Implementation of Bargaining Sets via Simple Mechanisms*

Transcription

1 Games ad Ecoomic Behavior 31, Ž 2000 doi: game , available olie at http: wwwidealibrarycom o Implemetatio of Bargaiig Sets via Simple Mechaisms* David Perez-Castrillo Departamet d Ecoomia i d Historia ` Ecoomica ` & CODE, Ui ersitat Autooma ` de Barceloa, Bellaterra, Barceloa, Spai ad David Wettstei Departmet of Ecoomics, Moaster Ceter for Ecoomic Research, Be-Gurio Ui ersity of the Nege, Beer-She a 8105, Israel Received March 10, 1998 We propose two simple mechaisms that implemet two bargaiig sets i super-additive eviromets The first bargaiig set is a close variatio of the oe proposed by L Zhou Ž 199, Games Ecoom Beha 6, , ad the secod is the Pareto optimum payoffs of the A Mas-Colell Ž1989, J Math Ecoom 18, bargaiig set We adopt a simple framework i which the cooperative outcomes are realized as o-cooperative subgame perfect equilibria i pure strategies of a two-stage game played by a auxiliary set of idividuals competig over the cooperative agets Joural of Ecoomic Literature Classificatio Numbers: C71, C Academic Press Key Words: bargaiig set; implemetatio; simple mechaism 1 INTRODUCTION Cooperative game theory models the iteractio existig i ecoomic ad social eviromets, emphasizig the advatages of cooperatio The differet solutio cocepts put forward i this theory suggest reasoable *We are grateful to Adreu Mas-Colell, Matthew Jackso, Michael Maschler, ad a aoymous referee for very helpful commets Perez-Castrillo also ackowledges the fiacial support by DGES uder Projects Nos PB ad PB ad the SGR umber Wettstei gratefully ackowledges support from the Moaster Ceter for Ecoomic Research Part of this research was coducted while Wettstei was visitig the Uiversitat Autooma ` de Barceloa, with a grat from the Geeralitat de Cataluya $3500 Copyright 2000 by Academic Press All rights of reproductio i ay form reserved 106

2 IMPLEMENTATION OF BARGAINING SETS 107 ways for the split of the surplus of such cooperatio amog the various parties The most popular cooperative solutio cocept is the core, which selects those allocatios from which o coalitio of players ca profitably deviate However, the core igores subsequet reactios that the deviatios ca geerate These reactios may reder the iitial deviatio oprofitable A subclass of solutios takig ito accout the future cosequeces of curret actios cosists of bargaiig sets The first defiitio of a bargaiig set was suggested by Auma ad Maschler Ž 196 I recet years, several other bargaiig sets have bee defied ad aalyzed by Mas-Colell Ž 1989, Vohra Ž 1991, Vid Ž 1992, ad Zhou Ž 199 A excellet survey of the literature ca be foud i Maschler Ž 1992 The differet defiitios of the bargaiig set share the commo feature of beig based o the cocepts of objectio Ž deviatio ad couter-objectio Žreactio to the deviatio However, they propose slightly differet meaigs for these cocepts I cotrast to the core, the bargaiig sets are o-empty for a large class of games A shortcomig commo to cooperative solutio cocepts is the lack of a explicit framework by which they ca be reached The area of research kow as implemetatio theory rigorously ivestigates the correspodece betwee ormative goals ad istitutios desiged to achieve Žim- 1 plemet these goals Several papers have recetly addressed this issue with regard to the core ŽPerez-Castrillo Ž 199, Perry ad Rey Ž 199, Serrao Ž 1995, ad Serrao ad Vohra Ž 1997 To the best of our kowledge, oly Eiy ad Wettstei Ž 1999 cosider the implemetatio of bargaiig sets Stears Ž 1968 ad Maschler ad Peleg Ž 1976 costruct dyamical systems that coverge to poits i the Auma ad Maschler bargaiig set However, these systems igore ay strategic cosideratio o the part of the players ad ca be maipulated by strategic agets I this paper, we propose two simple mechaisms that implemet two bargaiig sets i super-additive eviromets The simplicity of these mechaisms is i cotrast to the mechaisms suggested i most works These traditioal mechaisms are quite itricate ad require a large degree of sophisticatio o the part of the desiger ad the participatig agets 2 We adopt a rather simple framework as suggested by Perez- Castrillo Ž 199, i which core outcomes are realized as o-cooperative 1 Quoted from Palfrey Ž See, for example, the mechaisms proposed by Moore ad Repullo Ž 1988 ad Abreu ad Se Ž 1990, which apply to a much larger class of problems ad eviromets

