Efficient Combinatorial Exchanges

Transcription

1 CIRJE-F-826 Effcent Combnatoral Exchanges Htosh Matsushma Unversty of Tokyo November 2011 CIRJE Dscusson Papers can be downloaded wthout charge from: Dscusson Papers are a seres of manuscrpts n ther draft form. They are not ntended for crculaton or dstrbuton except as ndcated by the author. For that reason Dscusson Papers may not be reproduced or dstrbuted wthout the wrtten consent of the author.

2 1 Effcent Combnatoral Exchanges 1 Htosh Matsushma 2 Department of Economcs, Unversty of Tokyo Frst Verson: November 11, 2011 Ths Verson: November 14, 2011 Abstract We nvestgate combnatoral exchanges as a generalzaton of combnatoral auctons and blateral trades, where the multple commodtes to be traded are possessed by partcpants and a central planner as endowments. Prvate values, rsk neutralty, and ndependent types are assumed. Effcency, Bayesan Incentve Compatblty, and Interm Indvdual Ratonalty are requred. We characterze the least upper bound of the central planner s expected revenue. We ntroduce a stablty noton, namely, the margnal core, to the assumpton that the central planner s endowment s unprotected. We show that the central planner has a defct n expectaton f and only f the margnal core s non-empty. Keywords: Combnatoral Exchanges, Groves Mechansms, Outsde Opportunty, Effcent Endowment, Defct, Margnal Core JEL Classfcaton Numbers: D44, D61, D82 1 Ths study was supported by a grant-n-ad for scentfc research (KAKENHI ) from the Japan Socety for the Promoton of Scence (JSPS) and the Mnstry of Educaton, Culture, Sports, Scence and Technology (MEXT) of the Japanese government. 2 Department of Economcs, Unversty of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan. E-mal: htosh[at]e.u-tokyo.ac.jp

3 2 1. Introducton Ths paper nvestgates collectve decson problems that have ncomplete nformaton, namely, combnatoral exchanges. Combnatoral exchanges are regarded to unfy and generalze both cases of blateral trades concernng a barganng aspect of tradng, such as those nvestgated by Myerson and Satterthwate (1983), 3 and combnatoral auctons, whch have been explored by several authors such as Rassent, Bulfn, and Smth (1982), Kelso and Crawford (1982), and Ausubel and Mlgrom (2002). In the same manner as combnatoral auctons, multple heterogeneous commodtes are traded altogether such as spectrums; these commodtes are dvded nto multple packages to be allocated to partcpants (players), accordng to a specfed revelaton mechansm wth sde payments, along wth these partcpants announcements. Combnatoral auctons generally assume that the central planner (medator or government) possesses all the commodtes to be traded as hs (or her) ntal endowment. In realstc stuatons such as spectrum allocatons, however, each partcpant s valuatons of these commodtes s crucally dependent on hs valuatons of those commodtes that are possessed by other partcpants or hmself as ther ntal endowments, whch are regarded as substtutes and complements. Hence, the central planner expects to mprove welfare further by exchangng ther ntal endowments wth each other and allocatng the central planner s ntal endowment at the same tme. The framework of combnatoral exchanges does allow tradable commodtes to be possessed not only by the central planner but also by players as ther ntal endowments; each partcpant sells hs ntal endowment and purchases another package of commodtes at the same tme. 4 However, each player has the outsde opportunty not to partcpate n the collectve decson problem and nstead to consume hs ntal endowment by hmself; he could thus have sgnfcant barganng power over the central planner n ths case. Consequently, n order to mplement effcent allocatons 3 For related studes, such as double auctons, see Chatterjee and Samuelson (1983), Wlson (1985), and Matsushma (2008), for nstance. 4 For the argument about the mportance of combnatoral exchanges, see Mlgrom (2007). See also Chapter 1 of Mlgrom (2004).

4 3 n an ncentve-compatble manner, the central planner has to make consderable subsdes that fulfll ther nformatonal rents. For nstance, as Myerson and Satterthwate (1983) ponted out, n the opposng case of combnatoral exchanges such as blateral trades, where the central planner has no ntal endowment to be traded, t mght be nevtable for the central planner to have a defct n expected revenue. Ths contrasts wth the case of combnatoral auctons, whch guarantees the postvty of the central planner s expected revenue. Based on these observatons, the purpose of ths paper s to clarfy the degree of fnancal burden on the central planner when mplementng effcent allocatons n the context of combnatoral exchanges n a manner that s consstent not only wth Bayesan Incentve Compatblty (BIC) but also wth Interm Indvdual Ratonalty (IIR). IIR requres each player s nterm expected payoff to be at least the same as hs type-dependent outsde opportunty value. In partcular, the man concern of ths paper s to clarfy what s the necessary and suffcent condton under whch the central planner has a defct n expected revenue. Ths paper permts each player s consumpton to have an externalty effect on other players welfare. We assume quas-lnearty, rsk-neutralty, prvate values, and an ndependent dstrbuton of types. We also assume the payoff/revenue equvalence property n terms of a Bayesan Nash equlbrum, accordng to whch, along wth effcency, we can focus only on Groves mechansms that are consstent wth IIR. We derve the least upper bound of the central planner s expected revenue n general collectve decson problems. We then show a full characterzaton of the case that the central planner has a defct n expected revenue n the context of combnatoral exchanges from the vewpont of stablty. We ntroduce a new concept that s a weaker verson of the core, namely the margnal core, whch s defned as the collecton of all effcent mputatons that are unblocked by any coalton that conssts of all players but a sngle player,.e., are margnally unblocked. Besdes the restrcton on possble blockng coaltons, there s a substantal dfference from the standard defnton of the core n that the mputaton for the central planner s assumed to be zero n our defnton; t was assumed n our defnton that each player s ntal endowment s protected by hs prvate property rght, whle the central planner s ntal endowment s unprotected. Ths assumpton excludes an aspect

