Incentive Compatible Market Design with an Application to. Matching with Wages

Transcription

1 Incentve Compatble Market Desgn wth an Applcaton to Matchng wth Wages M. Bumn Yenmez Stanford Graduate School of Busness Job Market Paper November 1, 2009 Abstract: Ths paper studes markets for heterogeneous goods usng mechansmdesgn theory. For each combnaton of desrable propertes, I derve an assgnment process wth these propertes n the form of a correspondng drect-revelaton game, or I show that t does not exst. Each partcpant s preferences are prvately known, quaslnear n money, and depend upon the allocaton that he gets thus, a partcpant s type s multdmensonal. The key propertes are ndvdual ratonalty, ncentve compatblty, budget balance, effcency, and stablty aganst coaltonal devatons. The man results characterze mechansms that are ex post ncentve compatble n combnaton wth other propertes. JEL: C71, C78, D82, D44. Keywords: Auctons, Ex Post Incentve Compatblty, Incomplete Informaton, Matchng, Multdmensonal Types, Stablty. I am grateful to my advsors Mchael Ostrovsky, Andrzej Skrzypacz, and Robert Wlson for contnuous gudance and support. I thank Jeremy Bulow, Yoss Fenberg, Alexander Frankel, John Hatfeld, Onur Kesten, John Lazarev, Hongy L, Naz Nam, Mchael Schwarz, Murel Nederle, Danel Taylor, and Al Yurukoglu for helpful comments. Contact: byenmez@stanford.edu. The author acknowledges the fnancal support of the SIEPR Leonard W. Ely & Shrley R. Ely Graduate Student Fellowshp. 1

2 1 Introducton In ths paper, we study a general heterogenous goods market wth transfers. Ths envronment encompasses several classc models consdered n market desgn lterature: one-to-one matchng markets, housng markets, roommate problems, seller-buyer markets wth dscrete heterogeneous goods, and partnershp dssoluton problems. Each agent s preferences are prvately known, and quas-lnear n money. In most of the paper, we focus on drect revelaton mechansms, and we are nterested n fndng mechansms that satsfy the followng propertes at the ex post stage. Agents fnd t optmal to report ther types truthfully (ncentve compatblty). The mechansm does not run a defct (budget balance). The mechansm does not run a defct or create a surplus (exact budget balance). The sum of allocaton utltes s maxmzed (effcency). Agents beneft from partcpaton (ndvdual ratonalty). Our man results characterze necessary and suffcent condtons for the exstence of ex post ncentve compatble mechansms n combnaton wth the propertes lsted above. Ex post ncentve compatblty states that each agent prefers truth-tellng even when the agent knows the reports of other agents assumng these agents report truthfully. Ths s equvalent to an ex post no regret property, as no agent would lke to change her report even f she were to know the reports of other agents. Ths property s desrable for at least three reasons. Frst, t allows the mechansm to perform well even when agents do not have common knowledge of the dstrbuton of types (whch s crucal for mechansms that are only nterm ncentve compatble, and n many applcatons, t s an unrealstc assumpton). Second, t allows desgnng the mechansm wth mnmal assumptons about the dstrbuton of types. Thrd, t removes any ncentves to game the mechansm by ether spyng on other agents (to learn what they plan to report) or tryng to delay reportng. In the case of prvate values, ex post ncentve compatblty s equvalent to domnant strategy ncentve compatblty for drect mechansms. Domnant strategy ncentve compatblty states that each agent prefers truth-tellng regardless of what other agents report. Ths s an addtonal 2

3 desrable property of mechansms snce t allows agents to optmze wthout formng any belefs about the behavor of others. Our frst result (Theorem 1) provdes a necessary and suffcent condton for the exstence of an ex post ncentve compatble, ndvdually ratonal, and budget balanced mechansm. 1 Theorem 1 characterzes when t s possble to mplement any gven allocaton rule, not necessarly effcent, wth a mechansm that satsfes these propertes. Prevous results for partcular models have shown that n markets wth asymmetrc nformaton, effcent mechansms may requre ether runnng a defct (.e., havng the socal planner subsdze the mechansm) or creatng a surplus (to be collected by the socal planner or a thrd party). For example, Myerson and Satterthwate (1983) show that n a seller-buyer market wth an overlap n valuatons, t s necessary to subsdze the mechansm to acheve effcency. On the other hand, t s well known that n a publc goods model, an effcent mechansm may run a surplus (Green and Laffont (1979)). Theorem 1 may be vewed as a general characterzaton of envronments n whch a mechansm can acheve the desred propertes wthout requrng a subsdy, and covers all allocaton rules (not only the effcent ones). Our second result (Theorem 2) provdes a necessary and suffcent condton for the exstence of a mechansm that s ex post ncentve compatble and exact budget balanced wthout the ndvdual ratonalty constrant. To explan further, Theorem 2 characterzes when t s possble to mplement any gven allocaton rule wth a mechansm that satsfes these propertes. These two general results apply to both models wth prvate and nterdependent values, and they accommodate arbtrary correlaton of types. However, the condtons we provde may be dffcult to verfy n a general nterdependent values envronment. It turns out that f agents have prvate values, then we can use exstng results about Vckrey-Clarke-Groves mechansms to rewrte the condtons for the effcent allocaton rule. These condtons are easer to check for some applcatons. In the second part of the paper, we apply our general results to a labor market n whch a set of frms wth one job openng each s matched wth a set of workers. Frms care who they hre, and lkewse, workers care what postons they get. We assume that n ths market, each match may be 1 When we wrte ndvdual ratonalty, budget balance, and exact budget balance wthout any qualfer, we mean ex post. 3

