Cahier Robust Design in Monotonic Matching Markets : A Case for Firm-Proposing Deferred-Acceptance. Lars Ehlers and Jordi Massó

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1 Cahier Robust Design in Monotonic Matching Markets : A Case or Firm-Proposing Deerred-Acceptance Lars Ehlers and Jordi Massó

2 Le Centre interuniversitaire de recherche en économie quantitative (CIREQ) regroupe des chercheurs dans les domaines de l'économétrie, la théorie de la décision, la macroéconomie et les marchés inanciers, la microéconomie appliquée ainsi que l'économie de l'environnement et des ressources naturelles. Ils proviennent principalement des universités de Montréal, McGill et Concordia. Le CIREQ ore un milieu dynamique de recherche en économie quantitative grâce au grand nombre d'activités qu'il organise (séminaires, ateliers, colloques) et de collaborateurs qu'il reçoit chaque année. The Center or Interuniversity Research in Quantitative Economics (CIREQ) regroups researchers in the ields o econometrics, decision theory, macroeconomics and inancial markets, applied microeconomics as well as environmental and natural resources economics. They come mainly rom the Université de Montréal, McGill University and Concordia University. CIREQ oers a dynamic environment o research in quantitative economics thanks to the large number o activities that it organizes (seminars, workshops, conerences) and to the visitors it receives every year. Cahier Robust Design in Monotonic Matching Markets : A Case or Firm-Proposing Deerred-Acceptance Lars EHLERS and Jordi MASSÓ CIREQ, Université de Montréal C.P. 628, succursale Centre-ville Montréal (Québec) H3C 3J7 Canada Téléphone : (54) Télécopieur : (54) cireq@umontreal.ca

3 Ce cahier a également été publié par le Département de sciences économiques de l Université de Montréal sous le numéro (208-02). This working paper was also published by the Department o Economics o the University o Montreal under number (208-02). Dépôt légal - Bibliothèque nationale du Canada, 208, ISSN Dépôt légal - Bibliothèque et Archives nationales du Québec, 208 ISBN-3 :

4 Robust Design in Monotonic Matching Markets: A Case or Firm-Proposing Deerred-Acceptance Lars Ehlers Jordi Massó May 208 Abstract We study two-sided matching markets among workers and irms. Workers seek one position at a irm but irms may employ several workers. In many applications those markets are monotonic: leaving positions unilled is costly as or instance, or hospitals this means not being able to provide ull service to its patients. A huge literature has advocated the use o stable mechanisms or clearinghouses. The interests among workers and irms are polarized among stable mechanisms, most amously the irm-proposing DA and the worker-proposing DA. We show that or the irmproposing DA ex-ante incentive compatibility and ex-post incentive compatibility are equivalent whereas this is not necessarily true or the worker-proposing DA. The irm-proposing DA turns out to be more robust than the worker-proposing DA under incomplete inormation when incentives o both sides o the market are important. JEL classiication number: C78; D8; J44. Keywords: Many-to-one matching market; Stability; Incomplete inormation; Monotonic responsive extensions; Robust mechanism design. L. Ehlers acknowledges inancial support rom the FRQSC (Québec) and 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. Département de Sciences Économiques and CIREQ, Université de Montréal. Montréal, Québec H3C 3J7, Canada; lars.ehlers@umontreal.ca Departament d Economia i d Història Econòmica, Universitat Autònoma de Barcelona, and Barcelona GSE Cerdanyola del Vallès (Barcelona), Spain; jordi.masso@uab.es

5 Introduction Centralized many-to-one matching markets operate as ollows: each participant submits a ranked list o potential partners and a mechanism matches irms and workers on the basis o the submitted ranked lists. The participants belong to two sides: the irms (colleges, hospitals, schools, etc.) and the workers (students, medical interns, children, etc.). In applications many o the successul mechanisms are stable.,2 Most markets are monotonic meaning it is (very) costly to leave positions unilled and a irm desires to ill as many positions as possible with acceptable workers. 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 and in school choice unding o a school may depend on the number o students admitted. Stability o a matching (in the sense that all agents are matched to acceptable partners and no unmatched pair o a irm and a worker preer each other to their proposed partners) has been considered to be the main eature in order to survive. 3 This is puzzling because there exists no stable mechanism which is ex-post incentive compatible on the ull domain o preerences (Roth, 982). Thereore, an agent s (submitted) ranked lists o potential partners are not necessarily his true ones and the implemented matching may not be stable or the true proile. 4 Complete inormation is unrealistic and we use the (ordinally) Bayesian approach where nature selects a preerence proile according to a common belie. Since real-lie matching markets require to report ranked lists and not their speciic utility representations, we ollow the ordinal setting. Probability distributions are evaluated according to the irstorder stochastic dominance criterion. Then a mechanism is monotonic ordinal Bayesian incentive compatible (monotonic OBIC) i and only i truth-telling is a monotonic ordinal Bayesian Nash equilibrium (monotonic OBNE) or any realized preerence proile. 5 Our main result shows that or the irm-proposing deerred-acceptance mechanism (irm-proposing DA) monotonic OBIC is equivalent to ex-post incentive compatibility on the support o the common belie. Hence, Bayesian incentive compatibility o the irm- See Roth (984a), Roth and Peranson (999), and Roth (2002) or a careul description and analysis o the American entry-level medical market. Roth (99), Kesten (2005), Ünver (2005), and Ehlers (2008) describe and analyze the equivalent UK markets. 2 Abdulkadiroğlu and Sönmez (2003) introduce school choice which is urther studied by Chen and Sönmez (2006), Abdulkadiroğlu, Pathak, and Roth (2009) and Abdulkadiroğlu, Che, and Yosuda (20). For college admissions, see Roth and Sotomayor (989, 990) and recently Chen and Kesten (207). 3 See, or instance, Roth (984a). 4 The literature has studied intensively Nash equilibria o direct preerence revelation games induced by dierent stable mechanisms under complete inormation. See Dubins and Freedman (98), Roth (982, 984b, 985), Shin and Suh (996), Sönmez (997), Ma (995, 2002), and Alcalde (996). 5 This means that or every von Neumann Morgenstern (vnm)-utility unction o an agent s preerence ordering (his type), submitting the true ranking maximizes his expected utility in the direct preerence revelation game induced by the common prior and the mechanism given that all other agents truth-tell. This notion was introduced by d Aspremont and Peleg (988). Majumdar and Sen (2004) use it to relax strategy-prooness in the Gibbard-Satterthwaite Theorem and Ehlers and Massó (2007, 205) use it to study strategic behavior in matching markets under incomplete inormation. 2

