ∈ P w ; and (n2) for all f ∈ F and all P * A common prior is a probability distributionP over P. Given P ∈ P, let Pr{P = P } denote the probability thatP assigns to P . Given v ∈ V , letP v denote the marginal distribution ofP over P v . Given a common priorP and P v ∈ P v , letP −v | Pv denote the probability distribution whichP induces over P −v conditional on P v . It describes agent v's (Bayesian) uncertainty about the preferences of the other agents, given that his preference ordering is P v . 7 A random matchingη is a probability distribution over the set of matchings 7 This formulation does not require symmetry nor independence of priors; conditional priors might be very correlated if agents use similar sources to form them (i.e., rankings, grades, recommendation letters, etc.).
Letη(w) denote the distribution whichη induces over w's set of potential partners F ∪ {∅}, and letη(f ) denote the distribution whichη induces over f 's set of potential partners 2 W . Given two random matchingsη andη , (fo1) for w ∈ W and P w ∈ P w we say thatη(w) first-order stochastically P w −dominatesη (w), de- A mechanism ϕ and a common priorP define a direct (ordinal) preference revelation game under incomplete information. A strategy of agent v is a function s v : P v → P v specifying for each type P v of v a list that v submits to the mechanism, s v (P v ). We restrict our analysis to pure strategies in the main text. The Appendix generalizes our main result to mixed strategies and random mechanisms. A strategy profile is a list s = (s v ) v∈V of strategies specifying for each true profile P a submitted profile s(P ). Given a mechanism ϕ : P → M and a common priorP over P, a strategy profile s : P → P induces a random matching ϕ[s(P )] in the following way: Using Bayesian updating, the relevant random matching for agent v, given his type P v and a strategy profile Definition 4 (Ordinal Bayesian Nash Equilibrium) LetP be a common prior. 8 Observe that this definition requires thatη first-order stochastically dominatesη according to all responsive extensions of P f . 9 It is well-known that (fo1) is equivalent to that for any vNM-representation of P w the expected utility ofη is greater than or equal to the expected utility ofη (and similarly for (fo2) and all vNMrepresentations of any responsive extension of P f Truth-telling is an OBNE in the mechanism ϕ under incomplete informationP if and only if for all v ∈ V and all P v ∈ P v such that In general for arbitrary mechanisms there is no connection between NE under complete information and OBNE under incomplete information. For instance, suppose that the common priorP u is uniform in the sense that it puts equal probability on all preference profiles. Furthermore, suppose that the mechanism ϕ matches a worker and a firm if and only if they rank each other as their most preferred choice (and ϕ leaves all other positions unfilled and all other workers unmatched). Then it is easy to verify that truth-telling is an OBNE in the mechanism ϕ under the uniform priorP u . 11 However, truth-telling is not always a NE in the mechanism ϕ under complete information since for some profiles, a firm may rank a worker first and a worker the firm second, and if the worker is unmatched, then she profitably manipulates by moving the firm to the first position of her submitted ranking. Our main result will show that such a disconnection between NE and OBNE is only possible for unstable mechanisms. 10 In the definition of OBNE optimal behavior of agent v is only required for the preferences of v which arise with positive probability underP . If P v ∈ P v is such that Pr{P v = P v } = 0, then the conditional priorP −v | Pv cannot be derived fromP . However, we could complete the prior of v in the following way: letP −v | Pv put probability one on a profile where all other agents submit lists which do not contain v. 11 For any agent v and any P v ∈ P v ,P u −v | Pv is uniform over P −v . For all agents belonging to the opposite side of the market, the probability that she ranks v first is identical. Hence, v cannot do better than submitting the true preference relation.

6
The support of a common priorP is the set of profiles on whichP puts positive probability: P ∈ P belongs to the support ofP if and only if Pr{P = P } > 0.
Theorem 1 LetP be a common prior, s be a strategy profile, and ϕ be a stable mechanism. Then, s is an OBNE in the stable mechanism ϕ under incomplete informatioñ P if and only if for any profile P in the support ofP , s(P ) is a Nash equilibrium in the stable mechanism ϕ under complete information P .
Theorem 1 has several consequences and applications. One immediate consequence is that for determining whether a strategy profile is an OBNE, we only need to check whether for each realization of the common prior the submitted preference orderings constitute a Nash equilibrium under complete information. This means that the uniquely relevant information for an OBNE is the support of the common prior and no calculations of probabilities are necessary. This consequence is very important for applications because we need to check equilibrium play only for the realized (or observed) profiles. Furthermore, by Theorem 1, we can use properties of NE (under complete information) to deduce characteristics of OBNE.