4 benefcal to any frm-worker par, that s, for all types, there exsts a wage level such that these agents prefer to be matched at that wage rather than not partcpatng, the so called gap case. As an applcaton of Theorem 1, we show that there exsts a domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced mechansm for the labor market (Theorem 3). Moreover, we construct such a mechansm n Proposton 1 to serve as a benchmark. Ths result seems contrary to the result of Myerson and Satterthwate (1983) who state that n a two-agent market t s necessary to subsdze the mechansm to obtan effcency. The dfference stems from our gap case assumpton. Our next result (Theorem 4) shows that f the type space s rch, then t s mpossble to strengthen the prevous result by requrng exact budget balance, even wthout ndvdual ratonalty. Ths result may seem surprsng because a second-prce aucton s a mechansm that has all the desred propertes n a labor market wth one frm (whch s a specal case of our model), f the frm cares only about the wage and consders all the workers to be equally desrable. However, ths s possble because the seller s ndfferent between buyers, whch volates the rch type space assumpton. More generally, we show that f the set of agents can be dvded nto two subsets, whch satsfy the condton n Theorem 1, such that types of the agents n one subset does not change the effcent allocaton of the agents on the other subset, then there exsts a mechansm that s domnant strategy ncentve compatble, ndvdually ratonal, effcent, and exact budget balanced. In other words, f the type space s suffcently restrcted, t may be possble to obtan exact budget balance together wth the other propertes. For the labor market, we consder an addtonal property called no blockng that rules out the exstence of a frm-worker par who can do better as a coalton of two rather than partcpatng n the mechansm. Stablty requres both ndvdual ratonalty and no blockng, so that nether an agent nor a par of agents blocks the mechansm. 2 Stablty s ex post n nature. Agents can form blockng pars at the ex post stage after they learn ther types and ther assgned matches. It turns out that ex post stablty s too strong a condton to be satsfed. We prove that f there are more than two agents, and f the type space s rch, then there exsts no mechansm whch s domnant strategy ncentve compatble, ex post stable, and budget balanced. 2 Stablty has been studed extensvely by the matchng lterature, see Roth and Sotomayor (1990). 4

5 No Blockng IR IC BB exact BB EFF Yes Yes (Ex Ante) (Ex Ante) No No Table 1: Summary of Results for the Matchng Problem wth Wages. Each column refers to a property (IR for ndvdual ratonalty, IC for ncentve compatblty, BB for Budget Balance, and EFF for effcency), whereas each row represents a result: the frst entry states whether the result s postve or negatve, other entres show the set of propertes requred for the result. Ex post stablty, albet desrable, does not need to be necessary for a good performance of a mechansm n practce, f agents are not able to block the mechansm after they learn the outcome. For example, n some countres, t s customary for unverstes to hre only ther own former graduate students as assstant professors. Thus, agents can only form blockng pars at the ex ante stage when the qualtes of students are not yet known. We defne ex ante stablty for such envronments: pars of agents can block the mechansm at the ex ante stage before types are realzed. Moreover, any agent can stll block the mechansm ex post by quttng. If workers and frms are each symmetrc ex ante, we construct a domnant strategy ncentve compatble and ex ante stable mechansm, whch also balances the budget on average (Theorem 5). In Table 1, we present a summary of the results for the labor market. 3 In the last part of the paper, we consder other applcatons of the general theory. The frst applcaton s a seller-buyer market for multple heterogeneous goods. Buyers have general preferences over bundles of goods allowng substtutes or complements. In ths market, there exsts a domnant strategy ncentve compatble, ndvdually ratonal, effcent, and exact budget balanced mechansm. The second applcaton s a housng market, where each agent has a unque good to trade wth unt demand. For the housng market, we show that there does not exst a domnant strategy ncentve compatble, effcent, and budget balanced mechansm. The last example s a roommates problem wth transfers, where tradng s restrcted to pars of agents. We show that f agents prefer to trade wth anyone rather than keepng ther endowments, then there exsts a domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced mechansm. 3 We also defne nterm stablty for envronments where a blockng par may form after types are realzed, but before the outcome s known. We show that f there s only one agent on the short sde of the market, then a domnant strategy ncentve compatble, nterm stable, effcent, and budget balanced mechansm exsts. 5

6 Most of the matchng lterature studes markets wth complete nformaton. For example, Gale and Shapley (1962) study stablty n the marrage model, a one-to-one matchng market wthout transfers under complete nformaton. Smlarly, Shapley and Shubk (1971) add the possblty of makng transfers n ther assgnment game, a one-to-one matchng model. In ther setup, the magntude of benefts from trade s commonly known. Roth (1982), Roth (1989), Majumdar (2003), Ehlers and Massó (2007), Lee and Schwarz (2008), Chakraborty et al. (2009), and Nederle and Yarv (2009) study ex post stablty and varous notons of ncentve compatblty n dfferent matchng markets wthout transfers. They ether show that ex post stable and ncentve compatble mechansms do not exst, or make harsh assumptons to get exstence. Our paper s also related to the ex post mplementaton lterature whch tres to reduce the common knowledge assumptons n the sprt of Wlson Doctrne (Wlson (1987)). In partcular, Bergemann and Morrs (2005) study when ex post mplementaton s equvalent to nterm mplementaton for all type spaces. Bergemann and Morrs (2008) study the full ex post mplementaton problem. In Appendx 1, we consder the specal case that there s one agent on the short sde of the market. In auctons, there s usually one seller, so auctons are specfc examples of such markets. For ths specal case, we characterze the mechansm that maxmzes the expected revenue of the solo agent. We also ntroduce new ascendng auctons to mplement ths optmal mechansm and the benchmark effcent mechansm. 2 General Model In a heterogenous goods market wth transfers, there s a fnte set of agents N and k types of goods. Each agent N has an endowment e Z k + whch specfes non-negatve nteger quanttes for all types of goods. Let e be the profle of endowments consstng of e for all N. Agent s type s θ Θ where Θ s a compact and connected subset of some Eucldean space. Suppose that θ θ s dstrbuted accordng to dstrbuton D. Ths formulaton allows N types to be correlated. Each agent s utlty functon s quas-lnear over allocaton and monetary transfer. Hence, f agent receves x Z k + and makes a payment of t, her utlty s u (x, θ) + t, where u (x, θ) s contnuous n θ. Therefore, we allow values to be nterdependent,.e., they 6

7 may depend on sgnals of other agents. Let A f denote a fnte set of feasble allocatons gven exogenously. For example, A f can be the set of allocatons where each agent gets one good at most. 4 A drect revelaton mechansm for our formulaton s a par (µ, t) where µ : Θ A f s an allocaton functon and t : Θ R N s a transfer functon. For a gven type profle θ, µ (θ) Z k + s the allocaton of agent and t (θ) s the transfer to agent. The net utlty of agent from partcpaton s, v (θ) [u (µ (θ), θ) u (e, θ)] + t (θ). By the revelaton prncple, t s suffcent to consder drect revelaton mechansms to check the exstence of mechansms wth the desred propertes lsted below. Defnton 1. A drect revelaton mechansm (µ, t) satsfes ex post ncentve compatblty f, for all N and θ Θ, for all θ Θ. v (θ) [u (µ (θ, θ ), θ) u (e, θ)] + t (θ, θ ) Ex post ncentve compatblty states that each agent prefers truth-tellng even when the agent knows the reports of other agents assumng these agents report truthfully. Ths s equvalent to an ex post no regret property, as no agent would lke to change her report even f she were to know the reports of other agents. 5 If (µ, t) s ex post ncentve compatble, then we say that ths mechansm mplements µ (wth ex post ncentves). Defnton 2. A drect revelaton mechansm (µ, t) satsfes (ex post) ndvdual ratonalty f, for all N, for all θ Θ. v (θ) 0 That s, a mechansm s ndvdually ratonal f agents have non-negatve net utltes for all realzed types. If ths s volated for agent, we say that blocks (µ, t). 4 The choce of modelng s nspred by Sonmez (1999). He studes a general ndvsble goods exchange problem wthout transfers. 5 Ex post ncentve compatblty was frst ntroduced as unform equlbrum n d Aspremont and Gérard-Varet (1979) and as unform ncentve compatblty n Holmstrom and Myerson (1983). 7