5 4 of the functonng of the competton among players, makng the non-emptness of the margnal core dffcult to be satsfed whenever the central planner possesses suffcent commodtes. Based on these observatons, we ntroduce a key condton named Effcent Endowment (EE). Ths condton mples that for each player, there s a partcular type wth whch the consumpton of hs ntal endowment by hmself s valuable to the pont that the effcent allocaton rule wll assgn t to hm, rrespectve of other players types. The condton of EE makes each player s barganng power over the central planner the strongest. Under EE, we show that the margnal core s non-empty f and only f the central planner has a defct n expected revenue. Ths characterzaton result unfes and generalzes the cases of combnatoral auctons and blateral trades. In combnatoral auctons, where the central planner possesses all commodtes, t s nevtable that the margnal core s empty, whch automatcally mples that the central planner can earn nonnegatve expected revenue. By contrast, Myerson and Satterthwate (1983) nvestgated blateral trades between a sngle seller and a sngle buyer, where the seller possesses a sngle unt of an ndvsble commodty, whle the buyer and central planner have no ntal endowments. They showed that no effcent mechansm satsfes BIC, IIR, or the balanced budgets across partcpants, mplyng that t s nevtable for the central planner to have a defct n expected revenue. The model of Myerson and Satterthwate could be regarded as an example of a specal case of combnatoral exchanges n whch the central planner possesses no ntal endowment. Under the condton of EE, t s nevtable that the margnal core s non-empty, automatcally mplyng that the central planner has a defct n expected revenue n any effcent mechansm that s consstent wth BIC and IIR. In ths case, the central planner loses the amount of money equvalent to the maxmal net expected surplus n the entre economy. Based on ths characterzaton, t s shown to be generally mpossble to make stablty n terms of the margnal core compatble wth BIC and IIR. Whenever a player possesses a suffcent ntal endowment, the excluson of hm from the collectve decson problem results n a decrease n other players welfare. By excludng ths player, they consequently lose the valuable chance to wn the commodtes that ths excluded player possessed. Ths makes the margnal core unlkely to be empty, but, at the same

6 5 tme, allows players to have sgnfcant barganng powers over the central planner, makng hs expected revenue negatve. Our characterzaton mples that the non-emptness of the margnal core s equvalent to the negatvty n the central planner s expected revenue. The standard defnton of the coaltonal game has been ntensvely consdered n prevous studes of combnatoral auctons such as Bernhem and Whnston (1986), Ausubel and Mlgrom (2002), Mlgrom (2007), Day and Raghavan (2007), and Day and Mlgrom (2008). Day and Raghavan (2007) and Day and Mlgrom (2008) nvestgated so-called core-selectng mechansms that have the advantage of stablty over a Groves mechansm; a core-selectng mechansm assgns to each type profle an mputaton that s ncluded n the core assocated wth the standard coaltonal game. These works commonly defned the stablty noton as the robustness of an mputaton n terms of the possblty of any coalton persuadng the central planner to allow ts members to consume hs endowment exclusvely. Consequently, the requrement of ths stablty mght make the central planner s revenue greater than that for a Groves mechansm. By contrast, the present paper dfferently defnes a stablty noton as the exstence of an mputaton that s robust to the possblty of any sze ( n 1) coalton consprng to steal the central planner s ntal endowment wthout hs allowance by removng the other player. The central planner s ntal endowment s assumed to be unprotected by hs property rght, and he has no means of retalatng for theft, mplyng that hs mputaton can never be postve. However, n terms of possble retalaton measures, the removed player can cancel hs partcpaton by wthdrawng hs ntal endowment from the collectve decson problem, removng the opportunty of ts exchange from all members of the coalton. Hence, any mputaton could be regarded as beng stable f any sze ( n 1) coalton hestates to conspre to steal the central planner s ntal endowment because they are afrad of the removed player s subsequent retalaton. 5 We further nvestgate a specal case where a sngle player possesses all commodtes as hs ntal endowment. Wth mnor restrctons, the central planner s expected defct s the worst of all possble dstrbutons of ntal endowments; the loss of the central planner s expected revenue n ths sngle seller case, compared wth n the 5 We note that n ths case these players cannot enjoy the postve externalty effect nduced by the removed player s consumpton.

7 6 combnatoral aucton case, could be equal to the gross surplus nduced by effcent allocatons. We further nvestgate mechansms that are not of Groves type, and show an mportant result mplyng that wth EE, the emptness of the margnal core s a necessary and suffcent condton under whch there exsts a Bayesan ncentve compatble mechansm wth Interm Indvdual Ratonalty that makes the central planner s ex post revenue nonnegatve at all tmes. Several works n the mechansm desgn lterature, such as Cremer and McLean (1985, 1988), Matsushma (1990a, 1990b, 2007), and Aoyag (1999), have nvestgated the correlated types dstrbuton. In partcular, Matsushma (2007) showed a suffcent condton for the exstence of effcent mechansms that satsfy BIC, IIR, and the balanced budgets, mplyng that the central planner s expected revenue can be nonnegatve. In contrast to these works, the present paper assumes the ndependent types dstrbuton rather than correlated types. Cramton, Gbbons, and Klemperer (1987) nvestgated the problem of dssolvng partnershps as a specal case of blateral trades; players have nearly equal shares n ther partnershp and trade these shares wth each other. They showed that the achevement of effcency can be compatble wth BIC, IIR, and the balanced budgets. Ther case, however, does not satsfy EE; the effcent allocaton rule always assgns the total share to the player who apprecates the value of ther partnershp more than does the other player, contradctng EE. Several works, such as Jehel and Moldovanu (1996), Jehel, Moldovanu, and Stacchett (1999), and Fgueroa and Skreta (2009), have nvestgated auctons that have externalty. Fgueroa and Skreta (2009) nvestgated a sngle-unt aucton that has externalty where each player s outsde opportunty depends on hs type. They assumed that the central planner can make a bndng commtment to make neffcent allocatons as a devce for threatenng any player who consders not partcpatng n ths aucton. In contrast to ther work, the present paper does not allow any such commtment devce; t s assumed that the central planner nvarably mplements effcent allocatons for the members who actually partcpated, regardless of whether a partcular player decded not to partcpate. The organzaton of the remander of ths paper s as follows. Secton 2 descrbes a

8 7 basc model for general collectve decson problems and demonstrates a calculaton method for the least upper bound of the central planner s expected revenue. Secton 3 explans combnatoral exchanges and EE, and descrbes a tractable characterzaton of the least upper bound. Secton 4 descrbes a man theorem that under EE, the non-emptness of the margnal core s necessary and suffcent for the central planner s defct n expected revenue. Secton 5 consders specal cases such as blateral trades, combnatoral auctons, and sngle seller cases. Secton 6 gves several dscussons about equal endowment dstrbutons, ncompatblty wth stablty, ex post revenue and defct, and an ssue concernng complexty and prvacy. Secton 7 concludes.

