Games and Economic Behavior

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1 Games and Economc Behavor Contents lsts avalable at ScVerse ScenceDrect Games and Economc Behavor Markov equlbra n a model of barganng n networks Dlp Abreu a, Mha Manea b, a Department of Economcs, Prnceton Unversty, Unted States b Department of Economcs, MIT, Unted States artcle nfo abstract Artcle hstory: Receved 7 May 200 Avalable onlne 22 September 20 JEL classfcaton: C7 D6 D85 L4 We study the Markov perfect equlbra MPEs of an nfnte horzon game n whch pars of players connected n a network are randomly matched to bargan. Players who reach agreement are removed from the network wthout replacement. We establsh the exstence of MPEs and show that MPE payoffs are not necessarly unque. A method for constructng pure strategy MPEs for hgh dscount factors s developed. For some networks, we fnd that all MPEs are asymptotcally neffcent as players become patent. 20 Elsever Inc. All rghts reserved. Keywords: Barganng Decentralzed markets Equlbrum exstence Ineffcency Markov perfect equlbrum Networks Random matchng. Introducton Many markets nvolve buyers and sellers of relatonshp specfc products and servces. The partculartes of these relatonshps may derve from locaton, technologcal compatblty, ont busness opportuntes, free trade agreements, socal contacts, etc. Such markets are naturally modeled as networks and the structure of the network determnes the nature of economc nteracton between the agents who form the nodes of the network. For example, magne a group of employers who have needs for dfferent tasks and a group of workers wth dstnct sets of sklls. The lnks between workers and employers depend on how sklls translate nto the necessary tasks and other factors such as physcal locaton and socal relatonshps Granovetter, 973. In another applcaton, a group of supplers for nstance, laptop component manufacturers offer exclusve commtments to a group of upstream producers. Another s the case of lcensng arrangements beng negotated between basc technology provders dfferent platforms for developng computer software or smart phone applcatons, for example and fnal product desgners. Compatblty ssues lmt the connectons between these groups and lead to non-trval, ncomplete networks. Our formal settng s as follows. We consder a network where each par of players connected by a lnk can ontly produce a unt surplus. The network generates the followng nfnte horzon dscrete tme barganng game. Each perod a lnk s selected accordng to some probablty dstrbuton, and one of the two matched players s randomly chosen to make We thank Drew Fudenberg for extensve dscussons and Fuhto Koma and Al Roth for helpful comments. We also thank the edtor and referees for useful suggestons. * Correspondng author. E-mal addresses: dabreu@prnceton.edu D. Abreu, manea@mt.edu M. Manea /$ see front matter 20 Elsever Inc. All rghts reserved. do:0.06/.geb

2 2 D. Abreu, M. Manea / Games and Economc Behavor an offer to the other player specfyng a dvson of the unt surplus between themselves. If the offer s accepted, the two players ext the game wth the shares agreed on. If the offer s reected, the two players reman n the game for the next perod. In the next perod the game s repeated on the subnetwork nduced by the set of remanng players. All players have a common dscount factor. We assume that players have perfect nformaton of the events precedng any of ther decson nodes n the game. In ths envronment the followng questons arse: How are the relatve strengths of the frms affected by the pattern of compatbltes that s, the network structure? Whch partnershps are possble n equlbrum and on what terms? Is an effcent allocaton of the processes achevable n equlbrum? We address these ssues n the context of Markov perfect equlbra MPEs. Fudenberg and Trole 99 and Maskn and Trole 200 present arguments for the relevance of MPEs. The natural noton of a Markov state n the class of models we consder s the network nduced by players who have not reached agreement, along wth the selecton of a lnk and a proposer. We prove that an MPE always exsts and demonstrate by example that MPE payoffs are not necessarly unque. Exstence of MPEs s establshed va a fxed-pont argument rather than by explct constructon. Fndng one MPE for a gven network structure s typcally a complex exercse due to the smultaneous determnaton of the pars of players reachng agreement n equlbrum f matched to bargan n the frst perod of the game and the evoluton of the network structure as agreements are realzed and play proceeds. We provde a method to construct pure strategy MPEs for hgh dscount factors based on conectures about the set of lnks across whch agreement may obtan n every subnetwork. We offer an example where no MPE of the barganng game s effcent even asymptotcally as players become patent. Ths leads naturally to the queston of whether an asymptotcally effcent non-markovan subgame perfect equlbrum always exsts. In a companon paper, Abreu and Manea forthcomng, we answer the queston affrmatvely. Even though MPEs may be neffcent, Markov strateges are stll essental to our constructon of asymptotcally effcent subgame perfect equlbra. The buldng block of the constructon s an MPE of a modfed game that dffers from the orgnal one prmarly n prohbtng neffcent agreements. Manea 20 assumes that players who reach agreement are replaced by new players at the same postons n the network. The barganng protocol s dentcal to the one of the present paper. The two models dffer n strategc complexty. In the model of Manea 20 barganng opportuntes are statonary over tme. In the current paper the evoluton of the network, occasoned by agreements and the departure of nvolved partes, plays a central role. Our