8 Defnton 3. A drect revelaton mechansm (µ, t) satsfes (ex post) effcency f for all θ Θ. µ(θ) arg max x A f u (x, θ), In words, a mechansm s effcent f the allocaton functon maxmzes the sum of allocaton utltes over the set of feasble allocatons. Defnton 4. A drect revelaton mechansm (µ, t) satsfes (ex post) budget balance f for all θ Θ. t (θ) 0 Budget balance means that the mechansm does not run a defct. We only use the above defntons of ndvdual ratonalty, effcency, and budget balance. Therefore, when we use them, we do not wrte ex post n front of the noton. Defnton 5. A drect revelaton mechansm (µ, t) satsfes (ex post) exact budget balance f for all θ Θ. t (θ) = 0 Defnton 6. A drect revelaton mechansm (µ, t) satsfes ex ante exact budget balance f E θ [t (θ)] = 0. Two separate exact budget balance notons are consdered. Ex post exact budget balance means that the sum of transfers s zero for all realzed types, whereas ex ante exact budget balance means the sum of transfers s zero on average. When we use the former noton, we do not wrte ex post explctly, whereas f we use the latter, we wrte ex ante. 3 General Results Let (µ, t) be an ex post ncentve compatble drect revelaton mechansm. In ths secton, we provde necessary and suffcent condtons for the exstence of a transfer functon t such that 8

9 (µ, t ) s 1) ex post ncentve compatble, ndvdually ratonal, budget balanced and 2) ex post ncentve compatble, and exact budget balanced. We use the followng lemma to prove these results. Lemma 1. Suppose (µ, t) s an ex post ncentve compatble drect revelaton mechansm. Then (µ, t ) s an ex post ncentve compatble drect revelaton mechansm f and only f t (θ) = t (θ) + g (θ ) for some functon g for all. All omtted proofs are n Appendx 2. The lemma tells us that f two mechansms are ex post ncentve compatble wth the same allocaton functon, then the dfference n transfers for any agent n two mechansms cannot depend on the type of ths agent. Moreover, t also states that changng the transfer functon of an agent by a functon whch depends on the types of other agents preserves ex post ncentve compatblty. Defnton 7. Let (µ, t) be an ex post ncentve compatble drect revelaton mechansm. Then, θ (θ ) s the type of agent whch mnmzes v (θ, θ ), called the worst-off type for agent gven θ. We need to prove that the worst-off type exsts n general. The next lemma establshes the exstence. Lemma 2. Suppose that (µ, t) s ex post ncentve compatble. Then there exsts a worst-off type, for all N and θ Θ. By Lemma 1, f (µ, t) and (µ, t ) are ex post ncentve compatble, then the worst-off types of agent gven θ for both mechansms are the same. Therefore, the worst-off type does not depend on the partcular transfer functon as long as t mplements the allocaton functon wth ex post ncentves. The exstence of a worst-off type s not straghtforward from defnton. However, ex post ncentve compatblty guarantees ts exstence. If there exsts an ex post ncentve compatble mechansm (µ, t) whch s not necessarly ndvdually ratonal or budget balanced, the next theorem gves a necessary and suffcent condton usng the utltes derved from (µ, t) for the exstence of a transfer functon t such that (µ, t ) s ex post ncentve compatble, ndvdually ratonal, and budget balanced. 9

10 Theorem 1. Suppose that (µ, t) s an ex post ncentve compatble drect revelaton mechansm. Let v (θ) be the net utlty of agent and θ (θ ) be the worst-off type n ths mechansm. Then there exsts a transfer functon t such that (µ, t ) s ex post ncentve compatble, ndvdually ratonal, and budget balanced f and only f the followng nequalty holds: v (θ (θ ), θ ) t (θ), θ Θ. (1) Proof. If part: Let t (θ) = t (θ) v (θ (θ ), θ ). By Lemma 1, (µ, t ) s ex post ncentve compatble. Add u (µ (θ), θ) u (e, θ) to both sdes of ths equaton to get v (θ) [u (µ (θ), θ) u (e, θ)] + t (θ) = v (θ) v (θ (θ ), θ ) whch must be non-negatve by defnton of the worst-off type, so ndvdual ratonalty s also satsfed. Fnally, t (θ) = [t (θ) v (θ (θ ), θ )] whch s less than or equal to zero by assumpton. Therefore, (µ, t ) s budget balanced as well as ncentve compatble and ndvdually ratonal. Only f part: Suppose that (µ, t ) s ex post ncentve compatble, ndvdually ratonal, and budget balanced. By Lemma 1, t (θ) = t (θ) + g (θ ). Add u (µ (θ), θ) u (e, θ) to both sdes to get v (θ) = v (θ) + g (θ ). Snce (µ, t ) s ex post ndvdually ratonal, g (θ ) v (θ (θ ), θ ). Fnally, ex post budget balance mples 0 t (θ) = [t (θ)+g (θ )] [t (θ) v (θ (θ ), θ )]. Therefore, we get v (θ (θ ), θ ) t (θ). The ntuton for ths result s as follows. The hghest amount that we can charge an agent cannot depend on her type because of ex post ncentve compatblty. Therefore, the hghest amount we can charge preservng ex post ndvdual ratonalty s the utlty that she gets f she were to be the worst-off type, whch depends on types of other agents. If the sum of these charges can cover the sum of transfers that we have to make to agents, then there exsts such a mechansm. Otherwse, there exsts none. That s exactly what (1) checks. Theorem 2. Let (µ, t) be an ex post ncentve compatble drect revelaton mechansm and τ(θ) be the sum of transfers,.e., τ(θ) = t (θ). For each fx a type θ Θ. Then there exsts a transfer functon t such that (µ, t ) s ex post ncentve compatble and exact budget balanced f and only f 10