9 8 2. The Basc Model Let us consder a collectve decson problem that has ncomplete nformaton n the followng manner. Let N { 1,2,..., n} denote the fnte set of players (traders or agents), where n 2. Each player N has a type that s unknown to ether other players or the central planner (medator or government), where set of possble types for player. Let N and jn /{} denotes the j. The types are ndependently dstrbuted across players accordng to a probablty measure that have the full support of. Let A denote the set of all alternatves that have typcal element a. Each player s payoff functon has a quas-lnear and rsk-neutral form wth prvate values,.e., v(, a ) t, where t R denotes the monetary transfer from the central planner to hm and v : A A s hs type-dependent valuaton functon for the alternatves. For every N, let U : R denote player s ' outsde opportunty functon, where the outsde opportunty for player that has type s gven by U ( ) R, mplyng the nterm expected payoff that he can receve when he does not partcpate n the collectve decson problem. Let UN ( U ) N. A drect mechansm s defned as ( f, x ), where f : A s an allocaton rule, n x : R s a payment rule, x ( ), and x : R. When each player N x N announces, the central planner selects the alternatve f ( ) A and makes the transfer payment to each player,.e., x ( ) R, where we denote n ( ) and x( ) ( x( )) R. We assume that the allocaton rule f s N N effcent n the sense that for every, the correspondng allocaton f ( ) A maxmzes the sum of players valuatons n the ex-post term,.e., v( f( ), ) max v( a, ). N aa N In order to make the collectve decson problem non-trval, we assume that

10 9 (1). 6 E v f E U [ ( ( ), )] [ ( )] 0 N N Ths assumpton mples that the effcent allocaton rule f nduces a postve net expected surplus n the ex-ante term, whch s expressed by the left-hand sde of (1), mplyng the dfference between the expected aggregate value nduced by the allocaton rule,.e., E[ v( f( ), )], and the sum of players expected outsde opportuntes, N.e., E U. [ ( )] N BIC (Bayesan Incentve Compatblty): A drect mechansm ( f, x ) satsfes BIC f for every N, every, and every m, Ev [ ( f( ), ) x( ) ] Ev [ ( f( m, ), ) x( m, ) ]. IIR (Interm Indvdual Ratonalty): A drect mechansm ( f, x ) and a profle of outsde opportunty functons U N satsfy IIR f for every N and every, Ev f x U. [ ( ( )), ) ( ) ] ( ) BIC mples that truth-tellng s a Bayesan Nash equlbrum n the collectve decson problem. IIR mples that each player has the ncentve to partcpate n the collectve decson problem, rrespectve of hs type. The revenue for the central planner s defned as the sum of the transfers from all players to the central planner,.e., x ( ). Gven the effcent allocaton rule f, N the central planner s purpose s to desgn a payment rule x such that the assocated drect mechansm ( f, x ) maxmzes hs ex-ante expected revenue E[ x ( )] under the constrants of effcency, BIC, and IIR. We mplctly assume that the central planner s preference has a lexcographc order n the sense that the achevement of 6 E[] denotes the ex-ante expectaton operator n terms of. E[ ] denotes the nterm expectaton operator n terms of condtonal on. N

11 10 effcency s the frst am and revenue maxmzaton s the second am. Let X denote the set of all payment rules. A payment rule x X s sad to be a Groves payment rule for an effcent allocaton rule f f there exsts a functon h : R for each N such that x ( ) v ( f( ), ) h( ) for all. j j jn\{ } Let X( f ) X denote the set of all Groves payment rules for f. In the mechansm desgn lterature, a drect mechansm ( f, x ) s called a Groves mechansm f and only f x X ( f ). It s evdent that any Groves mechansm ( f, x ) satsfes strategy-proofness n the sense that for every N and every, v( f( ), ) x( ) v( f( m, ), ) x( m, ) for all m. 7 It s evdent that strategy-proofness mples BIC. Ths paper mplctly assumes the payoff equvalence property 8 n that for every payment rule x X, f ( f, x ) satsfes BIC, then there exsts a Groves payment rule y X ( f ) that nduces the same nterm expected values of transfer payment,.e., satsfes that for every N, Ex [ ( ) ] Ey [ ( ) ] for all. Ths also mples the revenue equvalence property n that. N N E[ x( )] E [ y ( )] Hence, we confne our attenton to the subset X( f ). Let us denote by X fu (, N ) X( f) the set of all Groves payment rules x X ( f ) such that ( f, x ) and U N satsfy IIR. Let us denote by r r U R ( ) 0 0 N the least upper bound of the expected revenue for the central planner under the 7 See Vckrey (1961), Clarke (1971), Groves (1973), Green and Laffont (1977), Holmstrom (1979), and Mlgrom (2004). A Groves mechansm s sometmes called a VCG (Vckery Clarke Groves) mechansm. 8 Krshna and Maenner (2001) showed mld condtons such as convexty and regular Lpschtzan that are suffcent for the payoff equvalence property n a broad class of envronments that have multdmensonal types. See also Krshna and Perry (2000), Mlgrom and Segal (2002), and Bkhchandan et al. (2006). For works related to multdmensonal types, see also Rochet and Stole (2003) and Pavan, Segal, and Tokka (2011).