6 proposing DA does not depend on exact probabilities o the common belie or proiles in the support (and is detail-ree à la Wilson s doctrine). We show that this does not necessarily hold or the worker-proposing DA, and the irm-proposing DA is more robust than the worker-proposing DA when incentives o both sides o the market are taken into account. The paper is organized as ollows. Section 2 describes monotonic many-to-one matching markets with incomplete inormation. Section 3 states our main result, Theorem. It shows that Theorem does not hold or the worker-proposing DA and that the irm-proposing DA is more robust. Section 4 concludes and the Appendix contains all proos. 2 Monotonic Many-To-One Matching Markets Let F denote the set o irms, W denote the set o workers, and V F W denote the set o agents. For each irm, there is a maximum number q o workers that may hire, s quota. Let q = (q ) F denote the vector o quotas. Each worker w has a strict preerence ordering P w over F { }, where stands or being unmatched. Each irm has a strict preerence ordering P over W { }, where stands or leaving a position unilled. A proile P = (P v ) v V is a list o preerence orderings. Given S V, we sometimes write (P S, P S ) instead o P. Let P v be the set o all preerence orderings o agent v. Let P = v V P v be the set o all proiles and let P v = v V \{v}p v. Let R v denote the weak preerence associated with P v. Given w W, P w P w, and v F { }, let B(v, P w ) denote the weak upper contour set o P w at v; i.e., B(v, P w ) = {v F { } v R w v}. Let A(P w ) be the set o acceptable irms or w according to P w ; i.e., A(P w ) = { F P w }. Given a subset S F { }, let P w S denote the restriction o P w to S. Similarly, given P P, v W { }, and S W { }, we deine B(v, P ), A(P ), and P S. Given P and w, w W, let P w w stand or s ordering where w and w switch positions in P. 6 For example, i P : w w 2 w 3 w 4, then P w w 3 : w 3 w 2 w w 4. 7 A many-to-one matching market (or college admissions problem) is a quadruple (F, W, q, P ). Because F, W and q remain ixed, a problem is simply a proile P P. I q = or all F, (F, W, q, P ) is called a one-to-one matching market. A matching is a unction µ : V 2 V satisying the ollowing: (m) or all w W, µ(w) F and µ(w) ; (m2) or all F, µ() W and µ() q ; and (m3) µ(w) = {} i and only i w µ(). We will write µ(w) = instead o µ(w) = {}. I µ(w) =, we say that w is unmatched at µ. I µ() < q, we say that has q µ() unilled positions at µ. Let M denote the set o all matchings. Given P P and µ M, µ is stable (at P ) i (s) or all v V, µ(v) A(P v ) (individual rationality); and (s2) there exists no pair (w, ) W F such that P w µ(w) and either [wp and µ() < q ] or [wp w or some w µ()] (pairwise stability). Let C(P ) denote the set o stable 6 Formally, (i) P w w v W { }\{w, w }, vp w i vp w w W { }\{w, w } = P W { }\{w, w }, (ii) wp w i w P w w w, (iii) or all w, and (iv) or all v W { }\{w, w }, vp w i vp w w w. 7 We will use the convention that P : w w 2 w 3 w 4 means w P w 2 P w 3 P w 4 P. 3

7 matchings at P (or the core o P ). A (direct) mechanism is a unction ϕ : P M. A mechanism ϕ is stable i or all P P, ϕ[p ] is stable at P. The most popular stable mechanisms are the deerred-acceptance algorithms (DA) (Gale and Shapley, 962): the irm-proposing DA is denoted by DA F and the worker-proposing DA is denoted by DA W. A mechanism matches each irm to a set o workers, taking into account only s preerence ordering P over individual workers. To study irms incentives, preerence orderings o irms over individual workers have to be extended to preerence orderings over subsets o workers. The preerence extension P over 2W is monotonic responsive to P P i or all S 2 W, all w S, and all w / S: (r) S {w }P S S < q and w P ; (r2) (S\{w}) {w }P S w P w; and (r3) SP S i S < S q and S A(P ). Let R denote the weak preerence associated with P and mresp(p ) denote the set o all monotonic responsive extensions o P. Moreover, given S 2 W, let B(S, P ) be the weak upper contour set o P at S; i.e., B(S, P ) = {S 2 W S R S}. Any mechanism and any true proile deine a direct (ordinal) preerence revelation game under complete inormation or which we can deine the natural (ordinal) notion o Nash equilibrium. We denote such a game by (P, ϕ, P ) where P is the true proile, ϕ is a mechanism and any agent v s set o strategies is P v. Given a mechanism ϕ and P, P P, P is a monotonic Nash equilibrium (monotonic NE) in the mechanism ϕ under complete inormation P i (n) or all w W, ϕ[p ](w)r w ϕ[ ˆP w, P w](w) or all ˆPw P w ; and (n2) or all F and all P mresp(p ), ϕ[p ]()R ϕ[ ˆP, P ]() or all ˆP P. Truth-telling is a monotonic NE in ϕ under P i P is a monotonic NE in ϕ under P. A common prior is a probability distribution P over P. Given P P, let Pr{ P = P } denote the probability that P assigns to P. Given v V, let P v denote the marginal distribution o P over P v. Given a common prior P and P v P v, let P v Pv denote the probability distribution which P induces over P v conditional on P v. It describes agent v s (Bayesian) uncertainty about the preerences o the other agents, given that v s preerence ordering is P v. 8 A random matching η is a probability distribution over the set o matchings M. Given µ M, let Pr{ η = µ} denote the probability that η assigns to µ. Let η(w) denote the distribution which η induces over w s set o potential partners F { }, and let η() denote the distribution which η induces over s set o potential partners 2 W. Given two random matchings η and η, (o) or w W and P w P w we say that η(w) irst-order stochastically P w dominates η (w), denoted by η (w) Pw sd η (w), i or all v F { }, v F { }:v R Pr{ η (w) = wv v } v F { }:v R wv Pr{ η (w) = v }; and (o2) or F and P P, η() irst-order stochastically P dominates η (), denoted by η () P sd η (), i or all P mresp(p ) and all S 2 W, S 2 W :S R S Pr{ η () = S } S 2 W :S R S Pr{ η () = S }. 9 8 This ormulation does not require symmetry nor independence o priors; conditional priors might be very correlated i agents use similar sources to orm them (i.e., rankings, grades, recommendation letters, etc.). 9 It is well-known that (o) is equivalent to that or any vnm-representation o P w the expected utility o η is greater than or equal to the expected utility o η (and similarly or (o2) and all vnm-representations 4

