A Noncooperative View of Consistent Bankruptcy Rules*

Transcription

1 Ž. GAMES AND ECONOMIC BEHAVIOR 18, ARTICLE NO. GA A Noncooperatve Vew of Consstent Bankruptcy Rules* Nr Dagan Department of Economcs, Unerstat Pompeu Fabra, Barcelona, Span and Roberto Serrano and Oscar Volj Department of Economcs, Brown Unersty, Prodence, Rhode Island Receved February 7, 1995 We ntroduce a game form that captures a noncooperatve dmenson of the consstency property of bankruptcy rules. Any consstent and monotone rule s fully characterzed by a blateral prncple and consstency. Lke the consstency axom, our game form, together wth a blateral prncple, yelds the correspondng consstent bankruptcy rule as a result of a unque outcome of Nash equlbra. The result holds for a large class of consstent and monotone rules, ncludng the Constraned Equal Award, the Propostonal Rule, and many other well known rules. Moreover, all of the subgame perefect equlbra are coalton-proof n the assocated game n strategc form. Journal of Economc Lterature Classfcaton Numbers: C72 and D Academc Press 1. INTRODUCTION The consstency property has proved very powerful n characterzng some of the most mportant soluton concepts n cooperatve game theory Žsee, for example, the characterzatons of the core and the pre-kernel by * A large part of ths work was done whle Dagan was at the Hebrew Unversty of Jerusalem and Unversdad Carlos III de Madrd, and whle Volj was at the Hebrew Unversty and CentER of Economc Research at Tlburg. Serrano gratefully acknowledges the hosptalty of the Center of Ratonalty at the Hebrew Unversty of Jerusalem and Unverstat Pompeu Fabra of Barcelona. We are grateful to Mchael Maschler for very useful comments $25.00 Copyrght 1997 by Academc Press All rghts of reproducton n any form reserved.

2 56 DAGAN, SERRANO, AND VOLIJ Peleg Ž and of the Nash barganng soluton by Lensberg Ž However, consstency alone does not solate a unque rule n bankruptcy problems, even after restrctng attenton to symmetrc, scale-nvarant, and monotone rules. On the other hand, a smple extenson of a result of Aumann and Maschler s Ž shows that any monotone and consstent rule s completely characterzed by a two-person rule and consstency. Consstency has also been suggested as a valuable gude n desgnng noncooperatve mechansms that mplement some cooperatve solutons Ž see, for example, Krshna and Serrano, Namely, extensve forms can be constructed whose subgames relate to the respectve reduced cooperatve problems. By concentratng on the subgame-perfect equlbra of such mechansms, one can hope to mplement the underlyng consstent soluton. Ths paper provdes addtonal support for the dea that consstency s a useful tool n the Nash program for cooperatve games. We ntroduce a game n extensve form that captures a noncooperatve dmenson of the consstency property of bankruptcy rules. In the game one of the credtors wth the hghest clam must make a proposal about how to splt the estate. Those credtors who accept the proposal receve ther shares, and those who reject may appeal to the blateral court that stands as an outsde opton. Our game form generates a large famly of consstent and monotone bankruptcy rules presented n the axomatc theory. It takes a two-person rule as an nput and yelds the unque consstent generalzaton of that rule as an output. The unque equlbrum outcome of the game assocated wth a specfc two-person rule s the allocaton recommended by the unque consstent generalzaton of that rule. In ths sense our game form operates lke the consstency property n the axomatc approach, capturng a noncooperatve dmenson of consstency n the framework of bankruptcy problems. That s, by replacng the consstency axom n Aumann and Maschler s Ž result, our game form provdes ts noncooperatve counterpart. Lke other games based on consstency, our game allows for partal agreements, where a player cannot be prevented from gettng hs offered share f he s happy wth t. The queston arses of whether such equlbra are coaltonally stable. Could the proposer offer a larger fracton of the pe to a credtor and then splt t wth hm? When devatons are coaltonally credble, the answer s negatve: although they are not strong Nash, we show that all the subgame-perfect equlbra of the game are coalton-proof. We wll assume throughout that the clams are known by everybody Ž ncludng the court.. As dscussed above, our focus s the noncooperatve dmenson of the consstency axom n bankruptcy problems. In Dagan et 1 Ž. Ž. For a good survey, see Thomson 1990 or Thomson 1996b.