12 11 constrants of x X f U N (, ); r max E [ x ( )] 0. xx( f, UN ) N The followng theorem characterzes ths least upper bound. Theorem 1: It holds that (2). r n E v f U E v f 0 ( 1) [ ( ( ), )] max{ ( ) [ j( ( ), j) ]} N N jn Proof: Let us consder an arbtrary Groves payment rule wth IIR, x X f U N (, ). For every N and every, Ev [ ( f( )), ) x( ) ] Ev [ ( f( ), ) v( f( ), ) h( ) ] j j j jn\{ } E[ vj( f( ), j) ] E[ h( )]. jn Hence, IIR s equvalent to the nequaltes gven by Eh U E v f for all N. [ ( )] max{ ( ) [ j( ( ), j) ]} jn Hence, Ex [ ( ) ] E[ v( f( ), ) ] j j jn\{ } Ths mples that for every Hence, t follows that for all N. max{ U ( ) E[ vj( f( ), j) ]} jn x X f U N (, ), E[ x( )] ( n1) E[ v( f( ), )] N N. max{ U ( ) E[ vj( f( ), j) ]} N jn. r n E v f U E v f 0 ( 1) [ ( ( ), )] max{ ( ) [ j( ( ), j) ]} N N jn For every N, let us specfy h n a manner that

13 12 Eh U E v f, [ ( )] max{ ( ) [ j( ( ), j) ]} jn that s,. Ex [ ( )] E[ v ( f( ), ) ] max{ U ( ) E[ v ( f( ), ) ]} j j j j j N\{ } jn It s clear that the specfed payment rule x satsfes IIR, and E[ x ( )] ( n1) E[ v ( f( ), )] max{ U ( ) E[ v ( f( ), ) ]} N, j j N N jn whch mples (2). Q.E.D. From Theorem 1, t follows that for every N and every par of profles of outsde opportunty functons ( N, N) U U, f then t holds that U ( ) U ( ) for all N and all, r ( U ) r ( U ); the hgher players outsde opportuntes are, the 0 N 0 N lesser the central planner s expected revenue s. The proof of Theorem 1 showed that whenever a Groves payment rule x X ( f ) nduces the least upper bound of the central planner s expected revenue, the correspondng ex-ante expected payoff for each player N, denoted by r r( UN), s gven by (3). r E v f U E v f [ ( ( ), )] max{ ( ) [ j( ( ), j) ]} N jn

14 13 3. Combnatoral Exchanges From ths secton on, we focus on combnatoral exchanges as a specal case of the collectve decson problem, n whch players (moble phone companes, for nstance) and the central planner (e.g., the government) possess multple commodtes (spectrums) as ther ntal endowments and trade these objects altogether wth each other at the same tme. 3.1 The Model There exst L heterogeneous tems. For each l {1,..., L}, the total amount of the l l th tem to be traded s gven by a postve nteger e 0. Let e( e l ) L L 1 R. We specfy the set of all alternatves A as the set of all nl dmensonal vectors of nonnegatve ntegers a ( ) satsfyng that for every l {1,..., L}, a N l l l a e, and 0 N a for all N, where we denote a ( l ) L a l Let a ( a ) N A and a ( l ) L a l 1, where a l R mples the amount of the l th tem that s allocated to player. Let us denote S f( ) ( f ( )) N. For every non-empty subset S N, we denote a ( a) R. S S l L Let an L dmensonal vector of nonnegatve ntegers e ( e ) 1 0 denote the l ntal endowment for player N. Let us denote by e ( ) the profle of ther ntal endowments, where we assume that l l e e for all l {1,..., L}. N N e N l l Note that the central planner possesses e e 0 amount of the l th tem for N l 9 It s an rrelevant assumpton that the set of alternatves s dscrete. We can make the same arguments even f we replace t wth a subset of multdmensonal Eucldean space. For the case of dvsble commodtes, see Wlson (1979) and Back and Zender (1993), for nstance. 10 We can make bascally the same argument even f we specfy A as a non-empty proper subset of such nl dmensonal vectors.

15 14 each tem l {1,..., L} as hs ntal endowment; the ntal endowments possessed by the players and central planner are traded altogether. For every non-empty subset (coalton) S N, let us defne a subset A( S) A as the set of all alternatves a A such that a e for all S, mplyng that any player who belongs to the coalton S, S, does not partcpate n the collectve decson problem and consumes (or utlzes) hs ntal endowment e by S hmself. Let us specfy a functon f : N\ S AS ( ), whch s regarded as the effcent allocaton rule for the dfference coalton N \ S, n a manner that for every, N \ S N\ S v( f ( ), ) max v( a, ), S N\ S aa( S) N\ S N\ S where we denote N \ S and NS \ ( ) NS \ NS \. Accordng to N\ S f S, the central planner mplements effcent allocatons for partcpants,.e., players who belong to N \ S, provded that non-partcpants,.e., players who belong to S, consume ther respectve ntal endowments. We permt each player s consumpton to have an externalty effect on other players welfare; v(, a ) depends on non-decreasng wth respect to (4) We assume that a. 11 We also assume free dsposal n that v(, a ) s a. {} U( ) E[ v( f ( ), ) ] for all N and all. Each player has the outsde opportunty not to partcpate n the collectve decson problem and nstead to consume hs ntal endowment e by hmself; n ths case, the central planner allocates the remanng commodtes e e n order to maxmze the sum of other players (partcpants ) expected payoffs and hs expected revenue. Under Assumpton (4), we can rewrte r 0 r 0 ( e N ) and r r( en) nstead of r r U N 0 ( ) 0 11 It s mplct n ths paper to assume that the market for the players after the combnatoral exchange s well regulated so that these players aggregate welfare s postvely correlated to the total surplus ncludng consumers welfare. Secton 7 gves further dscussons.

16 15 and r r( UN). Theorem 2: It holds that (5) and for each N, (6) r n E v f E v f 0. {} ( 1) [ ( ( ), )] [ j( ( ), j)] N N jn\{ }. r E v f E v f {} [ ( ( ), )] [ j( ( ), j)] N jn\{ } Proof: From (4) and the defnton of U ( ) E[ v ( f( ), ) ] j j jn U ( ) E[max vj( a, j) ] aa j N U ( ) E[max v( a, ) ] aa({ }) N\ S U ( ) E[ v ( f ( ), ) ] { } j j jn E v f, {} [ j( ( ), j) ] jn\{ } {} f, t follows that for every, whch along wth Theorem 1 and the arguments n Secton 2 mples (5) and (6). Q.E.D Effcent Endowment A key condton for ths paper, EE (Effcent Endowment), can be descrbed as follows. EE (Effcent Endowment): For every N, there exsts such that f (, ) e for all. EE mples that for each player N, there s a partcular type wth