8 A mechanism ϕ and a common prior P deine a direct (ordinal) preerence revelation game under incomplete inormation. Given a mechanism ϕ : P M and a common prior P over P, truth-telling induces a random matching ϕ[ P ] in the ollowing way: or all µ M, Pr{ϕ[ P ] = µ} = Pr{ P = P }. P P:ϕ[P ]=µ Using Bayesian updating, the relevant random matching or agent v, given his type P v and report P v, is ϕ[p v, P v Pv ]; namely Pr{ϕ[P v, P v Pv ] = µ} = P v P v :ϕ[p v,p v ]=µ Pr{ P v Pv = P v }. Deinition (Monotonic OBIC) Let P be a common prior. Then mechanism ϕ is monotonic ordinal Bayesian incentive compatible (monotonic OBIC) under incomplete inormation P i and only i truth-telling is a monotonic ordinal Bayesian Nash equilibrium (monotonic OBNE) under incomplete inormation P, namely or all v V and all P v P v such that Pr{ P v = P v } > 0, 0 ϕ[p v, P v Pv ](v)p sd v ϕ[p v, P v Pv ](v) or all P v P v. () 3 Ex-ante and Ex-post Incentive Compatibility The support o a common prior P is the set o proiles on which P puts positive probability: P P belongs to the support o P i and only i Pr{ P = P } > 0. Our main result shows that or monotonic matching markets there is a strong link between complete and incomplete inormation or the irm-proposing DA. Theorem. Let P be a common belie. Then DA F is monotonic OBIC under incomplete inormation P i and only i the support o P is contained in the set o all proiles where truth-telling is a monotonic NE in DA F under complete inormation. For DA F to be ex-ante incentive compatible under a common belie, DA F restricted to the support o the common belie must be ex-post incentive compatible. This in turn implies that exact probabilities o the common belie do not matter and any agent may have a private belie as long as its support is contained in the common support. This means that or DA F to be monotonic OBIC under common belie is robust subject to perturbations o the probabilities as along as the support(s) o the private belies coincide. o any monotonic responsive extension o P ). See or instance, Theorem 3. in d Aspremont and Peleg (988). 0 In Deinition optimal behavior o agent v is only required or the preerences o v which arise with positive probability under P. I P v P v is such that Pr{ P v = P v } = 0, then the conditional prior P v Pv cannot be derived rom P. However, we could complete the prior o v in the ollowing way: let P v Pv put probability one on a proile where all other agents submit lists which do not contain v. 5

9 3. Truth-telling under Stable Mechanisms For one-to-one matching markets, Ehlers and Massó (2007, Theorem ) implies that a stable mechanism is monotonic OBIC under P i and only i the support o P is contained in the set o proiles with singleton core. Theorem implies that i DA F is monotonic OBIC under common prior P, then the support o P must be contained in the set o preerence proiles having a singleton core. We show that or any stable mechanism to be monotonic OBIC, it is necessary or the common belie to put positive probability only on proiles with singleton core. In monotonic matching markets the core is unique or any realized (or observed) proile under ex-ante incentive compatibility. Theorem 2. Let P be a common belie. I the stable mechanism ϕ is monotonic OBIC under incomplete inormation P, then the support o P is contained in the set o all proiles with a singleton core. It is natural to ask whether the link in Theorem breaks or stable mechanisms other than DA F. We provide an example o a market in which our main result is not true. Recall that Roth (985) exhibits an example o a proile in a many-to-one matching market with a singleton core where a irm proitably manipulates any stable mechanism and truth-telling is not a monotonic NE under complete inormation. Below we exhibit an example o a many-to-one matching market where (i) DA W is monotonic OBIC under the common belie (and any proile in the support has a singleton core) and (ii) there is a proile belonging to the support o the common belie at which truth-telling is not a monotonic NE under complete inormation or DA W. Thereore, parallel to Roth (985), the link in Theorem is broken in one direction 2 when all irms have only monotonic responsive extensions and one irm has a capacity o at least two. Example. Consider a many-to-one matching market with three irms F = {, 2, 3 } and our workers W = {w, w 2, w 3, w 4 }. Firm has capacity q = 2 and irms 2 and 3 have capacity q 2 = q 3 =. Consider the common belie P with Pr{ P = P } = p and Pr{ P = P } = p, where p < /2, and P and P are the ollowing proiles:. P P 2 P 3 P w P w2 P w3 P w4 P P2 P3 Pw Pw2 Pw3 Pw4 w w w w w 2 w w 2 w 2 w 3 2 w 2 w 3 w 3 w w 3 w 4 w 4 w 4 w 4 I or some P belonging to the support o P, we have C(P ) 2, then there exist µ C(P ) and w W such that µ(w)p w DA F [P ](w). Then µ(w) w. Let P w P w be such that A(P w) = {µ(w)}. Then µ C(P w, P w ) and by the act that the set o unmatched workers is identical or any two stable matchings, µ(w) = DA F [P w, P w ](w)p w DA F [P ](w). Hence, truth-telling is not a monotonic NE under P. 2 It is clear that the other direction is true or any arbitrary game o incomplete inormation. 6

10 Note that P = P. It is straightorward to veriy that both proiles have a singleton core and C(P ) = {µ} and C( P ) = { µ}, where ( ) ( ) µ = 2 3 and µ = 2 3. {w 3, w 4 } {w 2 } {w } {w, w 3 } {w 2 } {w 4 } Let ϕ be a stable mechanism. Thus, by stability o ϕ, ϕ[p ] = µ and ϕ[ P ] = µ. On the one hand, we will show that truth-telling is not a monotonic NE under complete inormation P. Let P P be such that P : w w 2 w 4 w 3. Then C(P, P ) = {µ } where ( ) µ = 2 3. {w, w 4 } {w 2 } {w 3 } Hence, by stability o ϕ, ϕ[p, P ] = µ. Obviously, or all monotonic responsive extensions P o P we have {w, w 4 }P {w 3, w 4 }, which is equivalent to ϕ[p, P ]( )P ϕ[p ]( ). Thereore, truth-telling is not a monotonic NE in any stable mechanism ϕ under complete inormation P (and proile P belongs to the support o the common belie P ). On the other hand, we will show that DA W is monotonic OBIC under incomplete inormation P. Note that or all v V \{ }, 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 irms 2 and 3 have quota one, it ollows that v cannot gain by a deviation. Next consider irm. All arguments except or the last one apply to any stable mechanism ϕ. Observe that P = P and the random matching ϕ[p, P P ] assigns to the set {w 3, w 4 } with probability p and the set {w, w 3 } with probability p. Let P mresp(p ) and P P be arbitrary. We show ϕ[p, P P ]( )P sd ϕ[p, P P ]( ). (2) We distinguish two cases. First, suppose that ϕ[p, P ]( ) = or ϕ[p, P ]( ) =. Now i (2) does not hold, then by monotonicity o P and the act that when submitting P, is matched to the set {w 3, w 4 } with probability p and to the set {w, w 3 } with probability p (where p > /2), we must have that ϕ[p, P ]( )P {w, w 3 } or ϕ[p, P ]( )P {w, w 3 }. Obviously, rom the deinition o P, the last is impossible. Thus, ϕ[p, P ]( )P {w, w 3 } and, by responsiveness o P, we must have ϕ[p, P ]( ) = {w, w 2 }. But then, without loss o generality, we would have DA F [P, P ]( ) = {w, w 2 } (because DA F chooses the most preerred stable matching rom the irms point o view). 3 Since we have C(P ) = {µ} and DA F [P, P ]( ) = {w 3, w 4 }, this would imply that in the corresponding one-to-one matching problem DA F is group manipulable by the two copies o (with each copy gaining strictly), a contradiction to the result o Dubins and Freedman (98). 4 3 I DA F [P, P ]( ) {w, w 2 }, then choose P P such that A(P ) = {w, w 2 }. Then we obtain DA F [P, P ]( ) = {w, w 2 }. 4 Their result says that in a marriage market no group o irms can proitably manipulate DA F at the true proile under complete inormation (with strict preerence holding or all irms belonging to the group). 7