Observe that, given a common beliefP , the set of OBNE in a stable mechanism is non-empty. For instance, imagine that the workers and the firms are divided into "local" matching markets as follows: let (W f ) f ∈F be a partition of the set of workers (allowing W f = ∅ for some firms f ) where W f denotes the set of workers belonging to the "local" market of f . In words, under the following strategy profile, if worker w belongs to the local market of firm f , then w ranks f uniquely acceptable if f is preferred to being unmatched and otherwise w ranks no firm acceptable. Any firm f ranks as acceptable (and in the true order) all workers which both belong to its local market and are acceptable according to its true preference relation. Let the strategy profile s be defined in the following way: (i) for any w ∈ W and any P w ∈ P w , Then for any stable mechanism and any profile P , s(P ) is a NE under complete information. Hence, s is an OBNE in any stable mechanism under any common belief. In the special case where W f = W for some firm f , firm f has a monopolistic market.
Restricting ourselves to truth-telling (where s(P ) = P for all P ∈ P), Theorem 1 shows that truth-telling is an OBNE in a stable mechanism ϕ if and only if for any profile in the support of the common belief, truth-telling is a NE in ϕ under complete information for any profile in the support of the common prior. In other words, in many-to-one matching a stable mechanism ϕ is ordinally Bayesian incentive compatible under the common priorP if and only if ϕ restricted to the support ofP is incentive compatible. Furthermore, it can be easily seen that P is a NE in the stable mechanism ϕ under complete information P if and only if P is a NE in any stable mechanism under complete information P . All what matters is the stability of the mechanism (and not which specific stable matching is chosen).
While the proof of Theorem 1's (If)-part is straightforward, its (Only if)-part proceeds roughly as follows. If for some profile P in the support ofP , s(P ) does not constitute a NE, then some agent v has a profitably deviation from s(P ) under complete information P . Using this deviation we then construct another profitable deviation for v from s(P ) under complete information P such that agent v strictly increases the probability of the weak upper contour set (at his type P v ) of assigned partner(s) of the deviation under P . This implies that strategy profile s cannot be an OBNE. The construction and the proof use repeatedly the following peculiarities of stable matchings in many-to-one matching markets (see Appendix A. and (2) comparative statics: starting from any many-to-one matching market and its workers-optimal matching, when new workers become available all firms weakly prefer any matching, which is stable for the enlarged market, to the workers-optimal stable matching of the smaller market.
Below we turn to the applications of Theorem 1.

Application I: Structure of OBNE
By Theorem 1, a strategy profile is an OBNE if and only if the agents play a Nash equilibrium for any profile in the support of the common prior. Therefore, (a) the set of OBNE is identical for any two common priors with equal support and (b) the set of OBNE shrinks if the support of the common prior becomes larger. We state these two facts as Corollary 1 below.
Corollary 1 (Invariance) Let s be a strategy profile and ϕ be a stable mechanism.
(a) LetP andP be two common priors with equal support. Then, s is an OBNE in the stable mechanism ϕ underP if and only if s is an OBNE in the stable mechanism ϕ underP .
(b) LetP andP be two common priors such that the support ofP is contained in the support ofP . If s is an OBNE in the stable mechanism ϕ underP , then s is an OBNE in the stable mechanism ϕ underP . Now by (a) of Corollary 1, for stable mechanisms any OBNE is robust to perturbations of the common prior which leave its support unchanged. Therefore, any OBNE remains an equilibrium if agents have different priors with equal support, i.e. each agent v may have a private priorP v but all private priors have identical (or common) support. 12 This consequence is especially important for applications since for many of them, the common prior assumption might be too strong. 12 Then in Definition 4 of OBNE the common priorP is replaced for each agent v by his private priorP v . Theorem 1 and its proof show that, a strategy profile s is an OBNE in a stable mechanism ϕ under private priors (P v ) v∈V if and only if for all v ∈ V and any profile P in the support of In other words, By (b) of Corollary 1, the set of OBNE with full support (i.e. all common priors which put positive probability on all profiles) is contained in the set of OBNE of any arbitrary common prior (or support). Therefore, any OBNE for a common prior with full support is an OBNE for any arbitrary prior. Hence, such OBNE are invariant with respect to the common prior and remain OBNE if the agents' priors are not necessarily derived from the same common prior (and the "local" markets example is an OBNE in any stable mechanism under any priors).