17 16 whch the consumpton of hs ntal endowment e by hmself s valuable to the pont that the effcent allocaton rule f assgns t to hm, rrespectve of other players types. EE excludes the case of the dssoluton of partnershps nvestgated by Cramton, Gbbons, and Klemperer (1987); the achevement of effcency was compatble wth BIC, IIR, and the balanced budgets n ther case. Under EE, we can replace Theorem 1 and ts related arguments wth the followng theorem, demonstratng a full characterzaton of the least upper bound of the central planner s expected revenue and the correspondng ex-ante expected payoff for each player. Theorem 3: Under EE, t holds that (7) and for each N, (8) r n E v f E v f 0, {} ( 1) [ ( ( ), )] [ j( ( ), j)] N N jn\{ }. r E v f E v f {} [ ( ( ), )] [ j( ( ), j)] N jn\{ } Proof: EE, along wth the effcency of f, mples that E v f E v f, {} [ j( ( ), j) ] [ j( ( ), j)] jn jn whch along wth the proof of Theorem 2 mples that U E v f max{ ( ) [ j( ( ), j) ]} jn E v f. {} [ j( ( ), j)] jn\{ } Ths along wth (2) and (3) mples (7) and (8). Q.E.D. The gross surplus nduced by the effcent allocaton n the economy wthout player,.e., the value of jn\{ } v f {} j( ( ), j), can be regarded as player s ' barganng power over the central planner; the larger hs ntal endowment s, the lesser the gross surplus wthout hm s.

18 17 Wthout EE, t mght be the case that the least upper bound r 0 s greater than the rght-hand sde of (5). EE,.e., the presence of a partcular type for each player wth whch the consumpton of hs ntal endowment e by hmself s suffcently valuable, makes each player s barganng power over the central planner the strongest,.e., makes the central planner s fnancal burden caused by the nformatonal ncompleteness on players types the heavest.

19 18 4. Unprotected Endowment and Margnal Core Ths secton demonstrates a full characterzaton of the case that the central planner has a defct n expected revenue from the followng vewpont of stablty. Let us defne N the coaltonal game :2 \{ } R as assgnng to each proper coalton S N the maxmal expected gross surplus n the economy wthout all players who belong to N \ S,.e., where we must note that N S E v f for all S 2 \{ }. N\ S ( ) [ j( ( S), j)] js ( N) E[ v ( f( ), )]. jn j j Theorem 4: It holds that r0 ( n1) ( N) ( N \{}), and for every N, N r ( N) ( N \{}). Under EE, t holds that r0 ( n1) ( N) ( N \{}), and for every N, N r ( N) ( N \{}). Proof: From the defnton of, Theorem 2, and Theorem 3, t s clear that ths theorem s correct. Q.E.D. n Let us call any n dmensonal vector ( ) N R an mputaton, where mples player s ' ex-ante expected payoff. It s mplctly assumed that any mputaton assgns to the central planner zero expected revenue. We defne the margnal core as the collecton of all mputatons satsfyng that

20 19 (9) ( N), N and for every N, (10) ( N \{}). S The margnal core mples the collecton of all effcent mputatons that are margnally unblocked,.e., unblocked by any sze ( n 1) coalton. The gven defnton of a coaltonal game s substantally dfferent from the standard defnton n related studes 12 because the mputaton for the central planner s assumed to be zero n our defnton. Ths assumpton excludes the effect of players competng wth each other over the central planner s ntal endowment on the stablty of allocaton, threatenng the non-emptness of the margnal core. An nterpretaton of the coaltonal game and margnal core follows. The ntal endowment of each player N,.e., e, s protected by ths player s prvate property rght, whle the ntal endowment of the central planner,.e., e e, s unprotected; any sze ( n 1) coalton can conspre to steal the central planner s ntal endowment by removng the other player. In terms of possble retalaton measures, ths removed player can cancel hs partcpaton by wthdrawng hs ntal endowment from the collectve decson problem, removng the opportunty of ts exchange from all members of the coalton. Hence, any mputaton can be regarded as beng stable f any sze ( n 1 ) coalton hestates to conspre to steal the central planner s ntal endowment because they are afrad of the removed player s subsequent retalaton. The more the central planner possesses hs ntal endowment, the more lkely the margnal core s to be empty. It s evdent that the margnal core s empty whenever the central planner possesses all commodtes as hs ntal endowment, as shown n the next secton. By contrast, the margnal core can be non-empty f the central planner s ntal endowment s suffcently small. N 12 See Bernhem and Whnston (1986), Ausubel and Mlgrom (2002), and Mlgrom (2007). For a defnton of core-selectng mechansms, see Day and Raghavan (2007) and Day and Mlgrom (2008).

21 20 Theorem 5: The margnal core s non-empty f and only f (11) ( n1) ( N) ( N \{}). N Proof: Suppose that the margnal core s non-empty. Then, there exsts satsfes (9) and (10). Then, 1 ( N) j N n 1 N jn\{ } 1 ( N \{}) n, 1 N n R that whch mples (11). n Suppose that (11) holds. Then, there exsts ( ) NR satsfyng that (12) j ( N \{}) for all N. jn\{ } n Let us specfy ( ) by N R ( N) j jn for all N. n It s evdent that satsfes (9). From (11) and (12), t follows that ( N) j 0, jn and therefore, for all N, whch along wth (12) mples (10). Hence, the margnal core s non-empty. Q.E.D. The followng theorem demonstrates the full characterzaton as the man result of ths paper; under EE, the non-emptness of the margnal core s necessary and suffcent for the central planner to have a defct n expected revenue. Theorem 6: If r0 0, then the margnal core s non-empty. Under EE, the margnal core s non-empty f and only f r0 0. Proof: From Theorem 4, t s evdent that r0 0 mples (11), and that under EE,

22 21 r0 0 s equvalent to (11). Ths observaton along wth Theorem 5 mples that ths theorem s correct. Q.E.D.