3 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 57 al. Ž 1995., we analyze the related problem of mplementng bankruptcy rules when the clams are unknown to the court. The paper s organzed as follows: Secton 2 s devoted to the axomatc treatment of bankruptcy problems. Secton 3 dscusses the relaton between blateral prncples of justce and consstency. The multlateral noncooperatve model and the man result are presented n Secton 4. Coalton-proofness s dscussed n Secton 5. A result concernng strctly monotone rules s the object of Secton 6, and Secton 7 concludes. 2. THE AXIOMATIC BANKRUPTCY MODEL Ž. I A bankruptcy problem s a par E; d where d s a vector of non-negatve real numbers Ž clams., ndexed by some fnte non-empty subset I of natural numbers Ž credtors., and 0 E Ý I d D. E s the estate to be allocated, and D s the sum of the clams. Ž. I An allocaton n E; d s a vector x such that Ý I x E and x d for all I. The set of all allocatons n Ž E; d. s denoted by AŽ E; d.. Remark. For any lst of clams d I, any vector x I wth x d s an allocaton of the bankruptcy problem Ž Ý x ; d. I. Therefore, when there s no danger of confuson, we shall call any such vector x an allocaton wthout specfyng the bankruptcy problem to whch t refers. A rule s a functon that assgns to each bankruptcy problem a unque allocaton. EXAMPLES. Ž. a The proportonal rule, PrŽ E; d. d, where D E. The proportonal rule, wdely appled nowadays, allocates awards n proporton to clam sze. The proportonalty prncple was favored by the phlosophers of ancent Greece, and Arstotle even consdered t as equvalent to justce. Ž. Ž. b The constraned equal award CEA rule: CEAŽ E; d. x Ž. Ž. 2 where x mn, d and solves the equaton Ý I mn, d E. Ths rule assgns the same sum to all credtors as long as t does not exceed 2 Ths equaton has a unque soluton when D E. If DE, any soluton s greater than or equal to the maxmum clam and therefore x d for all.

4 58 DAGAN, SERRANO, AND VOLIJ each credtor s clam. Ths rule s also very ancent, and was adopted by mportant rabbncal legslators, ncludng Mamondes. Ž. c The constraned equal loss Ž CEL. rule: CELŽ E; d. x where x maxž 0, d. and solves the equaton Ý maxž 0, d. I 3 E. Ths rule assgns losses Ž d x. n the same manner as the CEA assgns awards. Ž d. The Pneles Rule: PnŽ E; d. CEAŽ mnd2, E 4; d2. CEAŽ maxe D2,0 4; d2.. When the estate does not exceed half the sum of the clams, the Pneles rule assgns each credtor a fxed amount, as long as t does not exceed half hs clam Ž otherwse, t assgns hm half hs clam.. When the estate exceeds half the sum of the clams, t frst gves each credtor half hs clam and then dvdes the remander Žwhch, by defnton, cannot exceed half the sum of the clams. accordng to the procedure descrbed n the prevous sentence. Ths rule appears n Pneles Ž 1861, p. 64., and s an nterpretaton of a controversal mshna Ž Ketuboth 93.. Ž. e The Contested Garment Consstent Ž CGC. Rule: CGCŽ E; d. CEAŽ mnd2, E 4; d2. CELŽ maxe D2; 04; d2.. Ths rule was proposed by Aumann and Maschler Ž as an alternatve nterpretaton of the mshna mentoned above. Ž. f Equal Sacrfce Rules: Let U: be a contnuous and strctly ncreasng functon that satsfes lm x 0 U Ž x.. The equal sacrfce rule relatve to U satsfes Ž E; d. x c 0 such that I wth d 0, UŽ d. UŽ x. c, when E 0. These rules assgn awards so as to equalze absolute sacrfce evaluated accordng to a prespecfed utlty functon. Note that the equal sacrfce rule wth respect to the logarthmc functon s the proportonal rule. The equal sacrfce prncple n taxaton appears n Mll Ž 1848, Book V. and was axomatcally derved by Young Ž Wth a few exceptons that wll be ndcated, all these rules satsfy the propertes dscussed below. We begn wth some basc ones and devote the next secton to propertes concernng the concept of consstency. 3 Ths equaton has a unque soluton when E 0. If E 0, any soluton s greater than or equal to the maxmum clam and therefore x 0 for all.