Application II: Truth-Telling under Correlated Preferences
In empirical applications the preferences of one side of the market are often perfectly correlated. For example, each firm may rank all workers according to an objective criterion such as their degree of qualifications or each college may rank all students according to their grades. Furthermore, it is common in labor economics or search theory to often assume that all workers have identical preferences over firms. 13 Onesided perfect correlation is an extreme case of interdependence of preferences where an agent's preference may depend on the preferences of the other agents on his side.
We say that a common priorP is F -correlated if for any profile P in the support ofP , all firms have identical preferences. 14 Similarly we say that a priorP is W -correlated if for any profile P in the support ofP , all workers have identical preferences. Theorem 1 also helps us to prove the following result under F -correlated or each agent's strategy s v chooses a best response to the other reported preferences for any profile belonging to the support of his private prior. If all private priors have equal support, then it follows that a strategy profile s is an OBNE with private priors (with common support) if and only if for any profile P in the common support, s(P ) is a Nash equilibrium in the mechanism ϕ under complete information P . 13 For instance, Shi (2002) provides a long list of papers on directed search models in labor markets where at least one side of the market is homogenous.
14 Formally this means for all f, f ∈ F , A(P f ) = A(P f ) and P f |W = P f |W .
Proposition 1 LetP be a common prior.
(a) IfP is F -correlated or W -correlated, then truth-telling is an OBNE in any stable mechanism under incomplete informationP .
(b) Let s be a strategy profile such that s w (P w ) = P w for all w ∈ W and all P w ∈ P w . IfP is W -correlated and s is an OBNE in the stable mechanism DA W under incomplete informationP , then for all profiles P in the support ofP ,

Variations
Recall for truth-telling to be an OBNE for a common belief, it must be that for any firm and any of its realized preference over firms, truth-telling first order stochastically dominates submitting any other ranking for all responsive extensions of the true ranking. It is natural to ask whether Theorem 1 breaks down when we restrict the set of responsive extensions firms may have.
First, it is easy to see that the proof of Theorem 1 remains true if firms responsive extensions are additive, i.e. where a firm has a numerical value for each worker and the value of a set of workers is the sum of the values of the hired workers. 15 Second, we show that Theorem 1 depends on firms having responsive extensions which are not monotonic: given P f ∈ P f and P * f ∈ resp(P f ), P * f is monotonic if for all S, S ∈ 2 W such that |S | < |S| ≤ q f and S ⊆ A(P f ), we have SP * f S . Let mresp(P f ) denote the set of all monotonic responsive extensions of P f . We will call a strategy profile a monotonic OBNE in a mechanism under incomplete information if (fo2) holds for all monotonic responsive extensions of any firm's preference relation over individual workers. In the example below we show that truth-telling is a monotonic OBNE in DA W , while, for some preference profile P in the support of the common belief, truth-telling is not a Nash equilibrium under complete information P in the direct preference revelation game induced by DA W .
Example 1 Consider a many-to-one matching market with three firms F = {f 1 , f 2 , f 3 } and four workers W = {w 1 , w 2 , w 3 , w 4 }. Firm f 1 has capacity q f 1 = 2 and firms f 2 and f 3 have capacity q f 2 = q f 3 = 1. Consider the common beliefP with Pr{P = P } = p and Pr{P =P } = 1 − p, where p < 1/2, and P andP are the following profiles: .
Note that P f 1 =P f 1 . It is straightforward to verify that both profiles have a singleton 15 A responsive preference ordering P * f is additive if there exists an injective function g : W → R\{0} such that for all S, S ∈ 2 W with |S| ≤ q f and |S | ≤ q f , we have SP * f S ⇔ w∈S g(w) > w ∈S g(w ). In footnotes we show that any responsive extension in the proof of Theorem 1 can be chosen to be additive. core and C(P ) = {µ} and C(P ) = {μ}, where Let ϕ be a stable mechanism. Thus, by stability of ϕ, ϕ[P ] = µ and ϕ[P ] =μ.
First we will show that for the profile P truth-telling is not a Nash equilibrium under complete information P in the direct preference revelation game induced by DA W .