11 Second, suppose that ϕ[p, P ]( ) = 2 and ϕ[p, P ]( ) = 2. Then by deinition o P and ϕ[p, P ]( ) = 2, we must have ϕ[p, P ]( ) = {w, w 3 }. Thus, by stability o ϕ, {w, w 3 } A(P ). I or all µ C(P, P ), µ (w 4 ) =, then w 4 / A(P ). By deinition o P and w 3 A(P ), DA W [P, P ]( ) = {w 3 }. Then does not ill all its positions at the worker-optimal matching and by Roth and Sotomayor (990), is matched to the same set o workers at all stable matchings. Hence, ϕ[p, P ]( ) = {w 3 } and (2) holds. To see that observe irst that when submitting P, irm is matched with probability p to {w 3 } and with probability p to {w, w 3 } while when submitting P, irm is matched with probability p to {w 3, w 4 } and with probability p to {w, w 3 }. Since {w 3, w 4 }P {w 3 } or all monotonic responsive extensions o P, (2) holds. I or some µ C(P, P ), µ (w 4 ), then by deinition o P, µ (w 4 ) = ; otherwise the pair (w 2, 2 ) would block µ at (P, P ) i µ (w 4 ) = 2 and the pair (w, 3 ) would block µ at (P, P ) i µ (w 4 ) = 3. Thus, by {w 3, w 4 } A(P ), µ C(P, P ) and DA W [P, P ] = µ. Thereore, the probability distribution DA W [P, P P ]( ) and DA W [P, P P ]( ) coincide with the distribution in which is matched to {w 3, w 4 } with probability p and to {w, w 3 } with probability p. Hence, (2) holds or DA W. Observe that in Example DA F is not monotonic OBIC under incomplete inormation P : or the preerence P : w w 2 w 4 w 3 in Example we have ( DA F [P, P ] = Thus, 2 3 {w, w 4 } {w 2 } {w 3 } ) and DA F [P, P ] = Pr{DA F [P, P P ]( ) B({w, w 4 }, P )} = Pr{DA F [P, P P ]( ) B({w, w 4 }, P )} = p ( 2 3 {w, w 3 } {w 2 } {w 4 } or all monotonic responsive extensions P o P (i.e. this is an unambiguous deviation or irm because it does not depend on the choice o the responsive extension o P ). Thus, we do not have DA F [P, P P ]( )P sd DA F [P, P P ]( ) and DA F is not monotonic OBIC under P. Roughly speaking, in Example irm manipulates DA F by reordering its acceptable workers whereas this is not possible or DA W (when keeping the same set o acceptable workers). 3.2 Robust Design For monotonic matching markets we establish an equivalence result or (payo) type spaces on robust mechanism design à la Bergemann and Morris (2005). 5 Instead o the terminology o payo type spaces, in (ordinal) matching markets it is natural to use the term preerence type spaces. 5 We reer the interested reader to Bergemann and Morris (202) or a comprehensive introduction to robust mechanism design. 8 ).

12 Let Q P denote a set o possible preerence type proiles. Let Q v = {P v P Q} be the set o agent v s possible preerence types in Q. Let Q v Pv = {P v (P v, P v ) Q} denote the set o the other agents preerence types in Q when v s preerence type is P v. Let (Q v Pv ) denote the set o all probability distributions on Q v Pv. In our setting a preerence type space is simply given by (Q, (ˆπ v ) v V ) where Q denotes the agents possible preerence proiles and ˆπ v describes agent v s priors, i.e. or any P v Q v, ˆπ v (P v ) (Q v Pv ) is agent v s prior about the other agents preerence types when v s preerence type is P v. A preerence type space (Q, (ˆπ v ) v V ) is a product space i Q = v V Q v. Although Bergemann and Morris (2005) only ocussed on product preerence type (or payo type) spaces where Q v = P v or all agents v, we allow or non-product preerence type spaces. In applications (such as in matching markets), preerence type proiles may be correlated and not necessarily be independent rom each other, i.e. some preerence type proiles might be regarded as impossible a priori. Furthermore, Q v may be a strict subset o P v or an agent v and thus, agent v s set o possible true preerence types may be a strict subset o the rankings agent v may report to the mechanism. A preerence type space (Q, (ˆπ v ) v V ) is said to be common prior i there exists a common prior P on Q such that ˆπ v (P v ) = P v Pv or all v V and all P v Q v. Obviously, or ˆπ v to be well-deined, we must have Pr{ P v = P v } > 0 or all P v Q v because by Bayes rule we have or all P v Q v and all P v Q v Pv, ˆπ v (P v )[P v ] = Pr{ P = (P v, P v )} Pr{ P. v = P v } We denote a common prior preerence type space simply by (Q, P ). We adapt Bergemann and Morris (2005) s deinitions o interim incentive compatibility and ex-post incentive compatibility to our ordinal matching environment. Deinition 7 (Interim Incentive Compatibility) Let (Q, (ˆπ v ) v V ) be a preerence type space. A mechanism ϕ : P M is interim incentive compatible on (Q, (ˆπ v ) v V ) i or all v V and all P v Q v, ϕ[p v, ˆπ v (P v )](v)p sd v ϕ[p v, ˆπ v (P v )](v) or all P v P v. (3) Note that interim incentive compatibility reduces to monotonic OBIC or common prior preerence type spaces. Instead o deining ex-post incentive compatibility or all possible proiles (or all types), we deine ex-post incentive compatibility or subdomains o the set o all proiles. Deinition 8 (Ex-post Incentive Compatibility on Subdomains) Let Q P. A mechanism ϕ : P M is ex-post incentive compatible on Q i or each proile P Q, we have or all w W, ϕ[p w, P w ](w)r w ϕ[p w, P w ](w) or all P w P w, and or all F and all P mresp(p ), ϕ[p, P ]()R ϕ[p, P ]() or all P P. Note that ex-post incentive compatibility on P is incentive compatibility. When Q P, then both interim incentive compatibility and ex-post incentive compatibility are stronger 9