modelng strategy has been to allow for full generalty of the network structure whle keepng other elements of the model relatvely smple. Nevertheless, two aspects of the model deserve dscusson. One s the assumpton that the surplus any par of players can generate s ether zero or one. In fact t would not be dffcult to work wth a more general and less symmetrc model. However, the assumpton that all lnks have the same value s useful n analyzng partcular examples and allows us to characterze relatve barganng strengths n terms of the network structure. Another restrctve assumpton s that only one lnk s chosen for barganng n every round. We provde ustfcaton for ths assumpton below. Nevertheless, the assumpton may also be relaxed. Our man results generalze to settngs wth varyng gans from trade and multple smultaneous matches. There s an extensve lterature on barganng n markets startng wth Rubnsten and Wolnsky 985. Important subsequent papers nclude Gale 987, Bnmore and Herrero 988 and Rubnsten and Wolnsky 990. The focus s on the relatonshp between the equlbrum outcomes of varous decentralzed barganng procedures and the compettve equlbrum prce as the costs of search and delay become neglgble. The varous stochastc matchng processes consdered n ths lterature treat all buyers and respectvely all sellers anonymously. The analogue of ths modelng assumpton n our settng s the specal case of buyer seller networks n whch every buyer s connected to every seller. For such networks, the payoffs n any MPE of our barganng game converge to the compettve equlbrum outcome, as players become patent. However, n our analyss the network s arbtrary. In partcular, some pars of buyers and sellers are not connected and cannot trade. Snce barganng encounters are restrcted by network connectons, the compettve equlbrum analyss does not apply. Polansk 2007 studes a model smlar to ours, but wth a fundamentally dfferent matchng technology. He assumes that a maxmum number of pars of connected players are selected to bargan every perod. In that settng Polansk obtans a payoff characterzaton whch s neatly related to the classcal Galla Edmonds decomposton. As a consequence of the maxmum matchng assumpton, effcency s not an ssue n Polansk s model n contrast to our work and furthermore, n equlbrum, all matched pars reach agreement mmedately. In our completely decentralzed matchng process a fundamental tenson emerges between the global structure of effcent matchngs n a network and the local nature of ncentves for trade. Even n smple examples, asymptotcally neffcent outcomes arse n equlbrum. We also obtan rcher dynamcs for the evoluton of network structure due to the fact that not all matches lead to trade n equlbrum. As mentoned earler, the tools we develop can be extended to deal wth settngs where more than one lnk s chosen for barganng n every round. An alternatve ratonale for the one-match-per-perod assumpton s as follows. In terms of the essental analytcs, what matters s that multple agreements are not reached at the same nstant. If we take the underlyng temporal realty to be contnuous and consequently assume that matchng takes place n contnuous tme then the probablty that several matches occur smultaneously s zero under natural condtons on the matchng process. In ths vew our assumpton s ndeed approprate. Polansk and Wnter 200 consder a model where buyers and sellers connected by a network are matched to bargan accordng to a protocol smlar to ours. The crtcal dfference s that players do not ext the game upon reachng agreements.

3 D. Abreu, M. Manea / Games and Economc Behavor Although every player can make several transactons over tme, players are assumed to behave as f they derve utlty only from ther next transacton. Coromnas-Bosch 2004 analyzes a model n whch buyers and sellers alternate n makng publc offers that may be accepted by any of the responders connected to a specfc proposer. As n Polansk 2007, the matchng process specfes that when there are multple possbltes to match buyers and sellers that s, there are multple agents proposng or acceptng dentcal prces the maxmum number of transactons takes place. Kranton and Mnehart 200 study trade n networks n a model based on centralzed smultaneous auctons. The rest of the paper s organzed as follows. In Secton 2 we defne the model and establsh exstence of MPEs. Secton 3 provdes examples of MPEs n some smple networks. Secton 4 suggests an approach to computng MPEs. We show that the MPEs are not necessarly payoff equvalent and that asymptotcally effcent MPEs do not always exst n Sectons 5 and 6, respectvely. Secton 7 concludes. 2. Framework Let N denote the set of n players, N ={, 2,...,n}. Anetwork s an undrected graph H = V, E wth set of vertces V N and set of edges also called lnks E {, V } such that, E whenever, E. We dentfy the lnks, and,, and use the shorthand or nstead. We say that player s connected n H to player f E. Weoften abuse notaton and wrte H for V and H for E. Aplayerssolated n H f he has no lnks n H. Anetwork H = V, E s a subnetwork of H f V V and E E. AnetworkH = V, E s the subnetwork of H nduced by V f E = E V V.WewrteH V for the subnetwork of H nduced by the vertces n V \ V.EverynetworkH has an assocated probablty dstrbuton over lnks p H H wth p H>0, H whch descrbes the matchng process. Let G be a fxed network wth vertex set N. Alnk n G s nterpreted as the ablty of players and to ontly generate a unt surplus. 