5 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 59 An allocaton x n Ž E; d. s sad to be symmetrc f whenever d d j, xx j. A rule s symmetrc f t always assgns symmetrc allocatons. A rule s consstent f for any fnte non-empty set I of credtors, for all Ž E; d., d I, for all J I, ž Ý / J Ž E; d. xxj x, dj Ž 1. I J where when y, yjs the projecton of y on. A weaker condton s blateral consstency, whch requres Ž. 1 only for subsets J contanng exactly two credtors. The nterpretaton of consstency s as follows. Suppose that a rule assgns allocaton x to the bankruptcy problem Ž E; d.. Suppose also that some subset of credtors wants to reallocate the total amount Ý J x assgned to them. If we apply the same rule to allocate ths amount among these credtors, each wll get the amount orgnally assgned to hm, provded s consstent. Consstency n the setup of bankruptcy problems was frst dscussed by Aumann and Maschler Ž and further analyzed by Young Ž 1987, For an extensve survey, see Thomson Ž 1996a.. A rule s monotone f for all bankruptcy problems Ž E; d. and 0EE, Ž E; d. Ž E; d.. Monotoncty says that a decrease n the estate does not beneft any credtor. A rule s strctly monotone f for all bankruptcy problems Ž E; d. and 0 E E, f d0 then Ž E; d. Ž E; d.. Strct monotoncty says that a decrease n the estate leaves every non-zero credtor worse off. The rules n the above examples, wth the excepton of the proportonal and equal sacrfce rules, do not satsfy strct monotoncty. A rule s supermodular f for all Ž E; d. and 0 E E, f dd j then Ž E; d. Ž E; d. Ž E; d. Ž E; d. j j. A supermodular rule allocates each addtonal dollar n an order preservng manner. 4 A rule s anonymous f for every bankruptcy problem ŽE, Ž d,...,d.. 1 n and for all permutatons of the set of credtors, we have Ž. E, d,...,d E, Ž d,...,d. 1,2,...,n. Ž1. Žn. Ž. 1 n Anonymty requres that the rule be ndependent of the names of the credtors. 4 Equal sacrfce rules relatve to non-concave utlty functons are not necessarly supermodular.

6 60 DAGAN, SERRANO, AND VOLIJ The followng lemmas wll be useful n the rest of the paper. LEMMA 2.1. Any supermodular rule s symmetrc. Proof. Left to the reader. LEMMA 2.2. Let Ž E; d. be a bankruptcy problem and let be a credtor wth the hghest clam. If s supermodular and 0 E E, then Ž E; d. Ž E; d.. That s, s award s strctly monotone n the estate. Proof. Left to the reader. 3. ON BILATERAL COMPARISONS, JUSTICE, AND CONSISTENCY Snce every bankruptcy problem s a legal problem, ts solutons should be guded by the prncple of justce. Whatever form ths prncple may take, t should enable us to determne whether any one credtor receved better or worse treatment than another at any gven allocaton. For example, f we beleved, lke Arstotle, that justce s proportonalty, then we would say that s treated better than j at allocaton x f receves a larger proporton of ths clam than j does. Accordng to ths prncple of justce, an allocaton wll treat and j equally f they receve the same proporton of ther clams. Obvously, we can thnk of other notons of justce, but n order to make these parwse comparsons we clearly need only a blateral prncple. A blateral prncple s a functon that assgns a unque allocaton to every two-person bankruptcy problem. We nterpret ths unque allocaton as the just soluton to the blateral problem. We shall say that any other allocaton n a two-person problem treats one credtor better than the other snce t awards one credtor more than hs far share. Any rule nduces a blateral prncple, when t s projected on the class of 2-person bankruptcy problems. We denote a generc allocaton rule by and ts nduced blateral prncple by f. Monotoncty, anonymty, and supermodularty of blateral prncples are defned n an obvous way. Gven a bankruptcy problem Ž E; d. and a blateral prncple f, we shall say that an allocaton x treats and j f-equally f Ž x, x. f j xx j; Ž d, d.. An allocaton n Ž E; d. j s sad to be f-just f t treats every two credtors f-equally. Aumann and Maschler Ž showed that f a blateral prncple f s monotone, then there s at most one f-just allocaton for each bankruptcy problem. If a unque f-just allocaton exsts for any bankruptcy problem, then we can defne the f-just rule to be the rule that assgns to each bankruptcy problem ts unque f-just allocaton. Condtons

7 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 61 on a blateral prncple that guarantee the exstence of the assocated f-just rule can be found n Dagan and Volj Ž We explore some relatons between f-justce and propertes of allocaton rules. LEMMA 3.1. Let f be a monotone blateral prncple and let be the f-just rule, then s consstent. Proof. See Aumann and Maschler Ž 1985, Corollary LEMMA 3.2. Let be a monotone and blateral consstent rule, and let f be the blateral prncple nduced by. Then s the f-just rule. Proof. Left to the reader. COROLLARY 3.3. Let f be a monotone blateral prncple. The f-just rule s the unque consstent rule that concdes wth the blateral prncple f n two-credtor problems. Moreoer, when s monotone, consstency of s equalent to blateral consstency of. Proof. The frst part follows drectly from Lemma 3.2. As for the second part, f s monotone and blaterally consstent, then by Lemma 3.2 t s the f-just rule, whch by Lemma 3.1 s consstent. The followng lemma shows that monotoncty and supermodularty of the blateral prncple f are propertes nherted by the correspondng f-just rule. LEMMA 3.4. Let f be a blateral prncple and let be the f-just rule. Ž a. If f s supermodular, then s supermodular as well. Ž b. If f s monotone, then s monotone as well. Ž c. If f s supermodular, then s anonymous. Proof. Let Ž E; d. be a bankruptcy problem and let 0 E E. Let xž E, d. and x Ž E, d. be ther correspondng f-just allocatons. Ž. a j j j Assume that d d. We need to show that x x x x.by defnton of and by supermodularty of f we have Ž Ž.. Ž Ž.. Ž Ž.. Ž Ž.. x x f x x ; d, d f x x ; d, d j j j j f x x ; d, d f x x ; d, d j j j j j j x x j j. Ž. b Assume by contradcton that for some credtor I, x x. Ž 2.