23 22 5. Specal Cases 5.1. Blateral Trades Let us consder an example of combnatoral exchanges, namely blateral trades, n whch we assume that n 2 and that the central planner possesses no ntal endowment,.e., e1 e2 e. The model of Myerson and Satterthwate (1983) s a specal case. They addtonally assumed a sngle object wth a sngle unt; EE automatcally holds when the type spaces are the same between the seller and buyer. In ths blateral trades case, ( N \{}) E[ U ( )] for each {1, 2}, whch along wth Theorem 4 mples that under EE, r0 E v f E U. [ ( ( ), )] [ ( )] {1,2} {1,2} Because of (1), under EE, t s nevtable that the central planner has a defct n expected revenue; the central planner loses the amount of money equvalent to the maxmal net expected surplus n the entre economy Combnatoral Auctons Let us specfy a profle of ntal endowments en en ( e ) N by e 0 for all N, whch corresponds to combnatoral auctons n whch the central planner possesses all commodtes as hs ntal endowment. Ths subsecton makes an assumpton that restrcts the postvty of the externalty effect n such a weak manner that for every N, v f v f, where e 0. {} j( ( ), j) j( ( ), j) jn\{ } jn\{ } Ths assumpton mples that players prefer to exclude a sngle player and consume all

24 23 commodtes by themselves. We show that the central planner can earn a postve expected revenue as follows. Snce t follows that A( S) whch mples that under EE, A for all S N, ( N \{}) max v ( a, ) for all N, aa: a 0 j N \{ } (13) r0 r0( e N ) E[ vj( f( ), j) max v( a, j)]. j aa: N jn\{ } a 0 jn\{ } The assumpton of ths subsecton mples that the rght-hand sde of (13) s postve Sngle Seller Let us specfy another profle of ntal endowments e ˆ ( ˆ N en e) N by ê1 e, and eˆ 0 for all N \{1}, whch mples that player 1 possesses the entre commodtes as hs ntal endowment. Ths subsecton makes an assumpton that restrcts the externalty effect n such a weak manner that for every N, every, and every a A, player s valuaton of the null package equals zero at all tmes,.e., v( a, ) 0 f a 0. We show that under EE, t s nevtable that the central planner has a defct n expected revenue as follows. Snce AN ( \{1}), and A( N \{}) A t follows from the assumpton of ths subsecton that Hence, {1} v( f ( 1), ) 0 for all N \{1}. ( N \{1}) 0, and for every N \{1}, ( N \{}) max v ( a, ), aa: a 0 j N \{ } j for all N \{1},

25 24 mplyng that under EE,, (14) r ˆ 0 r0( en ) E[ vj( f( ), j) max v( a, j)] aa: N\{1} jn a 0 jn\{ } ( n1) ( N) ( N \{}), N\{1} whch s negatve because f s effcent. The nterm expected payoff for player 1 wth each type 1 can be gven by (15) Ev1 f 1 x1 1 E vj f j 1 jn [ ( ( ), ) ( ) ] [ ( ( ), ) ], and the nterm expected payoff for each player N \{1} wth each type can be gven by (16) Ev [ ( f( ), ) x( ) ]. E[ v ( f( ), ) max v ( a, ) ] j j j aa: jn a 0 jn\{ } From (15), the nterm expected payoff for player 1 s equvalent to the maxmal gross expected surplus n the entre economy. Hence, player 1 prefers to nvte potental buyers to the collectve decson problem as many tmes as s possble. From (14) and (16), t follows that the least upper bound of the central planner s expected revenue s equvalent to the sum of the expected payoffs for the players other than player 1. The central planner s expected revenue does not necessarly ncrease as the number of players who partcpate n the collectve decson problem ncreases. The central planner mght not thnk postvely about nvtng new traders to the collectve decson problem. Wth the assumptons made n ths and prevous subsectons, t follows from (13) and (14) that under EE, re ( ) re ( ˆ ) E[max v( a, )], N N j j aa j N \{1} mplyng that by gvng all commodtes to player 1 grats, the central planner must suffer a decrease of E[max vj( a, j)] n expected revenue; the central planner aa j N \{1} loses the amount of money equvalent to the maxmal gross surplus n the combnatoral aucton that does not have player 1.

26 25 6. Dscussons 6.1. Equal Endowment Dstrbuton Ths subsecton assumes that the sum of the maxmal gross surpluses n economes wthout any sngle players, gven by, W e N E v f {} ( N) ( \{}) [ j( ( ), j)] N N jn\{ } s convex wth respect to e N. Ths assumpton can be mpled by the concavty of v(, a ) wth respect to a, provded that there s no externalty effect. It s evdent from ths assumpton that re ( N ) re ( N) en en r( ), 2 2 ndcatng that n symmetrc models of combnatoral exchanges, the central planner s defct can be suppressed when players ntal endowments are equally dstrbuted compared wth the case that the dstrbuton of the ntal endowment s dvded between few people such as the sngle-seller case. The central planner s expected defct n the sngle seller case mght be the worst of all possble dstrbutons of ntal endowments Incompatblty wth Stablty We can show that rrespectve of the profle of ntal endowments, the central planner cannot earn nonnegatve revenue n expectaton n a compatble manner wth stablty. Theorem 7: Under EE, there exsts no payment rule x X f U N (, ) such that (17) E[ x ( )] 0, and (18) N E[ { vj( f( ), j) xj( )}] ( N \{}) for all N. jn\{ }

27 26 Proof: From Theorem 6, t follows that under EE, whenever the margnal core s non-empty, then there s no x X f U N (, ) that satsfes (17). Ths mples that f a payment rule x X f U N (, ) satsfes (17), then t never satsfes (18), mplyng that Theorem 7 s correct. Q.E.D. Theorem 7 mples the general mpossblty n that under EE, rrespectve of the profle of ntal endowments, no margnally unblocked mputaton s nduced by any Groves mechansm that satsfes IIR and the nonnegatvty of the central planner s expected revenue. Ths supports the statement that partcpants cannot generally accomplsh effcency n voluntary manners Ex-Ante Reallocaton Ths subsecton descrbes an aspect of the relatonshp between an arbtrary par of profles of ntal endowments, e N and e N. Let us denote by U ( ) and U ( ) the outsde opportuntes for player wth type assocated wth hs ntal endowments e N and e N, respectvely. Suppose that a payment scheme x X f U N (, ) nduces the least upper bound r ( ) 0 e N. For every N, let us specfy a real number d d( en, en) R by d U E v f max{ ( ) [ j( ( ), j) ]} jn U E v f. By usng ths specfed vector ( d ) max{ ( ) [ j( ( ), j) ]} jn n a manner that for every N and every, x( ) x( ) d. n N R, let us specfy another payment rule x X It s evdent that x belongs to X fu and nduces the least upper bound re ( ). (, N ) N