2 Consder the followng nfnte horzon barganng game generated by the network G. LetG 0 = G. Each perod t = 0,,... a sngle lnk n G t s selected wth probablty p G t and one of the players the proposer and s chosen randomly wth equal condtonal probablty to make an offer to the other player the responder specfyng a dvson of the unt surplus between themselves. If the responder accepts the offer, the two players ext the game wth the shares agreed on. If the responder reects the offer, the two players reman n the game for the next perod. In perod t + the game s repeated wth the set of players from perod t, exceptfor and n case perod t ends n agreement, on the subnetwork G t+ nduced by ths set of players n G. HenceG t+ = G t {, } f players and reach an agreement n perod t, and G t+ = G t otherwse. We assume that all players have perfect nformaton of all the events precedng any of ther decson nodes n the game. 3 All players share a dscount factor δ 0,. The barganng game s denoted Γ δ G. There are three types of hstores. We denote by h t a hstory of the game up to not ncludng tme t, whch s a sequence of t pars of proposers and responders connected n G, wth correspondng proposals and responses. We call such hstores, and the subgames that follow them, complete. A complete hstory h t unquely determnes the set of players Nh t partcpatng n the game at the begnnng of perod t; denotebygh t the subnetwork of G nduced by Nh t.letg be the set of subnetworks of G nduced by the players remanng n any subgame, G = h t Gh t, and defne G 0 = G \{G}. Wedenotebyh t ; the hstory consstng of h t followed by nature selectng to propose to. Wedenote by h t ; ; x the hstory consstng of h t ; followed by offerng x [0, ] to. A strategy σ for player specfes, for all complete hstores h t and all players such that Gh t,theoffer σ h t ; that makes to after the hstory h t ;, and the response σ h t ; ; x that gves to after the hstory h t ; ; x. We allow for mxed strateges, hence σ h t ; and σ h t ; ; x are probablty dstrbutons over [0, ] and {Yes, No}, respectvely. A strategy profle σ = σ N s a subgame perfect equlbrum of Γ δ G f t nduces Nash equlbra n subgames followng every hstory h t ; and h t ; ; x. The equlbrum analyss s smplfed f we restrct attenton to Markov strateges. The state at a certan stage s gven by the subnetwork of players who dd not reach agreement by that stage, along wth the selecton of a lnk and a proposer. Then the only feature of a complete hstory of past barganng encounters that s relevant for future behavor s the network nduced by the remanng players followng that hstory. That s, for all complete hstores h t and all lnks Gh t,the offer σ h t ; that makes to depends only on Gh t,,, and s response σ h t ; ; x to the offer x from depends only on Gh t,,, x. 4 A Markov perfect equlbrum MPE s a subgame perfect equlbrum n Markov strateges. 5 We frst establsh exstence of MPEs. Note the flexblty of the matchng protocol. In one appealng specfcaton, all lnks are equally lkely to generate a match. In another specal case, each player s drawn wth equal probablty and then one of hs lnks s chosen unformly at random. 2 We do not exclude networks n whch some players are solated. 3 The requrements on the nformaton structure may be relaxed for the case of Markov perfect equlbra. 4 Formally, a Markov strategy profle σ satsfes the followng condtons σ h t ; = σ h t ;, σ h t ; ; x = σ h t ; ; x for all h t, h t wth Gh t = Gh t, forevery Gh t and x [0, ]. 5 In other accounts Maskn and Trole, 200; Mertens, 2002, the concepts defned here would be referred to as statonary Markov strateges and statonary Markov perfect equlbrum.

4 4 D. Abreu, M. Manea / Games and Economc Behavor Proposton. There exsts a Markov perfect equlbrum for the barganng game Γ δ G. For the proof, we frst provde a characterzaton of MPE payoffs, and then use t to show that an MPE always exsts. Fx δ 0,. For a set of networks H, a collecton of Markov strategy profles σ H H H for the respectve games Γ δ H H H s subgame consstent f for every par of networks H, H H, σ H and σ H nduce the same behavor n any par of dentcal subgames of Γ δ H and Γ δ H. 6 Suppose σ δ G s an MPE of Γ δ G. By the defnton of an MPE, t must be that σ δ G belongs to a subgame consstent collecton of MPEs σ δ G G G of the respectve games Γ δ G G G wth correspondng payoffs v δ G G G.Inpartcular, when Γ δ G s played accordng to σ δ G, everyplayerk has expected payoffs v δ G at the begnnng of any subgame k before whch no agreement has occurred, and v δ G {, } at the begnnng of any subgame before whch only and k reached an agreement k,. Fx a hstory h t ; along whch no agreement has been reached Gh t = G. In the subgame followng h t ;, t must be that the strategy σ δ G specfes that player accept any offer larger than δv δ G, and reect any offer smaller than δv δ G. Then t s not optmal for to make an offer x >δv δ G to, snce would be better off makng some offer n the nterval δv δ G, x nstead, as accepts such offers wth probablty. Hence, n equlbrum has to offer at most δv δ G wth probablty, and may accept wth postve probablty only offers of δv δ G. Letq be the probablty condtonal on h t ; of the ont event that offers δv δ G to and the offer s accepted. The payoff of any player k, at the begnnng of the next perod s v δ k G {, } n case and reach an agreement, and v δ k Therefore, the tme t expected payoff of k condtonal on the hstory h t ; s qδv δ k If δv δ G otherwse. G {, } + qδv δ k G. G+ v δ G <, when s chosen to propose to, t must be that n equlbrum offers δv δ G and agreement obtans wth probablty,.e., q =. For, f q < then s expected payoff condtonal on offerng δv δ G s q δv δ G+ qδv δ G< δv δ G, whle condtonal on offerng δv δ G + ε ε > 0 s δv δ offers greater than δv δ G wth probablty. But for small ε > 0, q δv δ t s not optmal for to offer δv δ G to. By the same token, offers smaller than δv δ are reected wth probablty and yeld expected payoff δv δ G< δv δ than δv δ G may be optmal for ether. Therefore, f δv δ G + v δ to s equlbrum strategy. We establshed that f δv δ then q = 0. If δv δ G + v δ G = thenq canbeanynumbernthenterval[0, ]. Consder the correspondence f :[0, ] n [0, ] n defned by f v = { q δv δ G {, }, δv,δv + qδv }{{}}{{}}{{}, G ε we argued that accepts G + qδv δ G< δv δ G ε. Hence G are not optmal for snce they G ε. We already argued that no offer greater G < andq <, then cannot have a best response G + v δ G < thenq =. Smlarly, f δv δ G + v δ G > q = 0 f δv + v < >, and q [0, ] f δv + v = }, 2. where δv δ G {, }, δv,δv }{{}}{{}}{{}, represents the vector n [0, ] n wth the k, coordnate equal to δv δ k G {, }, coordnate equal to δv, and coordnate equal to δv. Note that f v δ G s the set of possble tme t expected payoffs for player k condtonal on k the hstory h t ;, where the behavors of and are constraned by the equlbrum analyss above. Let f :[0, ] n [0, ] n be the correspondence defned by f v = { G} 2 p G f v. Let h t be a hstory along whch no agreement has occurred, and consder the resultng perod t subgame. Snce nature selects player to make an offer to player wth probablty p G/2 foreachlnk G, and condtonal on the selecton, f v δ G descrbes the tme t expected payoffs constraned by the equlbrum requrements, f v δ G s the set of expected payoffs at the begnnng of the subgame h t consstent wth our partal equlbrum analyss when players behave accordng to σ δ G. In equlbrum, the tme t expected payoff vector condtonal on the hstory h t s v δ G, hence More precsely, subgame consstency of σ H H H requres that σ Hh t ; = σ H h t ; and σ Hh t; ; x = σ H h t ; ; x for all pars of players,, alloffersx, all complete hstores h t and h t such that the players remanng n the subgame h t of Γ δ H and the subgame h t of Γ δ H nduce dentcal networks whch nclude the lnk, and all H, H H.

5 D. Abreu, M. Manea / Games and Economc Behavor v δ G f v δ G. Therefore, v δ G s a fxed pont of f. Conversely, we show n Appendx A that any fxed pont of f s an MPE payoff vector. Lemma. A vector v s a Markov perfect equlbrum payoff of Γ δ G f and only f there exsts a subgame consstent collecton of Markov perfect equlbra of the games Γ δ G G G 0 wth respectve payoffs v δ G G G 0 such that v s a fxed pont of the correspondence f defned by In Appendx A, we use a bootstrap approach to construct an MPE for any Γ δ G G G based on a subgame consstent famly of MPEs σ δ G {, } G for the barganng games Γ δ G {, } G. We establsh that the correspondence f derved from the payoffs of the latter famly of MPEs has a fxed pont, whch by Lemma translates nto an MPE of Γ δ G. The proof proceeds by nducton on the number of vertces n G. Remark. It s straghtforward to extend the proof of Proposton to a settng wth heterogeneous lnk values. Remark 2. We can also generalze the exstence result to the case n whch multple pars of players are matched to bargan smultaneously. In the general specfcaton of the matchng protocol, a collecton of parwse dsont proposer responder pars s drawn at each date from a probablty dstrbuton whch depends only on the underlyng network at that date. We assume that a publc randomzaton devce s avalable n ths settng. The addtonal steps necessary for the proof are outlned n Appendx A. 3. Examples of MPEs In ths secton we provde examples of MPEs for some smple networks. We assume throughout that all lnks are equally lkely to be selected for barganng n the ntal network and n any subnetwork that may arse n subgames. That s, the probablty dstrbuton ph s unform across the lnks n H for all networks H. We manly focus on equlbrum payoffs. Strateges may be constructed as n the proof of Lemma. Consder frst a star network, where one player controls the barganng opportuntes of all others. Formally, n the star of n network G star n player s connected to each of the players k = 2,...,n. Proposton 4 n Rubnsten and Wolnsky 990 shows that the barganng game Γ δ G star n has a unque subgame perfect equlbrum whch turns out to be Markovan. In the equlbrum, agreement s obtaned n the frst match. It s easy to see that the payoffs satsfy the equatons v = 2 δv 2 + δv v 2 = v k = v 2 The soluton s 2n δv + δv 2 k = 3,...,n. v δ G star n = n δ n2 δ 2 and v δ k G star n = δ n2 δ 2 for k = 2,...,n. As δ, the equlbrum payoffs converge to /2 forbothplayersfn = 2, and to for player and 0 for all other players when n 3. Consder next a lne network, n whch players are located on a lne and can only bargan wth ther mmedate neghbors. Formally, n the lne of n network G lne n player k s connected to player k + fork =,...,n. Computng MPEs of the barganng game for lne networks s feasble for two man reasons. Frst, all the connected components nduced by the players remanng n any subgame are lne networks. Second, the number of conectures about what frst perod agreements are possble n equlbrum s relatvely small because each player has at most 2 neghbors. The networks G lne 2 and G lne 3 are somorphc 8 to G star 2 and G star 3 respectvely. Consder now the barganng game on the lne of 4 network, Γ δ G lne 4. If players 2 and 3 reach the frst agreement, then and 4 are left dsconnected and receve zero payoffs. If players and 2 3 and 4 reach the frst agreement, then 3 and 4 and 2 nduce a subnetwork somorphc to G lne 2 n the ensung subgame, and obtan expected payoffs of /2 n the next perod. One can then easly show that n any MPE the pars of players, 2 and 3, 4 reach agreements wth probablty when matched to bargan n the frst perod. For low δ there s a unque MPE of Γ δ G lne 4. In any subgame, every match ends n agreement. By the proof of Proposton and by symmetry, the equlbrum payoffs solve the followng system, 7 Recall that G 0 denotes the set of subnetworks of G, dfferentfromg, nduced by the players remanng n any subgame of Γ δ G. 8 Two networks H = V, E and H = V, E are somorphc f there exsts a becton g : V V such that E gg E.