3 108 PEREZ-CASTRILLO AND WETTSTEIN equilibria of a game played by a auxiliary set of idividuals Žor istitutios competig over the agets Žthat are the players i the cooperative game The first bargaiig set that we implemet is a close variatio of the oe proposed by Zhou Ž 199 I both sets, the couter-objectio must ivolve some player from both the objectio ad the remaiig set The oly differece betwee the two sets is that, i cotrast to Zhou Ž 199, we do ot impose the coditio that a couter-objectio should ot cotai the objectio I the mechaism that we propose, three pricipals Žexogeous players that maximize profits compete over the agets of the game Competitio occurs i two stages I the first stage, pricipals 1 ad 2 make simultaeous offers to the agets Each aget is the provisioally assiged to the pricipal havig made the highest offer to her, where ties are broke i favor of pricipal 1 If oe pricipal hires the whole set of agets the assigmet becomes defiitive, salaries are paid, ad the game eds Otherwise, pricipal 3 has the possibility of submittig a ew offer This ew offer has preferetial status, i the sese that pricipal 3 oly has to match the maximal provisioal offer to attract a aget However, the fial set of workers hired by this pricipal must cotai, if it is ot empty, agets from the two groups provisioally formed at stage 1 After this, pricipals pay salaries to the agets that were assiged to them, receive the value of the coalitio composed of these agets, ad the game eds This mechaism implemets i subgame perfect equilibrium Ž SPE i pure strategies the bargaiig set that we propose We also implemet i SPE the set of efficiet allocatios i the Mas-Colell Ž 1989 bargaiig set through a mechaism similar to, although somewhat more complicated tha, the previous mechaism The mai differece betwee the two mechaisms is that i the secod, we assume that pricipals strictly prefer employig more rather tha less agets This ca be iterpreted as beig part of the desig of the game: the desiger asks the pricipals to maximize profits, but also, other thigs beig equal, to hire as may agets as possible The rules of the two games are straightforward ad rely to a large degree o the ituitio uderlyig the defiitio of bargaiig sets Usig exogeous players allows us to avoid both the use of modulo Ž iteger games, ad the problem of feasibility of the out-of-equilibrium outcomes This paper is orgaized as follows I Sectio 2 we preset the basic cooperative game I Sectio 3 we implemet a close variat of the Zhou Ž 199 bargaiig set, while i Sectio we implemet the Pareto optimal part of the Mas-Colell Ž 1989 bargaiig set Fially, we coclude i Sectio 5

4 IMPLEMENTATION OF BARGAINING SETS THE COOPERATIVE GAME A -perso cooperative game with trasferable utility is a pair Ž N, N where N 1, 2,, is the set of agets ad : 2 R is the charac- N teristic fuctio which satisfies 0 The members of 2 represet all the possible coalitios of agets The characteristic fuctio yields the maximal payoff ay coalitio ca obtai whe its members cooperate We assume that the game is super-additive, ie, Ž S T Ž S Ž T if S T Moreover, i order to simplify the proofs ad to avoid excessive otatio, we proceed uder the assumptio that the precedig iequality is strict if S T N ad S, N Oce we have preseted our results uder this strog super-additivity assumptio, we will idicate how the mechaisms ca be modified to obtai the same results uder the usual super-additivity assumptio For ay vector x x 1, x 2,, x R ad ay subset S N, we write xs Ý x A payoff vector for the game Ž N, i S i is ay vector c x R that satisfies x N N We deote by S N S the complemet of the subset S i N A solutio cocept for cooperative games associates a set Žpossibly empty of payoff vectors with each game The descriptio of the solutio does ot ivolve ay procedures through which the agets ca achieve a outcome i the solutio The solutio cocept merely prescribes a set of requiremets that payoff vectors must satisfy i order to costitute part of the solutio The basic compoets of the defiitio of a bargaiig set are the otios of objectio ad couter-objectio The differet bargaiig sets proposed i the literature diverge o the precise defiitios of these two cocepts We start by providig the basic set of defiitios we are goig to use Give a payoff vector x R, a pair S, y with S N ad y R is said to be a objectio to x if ys Ž S ad yi xi for each i i S Give a objectio Ž S, y to the payoff x, a pair Ž T, z with T N ad z R is said to be a couter-objectio to Ž S, y if zt Ž T ad the followig two coditios are satisfied: C BS1 T S ad T S c BS2 zi yi for each i T S ad zi xi for each i T S The core is the set of all payoff vectors agaist which there is o objectio It is well kow that the core may be empty Furthermore, the defiitio of the core igores subsequet actios It may well be that the objectio to a payoff vector ot i the core may set i motio a process that would hurt the objectig idividuals Such a process is the ratioale