8 62 DAGAN, SERRANO, AND VOLIJ Snce Ý ki there must exst a credtor j wth x k Ý ki x k x x. Ž 3. j j Case 1. x x x x. By defnton of and monotoncty of f, j j Ž Ž.. Ž Ž.. x f x x ; d, d f x x ; d, d x j j j j Ž. whch s a contradcton to 2. Case 2. x x x x. By defnton of and monotoncty of f j j Ž Ž.. Ž Ž.. x f x x ; d, d f x x ; d, d x j j j j j j j j whch contradcts Ž. 3. Ž. c Let ² E; Ž d,...,d.: 1 n be a bankruptcy problem and let : 1,...,n4 1,...,n4 be a permutaton. Consder the auxlary replca bankruptcy problem ² ne; Ž d,...,d,...,d,...,d : 11 1n n1 nn, where dk d d for all, j, k 1,...,n 4. Let Ž x,..., x,..., x,..., x. jk k 11 1n n1 nn be the correspondng allocaton recommended by. Snce f s supermodular, by part Ž. a and Lemma 2.1 s symmetrc. Therefore, xk x jk xk for all, j, k 1,...,n 4. Consequently, snce s a permutaton, n Ý k1 n Ý x x E. 1k k1 kžk. By Lemma 3.1, s consstent. Therefore, by the defnton of d consstency of, k and the E; Ž d,...,d. x ; Ž d,...,d. n Ý 1 n 1k 11 1n k1 Ž x 11,..., x1n. x,..., x. Ž. 1 n

9 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 63 Analogously, E; d,...,d x ; d,...,d n Ž Ž1. Žn.. Ý kžk. Ž 1Ž1. nžn.. k1 Ž x,..., x. 1Ž1. nžn. Ž x,..., x.. Ž1. But snce ² E; Ž d,...,d.: 1 n s an arbtrary problem and s an arbtrary permutaton, the above two equaltes mply that s anonymous. LEMMA 3.5. Let f be a monotone blateral prncple, let Ž E; d. be a bankruptcy problem, and let x* be ts f-just allocaton. Let x be an allocaton n Ž E; d. n whch there are two credtors and j wth x x and x j x j. Then, f x x ; Ž d, d. j j x. Moreoer, f both nequaltes are strct, then f x x ; Ž d, d.x. j j Proof. Case 1. x x x x j j. By monotoncty and f-justce of x*, Ž. f x x ; d, d f x x ; Ž d, d. x x. Hence, f j j j j x x j; Ž d, d.x. j Case 2. x x x x j j. By monotoncty and f-justce of x*, Ž. f x x ; d, d f x x ; Ž d, d. x x. Hence, f j j j j j j j j x x j; Ž d, d.x. j Ths proves the frst part of the clam. As for the second part, t s proved analogously and s left to the reader. Lemma 3.5 says that the f-just allocaton of a bankruptcy problem s a good benchmark for blateral comparsons: f at some allocaton x player j gets more than the f-just allocaton assgns to hm and f player gets less than hs f-just share, then j must be recevng better treatment than at the allocaton x. The next two lemmas have ndependent nterest. They relate f-justce and the allocaton prescrbed by a consstent bankruptcy rule. LEMMA 3.6. Let Ž E; d. be a bankruptcy problem, let be a consstent and monotone rule, and let x be an allocaton n Ž E; d.. If there exsts a credtor such that for all j, x x x ; Ž d, d., then x Ž E; d.. Žn. j j Proof. If x Ž E; d. then there exsts a credtor j wth x Ž E; d. j j. Hence, by Lemma 3.5, x x x ; Ž d, d. j j, contradctng the assump- ton of the lemma. Analogous arguments are used f x Ž E; d.. LEMMA 3.7. Let Ž E; d. be a bankruptcy problem, let be a consstent, monotone, and supermodular rule and let x be an allocaton n Ž E; d.. Let