6 6 D. Abreu, M. Manea / Games and Economc Behavor δv2 + δv δ/2 v = 3 2 v 2 = δv2 + δv v 3 = v 2, v 4 = v. The unque soluton s gven by v δ G lne 4 = v δ 4 G lne 4 = δv3 + δv δ/ δ 2δ2 23 δ, v δ 2 G lne 4 = v δ 3 G lne 4 = 2 + 3δ 2δ2. 23 δ There s an MPE wth payoffs as above only f the soluton satsfes δv δ 2 G lne 4 + v δ 3 G lne 4. The latter nequalty s equvalent to δ δ.945, where δ s the unque root n the nterval [0, ] of the polynomal 8 8x 3x 2 + 2x 3. For hgh δ, theresno MPE of Γ δ G lne 4 n whch players 2 and 3 agree wth probablty when matched to bargan wth each other. In such an equlbrum players and 4 would be weak recevng zero payoffs n subgames followng agreements between 2 and 3, makng the patent players 2 and 3 powerful to an extent that prevents them from reachng an agreement wth each other. Also, there exsts no MPE n whch players 2 and 3 dsagree wth probablty when matched to bargan. In such an equlbrum all players would receve payoffs smaller than /2, and players 2 and 3 would have ncentves to trade. For δ>δ, there exsts an MPE of Γ δ G lne 4 n whch players 2 and 3 reach agreement wth some probablty q δ 0, condtonal on ther lnk beng selected for barganng. 9 As n the proof of Proposton, we need the equlbrum payoffs of players 2 and 3 to satsfy δv δ 2 G lne 4 + v δ 3 G lne 4 =. By symmetry, the equlbrum payoffs solve the followng system, v = δv2 + δv + q δ δv δ/2 v 2 = δv2 + δv δv δ/2 δv 2 + v 3 =, v 3 = v 2, v 4 = v. The unque soluton s gven by v δ G lne 4 = v δ 4 G 6 + 5δ + 2δ2 lne 4 =, 2δ 2 v δ 2 G lne 4 = v δ 3 G lne 4 = 2δ q δ = 29 2δ + δ2 + 2δ 3. δ 6 + 5δ + 2δ 2 Note that, as players become patent, the condtonal probablty of agreement between players 2 and 3 converges to 0 and thempepayoffsconvergeto/2 for each player. The ntuton s that players 2 and 3 could obtan payoffs greater than /2 n the lmt only by extortng players and 4 va the threat of an agreement across the lnk 2, 3, whch would leave and 4 dsconnected. Yet, players 2 and 3 cannot reach an agreement f ther lmt equlbrum payoffs are larger than /2. Smlarly, there exsts an MPE of the barganng game on the lne of 6 network, Γ δ G lne 6, n whch as δ goes to, the common probablty of frst perod agreement across the lnks 2, 3 and 4, 5 vanshes, whle agreement obtans wth probablty across all other lnks. All players receve expected payoffs of /2 n the lmt. For the lne of 5, 7, 8, 9,... networks, and other more complex ones, computng MPE payoffs for the barganng game for every δ may be a dffcult task. For such networks, the next secton nvestgates lmt MPE payoffs and agreement probabltes as players become patent. 4. Lmt propertes of MPEs Fx a network G. A payoff vector v s a lmt MPE payoff of Γ δ G as δ f there exsts a famly of MPEs of the games Γ δ G δ 0, wth respectve payoffs v δ δ 0, such that v = lm δ v δ.thental agreement probabltes nduced by a Markov strategy σ are descrbed by q G, where q s the probablty that and reach agreement under σ condtonal on beng matched to bargan n the frst perod of the game wth ether player n the role of the proposer. A collecton q G represents lmt MPE ntal agreement probabltes for Γ δ G as δ f there exsts a famly of MPEs of the games Γ δ G δ 0, wth respectve ntal agreement probabltes q δ G such that q = lm δ q δ for all G. For varous network structures, we can use a bootstrap approach to drectly compute lmt MPE payoffs and agreement probabltes as players become patent. We then construct MPEs of Γ δ G for hgh δ that generate the determned lmt 9 The probabltes that 2 accepts an offer from 3 and that 3 accepts an offer from 2 are not pnned down by the MPE requrements. Only the average q δ of the two condtonal probabltes s relevant for MPE payoff computaton. There exst multple MPEs, all payoff equvalent, as explaned n footnote 23.