13 than the corresponding versions o Bergemann and Morris (2005) because any agent v s set o preerence types Q v may be a strict subset o agent v s set o possible reports P v (and not only Q v ) to the mechanism. Corollary. Let Q P. Then the ollowing are equivalent. (a) DA F is interim incentive compatible on all preerence type spaces (Q, (ˆπ v ) v V ). (b) DA F is interim incentive compatible on all common prior preerence type spaces (Q, P ). (c) DA F is interim incentive compatible on one common prior preerence type space (Q, (ˆπ v ) v V ). (d) DA F is ex-post incentive compatible on Q. Proo. (a) (b) ollows by deinition because we are asking or interim incentive compatibility on a smaller collection o preerence type spaces, and similarly or (b) (c). (c) (d) ollows rom Theorem by considering a common prior preerence type space (Q, P ) where P has support Q. (d) (a) is trivial or matching markets. Note that Corollary diers in the ollowing sense rom Bergemann and Morris (2005, Corollary ) or social choice unctions and all payo type spaces: i DA F is interim incentive compatible or one common prior preerence type space, 6 then DA F is interim incentive compatible on all preerence type spaces (with the same support). This is a strong robustness eature or DA F (and or DA W, Corollary does not hold by Example ). Two other important qualiications between Bergemann and Morris (2005, Corollary ) 7 and Corollary are that (i) our equivalence result allows or non-product preerence type spaces whereas their result does not and (ii) our incentive compatibility notions are stronger than theirs. Again, non-product preerence type spaces are especially important or matching markets. 4 Conclusion Ehlers and Massó (205) study matching markets where all responsive extensions are allowed (i.e. where only (r) and (r2) are imposed on the responsive preerence extension). 6 As one may check, instead we could have replaced (c) by DA F is interim incentive compatible on one preerence type space (Q, (ˆπ v ) v V ) such that or any v V and P v Q v, the support o ˆπ v (P v ) coincides with Q v Pv. 7 Bergemann and Morris (2005) allow or larger type spaces which can be given here or Q by (T v, ˆθ v, ˆπ v ) v V where or each v V, T v is a non-empty set, ˆθ v : T v Q v speciies or each t v T v a payo type ˆθ v (t v ) in Q v and a belie ˆπ v (t v ) (T v ) such that or any t v in the support o ˆπ v (t v ) we have ˆθ v (t v ) Q v ˆθv(t v). It is easy to see that in Corollary or a ixed Q, a stable mechanism ϕ is interim incentive compatible on all type spaces i and only i ϕ is interim incentive compatible on all preerence type spaces. The reason is that preerence type spaces are type spaces and larger type spaces can be replicated by a collection o preerence type spaces. 0

14 From Ehlers and Massó (205, Theorem ) it ollows that a stable mechanism ϕ is OBIC under incomplete inormation P i and only i the support o P is contained in the set o proiles where truth-telling is a NE under complete inormation (where all responsive extensions are considered). Ehlers and Massó (205, Theorem ) crucially depends on irms having responsive extensions which are not monotonic. Example does not contradict their result: when considering the non-monotonic responsive extension P such that {w }P {w 3, w 4 }, then irm gains by submitting the list ˆP where worker w is the unique acceptable worker (i.e. A( ˆP ) = {w }). Then, since C( ˆP, P ) = C( ˆP, P ) = {ˆµ} and ˆµ( ) = {w }, we have both ϕ[ ˆP, P ]( ) = {w } and ϕ[ ˆP, P ]( ) = {w }, which means that no stable mechanism ϕ is OBIC under incomplete inormation P. Oten in real-lie matching markets it is costly to leave positions unilled and 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. Once we ocus on monotonic OBIC, the irm-proposing DA turns out to be more robust than the workers-proposing DA when incentives o both sides o the market are important. APPENDIX. A Proo o Theorem Let P be a common belie. ( ) Suppose that or all proiles 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 } > 0. Then, or all w W and all P w P w, DA F [P ](w)r w DA F [P w, P w ](w) and or all F and all P P, DA F [P ]()R DA F [P, P ]() or all P mresp(p ). Hence, or all w W and all P w P w such that Pr{ P w = P w } > 0, we have that or all P w P w, DA F [P w, P w Pw ](w)p sd w DA F [P w, P w Pw )](w), and or all F and all P P such that Pr{ P = P } > 0, we have that or all P P, DA F [P, P P ]()P sd DA F [P, P P )](). That is, P is a monotonic OBNE in DA F under P, the desired conclusion. ( ) Let DA F be monotonic OBIC under incomplete inormation P. Let P be such that Pr{ P = P } > 0. By Theorem 2, C(P ) =. To obtain a contradiction suppose that P is not a monotonic NE in DA F under complete inormation P. We irst show in Lemma below that then some irm with quota o at least two has a proitable deviation such that (i) the irm is only matched to acceptable workers, (ii) the irm is matched to q workers and (iii) or some monotonic responsive extension the irm strictly preers the assigned workers to the set received under truth-telling.

15 Lemma. Let P be such that C(P ) =. I P is not a monotonic NE in DA F under complete inormation P, then there exists a irm F such that q 2 and or some ˆP we have (i) DA F [ ˆP, P ]() A(P ), (ii) DA F [ ˆP, P ]() = q = DA F [P ](), and (iii) DA F [ ˆP, P ()P DA F [P ]() or some P mresp(p ). Proo o Lemma. Let P P be such that C(P ) =. Then, using a similar argument to the one used in the suiciency proo o Ehlers and Massó (2007, Theorem ), it can be seen that no worker has a proitable deviation rom P in DA F. The same argument applies to any irm with quota one. Thus, i P is not a monotonic NE in DA F under complete inormation P, then or some F with q 2 and ˆP, we have DA F [ ˆP, P ]()P DA F [P ]() (4) or some P mresp(p ), which is (iii) o the lemma. Because P is responsive, DA F [ ˆP, P ]() A(P )R DA F [ ˆP, P ](). Thus, by (4) and monotonicity o P, DA F [ ˆP, P ]() A(P ) DA F [P ](). (5) But then we may suppose w.l.o.g. that DA F [ ˆP, P ]() A(P ). To see that, assume DA F [ ˆP, P ]() A(P ). Then, choose any preerence P P such that A(P ) = DA F [ ˆP, P ]() A(P ). Now consider the matching market where the set o workers Ŵ = DA F [ ˆP, P ]()\A(P ) is not present with proile (P, P Ŵ {}). Then, it is easy to see that DA F [P, P Ŵ {}]() = A(P ). Now letting workers in Ŵ reenter the market, irms should weakly gain meaning that DA F [P, P ]() = A(P ).8 Since DA F [P, P ]()P DA F [P ](), has a proitable deviation rom P in DA F where is only matched to acceptable workers. Hence, we can replace in (4) ˆP by P. Thus, DA F [ ˆP, P ]() A(P ) holds. But the inclusion is (i) o the lemma. Next we show that DA F [ ˆP, P ]() = DA F [P ](). To obtain a contradiction, assume otherwise. By (5) and (i), we have DA F [ ˆP, P ]() > DA F [P ](). Then at DA F [P ] irm has some positions unilled and the ranking o P over A(P ) is irrelevant or the stability o DA F [P ] under P. Let k = DA F [ ˆP, P ](). In particular, DA F [P ] is stable under P i irm s quota is reduced rom q to k. W.l.o.g. we may suppose that A( ˆP ) = DA F [ ˆP, P ](). Now again in any matching which is stable under ( ˆP, P ) irm is matched to DA F [ ˆP, P ](). Again irm s quota may be reduced to k. But now consider P such that A(P ) = A(P ) and DA F [ ˆP, P ]() are the irst k most 8 See Theorem 2.25 in Roth and Sotomayor (990). 2