10 64 DAGAN, SERRANO, AND VOLIJ be a credtor wth the hghest clam. If for all credtors j, x x x j; Ž d, d. then x Ž E; d.. j Proof. By Lemma 3.6, x Ž E; d.. Now assume there exsts a cred- tor j wth x Ž E; d. j j. By consstency, supermodularty, and Lemma 2.2, x x x ; Ž d, d. j j, contradctng the assumpton. So t must be that for all j, x Ž E; d. j j. Snce x s an allocaton, ths mples that x Ž E; d.. Lemma 3.7 says that f an allocaton s such that all credtors are treated f-equally to one wth a maxmum clam, then ths allocaton s the f-just allocaton. Thus, only n equatons Žthe n 1 condtons of f-equalty wth one of the hghest clamants and the effcency condton. are needed to calculate the f-just allocaton of any n-credtor bankruptcy problem. 4. A MULTILATERAL NONCOOPERATIVE MODEL Let Ž E; d. be a gven bankruptcy problem. We are nterested n defnng an extensve form game for each blateral prncple f. The game, denoted by G f Ž E; d. s defned as follows. Credtor 1 proposes an allocaton x n AŽ E; d.; followng ths proposal all the other credtors respond sequentally, ether by acceptng or rejectng the offer. The order of responses follow the protocol nduced by the credtors ndces, namely credtor 2 s the frst to respond and credtor n s the last. In order to defne the players payoffs, t s convenent to defne the followng state varable of nterm shares for the proposer. Gven a proposal x and a profle of responses to t, defne w x 1 1 and for t 2,...,n, wt1 f t accepted x wt Ž 4 ½. f 1 w t1 x t, Ž d 1, d t. f t rejected x An acceptng credtor t receves as a payoff zt x t, a rejectng credtor t receves z f Ž w x ; d, d., and the proposer receves z w. t t t1 t 1 t 1 n Note that credtor 1 s payoff s determned n several steps. The varable wt represents credtor 1 s nterm share of the total estate after a proposal s made and the payoffs to all responders wth ndces no greater than t are decded. That s, when credtor 1 proposes x he determnes w. To 1 determne w 2, we need to know credtor 2 s response to x. If credtor 2 accepts t, w w Žcredtor 2 fnds no grounds to appeal to the blateral 2 1

11 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 65 court about the unfarness of the proposal.; otherwse, w f 2 1 w1 x ; Ž d, d because credtor 2 renegotates wth credtor 1 over hs far share of w1 x 2, and so on. In sum, credtor 1 may be nterpreted as the admnstrator of the estate. He proposes an allocaton x and he s enttled to award the proposed shares to the acceptng credtors. However, he must be open to counterarguments made by those who reject the offer, who may hold hm responsble for hs proposal before the court Žby clamng ther far share of the amount w x. t1 t. That s, we assume that the blateral prncple f s commonly accepted n socety, or that resortng to ltgaton s an outsde opton for the credtors. The admnstrator then receves the remander of the estate after all the other credtors receved ther shares. Note that every play of the game G f Ž E, d. results n an allocaton of Ž E, d.. To see ths let be an arbtrary strategy profle, let Ž x, x,..., x. 1 2 n be the proposal, let w 4 t t1 n be the assocated nterm payoff sequence, and let Ž z, z,..., z. 1 2 n be the correspondng payoff vector. By the defnton of the payoff vector, 0 zt d t, for t 1, 2,..., n. Therefore t s enough to show that Ý n t1 zt E. Note that by the defnton of w t, for t 1, 2,..., n and of the payoff vector, zt wt1 wt xt for t 2,3,..., n. Therefore, n Ý t1 z n Ý z w t t n t2 n Ý Ž t1 t t. n t2 w w x w w n Ý x 1 t t2 n Ý x t t1 E. The ntroducton of the blateral prncple to calculate the proposer s payoff may seem arbtrary. However, we want to emphasze that our purpose s not to characterze a certan consstent rule or any blateral prncple; nstead, we are more nterested n the relatonshp between the blateral prncples and ther consstent generalzatons. For a model n whch the blateral prncple does not appear n the extensve form game see Serrano Ž 1995., who characterzed the contested garment consstent rule. Now we are ready to state the man result of ths paper.