7 D. Abreu, M. Manea / Games and Economc Behavor payoffs and agreements as δ. As n Proposton, we use known lmt MPE payoffs n subgames Γ δ G for G G 0 n order to characterze equlbrum behavor n Γ δ G. Suppose that for every δ 0, we specfed a subgame consstent famly of MPEs for the barganng games Γ δ G {, } G wth respectve payoffs v δ G {, } G. Fx a profle of ntal agreement probabltes q δ G for every dscount factor δ. We set out to construct an MPE for Γ δ G that generates the frst perod agreement probabltes q δ and leads to the payoffs v δ G {, } n subgames that nduce the subnetwork G {, }. By the proof of Proposton, the MPE payoffs solve the n n lnear system of equatons, 0 v k = { k G} k =,n. 2 p kq δ k δv + G {k} p q δ δv δ k G {, } + { k G} 2 p kq δ k G {k} p q δ δv k, Contrary to appearances, the equatons above do not assume that the probablty of an agreement between and k s splt evenly between the events that or k plays the role of the proposer. The splt s not unque only f q δ 0,, n whch case k the MPE payoffs should satsfy δv = δv k. Then the exact allocaton of the total probablty of agreement p k q δ k between the terms δv and δv k does not affect the expresson on the rght-hand sde. See also footnote 23. Assume that for all G, q δ and v δ G {, } converge to q and v G {, }, respectvely, as δ goes to. Consder the lnear system obtaned by takng the lmt δ n 4., v k = { k G} k =,n. 2 p kq k v + G {k} p q v k G {, } + { k G} 2 p kq k G {k} p q v k, The next result descrbes the relatonshp between the solutons of the two lnear systems and provdes suffcent condtons under whch solutons to the latter system consttute lmt MPE payoffs. Proposton 2. Suppose that lm δ q δ = q and lm δ v δ G {, } = v G {, } for all G. For parts 2 4, assume addtonally that q > 0 for at least two lnks G. Then the followng statements hold The system 4. has a unque soluton, denoted v δ,qδ. 2 The system 4.2 also has a unque soluton, denoted v q. 3 The solutons satsfy lm δ v δ,qδ = v q. 4 If q {0, } for all Gandv q satsfes the condtons v q + vq < f q = and v q + vq > f q = 0, then there exsts δ < such that for every δ δ, there s an MPE of Γ δ G wth payoffs v δ,q and ntal agreement probabltes q. The proof appears n Appendx A. Remarks and 2 also apply here. The next secton provdes an llustraton of Proposton 2. We have also appled the result to determne lmt MPE payoffs and ntal agreement probabltes for the barganng games on the lne of 5, 7, 8,...,2 networks. Fg. summarzes lmt MPE outcomes for all lne networks wth at most 2 players. In ths and subsequent dagrams, for every network, lmt MPE payoffs for each player are represented next to the correspondng node. Each lnk s drawn as a thn, dashed, or thck lne segment dependng on whether the probablty of frst perod agreement across that lnk n MPEs for hgh δ s 0, a number n 0, then the lmt probablty as δ s mentoned next to the lnk, 2 or, respectvely. Note that the propertes that lmt MPE payoffs are /2 for all players and that lmt probabltes of frst perod agreement across lnks k, k + are 0 and for k even and odd, respectvely, do not extend to lnes wth an even number of players greater than 6. In the lmt MPE for the lne of 8 network llustrated n Fg., players 4 and 5 do not reach an agreement when matched to bargan wth each other n the frst perod. However, the barganng game does not reduce to two ndependent lne of 4 games snce players 4 and 5 have ncentves to trade n subgames followng ntal agreements across the lnks 2, 3 and 6, 7. Indeed, ether of the latter agreements leaves 4 and 5 n a subnetwork somorphc to a lne of 5, where all matches result n mmedate agreement. A frst perod agreement between players 2 and 3 6 and 7 leads to lmt contnuaton payoffs of approxmately.72 and.793 for players 4 5 and 5 4, respectvely, and of.069 for player 6 3. Player 4 5 explots 3 6 s vulnerablty and obtans an expected lmt payoff greater than /2. Consequently, for hgh dscount factors, players 4 and 5 do not have ncentves to reach an agreement wth each other n the frst perod. Proposton 2 does not characterze lmt MPEs n whch the probablty of an agreement across some lnks dffers from 0 and. 3 Relatedly, the result does not cover the possblty that v q + v q = for some lnk G. Bypart3of 0 To smplfy notaton, we wrte p for p G. See, for example, the lnk 4, 5 n the lne of 8 network from Fg.. 2 For some lnks the ntal agreement probabltes for δ< may be postve, and converge to 0 as δ, as n the case of the lnk 2, 3 n G lne 4. 3 In Secton 3 we dscussed networks n whch the MPE probabltes of agreement for hgh δ are dfferent from 0 and, but converge to 0 or as δ. Secton 5 detals an example n whch for some lnks even the lmt MPE agreement probabltes belong to 0,.