5 110 PEREZ-CASTRILLO AND WETTSTEIN behid the couter-objectio Notice that the couter-objectio may geerate further deviatios, which are igored i our aalysis See Dutta, Ray, Segupta, ad Vohra Ž 1989 for the costructio of a bargaiig set that takes ito accout objectios of ay order To defie the bargaiig set Ž BS, we itroduce the defiitio of a justified objectio A objectio Ž S, y to the payoff x is said to be justified if there is o couter-objectio to Ž S, y The bargaiig set of the cooperative game Ž N,, deoted by BSŽ N,, is the set of all payoff cofiguratios x R agaist which there is o justified objectio The BS we defie is very close to the defiitio proposed by Zhou Ž 199 I his paper, the coditio Ž BS1 also requires that S T For ay give objectio, there are more couter-objectios satisfyig our coditio Ž BS1 tha couter-objectios satisfyig Zhou s defiitio Hece, objectig is more difficult i our settig ad therefore the BS defied here is larger Sice Zhou s BS is o-empty, ours is o-empty as well Note also that every payoff i the BS is efficiet, ie, xž N Ž N for all x BSŽ N, However, it is ot ecessarily the case that the payoffs i the BS are idividually ratioal, while Zhou s BS does satisfy this property For example, cosider the game N 1, 2, 3, Ž 1, Ž 2 Ž 3 0, Ž 1, 2 Ž 1, 3 16, Ž 2, 3 17, Ž 1, 2, 3 2 The payoff vector Ž 2, 11, 11 is i our BS However, it is ot idividually ratioal for aget 1, hece it is ot i Zhou s BS We ow defie the bargaiig set proposed by Mas-Colell Ž 1989 It uses differet otios of objectio ad couter-objectio Give a payoff vector x R, a pair S, y with S N ad y R is said to be a Mas-Colell objectio Ž M-objectio to x if ys Ž S ad yi xi for each i S with at least oe strict iequality Give a M-objectio Ž S, y to x, a pair T, z with T N ad z R is said to be a M-couter-objectio to Ž S, y if zt Ž T ad zi yi for all i i T S, ad zi xi for all i i T S c, with at least oe strict iequality for some i T A M-objectio Ž S, y to x is justified if there is o M-couter-objectio to it The Mas-Colell bargaiig set of the cooperative game Ž N,, deoted hereafter by MBSŽ N,, is the set of all payoff vectors agaist which there is o justified M-objectio The MBSŽ N, is o-empty for the class of cooperative games we cosider Furthermore it always cotais efficiet payoff vectors Žsee Eiy et al Ž 1999 We deote by POMBS the set of Pareto optimal payoff vectors i MBS 3 IMPLEMENTATION OF THE BARGAINING SET I this sectio we describe a o-cooperative mechaism whose equilibrium outcomes coicide with the BSŽ N, previously described The

6 IMPLEMENTATION OF BARGAINING SETS 111 mechaism is iteded to solve the problem of a desiger who does ot kow the characteristic fuctio ad yet wats to attai outcomes i the BS The mechaism is to be played by players who are differet from the agets described i Sectio 2 These ew players will be called pricipals There are three pricipals, 1, 2, ad 3, a typical pricipal will be deoted by j The pricipals will compete over the agets via wage offers made to the agets The payoff of a pricipal hirig a subset S of the agets is the differece betwee Ž S, that is, the value of the coalitio of the agets, ad the sum of wages paid out We iterpret this to mea that the desiger hires some maagers to play the game ad offers them the value of the coalitios they hire as paymet, but they have to pay the wages they offer to the agets hired The mechaism H is played as follows Stage 1 Each pricipal j, for j 1, 2, submits a offer x j R, where xi j is the amout pricipal j would pay to aget i if he were employed by her The offers are made simultaeously Each aget is provisioally assiged to the pricipal who submits the highest wage for that aget, Q 1 which is x Max x, x 2 i i i I case of idetical maximal offers, aget i is assiged to pricipal 1 At the ed of stage 1 we have two subsets of agets deoted by S 1 ad S 2, where S j is the set of agets assiged to pricipal j, for j 1, 2 If there exists a pricipal for which S N, the game eds with all agets employed by pricipal, the payoff to aget i is xi, the payoff to pricipal is Ž N x Ž N, ad the other pricipals obtai zero profits If both S 1 ad S 2 are o-empty, the the game moves to stage 2, with all pricipals fully iformed about the outcome of stage 1 Stage 2 Pricipal 3 ca submit a ew offer x 3 R Cosider the set 3 Q 1 2 K i N xi x i If K cotais elemets from both S ad S, the K is assiged to pricipal 3 Otherwise, pricipal 3 is ot assiged ay 3 agets Deote by T the set of agets assiged to pricipal 3 Žeither T K, ort The pricipals 1 ad 2 hire the sets T S T ad T 2 S 2 T 3 The payoff to aget i is xi 3 if i T 3 ad xi Q otherwise The payoffs to the pricipals are ŽT j x j ŽT j, for j 1, 2, 3 The mai purpose of this sectio is to show that the mechaism H implemets the BS i pure strategies That is, the set of subgame perfect equilibria i pure strategies Ž SPE of the o-cooperative mechaism H coicides with the BS Previous to the theorem, we preset the followig two properties of the BS that will be useful i the proof of the implemetatio result