11 the followng holds: N ( 1) j j=0 { 1,..., j } N τ(θ 1,..., θ j, θ 1,..., j ) = 0, θ Θ. 6 (2) The above statement s vald for any partcular choce of θ. Proof. If part: Suppose that (2) s satsfed. We construct t such that (µ, t ) s ex post ncentve compatble and exact budget balanced. N t 1(θ) = t 1 (θ) + ( 1) j j=1 1 { 1,..., j } N t 2(θ) = t 2 (θ) + ( 1) j τ(θ 1,..., θ j, θ 1,..., j ) j=1 1 { 1,..., j },2 { 1,..., j } τ(θ 1,..., θ j, θ 1,..., j ). t n(θ) = N t n (θ) + ( 1) j τ(θ 1,..., θ j, θ 1,..., j ) j=1 1,...,n 1 { 1,..., j },n { 1,..., j } By Lemma 1 (µ, t ) s ex post ncentve compatble. To show that t s also exact budget balanced, sum the above equaltes to get: Nt (θ) = N N t (θ) + ( 1) j j=1 { 1,..., j } N τ(θ 1,..., θ j, θ 1,..., j ). N By (2), ( 1) j τ(θ 1,..., θ j, θ 1,..., j ) = τ(θ). Moreover, t (θ) = τ(θ) by defnton. Therefore, we get that t (θ) = τ(θ) τ(θ) = j=1 { 1,..., j } N N 0. N Only f part: Suppose that (µ, t ) s ex post ncentve compatble and exact budget balanced. Then by Lemma 1, t (θ) = t (θ) + g (θ ). If we sum ths over all agents we get [t (θ)] = [t (θ) + g (θ )] whch s equvalent to τ(θ) = g (θ ). Defne g (1) (θ ) as follows: g (1) (θ ) = g (θ ) g (θ 1, θ 1, ) for 1 and g (1) 1 (θ 1) = g 1 (θ 1 ) + 6 Wth some abuse of notaton, θ 1,..., j s the vector of types for agents other than 1,..., j. For j = 0, θ 1,..., j = θ. 11

12 1g (θ 1, θ 1, ). Therefore, g (1) (θ ) = g (θ ). Hence, τ(θ) = g (1) (θ ). (3) Plug n θ 1 = θ 1 to get τ(θ 1, θ 1 ) = g (1) 1 (θ 1) + (θ 1, θ 1, ). By the above transformaton, 1 g (1) (θ 1, θ 1, ) = 0, 1. Therefore, we get that τ(θ 1, θ 1 ) = g (1) 1 (θ 1). Usng ths equalty, (3) can be rewrtten as g (1) τ(θ) τ(θ 1, θ 1 ) = 1 g (1) (θ ). g (j) j Defne g (j) (θ j ) = g (j 1) (θ ) nductvely on j by g (j) (θ ) = g (j 1) j (θ ) g (j 1) (θ j, θ,j ) for j and (θ j ) + g (j 1) j (θ j, θ,j ). By nducton on m, t s easy to see that τ(θ) j τ(θ 1,..., θ m, θ 1,..., m ) = g (m) (θ ) where N = { 1,..., m} N N\N τ(θ 1, θ 1 ) ( 1) m { 1 } N {1,..., m}. Take N = N to get: N ( 1) j j=0 { 1,..., j } N τ(θ 1,..., θ j, θ 1,..., j ) = 0. To acheve exact budget balance, we need to charge each agent an extra amount such that the sum of the charges s equal to the sum of transfers, τ(θ). However, for agent, ths extra charge can only be a functon of θ to preserve ex post ncentve compatblty. Therefore, we can acheve exact budget balance f and only f we can decompose τ(θ) addtvely nto N functons where functon depends only on θ. If such a decomposton s possble, then we can fnd a decomposton usng only τ by algebrac manpulatons as n the proof. Condton 2 gves the exact decomposton. For a general envronment, t may be hard to check (2). However, f one of the agents type has a degenerate dstrbuton,.e., when the support of the type has only one value, then the condton s satsfed. Thus, ths result can be appled to the mportant specal case of auctons, where t s usually assumed that the auctoneer has a constant type. Corollary 1. Let (µ, t) be an ex post ncentve compatble drect revelaton mechansm. Suppose 12

13 that one of the agents type has a degenerate dstrbuton. Then there exsts a transfer functon t such that (µ, t ) s ex post ncentve compatble and exact budget balanced. If agent s type has a degenerate dstrbuton, then ths agent does not face any ncentve ssues. Therefore, agent can be charged the resdual balance. Ths preserves ex post ncentve compatblty and acheves exact budget balance. 3.1 Effcent Allocaton wth Prvate Values In ths subsecton, we consder mplementng the effcent allocaton functon. Both Theorems 1 and 2 can be appled to envronments wth nterdependent values and correlated types. However, to apply these results, one needs to fnd a transfer functon to mplement the effcent allocaton functon wth ex post ncentves. Ths may be hard for envronments wth nterdependent values. However, f values are prvate, then we can use a Vckrey-Clarke-Groves (henceforth VCG) payment functon. 7 Ths enables us to rewrte the condtons gven n Theorems 1 and 2 for the effcent allocaton rule. Defnton 8. (Prvate Values) Agent N has prvate values f, for all θ Θ and x Z k +, u (x, θ) = u (x, (θ, θ )), for all θ Θ. When ths holds the utlty of agent s denoted by u (x, θ ). When values are prvate, ex post ncentve compatblty s equvalent to the stronger noton of domnant strategy ncentve compatblty that we defne below. Defnton 9. A drect revelaton mechansm (µ, t) satsfes domnant strategy ncentve compatblty f, for all N, θ Θ and θ Θ, [u (µ (θ, θ ), θ) u (e, θ)] + t (θ, θ ) [u (µ (θ ), θ) u (e, θ)] + t (θ ) for all θ Θ. Domnant strategy ncentve compatblty requres each agent to play truthfully regardless of what other players are dong. Suppose µ e s an effcent allocaton functon and SS(θ) be the maxmum value of sum of allocaton utltes when the type profle s θ,.e., SS(θ) = [u (µ e (θ), θ ) u (e, θ )]. Let t (θ) = 7 Thus, t follows the dea of effcent mechansm desgn by Vckrey (1961), Clarke (1971) and Groves (1973). 13