16 preerred workers under P. But then both DA F [ ˆP, P ] and DA F [P ] must be stable under (P, P ), which is a contradiction because DA F [ ˆP, P ]() DA F [P ]() and any irm is matched to the same number o workers at all stable matchings under the proile (P, P ). We have shown DA F [ ˆP, P ]() = DA F [P ](). Furthermore, i DA F [P ]() < q, then, by (i), both matchings DA F [P ] and DA F [ ˆP, P ] are stable under P. By (4), DA F [ ˆP, P ]() DA F [P ]() which means that C(P ) =, a contradiction. Hence, DA F [P ]() = q = DA F [ ˆP, P ]() (which is (ii)). We now proceed with the proo o Theorem. Let F, with q 2, and ˆP be the irm and its preerences identiied in Lemma, or which (i), (ii) and (iii) hold. Letting ˆµ = DA F [ ˆP, P ] and µ = DA F [P ], order the workers in ˆµ() and µ() according to P : let ˆµ() = {ŵ,..., ŵ q } and µ() = {w,..., w q }. Furthermore, because o (i)-(iii) and we are using DA F, we may assume w.l.o.g. that (a) A( ˆP ) = B(w q, P ) B(ŵ q, P ), (b) ˆP ˆµ() = P ˆµ(), (c) ˆP B(ŵ q, ˆP ) = P B(ŵ q, ˆP ), (d) ˆP A( ˆP )\B(ŵ q, ˆP ) = P A( ˆP )\B(ŵ q, ˆP ), and (e) ŵ ˆP w or all ŵ B(ŵ q, ˆP ) and all w A( ˆP )\B(ŵ q, ˆP ). Note that since ˆµ() B(ŵ q, ˆP ), (c) implies (b), which in turn implies ˆµ() B(ŵ q, P ). We distinguish two cases: ŵ q P w q and w q R ŵ q. Case : ŵ q P w q. Let W consist o the q workers which are P -least preerred in B(w q, P )\{w q }. Then choose the monotonic responsive extension P o P such that W R W i W contains q workers in B(w q, P )\{w q }. Let P be such that A(P ) = B(w q, P )\{w q } and P A(P ) = ˆP A(P ). Then ˆµ C(P, P ) and by construction, DA F [P, P ]() contains q workers in A(P ), but DA F [P ]() contains at most q workers in A(P ). Hence, by deinition o W and since w q / A(P ), DA F [P, P ]() R W P DA F [P, P ](). (6) To obtain a contradiction with the act that DA F is monotonic OBIC under incomplete inormation P, we consider any proile P such that Pr{ P = (P, P )} > 0 and restrict the attention to the upper contour set o P at W. We want to show that i DA F [P, P ]() R W, then DA F [P, P ]() R W. By the deinitions o W and P, DA F [P, P ]() R W implies that DA F [P, P ]() contains q workers in A(P ). I 3

17 DA F [P, P ]() contains at most q workers in A(P ), then the ranking does not matter and the matching DA F [P, P ] is also stable under (P, P ), where P is obtained by letting A(P ) = A(P ) and P A(P ) = P A(P ). Furthermore, DA F [P, P ] is stable under (P, P ) and DAF [P, P ] = q, by deinition o W. But then would be matched to dierent numbers o workers under dierent matchings belonging to C(P, P ), a contradiction. Thus, DA F [P, P ]() contains q workers in A(P ). Now by our choice o W and P, we obtain or any proile P in the support o P P, DA F [P, P ]() R W implies DA F [P, P ]() R W. Then, together with (6) and Pr{ P = (P, P )} > 0, this implication means that DA F is not monotonic OBIC under P because not DA F [P, P P ]()P sd DA F [P, P P ]() since or the monotonic responsive extension P o P, Pr{DA F [P, P P ]() B( W, P )} > Pr{DA F [P, P P ]() B( W, P )}. Case 2: w q R ŵ q. Let k be the irst index such that ŵ k P w k. By (iii) in Lemma, k exists. Note that k < q and ŵ k P w k R w q. Let P be such that A(P ) = B(ŵ q, P ), B(ŵ k, P ) ˆµ()P A(P )\(B(ŵ k, P ) ˆµ()), 9 P k, P ) ˆµ() = P B(ŵ k, P ) ˆµ(), and P A(P )\(B(ŵ k, P ) ˆµ()) = P A(P )\(B(ŵ k, P ) ˆµ()). We illustrate P below: P P B(ŵ k, P ) P ˆµ()\B(ŵ k, P ) P A(P )\(B(ŵ k, P ) ˆµ()) i.e. P ranks irst all elements in B(ŵ k, P ) according to P, then all elements in ˆµ()\B(ŵ k, P ) according to P, and then the remaining P -acceptable workers according to P. Because ŵ,..., ŵ k belong to B(ŵ k, P ), we have ˆµ()\B(ŵ k, P ) = {ŵ k+,..., ŵ q } (and this set consists o exactly q k workers). I is matched to ewer than q workers in any matching belonging to C(P, P ), then the dierences between ranking P and P do not matter and would be matched to ewer than q workers in DA F [P ](), the unique matching in C(P ), a contradiction. Thus, is matched to q workers in any matching belonging to C(P, P ). I ˆµ C(P, P ), then by construction and (4), DA F [P, P ]()R ˆµ()P µ() implying DA F [P, P ]()P µ(). (7) Suppose that ˆµ / C(P, P ). Then, some (w, ˆ) blocks ˆµ under (P, P ). By the deinition o P, ˆ = and w B(ŵk, P ). We show that DA F [P, P ]() contains at least k workers in B(ŵ k, P ). We show this with the ollowing two steps: Step : Let w be the P -highest ranked worker such that (w, ) orms a blocking pair o ˆµ under (P, P ). Let ˆµ 0 = ˆµ. Then match w to and consider ˆµ such that (i) 9 Here we use the convention SP T i sp t or all s S and all t T. 4,