12 66 DAGAN, SERRANO, AND VOLIJ THEOREM 4.1. Let Ž E; d. be a bankruptcy problem where credtor 1 has the hghest clam, let f be a monotone and supermodular blateral prncple, and let be the f-just rule. The unque Nash equlbrum outcome of G f Ž E; d. s Ž E; d.. Moreoer, t can be supported by a pure strategy subgame perfect equlbrum. Some remarks are n order. Ž. 1 The theorem holds for the case n whch the proposer s a credtor wth a hghest clam. Otherwse unqueness s not obtaned Žsee Example 6.1 n Secton 6.. On the other hand, as wll be shown n Secton 6, for strctly monotone rules the dentty of the proposer s of no mportance. Note also that the result s nvarant to the order of responders. Ž. 2 The result does not use any refnement of Nash equlbrum. Ths s n contrast to other models that relate noncooperatve models wth pure barganng problems. When the underlyng model s more complex, as n general cooperatve games, a refnement of subgame perfect equlbrum s usually needed n order to get unqueness Žsee, for example, Hart and Mas-Colell Ž and Gul Ž Ž. 3 Unlke some other models, whch provde a noncooperatve vew of a cooperatve soluton concept, our result yelds a noncooperatve vew of a large famly of allocaton rules for bankruptcy problems. The two crtcal propertes that characterze ths famly are consstency and monotoncty. These two propertes guarantee that the allocaton assgned by the consstent rule can be supported by a Nash equlbrum. These propertes are the ones that drve the results of other consstency-based non-cooperatve mechansms Žsee, for example, Krshna and Serrano Ž and Chae and Yang Ž Ž. 4 Our result holds for the whole famly of bankruptcy problems. Ths s n contrast to other models, such as those mentoned n the prevous remark, where the unque subgame-perfect equlbrum outcomes converge to the Nash barganng soluton agreement for dvdng a dollar barganng problems, n whch the Nash soluton s monotone Žsee Chun and Thomson, The reason why these models do not yeld a smlar result n all barganng problems becomes apparent: the Nash barganng soluton s not monotone n general. 5 Ž. 5 The unque equlbrum agreement s not acheved necessarly after unanmous agreement. Ths feature of the model agrees wth the 5 When the Nash soluton s not monotone, the strateges proposed by Krshna and Serrano Ž and by Chae and Yang Ž do not consttute even a Nash equlbrum. The proposer could fnd a proftable devaton by offerng more than hs equlbrum share to one of the responders, n the hope of beneftng from a bgger share n a smaller pe.

13 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 67 consstency prncple. As we know, after applyng a consstent rule any subset of agents s ndfferent between acceptng ther shares and renegotatng among themselves. Ž. 6 The possble emergence of an equlbrum wthout unanmous agreement s due to the fact that there s no cost of renegotaton, such as dscountng, fxed renegotaton fees, or the random elmnaton of players. Proof of Theorem 4.1. We denote Ž E; d. by x* and equlbrum outcomes by z. The proof follows from the followng steps. f Ž. STEP 1. In any equlbrum of G E; d, z x. 1 1 Proof. Credtor 1 can guarantee a payoff of x 1 smply by proposng x*. To see ths, note that f the proposal s x*, then w x, and f w x 1 1 t1 1 then by the defnton of w and consstency of, w x t t 1 ndependent of the responses. So n any equlbrum, the proposer gets at least x 1. f STEP 2. Let z be an equlbrum outcome of G Ž E; d.. Then f Ž 1 z1 z t; Ž d, d.. 1 t z1 for all t 1. Proof. Let be an equlbrum of G f Ž E; d. and let w 4 t t1 n be the nterm payoff sequence determned by the equlbrum path. Gven the equlbrum proposal x and the responses of the others, credtor t s payoff s ether f Žw x ; Ž d, d.. t t1 t 1 t or xt dependng on whether he rejects or accepts the proposal, respectvely. Therefore n equlbrum we must have and By Ž. 6 and the defnton of z Ž. 4 z max f w x ; Ž d, d., x Ž 5. t t t1 t 1 t t wt wt1 xt z t. Ž 6. 1 we have z1 wn wt for all t. Ž 7. Gven the dentty f Ž z z ; Ž d, d.. f Ž z z ; Ž d, d t 1 t t 1 t 1 t z1z t, t follows from Ž. 7 and monotoncty that Ž. Ž. f z z ; Ž d, d. f w z ; Ž d, d. z z. 1 1 t 1 t t t t 1 t 1 t By Ž 6., f Ž z z ; Ž d, d.. f Žw x ; Ž d, d.. z z and by Ž t 1 t t t1 t 1 t 1 t, f Žz z ; Ž d, d.. z. 1 1 t 1 t 1 STEP 3. z x. 1 1 Proof. If z x 1 1, snce z and x* are both allocatons, there must be some credtor t wth z x t t. Then, by Lemma 3.5 we must have f Žz z ; Ž d, d.. z, contradctng Step t 1 t 1