7 112 PEREZ-CASTRILLO AND WETTSTEIN LEMMA 1 S, y is a justified objectio to x R if ad oly if for e ery T N satisfyig coditio Ž BS1 : Ž T yž T S xž T S c Ž 1 Proof It is easy to see from the defiitio of a couter-objectio that there exists a couter-objectio to a objectio Ž S, y if ad oly if there exists a set T N satisfyig coditio Ž BS1 such that Ž T yt Ž S xt Ž S c QED LEMMA 2 If there exists a justified objectio S, y to x R, the there exists a justified objectio Ž S, y to x such that y Ž S Ž S Proof Take a justified objectio Ž S, y The umber of sets T N satisfyig coditio Ž BS1 is fiite ad Eq Ž 1 is satisfied for all of them Therefore, there exists a 0 such that Ž T y Ž T S xt Ž S c for every T N satisfyig coditio Ž BS1, where y i yi for all i N, ad y x for all i S Cosequetly, Ž S, y i i is a justified objectio for which y Ž S Ž S is satisfied QED We ow preset our first mai result THEOREM 1 The mechaism H implemets i SPE the set BSŽ N, Proof Ž a We first prove that ay payoff i BSŽ N, ca be reached as a SPE of H Let x BSŽ N, Cosider the followig strategies At stage 1, prici- 1 2 pals 1 ad 2 submit the offers x x x At stage 2 Žif this stage is reached pricipal 3 makes a offer that guaratees to her the highest possible profit, give the strategies actually played at stage 1 We claim that these strategies costitute a SPE whose outcome is that pricipal 1 hires the whole set N at salaries x ŽNote that if pricipals 1 ad 2 use the previous strategies, stage 2 is ot reached We prove the claim by cotradictio Suppose that there is a profitable deviatio by a pricipal h The deviatio must take place at stage 1, sice by defiitio o profitable deviatio exists for pricipal 3 Deote by y h R the deviatig offer Sice the deviatio gives a strictly positive profit to the pricipal ad xž N Ž N, pricipal h must ed up with a proper subset of N after stage 1, deoted by S h Moreover, yi h xi for all h h Ž h Ž h h h h i S ad y S S Let us take yi yi for all i S, with h Ž h Ž h Ž h h h small eough so that y S S The pair S, y satisfies yi xi h for all i S ad y h ŽS h ŽS h, hece it is a objectio to x Give Ž h that x BS N,, there exists a couter-objectio T, z to S, y h ; that Ž h hc is, T N satisfies BS1 ie, T S ad T S ad z R h h satisfies zt T, zi yi for each i T S ad zi xi for each i T S hc

8 IMPLEMENTATION OF BARGAINING SETS 113 If the deviatio takes place, both pricipals ed up hirig a proper subset of N at stage 1, ad the game goes to stage 2 Pricipal 3 the has the possibility of makig the offer x 3, with xi 3 zi for each i T S hc, 3 h 3 x z for each i T S, ad x for each i T c Ž i i i whe we write x we mea that x is lower tha ay other offer made to i 3 i aget i The strategy x guaratees strictly positive profits to pricipal 3, sice agets hired by pricipal 3 are exactly those i T ad x 3 Ž T Ž T However, it is ot ecessarily the best strategy for pricipal 3 Now, we claim that ay strategy that maximizes pricipal 3 s profits must leave pricipal h with o-positive profits Otherwise, pricipal 3 will obtai higher profits by hirig both of her group of agets ad the remaiig group of agets of pricipal h, at the correspodig salaries, ad she will obtai at least the same profits as the sum of the profits of both pricipals, sice the game is super-additive Therefore, pricipal h caot obtai strictly positive profits Hece, there is o profitable deviatio at stage 1 Ž b Now we prove that ay payoff that is the outcome of a SPE of H is i BSŽ N, We proceed by several steps Throughout the proof, we deote by x F R the wages actually paid to the agets That is, either F Q F Q x x, if the game eds at stage 1, or x Max x, x 3 i i i if the game reaches stage 2 Ž b1 I ay SPE, oe pricipal must hire all the agets We prove the statemet by cotradictio Suppose it is ot the case Sice profits must be o-egative at equilibrium Žotherwise, a pricipal with egative profits could deviate makig offers x i for i 1,,, ot hirig ay agets, it is the case that x F ŽT j ŽT j Žote that if there are more tha oe pricipal hirig agets, stage 2 has ecessarily bee reached Moreover, the super-additivity assumptios imply that Ž N ŽT 1 ŽT 2 ŽT 3, sice at least two sets are o-empty The, we claim that there is a profitable deviatio by pricipal 3: y 3 x F Followig this deviatio, pricipal 3 hires the whole set N, makig profits: Ž N x F Ž N Ž T 1 x F Ž T 1 Ž T 2 x F Ž T 2 Ž T 3 x F Ž T 3 Ž T 3 x F Ž T 3 Ž b2 I ay SPE, all pricipals make zero profits Suppose that it is ot the case, so the pricipal that, at equilibrium, hires all the agets makes positive profits, that is, Ž N x F Ž N Either pricipal 1 or 2 Žor both obtai zero profits at equilibrium Suppose it is pricipal 2 The, she could deviate from her iitial strategy makig a offer yi 2 xi F, for all i N, with 0 but small eough so that Ž N x F Ž N N This esures that pricipal 2 hires all the agets ad makes strictly positive profits

9 11 PEREZ-CASTRILLO AND WETTSTEIN F Q F F b3 I ay SPE, x x ad x N N To prove that x x Q ote, first, that x F x Q by defiitio of x F ad, secod, ay offer 3 Q 3 Q x x is domiated by y x O the other had, x F Ž N Ž N i i i i is a direct cosequece of the property that the pricipal that, at equilibrium, hires all the agets Ž b1 makes zero profits Ž b2 F b The payoff x BSŽ N, Suppose it is ot the case The, by F Lemma 2, there is a justified objectio S, y to x such that Ž S ys Cosider the followig deviatio by pricipal 1: yi 1 yi for all i S ad yi 1 for all i S c We claim that this ew strategy is a profitable deviatio from x 1, thus cotradictig the fact x F is a vector of fial wages F 1 F of a SPE First, sice S, y is a objectio to x, yi xi for all i S, ad the set S is hired by pricipal 1 at stage 1 Secod, pricipal 1 makes strictly positive profits if she keeps S after stage 2, sice Ž S y 1 Ž S Third, give that the deviatio Ž S, y is justified, there is o couter-objectio Ž T, z to Ž S, y ; that is, Ž T y 1 Ž T S x F ŽT S c for ay T satisfyig Ž BS1 Therefore, every offer i which pricipal 3 eds up with a o-empty subset of agets yields egative profits to this pricipal Hece, pricipal 1 keeps the set S at stage 2, so y 1 is a profitable deviatio We have prove that the salaries actually paid to the agets costitute a payoff i BSŽ N, Therefore, the equilibrium outcome costitutes a payoff i BSŽ N, QED The mechaism H is a simple mechaism that allows the desiger to reach the set of outcomes that are i the BS of the game The defiitio of the BS we use is differet from Ž eve if very closely related to other defiitios proposed i the literature The two features of our BS that are crucial for the implemetatio via mechaism H Ž described above are as follows: Ž i I the defiitio of a couter-objectio, it is sufficiet that the agets receive the same paymet that they are offered either i the mai proposal or i the objectio That is, i the coditio Ž BS2 there must be weak iequalities Otherwise, i mechaism H, there does ot exist a maximum for the program of the pricipal called to play at stage 2 Moreover, ad more importatly, this is the characteristic that makes Lemmas 1 ad 2 hold That is, it esures that ay objector has the optio to make strictly positive profits Ž ii The restrictios that the couter-objector faces must be symmetric i the sese that they apply i the same way to the objector ad to the origial proposer, so that it is ot ecessary to idetify the objector i order to operate the mechaism Determiig the idetity of the objector without kowig the characteristic fuctio Žwhich we assume is kow oly by the agets ad the pricipals ad ot by the desiger is very problematic ad is ot possible i mechaism H

10 IMPLEMENTATION OF BARGAINING SETS 115 These two features eable us to make the followig commet If we take a alterative defiitio of the bargaiig set that accords with our defiitio of a objectio, but for which coditio Ž BS1 is differet ad cosider a alterative coditio for Ž BS1 that is symmetric, i the sese that we ca exchage the roles of S ad S c without modifyig the coditio, a mechaism very similar to H implemets this alterative bargaiig set We do ot thik a mechaism alog the lies of mechaism H ca implemet a BS that fails to satisfy requiremet Ž ii Zhou s Ž 199 BS specifies o-symmetric restrictios Therefore we do ot implemet it, although it satisfies Ž i This is the reaso why we adopted a slightly differet defiitio O the other had, the BS proposed by Mas-Colell Ž 1989 does satisfy Ž ii, but it fails to satisfy Ž i I order to implemet the MBS, we first have to modify the secod step of the mechaism so that there is a maximum of the program of pricipal 3 ad secod, to itroduce ew icetives for the pricipals to deviate, eve if they will ot obtai profits Moreover, the fact that a M-objectio requires strict iequalities oly for oe of the deviatig agets also makes the task more difficult The ext sectio shows oe way to deal with the precedig problems IMPLEMENTATION OF THE PARETO OPTIMAL MAS-COLELL BARGAINING SET We ow costruct a ew mechaism H whose SPE outcomes coicide with the set of Pareto optimum payoff vectors i the Mas-Colell bargaiig set Ž POMBS As previously, the players are the pricipals who compete over the agets We ow assume that there are four pricipals We add oe further assumptio that pricipals, all other thigs beig equal, strictly prefer employig more rather tha less agets This ca be viewed as a tie-breakig rule that itroduces lexicographic prefereces whereby the pricipals first criterio is profit maximizatio ad the secod criterio is hirig the largest possible group of agets This assumptio is restrictive but it is ecessary to provide the motivatio for objectios sice a justified M-objectio always yields zero profits 3 3 Similar lexicographic prefereces ca also be foud i the automata literature For example, Abreu ad Rubistei Ž 1988 assume that prefereces over moetary payoffs ad complexity are lexicographic: machie A is strictly preferred to machie B if it yields higher moetary payoff or it yields the same payoff with lower complexity The treatmet of ties has also received attetio i the learig literature, where it is show that differet tie breakig rules lead to very differet outcomes Žsee, for example, Moderer ad Sela Ž 1996