14 [u j (µ e j (θ), θ j) u j (e j, θ j )]. Therefore, v (θ) = [u (µ e (θ), θ ) u (e, θ )]+t (θ) = [u j (µ e j (θ), θ j) j u j (e j, θ j )] = SS(θ). Moreover, t (θ) = ( N 1)SS(θ). Thus, Theorem 1 reduces to the followng. j N Corollary 2. Let SS(θ) be the sum of allocaton utltes for the effcent allocaton functon. If agents have prvate values, then there exsts a domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced mechansm f and only f the followng holds: SS(θ (θ ), θ ) ( N 1)SS(θ), θ Θ. (4) A smlar result s shown for nterm mplementaton wth exact budget balance n Krshna and Perry (1998) and Wllams (1999) where they have expectatons of terms n (4). For the effcent allocaton functon wth prvate values, Theorem 2 reduces to the followng. Corollary 3. Suppose that agents have prvate values. 1. For each fx a type θ Θ. Then there exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced mechansm f and only f the followng holds: N ( 1) j j=0 { 1,..., j } N SS(θ 1,..., θ j, θ 1,..., j ) = 0, θ Θ. (5) 2. Let µ e be an effcent allocaton functon. Suppose that for each agent there exsts an agent j such that for all θ Θ and θ j Θ j, µ e (θ) = µe (θ j, θ j). Then there exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced mechansm. Part 1 characterzes envronments wth prvate values for whch there exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced mechansm. Part 2 gves a suffcent condton for ths to hold. Ths condton s weaker than the condton gven n Corollary 1. We gnore the ndvdual ratonalty constrant n the corollary stated above. However, we can provde a suffcent condton for the exstence of a mechansm wth the propertes lsted above and ndvdual ratonalty. Corollary 4. Suppose the set of agents can be dvded nto two subsets N 1 and N 2 such that n each subset the effcent allocaton s a reallocaton of ther endowments, and that ths reallocaton 14

15 does not change wth the type profle of agents n the other subset. Suppose also that (4) holds for N 1 and N 2, separately. If agents have prvate values, then there exsts a mechansm whch s domnant strategy ncentve compatble, ndvdually ratonal, effcent, and exact budget balanced. By assumpton, there are two separate submarkets, and for each submarket (4) holds. By Corollary 2, there exsts a domnant strategy ncentve compatble, ndvdually ratonal, and effcent mechansm for each submarket that creates a surplus. We can dstrbute the surplus from one submarket to the agents on the other submarket n any way. Ths does not thwart ncentves, and ndvdual ratonalty s preserved. Moreover, the fnal allocaton s effcent for the whole economy by assumpton, and exact budget balance s satsfed. Now, let us consder some examples to see when (5) holds. Example 1. There are two agents, seller S and buyer B. The seller owns a good that she wants to sell. The seller s value for the good, θ S [0, 1], s her prvate value and the buyer s value, θ B [2, 3], s hs prvate value. Let θ S = 0 and θ B = 2. Hence, the left hand sde of (5) s equvalent to: SS(θ S, θ B ) SS(θ S, θ B ) SS(θ S, θ B ) + SS(θ S, θ B ) = (θ B θ S ) (θ B 0) (2 θ S ) + (2 0) = 0. Therefore, there exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced mechansm. In ths example, we can construct such a mechansm easly. Each agent reports a value. Regardless of the reports, we transfer the good at Ths mechansm satsfes all the propertes. In the frst example, both agents types do not change the effcent allocaton and (5) holds. Example 2. There are three agents, seller S and buyers B 1 and B 2. The seller owns a good. The seller s value for the good, θ S [0, 1], s her prvate nformaton. Smlarly, buyer B s value for the good, θ B [2, 3], s hs prvate nformaton, = 1, 2. Let θ S = 0, θ B1 = 2 and θ B2 = 3. The left 15

16 hand sde of (5) reduces to: SS(θ S, θ B1, θ B2 ) SS(θ S, θ B1, θ B2 ) SS(θ S, θ B1, θ B2 ) SS(θ S, θ B1, θ B2 ) + SS(θ S, θ B1, θ B2 ) + SS(θ S, θ B1, θ B2 ) + SS(θ S, θ B1, θ B2 ) SS(θ S, θ B1, θ B2 ) = max{θ B1 θ S, θ B2 θ S } max{θ B1, θ B2 } max{2 θ S, θ B2 θ S } max{θ B1 θ S, 3 θ S } + max{2, θ B2 } + max{θ B1, 3} + max{2 θ S, 3 θ S } max{2, 3} = max{θ B1, θ B2 } θ S max{θ B1, θ B2 } (θ B2 θ S ) (3 θ S ) + θ B (3 θ S ) 3 = 0. Therefore, there exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced mechansm. We can use a second-prce aucton to acheve these propertes. In the second example, the seller s type does not change the effcent allocaton and (5) holds. Example 3. There are three agents, seller S, buyer B 1, and buyer B 2. The seller owns a good that she does not value (or values at 0). Buyer B s value for the good, θ B [2, 3], s hs prvate nformaton, = 1, 2. The seller cares about who buys the good. Therefore, f B buys the good, then the socal surplus s θ S + θ B, where θ S denotes how much the seller wants B to buy the good. Thus, seller s prvate nformaton s θ S1 = (θ 11, θ 12 ) [0, 1] 2. Let (θ 11, θ 12 ) = (0, 0), θ B1 = 2 and θ B2 = 2. The left hand sde of (5) s: max{θ 11 + θ B1, θ 12 + θ B2 } max{θ B1, θ B2 } max{θ 11, θ 12 + θ B2 } max{θ 11 + θ B1, θ B2 } + θ B2 + θ B1 + max{θ 11, θ 12 }. For (θ 11, θ 12 ) = (0, 1/2), θ B1 = 2, and θ B2 = 2 3 4, the above expresson s equal to 1 4. Hence, (5) does not hold. Therefore, there does not exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced for ths example. In the thrd example, each agent s type changes the allocaton, and (5) fals. Example 4. There are sx agents, two sellers S 1 and S 2, and four buyers, B 1, B 2, B 3, and B 4. S 1 s sellng a Macntosh computer whereas S 2 s sellng a PC. B 1 and B 2 are only nterested n gettng a Mac and value the PC at zero. On the other hand, B 3 and B 4 are only nterested n gettng 16