18 ˆµ (w ) = and (ii) ˆµ (w) = ˆµ(w) or all w w. In other words, does not reject any worker and we allow to have capacity q + (i.e., ˆµ () = ˆµ 0 () {w }). I ˆµ does not contain any blocking pair, set µ = ˆµ and q = q + and then go to Step 2. Otherwise, ˆµ contains a blocking pair, say (w 2, 2 ). Again, let w 2 be the P 2-highest ranked worker such that (w 2, 2 ) orms a blocking pair o ˆµ. But then it must be that either (a) 2 = (which would mean that w 2 / ˆµ ()) or (b) 2 and w 2 ˆµ () or (c) 2 and w 2 / ˆµ (). Note that i 2, then we must have 2 = ˆµ(w ). Now consider ˆµ 2 such that (i) ˆµ 2 (w 2 ) = 2 and (ii) ˆµ 2 (w) = ˆµ (w) or all w w 2. For (b) 2 and w 2 ˆµ (), then ˆµ 2 is stable under (P, P ) and ˆµ 2 () contains at least k workers in B(ŵ k, P ) since ˆµ 2 () = (ˆµ 0 () {w })\{w 2 } and ˆµ 0 () contains at least k workers in B(ŵ k, P ). By construction, the same is true or DA F [P, P ]() since or worker w, the member o the original blocking pair, w R w and w B(ŵ k, P ) imply that w B(ŵ k, P ). Otherwise, i (a) 2 = and w 2 / ˆµ (), then ˆµ 2 () = ˆµ 0 () {w, w 2 }, we now allow to have capacity q + 2, and so on. For (c) 2 and w 2 / ˆµ (), we continue the above vacancy chain dynamics in the same manner until we go to Step 2 or we again reduce the capacity o to q or the worker belonging to the blocking pair was previously assigned to. In other words, we never reject a worker and only vacant positions are illed (where we allow overbooking or ). At the same time, we reduce s capacity by one whenever one o the workers assigned to leaves. Note that this process terminates as the workers preerence always weakly improves at each iteration (because rejects no worker). I at some point exactly q workers are matched to, then we must have ound a stable matching or (P, P ). Let µ denote the resulting matching, and say µ() = q. Then µ is stable under (P, P ) where has q q positions (instead o q ). By construction, µ() contains at least k workers in B(ŵ k, P ). Step 2 : Let q and µ be the outcomes o Step. Let µ() = { w,..., w q } where the workers are listed according to the ranking in P. Now consider the matching market where W = { w q +,..., w q } is not present. Let µ be deined by µ () = { w,..., w q } and µ ( ) = µ( ) or all. Then µ C(P, P W {}). Since µ contains at least k workers in B(ŵ k, P ) and by deinition o P, DA F [P, P W {}]() contains at least k workers in B(ŵ k, P ). Bringing back in { w q +,..., w q }, all irms must weakly beneit, i.e. DA F [P, P ]() must contain at least k workers in B(ŵ k, P ), the desired conclusion. Now let W consist o the k workers which are P -ranked least in B(ŵ k, P ) and the q k workers P -ranked least in B(ŵ q, P ). Obviously, ŵ k, ŵ q W. Let P be a monotonic responsive extension o P such that W R W i W B(ŵ q, P ), W = q and W contains at least k workers in B(ŵ k, P ). Now we have DA F [P, P ]() R W. By the deinition o k, ŵ k P w k. Hence, DA F [P, P ]() does not contain at least k workers in B(ŵ k, P ), which implies that DA F [P, P ]() R W P DA F [P, P ](). Summarizing, we have already showed in the case w q R ŵ q that there exist (P, P ) supp( P ), P P, W W and P mresp(p ) such that DA F [P, P ]() B( W, P ) and DA F [P, P ]() / B( W, P ) simultaneously hold. 5

19 Now consider (P, P ) supp( P ). Let µ = DA F [P, P ] and µ = DA F [P, P ]. Suppose that µ () R W. By construction, µ () contains at least k workers in B(ŵ k, P ), µ () = q and µ () B(ŵ q, P ). We need to show µ () R W. By construction, µ () B(ŵ q, P ) = A(P ). Furthermore, i µ () < q, then the ranking o P does not matter and we would have that µ C(P, P ). By Theorem 2, C(P, P ) = {µ } which is contradiction because µ () < q = µ (). Thus, µ () = q. It remains to be shown that µ () contains at least k workers in B(ŵ k, P ). Now i µ C(P, P ), then, by deinition o P, B(ŵ q, P ) = B(ŵ q, P ) and the act that µ () contains at least k workers in B(ŵ k, P ), µ () must contain at least k workers in B(ŵ k, P ), the desired conclusion. Suppose that µ / C(P, P ). Note that we cannot have µ () ˆµ() B(ŵ k, P ) as otherwise µ C(P, P ), P ˆµ() B(ŵ k, P ) = P ˆµ() B(ŵ k, P ), and ˆµ() B(ŵ k, P )P A(P )\(ˆµ() B(ŵ k, P )) would imply µ C(P, P ), a contradiction. 20 To obtain a contradiction, assume µ () does not contain at least k workers in B(ŵ k, P ). In particular, this means that µ () µ (). Then, since ˆµ()\B(ŵ k, P ) = q k, µ () B(ŵ k, P ) ˆµ() as otherwise is assigned k workers in B(ŵ k, P ). Let w denote the P -least preerred worker in µ (). By the above we have w / ˆµ() B(ŵ k, P ). Case 2.: w is the P -least preerred worker in µ (). Because Pr{ P = (P, P )} > 0, Theorem 2 and µ = DA F [P, P ] imply that C(P, P ) = {µ }. Hence, µ / C(P, P ). Then some pair ( w, ) blocks µ under (P, P ). Because w is the P -least worker in µ (), we have wp w. I w B(ŵ k, P ) ˆµ(), then by construction and w / ˆµ() B(ŵ k, P ) we have wp w and ( w, ) blocks µ under (P, P ), a contradiction to µ C(P, P ). I w / B(ŵ k, P ) ˆµ(), then by construction and w / ˆµ() B(ŵ k, P ) we have wp w and ( w, ) blocks µ under (P, P ), a contradiction to µ C(P, P ). Thus, Case 2. cannot occur. Case 2.2: w is not the P -least preerred worker in µ (). Let ŵ denote the P -least preerred worker in µ (). By ŵ w, and the act that by deinition o P, P W \(B(ŵ k, P ) ˆµ()) coincides with P W \(B(ŵ k, P ) ˆµ()), we have ŵ B(ŵ k, P ) ˆµ(). Furthermore, by w / B(ŵ k, P ) ˆµ() and w P ŵ, we cannot have ŵ B(ŵ k, P ). Then by construction we must have ŵ ˆµ(). But now we may just exchange the positions o ŵ and w and consider P ŵ w. We have µ C(P ŵ w, P ). In µ () the P -least preerred worker and the P ŵ w -least preerred worker coincide and is ŵ. But now we are in Case 2. and this is a contradiction (where ˆµ() is replaced by ˆµ() w w = (ˆµ()\{ŵ}) {w }). Thus, DA F [P, P ]() contains q workers in A(P ) and at least k workers in B(ŵ k, P ). Now by our choice o W and P we obtain that, or any proile P in the support o P P, DA F [P, P ]() R W implies DA F [P, P ]() R W. 20 Here we use again the convention SP T i sp t or all s S and all t T. 6