11 116 PEREZ-CASTRILLO AND WETTSTEIN The mechaism cosists of two stages I the first stage, pricipals 1 ad 2 submit wage offers ad, i the secod stage, pricipals 3 ad possibly bid for the right to make a secod proposal, possibly eticig a ew set of agets to work for them The formal descriptio of the mechaism H is as follows Stage 1 Pricipals 1 ad 2 submit simultaeous offers x 1, x 2 R Should oe offer weakly domiate aother, the weakly domiated offer is discarded Amog the remaiig offers, each aget is assiged to the pricipal who submitted the highest wage for that aget, amely x Q i 1 Max x, x 2 i i I case of idetical offers, aget i is assiged to the pricipal whose sum of offers is the smallest; i case of idetical sums, he is assiged to pricipal 1 At the ed of this stage there are two subsets of agets deoted by S 1 ad S 2, where S j is the set of agets assiged to pricipal j If there exists a pricipal for which S N, all agets are employed by pricipal, the payoff to aget i is xi, the payoff to pricipal is Ž N x Ž N, the other pricipals obtai zero profits, ad the game eds Otherwise, we go to stage 2 Ž 3 Stage 2 Pricipals 3 ad simultaeously submit pairs b, x 3 ad Ž 3 3 b, x, where b, b R ad x, x R Deote by the pricipal submittig the largest b j, where ties are broke i favor of pricipal 3 If b 0, pricipal is assiged the empty set I the case where b 0, pricipal pays b to aget 1 ad she is assiged the set K i N Q xi x i Deote by T the set hired by pricipal, either T K or T Pricipals 1 ad 2 employ the sets T j S j T, for j 1, 2 The Q payoff to aget i, for i 1,,, ismaxx, x i i, to which aget 1 adds Ž Ž the bid b The payoffs to the pricipals are T x T b for pricipal, zero for the pricipal differet from ad playig at stage 2, ad ŽT j x j ŽT j, for the pricipals j 1, 2 Prior to statig the theorem we itroduce oe additioal defiitio Give a allocatio x R, let BŽ x represet the maximal gai for a pricipal at stage 2, where x is the vector of maximal offers to the agets at stage 1 That is, BŽ x Sup Ž T zž T T N ad zi xi for all i T Notice that BŽ x 0, give that Ž xž 0 Also ote that if the supremum is achieved for coalitio T, z x for all i T, so BŽ x i i Sup Ž T xt T N The biddig moey ca go to ay aget, or to a third party The oly restrictio is that it should ot go to ay pricipal, sice this could iduce strategic behavior