17 a PC and value the Mac at zero. S 1 wants B 1 or B 2 to buy the Mac and S 2 wants B 3 or B 4 to buy the PC. Therefore, θ S1 = (θ 11, θ 12, θ 13, θ 14 ) where θ 11, θ 12 [0, 1] and θ 13 = θ 14 = 0. Smlarly, θ S2 = (θ 21, θ 22, θ 23, θ 24 ) where θ 21 = θ 22 = 0 and θ 23, θ 24 [0, 1]. θ B s the non-zero value that B has for the computer of preference stated above. For ths problem, the left hand sde of (5) has 64 terms, so we do not wrte them at all. However, all the terms cancel out because for each agent, there exsts another agent j such that the type of agent j does not change the allocaton of agent (Part 2 of Corollary 3). Therefore, there exsts a domnant strategy ncentve compatble, effcent, and exact budget balanced mechansm. In the last example, although each agent s type changes the effcent allocaton, (5) stll holds. The reason s that the market s separable n two submarkets. In general, t s hard to see when (5) holds. 4 Matchng wth Wages We now consder the followng specal case of our general model. In a matchng wth wages problem, the set of agents N can be dvded nto two dsjont sets, the set of frms, F = {f 1,..., f m }, and the set of workers, W = {w 1,..., w n }, (m, n 1) wth generc elements f and w. Each agent s endowed wth a unque good. The set of feasble allocatons A f s such that each agent keeps her endowment or gets a good from an agent on the other sde of the market. Moreover, f frm f gets the good owned by worker w, then worker w gets the good of frm f and vce versa. Formally, A f = {x = (x f1,..., x fm, x w1,..., x wn ) f F x f {e f, e w1,..., e wn }, w W x w {e w, e f1,..., e fm } and, j N x = e j x j = e }. We nterpret ths as a one-to-one matchng of frms and workers where agents may reman sngle, that s, unmatched. Each agent s type s the vector of prvate values for potental partners. Therefore, utlty functons return the value correspondng to the match. To ease notaton, we assume that worker w s type s a functon u w : F R + and frm f s type s a functon u f : W R +. Hence, the type of agent s the utlty functon u whch takes values on a connected doman Θ. If s a frm, then Θ R n +. Otherwse, f s a worker, then Θ R m +. Assume that dstrbuton D s such that u has postve support on Θ regardless of θ. Fnally, suppose that for agent N the utlty of 17

18 beng unmatched, whch s denoted by u (), s zero. Therefore, we mplctly assume that for all (f, w) F W, u f (w) 0 = u f (f) and u w (f) 0 = u w (w). 8 For any frm-worker par (f, w), φ(f, w) = u f (w) + u w (f) s the jont producton of ths par, nterpreted as the gan from trade of a contract between them. For ths problem, we show that (4) s satsfed. Theorem 3. There exsts a domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced mechansm for the matchng wth wages problem. The dervng force behnd ths theorem s the assumpton that utlty values cannot be negatve. If ths assumpton s volated, then by Myerson and Satterthwate (1983) we know that such a mechansm does not exst. Theorem 3 states the exstence of a mechansm. We can actually construct such a mechansm usng the f part of the proof of Theorem 1. Proposton 1. Let µ e (u) be a matchng functon whch maxmzes the sum of match utltes for all agents and SS(u) be the correspondng maxmum value functon. Then (µ e, t) s a domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced drect revelaton mechansm where t s gven by t (u) = [SS(u) u (µ e (u))] SS(0, u ). Moreover, for all N, t (u) 0, for all u Θ. Let us see how ths mechansm works n a smple example. Example 5. There are three agents n the economy, one frm, Captal, and two workers, Ann and Bob. Suppose t s common knowledge that Captal prefers to hre any worker for 50K rather than not hrng anyone, and each worker prefers to work at Captal for 50K rather than not workng. Each worker reports the wage whch makes them ndfferent between not workng and workng at that wage. Smlarly, Captal reports a wage for each worker such that t s ndfferent between not hrng and hrng that worker at the correspondng wage. Suppose that Captal reports 70K and 80K for Ann and Bob respectvely. Ann reports 30K, and Bob reports 20K. To determne 8 Ths assumpton holds, for example, f t s common knowledge that there exsts a wage p j for whch frm f s wllng to hre worker w j at p j and the worker s wllng to work for the frm at ths wage. 18

19 who s hred, we look at the total surplus created by hrng. If Ann s hred then the total surplus s 70K 30K = 40K, whereas f Bob s hred t s 80K 20K = 60K. Therefore, Captal should hre Bob. Ann stays sngle and does not make any transfers. The benchmark mechansm sets the wage pad by Captal so that ts net utlty s equal to the dfference of the socal surplus under the orgnal report and the mnmum report that t could have made, whch s 50K for both workers. The socal surplus under the orgnal report s 60K. If Captal had reported 50K for the workers, then the socal surplus would be 30K. Therefore, Captal s payment can be calculated by 60K 30K = 80K x, so x = 50K. However, Bob s wage s lower than ths amount whch can be calculated smlarly. If Bob had reported 50K, then the socal surplus would be 40K. Hence, Bob s wage can be found by 60K 40K = x 20K. Thus, Bob s wage s 40K. In ths nstance, the mechansm creates a surplus of 10K. Theorem 3 establshes a benchmark result. It shows that there exsts a mechansm whch s domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced. Budget balance means that the sum of transfers should not be postve. Therefore, the mechansm does not need any outsde subsdy to run but may create some surplus. Then an mportant queston s what happens wth the surplus. One possble answer s that the market manager receves the resdual money. For example, n a seller-buyer market the payment that the seller receves can be smaller than the prce that the buyer pays. The dfference goes to the market maker. Smlarly, a contractor workng n a company receves less than the company pays. The dfference goes to the managed servces provder. In the absence of such a person, one may try to redstrbute the surplus between agents provded that ths does not thwart ncentve compatblty. For the rest of ths secton, we show that the surplus cannot be redstrbuted among the agents wthout destroyng ncentve compatblty or effcency even f we gnore ndvdual ratonalty. Therefore, f we want to have a mechansm wth exact budget balance, then we need a clamant for the resdual money. Theorem 5.3 n Green and Laffont (1979) s a smlar result for a publc goods model. However, ther result cannot be appled here snce they requre valuaton functons to be unrestrcted over the whole allocaton. Ths assumpton s not satsfed here because our economy s a prvate goods 19