28 27 We can nterpret the above observatons as follows. Suppose that each player N possesses e as hs ntal endowment. At the pre-play stage, the central planner collects all commodtes possessed by players and then reallocates these collected commodtes as well as the commodtes that the central planner possesses to each player by gvng e and the fxed amount of money d. After reallocatng n ths manner, the central planner enforces the Groves mechansm ( f, x ). The Groves mechansm ( f, x ) assocated wth the profles of players ntal endowments e N s the payoff/revenue equvalent to the Groves mechansm ( f, x ) that follows the replacement of player N. e wth e accompaned wth a type-ndependent payment d to each 6.4. Ex Post Revenue and Defct We must note that even f the central planner s expected revenue s nonnegatve, t mght be the case that there exsts a type profle at whch the central planner has a defct n the ex post term. We, however, can easly suppress ths trouble by allowng the central planner to make an opton contract wth a rsk-neutral thrd party n a manner that whenever players announce any type profle, then the central planner gves ths thrd party an amount of money gven by. N N E[ x ( )] x ( ) Accordng to ths contract, the central planner s revenue s kept constant across possble type profles,.e., equal to E[ x ( )] at all tmes. Hence, wth the avalablty of N opton contractng, Theorem 6 also mples a characterzaton of the case that the central planner s ex-post revenue s nonnegatve at all tmes. The central planner can earn nonnegatve revenue n the ex post term at all tmes f the margnal core s empty. Under EE, the central planner can earn nonnegatve revenue n the ex post term at all tmes f and only f the margnal core s empty. It would be more mportant to note that wth the assumpton of payoff/revenue equvalence property, we can construct a non-groves-type mechansm that s Bayesan

29 28 ncentve compatble, s nterm ndvdually ratonal, and makes the central planner s revenue nonnegatve at all tmes f and only f the least upper bound of the central planner s expected revenue s nonnegatve. Theorem 8: There exsts a payment rule x X such that ( f, x ) satsfes BIC, ( f, x ) and U N satsfy IIR, and x ( ) 0 for all. N f and only f r0 0. Proof 13 : It s evdent from the revenue/payoff equvalence property that the only f part s correct. All we have to do s to prove the f part. Suppose that r0 0. Hence, there exsts a Grove payment rule wth IIR, x X f U N (, ), such that r0 E[ x ( )]. N From Arrow (1979) and d Aspremont and Gérard-Varet (1979), t s evdent that there exsts a payment rule x X such that ( f, x ) satsfes BIC and the balanced budgets,.e., x ( ) 0 for all. N From the payoff/revenue equvalence property, t s evdent that there exsts a n-dmensonal vector ( b) Note that n N R such that Ex [ ( ) ] Ex [ ( ) ] b for all N and all. b r0. N We specfy another payment rule x X by x( ) x( ) b for all N and all. It s evdent from ths specfcaton that ( f, x ) satsfes BIC, ( f, x ) and U N 13 The proof of Theorem 8 s closely related to Krshna and Perry (1998). See also Chapter 5 of Krshna (2010).

30 29 satsfy IIR, and x ( ) b r0 0 for all. N N Q.E.D. Even wthout the avalablty of opton contractng, t follows from Theorems 6 and 8 that the central planner can earn nonnegatve revenue n the ex post term at all tmes f the margnal core s empty. Under EE, the central planner can earn nonnegatve revenue n the ex post term at all tmes f and only f the margnal core s empty Complexty and Prvacy The present paper has nvestgated drect mechansms n whch each player reveals full nformaton about hs entre valuatons. However, drect mechansms have been crtczed from the practcal vewponts concernng complexty and prvacy. 14 In combnatoral auctons that have no externalty, several authors have attempted to replace the standard practce of such drect revelatons wth less complcated and more prvacy-preserved dynamcal protocols such as smultaneous ascendng/descendng clock (Japanese) auctons. 15 In such auctons, the auctoneer contnues to ask and adjust non-anonymous and non-lnear prce vectors to each player (buyer), and each player contnues to make hs demand correspondences as a prce taker. Such protocols must collect suffcent nformaton n order to acheve the allocatons of the orgnal drect mechansm whle preservng players prvacy. In combnatoral auctons that have no externalty, Lahae and Parkes (2004), Parkes (2006), and Mshra and Parkes (2007) have ntroduced the concept of a unversal compettve equlbrum, whch mples the compettve equlbrum propertes not only n the entre economy but also n economes wthout a sngle buyer. These studes showed that a pvot mechansm, whch s defned as a specal verson of a Groves 14 See Rothkopf, Tesberg, and Kahn (1990), Segal (2006), Ausubel and Mlgrom (2006), and Parkes (2006), for nstance. 15 See Kelso and Crawford (1982), Bkhchandan and Mamer (1997), Gul and Stacchett (1999, 2000), Parkes and Ungar (2002), Ausubel and Mlgrom (2002), Ausubel and Cramton (2004), Lahae and Parkes (2004), Ausubel (2004, 2006), Hatfeld and Mlgrom (2005), Ausubel, Cramton, and Mlgrom (2006), Parkes (2006), Mshra and Parkes (2007), and Matsushma (2011), for nstance.