12 IMPLEMENTATION OF BARGAINING SETS 117 We ow show that the SPE of mechaism H described above coicide with the set of payoff vectors i POMBS THEOREM 2 The mechaism H implemets i SPE the set POMBS N, Proof Ž a We first prove that ay payoff i POMBSŽ N, ca be reached as a SPE of H Let x POMBS We prove that the followig set of strategies costitutes a SPE At stage 1, x 1 x 2 x At stage 2, if 3 it is reached, pricipals 3 ad submit a umber b b BŽ x Q, where Q x is the maximum offer actually made to aget i at stage 1 If BŽ x Q i 0, the the wage offers are irrelevat sice pricipals biddig zero are assiged the empty set by defiitio Moreover, if BŽ x Q 0, each pricipal j, for j 3,, selects the vector x j i such a way that x j x Q ad if she is selected as pricipal, the T i N x x Q i i will be oe amog the largest sets that produce profit BŽ x Q If the previous strategies are followed, pricipal 1 will hire all the agets at stage 1 with zero profits ad the agets will be paid accordig to x To see that the previous strategies costitute a SPE of the mechaism H, let us first aalyze stage 2 The strategy of a pricipal j, for j 3,, i stage 2 Ž Ž Q j j cosists of sedig the message B x, x, where the offer x leaves pricipal j with oe of the coalitios guaratyig profits BŽ x Q The coordiates of x j over the coalitio would coicide with the wage offers each coalitio member holds from the previous stage To see that these messages comprise a Nash equilibrium, cosider a subgame where BŽ x Q 0 Žif BŽx Q 0, the ay deviatio would geerate either o chage or egative profits The outcome of the previous strategies is that pricipal 3 employs oe of the largest possible set of agets while still makig zero profits, sice the bid BŽ x Q is precisely the differece betwee the value of the coalitio hired by the pricipal ad the sum of the salaries paid to this coalitio A higher bid would geerate losses for ay pricipal A lower bid for pricipal 3 would result i pricipal 3 ot hirig ay agets ad thus would make the pricipal worse off, while it could ot chage the outcome if made by pricipal Let us ow aalyze stage 1 Suppose there is a deviatio y h which, whe aouced by pricipal h 1, 2 i stage 1, leads to a strictly preferred outcome for pricipal h This deviatio must leave pricipal h with positive profits if h 1, or with o-egative profits ad a o-empty set of employees if h 2 I either case, followig the deviatio, both pricipals employ o-empty sets of agets at the ed of stage 1 Hece, followig a potetially profitable deviatio, the mechaism moves ito stage 2 Deote by S h the set of agets provisioally employed by the deviatig pricipal at the ed of stage 1

13 118 PEREZ-CASTRILLO AND WETTSTEIN Ž Q Ž Q h h Q We ow show that B x 0 ote that xi yi if i S ad xi xi hc Ž h if i S If S, y h is a M-objectio, there exists a couter-objectio Ž Q h T, z with zt T ; hece B x 0 The deviatio y will ot be a M-objectio whe yi h xi for all i S However, if this is the case the deviator must be pricipal 1, sice pricipal 2 could ot have gaied from Ž 2 such a deviatio this is so because a offer by pricipal 2 with yi xi for all i S ad yi 2 xi for all i S c with at least oe strict iequality is weakly domiated by the offer of pricipal 1, which results i ot hirig ay agets If pricipal 1 is the deviator, this ca be profitable oly if ŽS 1 y 1 ŽS 1 ; hece BŽx Q 0 holds as well Therefore, the bid that pricipals 3 ad will submit at stage 2 is positive As a result of the Nash equilibrium for the game begiig at stage 2, the deviatig pricipal h i stage 1 caot be better off We show that either pricipal h has strictly egative profits or eds up employig o agets at all Suppose that at the ed of stage 2, h employs a o-empty h set T of agets with y h ŽT h ŽT h The, due to the super-additivity assumptio, it is ot possible that the wiig bid ad wage offer proposed i stage 2 were the best respose The coalitio could have bee elarged without decreasig profits Hece, i the case where pricipal h employs ay agets the profits are egative Therefore, there caot be a profitable deviatio i stage 1 Ž b We ow prove that ay payoff that is the outcome of a SPE of H is i the POMBSŽ N, Note that we ca mimic the proofs of steps Ž b1 ad Ž b2 i Theorem 1 to prove the followig two claims First, i ay SPE oe pricipal must hire all the agets Secod, i ay SPE all pricipals make zero profits Give the two previous properties, the sum of paymets by the pricipal who hires all the agets i equilibrium is Ž N, so we obtai efficiecy If this paymet vector does ot belog to the POMBS, there exists a justified objectio Ž S, y A pricipal h 1, 2 ot employig ay agets who deviate at stage 1 offerig yi h yi for i S ad yi h for i S c ca employ the coalitio S Žthe offer y is ot weakly domiated by x ad yž N xž N, so the agets i S are assiged to pricipal h If coalitio S after stage 2 is maitaied, the deviatig pricipal is better off Sice the objectio is justified, ay pricipal biddig b j 0 i stage 2 would make strictly egative profits Hece, the outcome of stage 1 remais, ad the deviatig pricipal has profited This is i cotradictio to the fact that we were at a SPE QED We have costructed the mechaisms implemetig the bargaiig sets uder a strog super-additivity assumptio This hypothesis simplifies the presetatio sice every efficiet outcome ivolves the formatio of the grad coalitio We ow idicate the modificatios eeded whe efficiet