20 economy. Theorem 4. Suppose that there are at least three agents n the matchng wth wages problem and that there exsts c > 0 such that for each frm f, Θ f [0, c] n, and for each worker w, Θ w [0, c] m. Then there exsts no matchng mechansm whch satsfes domnant strategy ncentve compatblty, effcency and exact budget balance. The argument of the proof s as follows. Assume that (5) s satsfed. Suppose that u Θ s such that no lnear relatonshps hold for any components of u, for example, 2u f1 (w 2 ) u w1 (f 2 )+u w2 (f 1 ). Then each component of u must appear an even number of tmes n (5), matchng the negatve and postve appearances. However, because of the rch type space assumpton, we can fnd u Θ such that one component does not appear an even number of tmes, wth the same property that no lnear relatonshps hold for any components of u. Ths gves a contradcton. 5 Ex Post Stablty In ths secton we study ex post stablty. At the ex post stage preferences of agents are common knowledge. Therefore, each agent can dentfy partners such that parng up wth that agent can be mutually benefcal. If such a par exsts, we say that the par blocks the matchng. Ex post no blockng rules out exstence of such pars. The formal defnton s as follows. Defnton 10. A drect revelaton mechansm (µ, t) satsfes ex post no blockng f, for all (f, w) F W, v f (u) + v w (u) φ(f, w) for all u Θ. A matchng mechansm s ex post stable f t satsfes ex post no blockng and ndvdual ratonalty. A hgh correlaton between ex post stablty and the success of mechansms has been documented: for example, n varous regons of the Brtsh Natonal Health Servces (see Nederle et al. (2008)). Ths has also been observed n controlled lab experments (Kagel and Roth (2000)). In lght of ths, central clearnghouses have been desgned for varous markets to delver stable outcomes. Examples nclude New York Cty hgh schools, Boston publc schools, kdney exchange 20

21 programs, and varous medcal specaltes usng the Natonal Resdent Matchng Program (see Roth (2008)). However, t turns out that ths property cannot be satsfed together wth domnant strategy ncentve compatblty and budget balance. Proposton 2. Suppose that there are at least three agents n the matchng wth wages problem, and that there exsts c > 0 such that for each frm f, Θ f [0, c] n and for each worker w, Θ w [0, c] m. Then there exsts no domnant strategy ncentve compatble, ex post stable, and budget balanced mechansm. In fact, even a weaker condton than ex post stablty s also mpossble to satsfy. Defnton 11. A drect revelaton mechansm (µ, t) satsfes no cross-subsdes f, for any matched par (for any (f, w) such that µ f (u) = w), t f (u) + t w (u) 0 for all u Θ. No cross-subsdes requres that pars who are matched do not subsdze the system. Proposton 3. Suppose that there are at least three agents n the matchng wth wages problem. Suppose also that there exsts c > 0 such that for each frm f, Θ f [0, c] n and for each worker w, Θ w [0, c] m. Then there exsts no matchng mechansm whch satsfes domnant strategy ncentve compatblty, ndvdual ratonalty, effcency, budget balance, and no cross-subsdes. Snce ex post stablty mples effcency, Proposton 3 s a stronger result than Proposton 2. Before provng Proposton 3, we present a lemma. Lemma 3. If (µ, t) satsfes budget balance, ndvdual ratonalty, and no cross-subsdes, then the followng holds for all u: 1. If µ (u) = then t (u) = 0, 2. If µ f (u) = w (and µ w (u) = f) then t f (u) + t w (u) = 0. 21

22 Ths lemma states that f a mechansm satsfes budget balance, ndvdual ratonalty, and no cross-subsdes, then each sngle agent makes no transfer and that the sum of transfers for a par s zero. Therefore, these condtons mply exact budget balance. On the other hand, by Theorem 4, we know that exact budget balance s not compatble wth domnant strategy ncentve compatblty, effcency, and the addtonal assumpton that the type space s rch. Snce the same assumpton s made n Proposton 3, the proof s complete. A more drect proof s n Appendx 2. If there are only two agents n the market, then the matchng whch matches them wthout any transfer satsfes domnant strategy ncentve compatblty, ex post stablty, and budget balance. Smlarly, f there are more than two agents and the type space s not rch enough t may be possble to construct a mechansm wth all the propertes. For such stuatons, see examples 1,2, and 4. Hence, the assumptons made n Proposton 3 are, n a sense, necessary. 6 Ex Ante Stablty In the prevous secton we show that no domnant strategy ncentve compatble, ex post stable, and budget balanced mechansm exsts. In ths secton we consder a weaker no blockng condton where agents can form blockng pars only at the ex ante stage. Defnton 12. A mechansm satsfes ex ante no blockng f, for all (f, w) F W, E u [v f (u)] + E u [v w (u)] E u [φ(f, w)]. In words, ex ante no blockng means that the sum of expected utltes of any frm-worker par from partcpatng n the mechansm s greater than or equal to the expected producton of the par. If ths condton holds, then no par can make a deal at the ex ante stage whch gves them both hgher expected utltes than the expected utlty they get from partcpaton n the mechansm. A matchng mechansm s ex ante stable f t satsfes ndvdual ratonalty and ex ante no blockng. We can acheve ex ante stablty when frms and workers are each ex ante symmetrc. Theorem 5. Suppose that u w (f) s drawn from a dstrbuton D W (wth densty d W and mean µ W < ) and u f (w) s drawn from a dstrbuton D F (wth densty d F and mean µ F < ) for all f and w ndependently. Then there exsts a mechansm whch satsfes domnant strategy ncentve 22

23 compatblty, effcency, ex ante stablty, and ex ante exact budget balance for the matchng wth wages problem. The dea of the proof s as follows. Consder the mechansm descrbed n Proposton 1. Ths mechansm s domnant strategy ncentve compatble, ndvdually ratonal, effcent, and budget balanced. Moreover, t creates an expected surplus. To get ex ante no blockng, we dstrbute the surplus across the agents as a lump sum transfer. Suppose that the dstrbuton s such that each agent gets the same extra payment. If the number of workers s dfferent than the number of frms, ex ante no blockng may not be satsfed. The reason s that the sum of addtonal payments that a frm and a worker gets s small, so they may fnd t benefcal to ex ante block. Snce we assume that workers and frms are each ex ante dentcal, the expected utltes are the same except the extra transfer whch comes from the dstrbuton of the surplus. Therefore, ex ante no blockng s satsfed f and only f the worker and frm who get the mnmum extra transfer do not form a blockng par. The best dstrbuton of the surplus, whch maxmzes the sum of transfers for these agents, s the one whch dstrbutes the total surplus equally among the agents on the short sde of the market. In the proof we show that ths constructon works. Proof. Frst, we show a weaker verson of the clam where nstead of ex ante exact budget balance we have that the sum of expected transfers s non-postve. Snce we can ncrease the transfer of any agent by a lump-sum transfer, we can attan exact budget balance wth the other propertes. Let µ e (u) be an effcent matchng. Let t f (u) = SS(u) SS(0, u f ) u f (µ e f (u)) + k F and t w (u) = SS(u) SS(0, u w ) u w (µ e w(u)) + k W. If we apply Lemma 1 to the mechansm descrbed n Proposton 1 and (µ e, t), we get that (µ e, t) s domnant strategy ncentve compatble. Moreover t s effcent. Now we analyze ex ante no blockng condton. Snce frms and workers are ex ante dentcal, ex ante no blockng condtons for any par of frms and workers are the same. Ths condton can be wrtten as: E u [v f (u)] + E u [v w (u)] = 2E u [SS(u)] E u w [SS(0, u w )] E u f [SS(0, u f )] + k W + k F µ F + µ W. 23