31 30 mechansm, can be mplemented by an arbtrary dynamcal protocol that always collects suffcent nformaton n order to dscover a unversal compettve equlbrum. Subsequently, Matsushma (2011) showed a tractable method for explanng whether an arbtrary dynamcal protocol can mplement a pvot mechansm and clarfyng ts degree of prvacy preservaton. Ths subsecton brefly shows that the above arguments can be extended to combnatoral exchanges that have no externalty, where v(, a ) s assumed to be ndependent of a ; thus, we wrte v( a, ) nstead of v(, a ). Let us specfy a Groves payment rule x x X( f ) by h( ) E[ max v ( a, )] for all N j j j aa({ }) N jn\{ } and. The correspondng drect mechansm ( f, x ), whch can be called a pvot mechansm, satsfes strategy-proofness as well as ex-post ndvdual ratonalty n the sense that for every N and every, v f x U. ( ( ), ) ( ) ( ) Ths nequalty holds wth equalty whenever f( ) e. Let us denote p : A R and p ( ) p N, the latter of whch s called a prce vector. A prce vector p s sad to be a unversal compettve equlbrum for f there exst a j a A j ({ }) for each j N, such that for every a A, p a, N ( ) p( a) N v a p a v a p a for all N, (, ) ( ) (, ) ( ) for every j N and every a A({ j}), and p a p a, j ( ) ( ) N\{ j} N\{ j} A, and Note that v a p a v a p a for all N \{ j}. j j (, ) ( ) (, ) ( ) v a for all a A, N (, ) v( a, ) N

32 31 and for every j N, mplyng that a v a v a for all a A({ }), j (, ) (, ) N\{ j} N\{ j} A s an effcent allocaton n the entre economy and j a A s an effcent allocaton n the economy that does not have player j. In the same manner as descrbed by Lahae and Parkes (2004), Parkes (2006), and Mshra and Parkes (2007), t can thus be shown that wthout externalty, the pvot mechansm can be mplemented usng an arbtrary dynamcal protocol f and only f ths protocol always dscovers a unversal compettve equlbrum. We can also extend the argument of Matsushma (2011) to combnatoral exchanges. Matsushma (2011) ntroduced the concept of the representatve valuaton functon for each player, whch assgns the mnmal relatve valuaton to each package that has been revealed durng the hstory of play. Ths representatve valuaton functon can easly be calculated from the hstory of play by makng mnor assumptons such as revealed preference actvty rules and connectedness and by descrbng the degree of players prvacy preservaton. In the same manner as shown n Matsushma (2011), n combnatoral exchanges that do not have externalty, the pvot mechansm can be mplemented by an arbtrary dynamcal protocol f and only f () there always exst effcent allocatons n the entre economy and n economes that do not have sngle players assocated wth the calculated representatve valuaton functon profle and () the packages that compose these allocatons have all been revealed durng the hstory of play. We can also show that whenever the dynamcal protocol mplements the pvot mechansm, the resultng representatve valuaton functon profle can be the unversal compettve equlbrum for ther true types.

33 32 7. Concluson and Future Researches The present paper nvestgated combnatoral exchanges that have ncomplete nformaton, n whch, n contrast to standard combnatoral auctons, the multple heterogeneous commodtes to be traded are possessed not only by the central planner but also by players as ther ntal endowments. Each player thus has the outsde opportunty not to partcpate n the collectve decson problem and nstead to consume hs ntal endowment by hmself. Accordng to the payoff/revenue equvalences assumpton, we focused on Groves mechansms that are compatble wth IIR. Compared wth standard combnatoral auctons, players n combnatoral exchanges that possess non-neglgble ntal endowments can have sgnfcant barganng powers over the central planner, makng t dffcult for the central planner to earn nonnegatve expected revenue. We ntroduced the key condton of EE, whch mples that for each player, there s a partcular type wth whch the consumpton of hs ntal endowment by hmself s valuable to the pont that the effcent allocaton rule wll assgn t to hm, rrespectve of other players types. Under EE, each player s barganng power over the central planner s at ts strongest. Accordng to the standard calculaton for Groves mechansms, we characterzed the least upper bound of the central planner s expected revenue. Subsequently, from the vewpont of stablty, we showed a full characterzaton of the case that the central planner had a defct n expected revenue. The margnal core, whch was defned as the collecton of all effcent mputatons across players that are margnally unblocked by any sze ( n 1) coalton, s empty f and only f the central planner can earn nonnegatve expected revenue. Based on ths characterzaton, t was shown to be generally mpossble to make stablty n terms of the margnal core compatble wth BIC and IIR. Whenever a player possesses a suffcent ntal endowment, the excluson of ths player from the collectve decson problem results n a decrease n other players welfare; by excludng ths player, they consequently lose the valuable chance to wn the commodtes that ths excluded player possessed. Ths makes the margnal core unlkely to be empty, but, at the same tme, allows players to have sgnfcant barganng powers over the central planner, makng hs expected revenue negatve. Our characterzaton mples that the

34 33 non-emptness of the margnal core s equvalent to the negatvty n the central planner s expected revenue. Ths paper assumed that the central planner s preference followed a lexcographc order n that the achevement of effcency s the frst am and revenue enhancement s the second am. Future research mght wsh to elmnate ths assumpton and nstead nvestgate the possblty that the central planner can ncrease hs expected revenue further by employng neffcent mechansms. Ths research avenue s needed n order to provde an nsght nto desgnng protocols for combnatoral exchanges that are optmal n terms of the central planner s expected revenue (see also Myerson (1981) and Rley and Samuelson (1981)). In realstc stuatons, the central planner should be constraned by the fact that all commodtes are sold out to thrd partes. It mght be also practcally mportant to consder the manner n whch the central planner collects the proporton of the ntal endowments possessed by players n the pre-play stage n order to reduce the defct at the expense of effcency. As footnote 10 ponted out, ths paper has mplctly assumed that the market for players after the combnatoral exchange s well regulated so that these players aggregate welfare s postvely correlated wth total surplus ncludng consumer welfare. Wthout ths assumpton, we would need to be more cautous about the central planner s objectve functon concernng both the central planner s revenue and total surplus (.e., consumer surplus as well as player welfare). Ths s another reason why future research should nvestgate neffcent allocaton rules. Ths paper has nvestgated only statc models of combnatoral exchanges. Thus, a hghly promsng future research avenue would be to extend our model to dynamcal contexts where players receve prvate nformaton over tme, where the populaton of partcpants could change over tme, and where commodtes could be resold. For related works, see Parkes and Sngh (2003), Bergemann and Välmäk (2010), and Bergemann and Sad (2010), for nstance. Fnally, t must be noted that ths paper assumed that players are fully ratonal and requre drect mechansms to satsfy BIC so that truth-tellng s exactly the best response for any player. Accordng to the semnal works of Parkes, Kalagnanam, and Eso (2002), Day and Mlgrom (2008), and Erdl and Klemperer (2010), future research mght am to weaken ths ratonalty assumpton and nstead nvestgate the case that the central