14 IMPLEMENTATION OF BARGAINING SETS 119 outcomes ca also be geerated by partitios other tha the grad coalitio Let us call a efficiet structure a partitio of the set of agets such that the sum of the value of the coalitios i the partitio equals the value of the grad coalitio Deote by D the maximum umber of coalitios a efficiet structure could possibly cotai We itroduce two modificatios i the mechaism H First, we assume that there are at least D 1 pricipals competig over the agets at stage 1 Secod, at stage 2, that is reached if there are at least two pricipals which are provisioally assiged agets at stage 1, a o-empty set hired by pricipal D 2 must itersect with all o-empty sets provisioally assiged to pricipals at stage 1 This mechaism implemets the BS As to mechaism H, if we assume that there are at least D 1 pricipals competig over agets at stage 1, the the mechaism implemets the POMBS 5 CONCLUSIONS We have costructed two simple mechaisms that implemet two bargaiig sets The first is a variatio o the Zhou Ž 199 bargaiig set ad the secod is the Pareto optimal subset of the Mas-Colell Ž 1989 bargaiig set These mechaisms are more straightforward tha those previously suggested by Eiy ad Wettstei Ž 1999 The simplicity is obtaied at the cost of itroducig a auxiliary set of idividuals Žas i Perez-Castrillo Ž 199 The implemetatio of the Pareto optimal subset of the Mas-Colell bargaiig set turs out to be less elegat tha that of the first bargaiig set due to the eed for the pricipals to use a explicit tie breakig rule While oe ca see this as a weakess of the implemetatio of the Mas-Colell bargaiig set, the eed for the rule is actually due to the fact that Mas-Colell justified objectios are always efficiet, that is, they all share the whole value of the coalitio amog its members Hece, a property that seems desirable from a cooperative poit of view iterferes with the o-cooperative implemetatio through our class of mechaisms Appropriately modified mechaisms could similarly implemet other bargaiig sets where the treatmet of objectios ad couter-objectios is symmetric It should be oted that the bargaiig sets of Zhou Ž 199 ad Auma Maschler Ž 196 do ot fall ito this class It might well be that i order to implemet these two bargaiig sets oe would have to resort to more sophisticated costructios

15 120 PEREZ-CASTRILLO AND WETTSTEIN REFERENCES Abreu, D, ad Rubistei, A Ž 1988 The Structure of Nash Equilibrium i Repeated Games with Fiite Automata, Ecoometrica 56, Abreu, D, ad Se, A Ž 1990 Subgame Perfect Implemetatio: A Necessary ad Almost Sufficiet Coditio, J Ecoom Theory 50, Auma, R, ad Maschler, M Ž 196 The Bargaiig Set for Cooperative Games, i Ad aces i Game Theory Ž M Dresher, L S Shapley, ad A W Tucker, Eds Priceto, NJ: Priceto Uiv Press Dutta, B, Ray, D, Segupta, K, ad Vohra, R Ž 1989 A Cosistet Bargaiig Set, J Ecoom Theory 9, Eiy, E, Holzma, R, ad Moderer, D Ž 1999 O the Least Core ad Bargaiig Sets, Games Ecoom Beha 28, Eiy, E, ad Wettstei, D Ž 1999 A No-Cooperative Iterpretatio of Bargaiig Sets, Re Ecoom Desig, to appear Mas-Colell, A Ž 1989 A Equivalece Theorem for a Bargaiig Set, J Math Ecoom 18, Maschler, M Ž 1992 The Bargaiig Set, Kerel, ad Nucleolus, i Hadbook of Game Theory Ž R J Auma ad S Hart, Eds, Vol 1 Amsterdam: Elsevier Sciece Publishers Maschler, M, ad Peleg, B Ž 1976 Stable Sets ad Stable Poits of Set-valued Dyamic Systems with Applicatios to Game Theory, SIAM J Cotrol Optim 1, Moderer, D, ad Sela, A Ž 1996 A 2 2 Game without the Fictitious Play Property, Games Ecoom Beha ior 1, 1 18 Moore, J, ad Repullo, R Ž 1988 Subgame Perfect Implemetatio, Ecoometrica 56, Palfrey, T R Ž 1995 Implemetatio Theory, i Hadbook of Game Theory ŽR J Auma ad S Hart, Eds, Vol 3 Amsterdam: Elsevier Sciece Publishers Perez-Castrillo, D Ž 199 Cooperative Outcomes through No-Cooperative Games, Games Ecoom Beha ior 7, 28 0 Perry, M, ad Rey, P Ž 199 A No-Cooperative View of Coalitio Formatio ad the Core, Ecoometrica 62, Serrao, R Ž 1995 A Market to Implemet the Core, J Ecoom Theory 67, Serrao, R, ad Vohra, R Ž 1997 No-Cooperative Implemetatio of the Core, Soc Choice Welf 1, Stears, R E Ž 1968 Coverget Trasfer Schemes for N-perso Games, Tras Amer Math Soc 13, 9 59 Vid, K Ž 1992 Two Characterizatios of Bargaiig Sets, J Math Ecoom 21, Vohra, R Ž 1991 A Existece Theorem for a Bargaiig Set, J Math Ecoom 20, 19 3 Zhou, L Ž 199 A New Bargaiig Set of a N-Perso Game ad Edogeous Coalitio Formatio, Games Ecoom Beha ior 6,