14 68 DAGAN, SERRANO, AND VOLIJ STEP 4. z x*. Proof. Assume by contradcton that z x*. Then there s a credtor t wth z x t t. By Steps 1 and 3 ths credtor cannot be the proposer. Hence, by Steps 2, 3, and 1 and by supermodularty of f Ž Lemma 2.2. we have that Ž f x x ; Ž d, d.. f Žz z ; Ž d, d t 1 t 1 1 t 1 t z1x 1. Ths s a contradcton to the f-justce of x*. As for exstence, t can easly be seen that the followng strategy profle consttutes a subgame perfect equlbrum: Ž. I Credtor 1 proposes x*; and Ž II. Credtor t accepts a proposal x f and only f x f t t wt1x t; Ž d, d., where w s defned n 4. 1 t t1 Ths concludes the proof of the theorem. 5. COALITIONAL STABILITY OF THE EQUILIBRIA In the game presented n Secton 4 credtors may ext wth the share proposed to them smply by acceptng the proposal. The queston arses of whether the equlbra are coaltonally stable. Could the proposer offer a larger share of the pe to a responder n the hope of proftng from a jont devaton? To answer ths queston we consder the game f Ž E; d. n strategc form that corresponds to the game G f Ž E; d. n extensve form. We frst consder any knd of coaltonal devaton and ask f the subgame perfect equlbra of our model are strong Nash n f Ž E; d. Ž Aumann, Ths requres that no coalton of players have a jont devaton whch leaves all ts members better off. We fnd that the equlbra of our model fal n general to be strong. Ths s llustrated n the followng example. EXAMPLE 5.1. Consder the bankruptcy problem problem Ž E; d. ² 99; Ž 100, 100, 100.:. Snce ths problem s symmetrc, for all symmetrc blateral prncples the game G f Ž E; d. s the same. Clearly, the f-just allocaton n ths problem, and hence the unque subgame perfect equlbrum outcome, s Ž 33, 33, 33.. Consder the followng devaton by the frst and thrd credtors: The proposer offers x Ž 0, 1, 98., and the thrd credtor, who was offered 98 dollars, rejects t. Ths devaton yelds more than 33 both to credtor 1 and credtor 3, no matter how credtor 2 responds. Therefore no equlbrum of G f Ž E, d. s a strong equlbrum. Note that although ths devaton mproves credtor 3 s payoff relatve to the f-just allocaton, he can do even better by devatng from hs jont devaton wth credtor 1 and acceptng 1 s offer. Ths makes the above

15 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 69 devaton unstable. Examples of ths sort motvated the alternatve concept of coaltonal stablty known as coalton-proof Nash equlbrum, ntroduced by Bernhem et al. Ž The followng defnton 6 refers to games n normal form, ŽI, Ž S., Žg.. I I, where I s the set of players, S s the strategy set of player, and g s the payoff functon of player. Let J be a coalton, that s, J I. We denote SJ J S. Also, f S I S s a lst of strateges, J denotes the restrcton of to coalton J. Gven a game G, a lst of strateges Ž one for each player., and a coalton J of players, an nternally consstent mproement of J upon s defned by nducton on J. If J4 for some n I, then S s an nternally consstent mprovement of J upon f g Ž,. g Ž..If J I4 1 then J SJ s an nternally consstent mprovement upon f Ž. g Ž,. g Ž. J I J for all n J, and Ž. no T J, T, has an nternally consstent mprovement upon Ž t,. J I J. The profle s a coalton-proof Nash equlbrum f no J I, J, has an nternally consstent mprovement upon. In contrast to strong Nash, the coalton-proof Nash equlbrum requres that no coalton should have a proftable and self-enforcng devaton. THEOREM 5.2. Let f be a monotone and supermodular blateral prncple and let be the f-just rule. For all bankruptcy problems Ž E; d., all the subgame-perfect equlbra of G f Ž E; d. are coalton-proof equlbra of f Ž E; d.. Proof. Let Ž E; d. be a bankruptcy problem and let f be a monotone and supermodular blateral prncple. Let be a subgame perfect equlbrum and suppose there s a coalton of credtors that has an nternally consstent mprovement upon. Assume that the proposer does not belong to the devatng coalton. Let k be the member of the devatng coalton who has the lowest ndex, that s, credtor k s the frst to respond among the devators. Consder the node of the devaton path n whch credtor k has to respond. Snce hs payoff s ndependent of the strateges of the credtors wth hgher ndces than k, the mere exstence of the assumed mprovement contradcts the fact that s a Nash equlbrum. Therefore, the proposer must belong to the devatng coalton. Thus, suppose that the proposer s a member of the devatng coalton and let x be the proposal made n the devaton. Denote by 4 t the sequence of nterm payoffs of the proposer determned by the jont 6 Ž. It s taken from Peleg and Tjs 1996.