Hence, by stability of ϕ, ϕ[P f 1 , P −f 1 ] = µ . Obviously, for all responsive extensions P * . Therefore, truth-telling is not a Nash equilibrium in any stable mechanism ϕ under complete information P (and profile P belongs to the support of the common belief P ).
On the other hand we will show that truth-telling is a monotonic OBNE in the stable mechanism DA W under incomplete informationP . Note that for all v ∈ V \{f 1 }, if v observes his preference relation, then v knows whether P was realized or P was realized. Since at both of P andP the core is a singleton and firms f 2 and f 3 have quota one, it follows from the proof of Theorem 1 in Ehlers and Massó (2007) that v cannot gain by deviating.
Next we consider firm f 1 . All arguments except for the last one apply to any stable mechanism ϕ. Observe that P f 1 =P f 1 and the random matching ϕ[P f 1 ,P −f 1 | P f 1 ] assigns to f 1 the set {w 3 , w 4 } with probability p and the set {w 1 , w 3 } with probability 1 − p. Let P * f 1 be a monotonic responsive extension of P f 1 and P f 1 ∈ P f 1 . We show that ( We distinguish two cases. First, suppose that |ϕ does not hold, then by monotonicity of P * f 1 and the fact that when sub-mitting P f 1 , f 1 is assigned the set {w 3 , w 4 } with probability p and the set {w 1 , w 3 } with  If for some µ ∈ C(P f 1 , P −f 1 ), µ (w 4 ) = ∅, then by definition of P −f 1 , µ (w 4 ) = f 1 ; otherwise the pair (w 2 , f 2 ) would block µ at (P f 1 , P −f 1 ) if µ (w 4 ) = f 2 and the pair 17 Their result says that in a one-to-one matching market no group of firms can profitably manipulate DA F at the true profile under complete information (with strict preference holding for all firms belonging to the group). 18 Note that Example 1 does not contradict Theorem 1. When considering the non-monotonic The above example has another important implication: suppose that firms submit preference orderings over sets of workers instead of submitting preference orderings over individual workers only and the common belief is a distribution over profiles where firms' preference orderings are over sets of workers. Now if the common belief puts only positive probability on profiles where all firms' preference orderings are responsive and monotonic, then the above example shows that truth-telling can be an OBNE while not necessarily at all profiles in the support truth-telling is a NE under complete information. This is partly due to the fact our main result is a statement for any common belief.
Once we put certain conditions on the common belief, our main result continues to hold even if firms submit preference orderings over sets of workers. Without going into details, let P * f denote the set of all responsive preference orderings of f over 2 W and P * = (× f ∈F P * f ) × (× w∈W P w ). Let the common beliefP * on P * be such that for all In words, if whenever the common belief puts positive probability on some profile, then for any firm and any other preference ordering which is responsive to the same ordering, the belief also puts positive probability on the profile where the firm's preference is replaced by this preference ordering (the support of the common belief does not distinguish preference orderings which are responsive to the same ordering over individual workers). The proof of Theorem 1 then shows that truth-telling is an OBNE in the stable mechanism ϕ if and only if for any profile in the support ofP * , truth-telling is a NE in the stable mechanism under complete information.
It would be interesting to identify other economic environments where a similar link between BNE under incomplete information and NE under complete information extension P * f1 such that {w 1 }P * f1 {w 3 , w 4 }, then firm w 1 can gain by submitting the listP f1 where worker w 1 is the unique acceptable worker (i.e. A(P f1 ) = {w 1 }). Then we have both ϕ[P f1 , P −f1 ](f 1 ) = {w 1 } and ϕ[P f1 ,P −f1 ](f 1 ) = {w 1 }, which means that truth-telling is not an OBNE in any stable mechanism ϕ under incomplete informationP .
holds. In those environments the strategic analysis under complete information is essential to undertake the corresponding analysis under incomplete information.

APPENDIX
Before we prove Theorem 1, we recall the following properties of the core of a many-to-one matching market. These properties will be used frequently in the proof. It will be convenient to write (F, W, P ; q) for any many-to-one matching market (F, W, q, P ) in which q f = 1 for all f ∈ F .