8 8 D. Abreu, M. Manea / Games and Economc Behavor Fg.. Lmt MPE payoffs and ntal agreements for the barganng games on the lne of 2, 3,..., 2 networks. Proposton 2, v q + v q = mples that lm δ δv δ,qδ techncal challenge s that n general we cannot nfer whether δv δ,qδ + v δ,qδ = for any famly q δ δ that converges to q as δ. The + v δ,qδ s smaller than, equal to, or greater than. As the proof of Proposton demonstrates, the latter comparson drves the ncentves for agreements n MPEs wth the structure assumed above. The followng example clarfes that the strct nequaltes from part 4 of Proposton 2 cannot be replaced by weak ones. Consder the 4-player network G tr+pont from Fg. 4 n Secton 6, and assume that all lnks are chosen for barganng wth equal probablty. Proposton 4 establshes that for every δ the game Γ δ G tr+pont has a unque MPE, n whch agreement obtans wth probablty across each lnk. Let q be the profle of ntal agreement probabltes gven by q 2 = q 34 = and q 23 = q 24 = 0. The correspondng lmt system 4.2 wth obvous specfcatons for lmt MPE payoffs n subgames followng an agreement has the unque soluton v q = vq 2 = vq 3 = vq 4 = /2. In partcular, vq + v q = forall G tr+pont. Ths s the equalty case left unaddressed by Proposton 2. Indeed, as Proposton 4 shows, q does not descrbe lmt MPE ntal agreement probabltes and v q does not defne lmt MPE payoffs for Γ δ G tr+pont. 5. Multple MPE payoffs Multple MPE payoffs may exst for the barganng game on some networks for hgh dscount factors. One example s the barganng game Γ δ G sq+lne 3, on the network G sq+lne 3 depcted n Fg. 2. Proposton 3. There exsts δ < such that for every δ δ, the game Γ δ G sq+lne 3 has at least three MPEs that are parwse payoff nequvalent. Proof. We ntend to use Proposton 2 to show that for hgh δ the game Γ δ G sq+lne 3 admts an MPE n whch the condtonal probablty of agreement n the frst perod s 0 across the lnk, 4 and for all other lnks. To defne the subgame consstent collecton of MPEs σ δ G sq+lne 3 {k, k + } k=,2,...,6 necessary for 4. and 4.2, note that the frst agreement may nduce the followng subgames. If players and 2 2 and 3 reach the ntal agreement, then the remanng players 3, 4, 5, 6, 7, 4, 5, 6, 7 nduce a subgame on a network somorphc to the lne of 5 network. If players 3 and 4 4 and 5 reach the frst agreement, the nduced subnetwork has two connected components, parttonng the set of remanng players nto {, 2} and {5, 6, 7} {, 2, 3} and {6, 7}. Players and 2 6 and 7 are then nvolved n a subgame smlar to the barganng game on the lne of 2 network, wth lower matchng frequences, snce they are not matched to bargan when the lnk 5, 6 or 6, 7, 2 or 2, 3 s selected for barganng. Smlarly players 5, 6, 7, 2, 3 are nvolved n a verson of the barganng game on the lne of 3 network wth dfferent matchng frequences. For both

9 D. Abreu, M. Manea / Games and Economc Behavor Fg. 2. Three sets of lmt MPE payoffs and ntal agreements for Γ δ G sq+lne 3. Fg. 3. Lmt MPE payoffs and ntal agreements for the barganng game on G sq top left, G sq+pont top rght, and a network somorphc to G lne 5. varatons of the barganng games on the lne of 2 and 3 networks the lmt MPE payoffs are dentcal to those n the respectve benchmark versons. If players 5 and 6 reach the frst agreement, then players, 2, 3, 4 nduce a subgame equvalent to the barganng game onthesquarenetwork,g sq, and player 7 s left dsconnected. If players 6 and 7 reach the ntal agreement, then players, 2, 3, 4, 5 nduce a subgame equvalent to the barganng game on the square plus pont network, G sq+pont. The lmt MPE payoffs and ntal agreements for G sq, G sq+pont, and a network somorphc to the lne of 5 are summarzed n Fg. 3. The lmt lnear system 4.2 for Γ δ G sq+lne 3 wth the conectured profle of ntal agreement probabltes s as follows, v = 7 2 v + v / / / v v 2 = 7 2 v 2 + v v 2 + v / / v 2 v 3 = 7 5/ v 3 + v v 3 + v / v 3 v 4 = 7 23/ / v 4 + v v 4 + v / v 4 v 5 = 7 2/ / v 5 + v v 5 + v v 5