24 Reorderng gves: k W + k F µ F + µ W 2E u [SS(u)] + E u w [SS(0, u w )] + E u f [SS(0, u f )]. If the RHS of the last nequalty s negatve, then we can set k F = k W = 0. In ths case, (µ e, t) concdes wth the mechansm n Proposton 1. Therefore, t s budget balanced whch mples that the sum of expected transfers to be non-postve. For ths case the proof s complete. Otherwse, suppose wthout loss of generalty that m n. To mnmze the sum of expected transfers, we set k F = 0 and k W = µ F + µ W 2 E u [SS(u)] + E u w [SS(0, u w )] + E u f [SS(0, u f )]. Therefore, (µ, t) satsfes domnant strategy ncentve compatblty, ndvdual ratonalty, effcency, and ex ante no blockng. Now, we prove that the sum of expected transfers s nonnegatve. E u [t (u)] = (m n 1)E u [SS(u)] (m n)e u f [SS(0, u f )] + n(µ F + µ W ) = (m n){e u [SS(u)] E u f [SS(0, u f )]} E u [SS(u)] + n(µ F + µ W ). Hence, the necessary and suffcent condton s: (m n){e u [SS(u)] E u f [SS(0, u f )]} E u [SS(u)] n(µ F + µ W ). (6) If m = n, (6) reduces to n(µ F + µ W ) E u [SS(u)]. Ths equaton s satsfed because n the effcent matchng there are n pars wth each par s producton exceedng µ F + µ W on average. Before proceedng wth the general case, we ntroduce some notaton. When there are l frms and j workers, let E u [SS l,j ] be the expected socal surplus. In (6), the socal surplus s calculated for m frms and n workers. Snce E u f [SS m,n (0, u f )] E u [SS m 1,n ], we get E u [SS m,n (u)] E u f [SS m,n (0, u f )] E u [SS m,n (u)] E u [SS m 1,n ]. 24

25 Therefore, nstead of (6), we can prove the followng: (m n){e u [SS m,n (u)] E u [SS m 1,n (u)]} E u [SS m,n (u)] n(µ F + µ W ). (7) We prove the general case by mathematcal nducton on m. To be more clear, fx n. Snce we have assumed m n + 1, m takes values n {n + 1, n + 2,...}. The base case corresponds to m = n + 1. Base Case (m=n+1): When m = n + 1 (7) reduces to n(µ f + µ W ) E u [SS n+1,n (u)] whch we argue to be true above. Inductve Step: Suppose that (7) holds for m = n + 1, n + 2,..., l. Now we prove t for m = l + 1. Consder the dfference E u [SS n+,n (u)] E u [SS n+ 1,n (u)] for 1. Ths dfference s the extra surplus that an addtonal frm contrbutes when there are more frms than workers n the economy. Snce all the workers are matched n the effcent matchng, ths contrbuton s smaller when there are more frms. Therefore, we get: E u [SS l+1,n (u)] E u [SS l,n (u)] E u [SS l,n (u)] E u [SS l 1,n (u)]. (8) Snce (7) holds for l, we have the followng: (l n){e u [SS l,n (u)] E u [SS l 1,n (u)]} E u [SS l,n (u)] n(µ F + µ W ). Multply both sdes of (8) by l n and use the nequalty above to get: (l n){e u [SS l+1,n (u)] E u [SS l,n (u)]} E u [SS l,n (u)] n(µ F + µ W ) Addng E u [SS l+1,n (u)] E u [SS l,n (u)] to both sdes of ths equaton gves: (l + 1 n){e u [SS l+1,n (u)] E u [SS l,n (u)]} E u [SS l+1,n (u)] n(µ F + µ W ). Therefore, (7) holds and the proof s complete. 7 Interm Stablty Agents may be able to form blockng coaltons before they agree to partcpate n the mechansm but after they learn ther values n some markets. Ths can happen f partcpaton n the mechansm 25

26 s a bndng commtment and agents know ther values before partcpaton. For example, n the Natonal Resdent Matchng Program, whch matches medcal school graduates to resdency programs, the outcome of the matchng mechansm s enforced. Hence, ex ante and ex post no blockng condtons may not be the approprate notons to study for such markets. In ths secton we ntroduce an nterm no blockng noton to solve ths problem. Interm no blockng noton s harder to defne than ex ante and ex post no blockng. The man reason s that agents have asymmetrc nformaton at the nterm stage. Therefore, f a frm and a worker agree to make a sde deal, then ths gves nformaton about the frm s valuaton to the worker and smlarly, nformaton about the worker s valuaton to the frm. Wth ths new nformaton they presumably update ther estmates of expected utltes from partcpatng n the mechansm. Therefore, t may no longer be benefcal for them to do the sde deal. 9 In our defnton of nterm no blockng, agents have belef domans about each other s valuaton. Moreover, f there are two agents who nterm block a mechansm, they are able to make a sde deal wthout nformaton updatng. Another reason for the dffculty s that there s also a barganng ssue snce agents wllngness to pay depend on ther prvate nformaton. We resolve the barganng problem by notcng that n a one frm-one worker case, there s a focal ncentve compatble, ndvdually ratonal, effcent, and budget balanced mechansm. In ths mechansm, the frm and the worker get matched wthout makng any transfers. Therefore, we assume that ths s the only deal that they make. Defnton 13. A drect revelaton mechansm satsfes nterm no blockng f there exsts no par (f, w) wth non-empty neghborhoods N f and N w such that E u f [v f (u) u w N w ] < u f (w) f and only f u f N f, E u w [v w (u) u f N f ] < u w (f) f and only f u w N w. In ths defnton, N w can be thought of as the belef doman of the frm about the worker s utlty functon. Smlarly, N f s the belef doman of the worker about the frm s utlty functon. If they can block the mechansm, then ther belef domans are accurate: the frm wants to make 9 Smlar dffcultes arse n auctons wth colluson lterature as well. See for example Laffont and Martmort (1997), Laffont and Martmort (2000) and Pavlov (2008). 26