A.1 Properties of the Core
The core of a many-to-one matching market has a special structure. The following well-known properties will be useful in the sequel: 19 (P1) For each profile P ∈ P, C(P ) contains two stable matchings, the firms-optimal stable matching µ F and the workers-optimal stable matching µ W , with the property that for all µ ∈ C(P ), µ W (w)R w µ(w)R w µ F (w) for all w ∈ W , and for all f ∈ F , (P2) For each profile P ∈ P and any responsive extensions P * F = (P * f ) f ∈F of P F = (P f ) f ∈F , C(P ) coincides with the set of group stable matchings at (P W , P * F ), where group stability corresponds to the usual cooperative game theoretical notion of weak blocking. 20 This means that the set of group stable matchings (relative to P ) is invariant with respect to any specific responsive extensions of P F .
(P3) For each P ∈ P, the set of unmatched agents is the same for all stable matchings (see Roth and Sotomayor, 1990, Theorems 5.12 and 5.13): for all µ, µ ∈ C(P ), and 19 See Roth and Sotomayor (1990) for a detailed presentation of these properties. 20 A matching µ is weakly blocked by coalition S ⊆ V under (P W , P * F ) if there exists a matching µ such that (b1) for all v ∈ S, µ (v) ⊆ S, (b2) for all w ∈ W ∩ S, µ (w)R w µ(w), and (b3) for all , with strict preference holding for at least one v ∈ S.
(P4) Consider a one-to-one matching market (F, W, P ; q) and suppose that new workers enter the market. Let (F, W , P ; q) be this new one-to-one matching market where W ⊆ W and P agrees with P over F and W . Let DA W [P ] = µ W . Then, for all (P5) Given (F, W, q, P ), split each firm f into q f identical copies of itself (all having the same preference ordering P f ) and let F be this new set of f ∈F q f splitted firms. Set q f = 1 for all f ∈ F and replace f by its copies in F (always in the same order) in each worker's preference relation P w . Then, (F , W, P ; q ) is a oneto-one matching market for which we can uniquely identify its matchings with the matchings of the original many-to-one matching market (F, W, q, P ), and vice versa (Roth and Sotomayor, 1990, Lemma 5.6). Then, and using this identification, we write C(F, W, q, P ) = C(F , W, P ; q ).

A.2 Proof of Theorem 1
Below we extend our result to random stable mechanisms 21 and mixed strategies.
Let ∆(M) denote the set of all probability distributions over M. A random mechanism is a functionφ : P → ∆(M) choosing for each profile P ∈ P a distributioñ ϕ[P ] over M. The random mechanismφ is stable if for all P ∈ P, the support of ϕ[P ] is contained in C(P ). Given v ∈ V , let ∆(P v ) denote the set of all probability distributions over P v . A mixed strategy of agent v is a function m v : Given a random mechanismφ, P ∈ P, and m, m(P ) is a NE inφ under complete As usual, if s is an OBNE in pure strategies in the (deterministic) mechanism ϕ under P , then s is an OBNE in mixed strategies in the mechanism ϕ (where ϕ is a random mechanism putting probability one on a unique matching for each profile).
Theorem 2 LetP be a common prior, m be a mixed strategy profile, andφ be a random stable mechanism. Then, m is an OBNE in the random stable mechanism ϕ under incomplete informationP if and only if for any profile P in the support of P , m(P ) is a Nash equilibrium in the random stable mechanismφ under complete information P .
Proof. LetP be a common prior, m be a mixed strategy profile andφ be a random stable mechanism. For any probability distribution D we denote by supp(D) its support (and e.g., supp(P ) is the support ofP ).
(⇐) Suppose that for any profile P in the support ofP , m(P ) is a Nash equilibrium in the mechanismφ under complete information P . Let v ∈ V and P v ∈ P v be such that Pr{P v = P v } > 0. By the previous fact, then we have for all P v ∈ P v and all and for any u v ∈ ∆(P v ), than any pure strategy P v ∈ P v .
Hence, m is an OBNE in mixed strategies inφ underP , the desired conclusion.
(⇒) Let m be an OBNE in mixed strategies in the random stable mechanismφ under P .
First we show that for all P ∈ P such that Pr{P = P } > 0, If for some P in the support ofP , for some v ∈ V , and some µ ∈ supp(φ[m(P )]), (the case v ∈ W is analogous and easier). We choose a responsive extension P which means that m is not an OBNE in the random stable mechanismφ underP , a contradiction. Hence, (5) holds.
Second, suppose that for some P ∈ supp(P ), m(P ) is not a NE inφ under P .