20 Since DA F [P, P ]() R W P DA F [P, P ]() and P supp( P ), we have Pr{DA F [P, P P ]() B( W, P )} > Pr{DA F [P, P P ]() B( W, P )}. Hence, not DA F [P, P P ]()P sdda F [P, P P ]() and DA F under P, a contradiction. is not monotonic OBIC B Proo o Theorem 2 Let ϕ be a stable mechanism and P be a common belie. To obtain a contradiction, suppose that there is some P P such that both Pr{ P = P } > 0 and C(P ) 2 hold. By stability o ϕ, ϕ[p ] C(P ). Let µ C(P )\{ϕ[p ]} be arbitrary. I there is some w W such that µ(w)p w ϕ[p ](w), then similarly to Ehlers and Massó (2007, Theorem ) we can show that ϕ is not monotonic OBIC under P. Thus, or all µ C(P )\{ϕ[p ]}, ϕ[p ](w)r w µ(w) (8) or all w W. Hence, and since DA W [P ] is the worker-optimal stable matching, ϕ[p ] = DA W [P ]. (9) Observe that (9) needs to hold or all proiles belonging to the support o P. Fix a matching µ C(P )\{ϕ[p ]}. Since µ ϕ[p ] and ϕ[p ] = DA W [P ], there exists some F such that µ() ϕ[p ](). Then, by Roth and Sotomayor (990, Theorems 5.2 and 5.3), µ() = ϕ[p ]() = q. Note that DA W [P ] matches each irm with s worst set o partners according to any monotonic responsive extension o P, among all partners that is matched to across all stable matchings. Thus, by (9), µ()p ϕ[p ]() (0) or all monotonic responsive extensions P o P. Let w µ() be such that wr w or all w µ(). Let P P be such that (i) A(P ) = B( w, P ) and (ii) P A(P ) = P A(P ). By construction o P, µ C(P, P ). Hence, either ϕ[p, P ]() = µ(), in which case, by (0), ϕ[p, P ]()P ϕ[p ]() or all monotonic responsive extensions P o P, or else ϕ[p, P ]() µ(). We want to show that ϕ[p, P ]()P µ() or all monotonic responsive extensions P o P. Suppose otherwise; since ϕ[p, P ]() µ(), this means that µ() ˆP ϕ[p, P ]() or some monotonic responsive extension ˆP o P. Then, by Roth and Sotomayor (989, Theorem 4), or all w µ() and all w ϕ[p, P ]()\µ(), wp w. () 7

21 Consider any w ϕ[p, P ]()\µ(). Since ϕ[p, P ] is individually rational under (P, P ), the deinition o P implies that w B( w, P ). Hence, by P B( w, P ) = P B( w, P ), w R w, which either contradicts () or else it means that ϕ[p, P ]() µ(). Then, ϕ[p, P ]() < µ(). But this contradicts Theorem 5.3 in Roth and Sotomayor (990). Thus, ϕ[p, P ]()P µ() or all monotonic responsive extensions P o P. By (0), ϕ[p, P ]()P ϕ[p ]() (2) or all monotonic responsive extensions P o P. We have already shown that there exist P P with Pr{ P = P } > 0, F and P P such that (2) holds or all P mresp(p ). To obtain a contradiction with truth-telling being a monotonic OBNE, we will ind a P mresp(p ) such that or all P P with Pr{ P = (P, P )} > 0 and ϕ[p, P ]() B(µ(), P ), we have that ϕ[p, P ]() B(µ(), P ). This would imply then that the inequality Pr{ϕ[P, P P ] () = S } Pr{ϕ[P, P P ] () = S } {S 2 W S R µ()} {S 2 W S R µ()} (3) holds. Since ϕ[p, P ]() B(µ(), P ), ϕ[p, P ]() / B(µ(), P ) and Pr{ P = P } > 0, we could conclude then that the inequality in (3) is strict and hence, ϕ[p, P P ] () does not irst order stochastically dominate ϕ[p, P P ] (). For showing this, let P be the monotonic responsive extension o P such that or all W 2 W, W P µ() i and only i (i) W = q (note that µ() = q ) and (ii) there is a one-to-one mapping h : W µ() such that w R h(w ) or all w W with strict preerence holding or some worker in W. In other words, W is strictly preerred to µ() under P only i W is unambiguously strictly preerred to µ() or all monotonic responsive extensions o P. Let P P be such that both Pr{ P = (P, P )} > 0 and ϕ[p, P ]() B(µ(), P ). We want to show that ϕ[p, P ]() B(µ(), P ). Let ˆµ = ϕ[p, P ] and µ = ϕ[p, P ] and assume ˆµ µ (otherwise, the statement ollows trivially). By the deinitions o P and P, we have that ˆµ() R µ(), ˆµ() A(P ) and ˆµ() = q. Thus, by A(P ) = B( w, P ) and P A(P ) = P A(P ), ˆµ C(P, P ). Hence, µ, ˆµ C(P, P ). We distinguish between two cases. Case : µ () P ˆµ(). Since ˆµ() B(µ(), P ), we have that µ () B(µ(), P ). Case 2: ˆµ() P µ (). But then µ () A(P ) = B( w, P ) A(P ) and by P A(P ) = P A(P ), we have µ C(P, P ). Note that (P, P ) is a proile belonging to the support o P and thus, by (9), ˆµ = ϕ[p, P ] = DA W [P, P ]. Hence, ˆµ(w)R wµ (w) or all w W, and again, by the same argument we used or (2), µ () P ˆµ(), a contradiction. Cases and 2 show that i P P is such that Pr{ P = (P, P )} > 0 and ϕ[p, P ]() B(µ(), P ), then ϕ[p, P ]() B(µ(), P ). Since Pr{ P P = 8

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