16 70 DAGAN, SERRANO, AND VOLIJ devaton and by w 4 t the sequence correspondng to the case n whch all responders follow ther equlbrum strateges after x has been proposed. We clam that t wt for all t. The proof s by nducton. By defnton 1w1x 1. Assume now that the equalty s true for all t s and consder the node where credtor s has to respond. If credtor s does not belong to the coalton of devators, then t s clear that ws s because s1 ws1 and he does not devate. If credtor s belongs to the set of devators, then snce the devaton s nternally consstent t means that, n partcular, credtor s cannot fnd t proftable to devate from the jont devaton. Therefore s w s. On the other hand, snce s a subgame perfect equlbrum and w, t must be the case that w. s1 s1 s s Hence ws s. Ths means that, followng the proposal x, and whether the devatng responders follow the devaton or not, the proposer s payoff s the same. But then, snce s a subgame perfect equlbrum, by Theorem 1 w x n n 1 whch contradcts the fact that the proposer belongs to a coalton that has an nternally consstent mprovement upon. Remark. In fact, the proof above also shows that all the subgame-perf fect equlbra of G Ž E; d. are perfectly coalton-proof equlbra of the same game. ŽFor a defnton of perfectly coalton-proof equlbrum, see Bernhem et al. Ž Ths result, however, s not surprsng snce Peleg Ž showed that for extensve form games wth perfect nformaton, the set of subgame perfect equlbra concdes wth the set of perfectly coalton-proof equlbra. 6. STRICTLY MONOTONE RULES The unqueness result n Theorem 4.1 s drven by the strct estate monotoncty of rule wth respect to the hghest clam Ž Lemma Ths s why t s mportant that the proposer should be a credtor wth the hghest clam. If the proposer s component 1 was not strctly monotone n the estate, multplcty of subgame-perfect equlbrum outcomes mght arse, as shown by the followng example: EXAMPLE 6.1. Let Ž E; d. ² 100; Ž 10, 100, 100.:. If s the constraned equal award rule, Ž E; d. Ž 10, 45, 45.. The reader can check that all the outcomes of the form Ž 10, 45 a,45a. for 35 a 35 can be supported by subgame perfect equlbra of the correspondng extensve form game. If we confne ourselves to consstent and strctly monotone rules, the man result can be generalzed to the case n whch the proposer s any credtor wth a postve clam. These results are stated formally n Theorem 6.2.

17 NONCOOPERATIVE VIEW OF BANKRUPTCY RULES 71 THEOREM 6.2. Let Ž E; d. be a bankruptcy problem, let f be a strctly monotone blateral prncple, and let be the f-just rule. Suppose credtor 1 has a poste clam. The unque subgame perfect equlbrum outcome of G f Ž E; d. s Ž E; d.. Moreoer, all subgame-perfect equlbra of G f Ž E; d. are coalton-proof equlbra of f Ž E; d.. Proof. The proof s dentcal to the Proofs of Theorems 4.1 and 5.2 wth the only excepton that n Step 4 of Theorem 4.1, the words supermodularty Ž Lemma 2.2. should be replaced by strct monotoncty. 7. CONCLUDING REMARKS By gvng a noncooperatve vew of a wde class of bankruptcy rules, we beleve we have provded addtonal support to the dea that the property of consstency s useful n the Nash Program for cooperatve games. On the other hand, consstency alone, wthout the assstance of monotoncty, s nsuffcent to reach the results. Thus, constructon of consstency based noncooperatve models that support consstent cooperatve soluton concepts whch are not monotone seems to us a dffcult task. Therefore there mght be problems n supportng the nucleolus or the Nash barganng soluton on general pes by means of a noncooperatve model. 7 In the bankruptcy model, however, monotoncty s a natural requrement. Moreover, t s almost mpled by consstency: Young Ž 1987, Lemma 1. showed that f a rule s symmetrc, contnuous, and consstent, then t s also monotone. REFERENCES Aumann, R. J. Ž Acceptable Ponts n General Cooperatve n-person Games, n Contrbutons to the Theory of Games, Vol. 4 Ž A. W. Tucker and R. D. Luce, Eds.., pp , Prnceton, NJ: Prnceton Unversty Press. Aumann, R. J., and Maschler, M. Ž Game Theoretc Analyss of a Bankruptcy Problem from the Talmud, J. Econom. Theory 36, Bernhem, B. D., Peleg, B., and Whnston, M. D. Ž Coalton-Proof Nash Equlbra I Concepts, J. Econ. Theory 43, 112. Chae, S., and Yang, J.-A. Ž An N-Person Pure Barganng Game, J. Econ. Theory 62, Chun, Y., and Thomson, W. Ž Monotoncty Propertes of Barganng Solutons when Appled to Economcs, Math. Soc. Sc. 15, Ž. Hart and Mas-Colell 1996 support the Nash barganng soluton for general pes va a noncooperatve model, but ther model s not consstency based.

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