Then, without loss of generality, there exists f ∈ F and P f ∈ P f such that Then there exists a responsive extension P * f of P f and 23 The responsive extension P * v of P v can be chosen to be additive by selecting g : W → R\{0} such that (i) for all w, w ∈ W , wP v w ⇔ g(w) > g(w ), (ii) for all w ∈ A(P v ), g(w) ∈ (0, 1), and (iii) for all w ∈ W \A(P v ), g(w) < −|W |. It is easy to see that for all W ∈ 2 W with |W | ≤ q v , firm with quota one has a profitable deviation.
We now construct from P f another deviation P f and from µ (f ) both a responsive extension P * * f of P f and a subset of workers W * , and prove that the random matchingφ We proceed by distinguishing between two mutually exclusive cases. Note that w k ∈ A(P f ) because w k P f w k and by (5), First we show that for all µ ∈ supp(φ[P f , P −f ]), µ (f ) contains at least k workers.
Note that any profile implicitly specifies the set of agents of the matching problem.
For the time being, below we specify both the profile and the quota of the matching problem.
Becauseφ is stable and µ ∈ supp(φ[P f , P −f ]), µ ∈ C(P f , P −f ; q). Let µ be the matching for the problem (F, W \{w k+1 , . . . , w |µ (f )| }, (k, q −f ), (P f , P −{f }∪{w k+1 ,...,w |µ (f )| } )) such that µ (f ) = {w 1 , . . . , w k } and µ (f ) = µ (f ) for all f ∈ F \{f }. Then from By our choice of P f , we have µ (f ) ⊆ A(P f ) and P f |A(P f ) = P f |A(P f ). Hence, we also have by (7), at least k copies of f must be also matched to a worker under any stable match- Second we choose a responsive extension P * * The following claim will be the key to the proof. We show that for any profile, if some stable matching is weakly preferred to W * under P * * f , then all matchings, which are stable under the profile where f 's preference ordering is replaced by P f , are weakly preferred to W * under P * * f .
Letμ ∈ C(P f ,P −f ). Letμ be the matching for the problem (F, W \(μ(f )\A(Ṗ f )), q, . Hence, in any matching belonging to this core firm f is matched to |μ workers. Now when considering the worker optimal matching in this core, we may A(P f ). Let P f ∈ P f be such that A(P f ) = B(w k+1 , P f ) and P f |A(P f ) = P f |A(P f ).
Since µ(f ) ⊆ B(w k+1 , P f ) = A(P f ) and µ(f ) does not fill the quota of firm f , we must have µ ∈ C(P f , P −f ; q). Hence, firm f is matched to k workers under any matching in C(P f , P −f ; q).
On the other hand, let µ be the matching for the problem (F, W \{w k+2 , . . . , w |µ ( in this core, we may split firm f into k + 1 copies (all having the same preference P f ) and each copy of firm f weakly prefers according to P f any matching in C(P f , P −f ; q) to this matching. Since at least k + 1 copies of f are matched to a worker under the worker optimal matching in C(P f , P −{f }∪{w k+2 ,...,w |µ (f )| } ; q), at least k + 1 copies of f must be also matched to a worker under any matching in C(P f , P −f ; q), which contradicts (13) and the fact that firm f is matched to the same number of workers under any matching in C(P f , P −f ; q). Hence, Case 2 cannot occur.
Theorem 1 is a corollary of Theorem 2 by restricting Theorem 2 and its proof to pure strategies and deterministic mechanisms.

A.3 Proof of Proposition 1
Proposition 1 LetP be a common prior.
(a) IfP is F -correlated or W -correlated, then truth-telling is an OBNE in any stable mechanism under incomplete informationP .
(b) Let s be a strategy profile such that s w (P w ) = P w for all w ∈ W and all P w ∈ P w . IfP is W -correlated and s is an OBNE in the stable mechanism DA W under incomplete informationP , then for all profiles P in the support ofP , DA W [s(P )] is stable with respect to P . The analogous statement is true for the stable mechanism DA F .
Proof. (a) Let ϕ be a stable mechanism andP be a common prior. Without loss of generality, letP be F -correlated. The case whereP is W -correlated is analogous to the case whereP is F -correlated and all firms have quota 1. Let P be in the support ofP . Because all firms' preferences are identical at P , we have |C(P )| = 1, say C(P ) = {µ}. By stability of ϕ, ϕ[P ] = µ. By Theorem 1, it suffices to show that P is a NE in the mechanism ϕ under complete information P .