10 0 D. Abreu, M. Manea / Games and Economc Behavor v 6 = 7 23/ / / v 6 + v v 6 + v v 6 v 7 = 7 5/ / / v 7 + v v 7. In each equaton the terms correspond n order to the selecton for barganng of the lnks k, k + for k =, 2,...,6, followed by the lnk, 4. The unque soluton s gven by 4 v.235, v 2.759, v 3.79, v 4.792, v 5.069, v 6.793, v The soluton satsfes the condtons from Proposton 2 v + v 4 > andv + v < foralllnks dfferent from, 4, so for hgh δ there exsts an MPE of Γ δ G sq+lne 3 wth the assumed agreement structure and payoffs approachng the values above as δ. The rough ntuton for ths equlbrum specfcaton s that odd labeled players are relatvely weak and even labeled players are relatvely strong n the barganng game Γ δ G sq+lne 3 for hgh δ. 5 However, the asymmetrc behavor of players and 3 n frst perod matches wth 4 places player n a better poston than 3. Odd labeled players are relatvely weak and even labeled players are relatvely strong n the barganng games on G lne 3, G lne 5, and G sq+pont.players2, 4, and 6 occupy the postons correspondng to even labels n the latter networks followng some ntal equlbrum agreements. The sgnfcant dfference between the payoff of player 2 and the almost dentcal payoffs of players 4 and 6 s due to the ntal agreement between players 3 and 4, whch undermnes player 2 s poston. When 3 and 4 reach the frst agreement, 2 s left n a blateral barganng game wth, whch leads to a lmt payoff of /2 for player 2. Ths dmnshes the effect of strong even postons for player 2 n the three types of subnetworks enumerated earler. Smlarly, player s better off than player 3. Although players and 3 have symmetrc postons n the network, s at an advantage over 3 snce ntal agreement obtans across the lnk 3, 4, but not across, 4. Player 7 s slghtly weaker than 3 because, as argued above, player 7 s only neghbor, player 6, s sgnfcantly stronger than one of player 3 s neghbors, player 2. Fnally, player 5 s the weakest of all odd labeled players because hs central poston s nferor to the perpheral postons of the other odd players n the subnetworks somorphc to the lne of 5 network nduced by ntal agreements across the lnks, 2 and 2, 3. Players and 4 are reluctant to reach the frst agreement wth each other because each of them can beneft from watng to be matched wth a weaker neghbor. It s possble for all other pars of payers to trade when matched to bargan n the frst perod snce no other two relatvely strong players n the constructed MPE wth odd and even labels are lnked n the network. The conectured agreement structure s self-enforcng and leads to an MPE for hgh δ. Note that players and 3 hold symmetrc postons n G sq+lne 3, but they play asymmetrc roles n the MPE constructed above. We can obtan another MPE for hgh δ by smply nterchangng the roles of players and 3 n the postulated agreement structure. The payoffs of players and 3 dffer between the two pure strategy MPEs for hgh δ. For suffcently hgh δ, Γ δ G sq+lne 3 has a thrd MPE, n whch there s a common probablty n the nterval 0, of frst perod agreement across the lnks, 4 and 3, 4. 6 The lmt MPE agreement probabltes are q 4 = q and q = for all other lnks.thelmtmpepayoffsare v = v 3.2, v 2.755, v 4.789, v 5.068, v 6.796, v Therefore, for hgh δ the barganng game Γ δ G sq+lne 3 has at least three MPEs. Note that the mxed strategy MPE s not payoff equvalent wth ether of the pure strategy MPEs for any player. 6. Ineffcent MPEs Let μg denote the maxmum total surplus that can be generated n the network G. That s, μg s the cardnalty of the largest collecton of parwse dsont lnks n G. 8 To generate the maxmum total surplus μg n Γ δ G as δ, pars of players connected by lnks that are neffcent n the nduced subnetworks n varous subgames need to refran from reachng agreements. However, provdng ncentves aganst agreements that are collectvely neffcent s dffcult. Some players may be concerned that passng up barganng opportuntes can lead to agreements nvolvng ther potental barganng partners whch undermne ther poston n the network n future barganng encounters. Indeed, one can fnd networks for whch all MPEs of the barganng game are asymptotcally neffcent as players become patent. Consder the network G tr+pont llustrated n Fg. 4, wth a unform probablty dstrbuton governng the selecton of lnks for barganng. Assume that δ s close to so that the welfare cost of delay between consecutve matches s neglgble. The maxmum total surplus n ths network s 2 and t can be acheved n the lmt as δ only f both pars, 2 4 The exact soluton nvolves rreducble fractons wth 8-dgt denomnators. 5 The words weak and strong vaguely mean payoffs sgnfcantly below and respectvely above /2. 6 The proof s smlar to that of Proposton 2. 7 The value of the lmt probablty s one of the four roots of an rreducble polynomal of degree 4. 8 Ths and related terms are defned formally n Abreu and Manea forthcomng.

11 D. Abreu, M. Manea / Games and Economc Behavor Fg. 4. Asymptotcally neffcent MPEs for the barganng game on G tr+pont. and 3, 4 reach agreement. It s clearly neffcent for player 2 to trade wth ether player 3 or 4 because ths would leave the remanng players solated and create only one unt of surplus. Proposton 4 below establshes that for any δ 0,, Γ δ G tr+pont has a unque MPE. In the MPE every par reaches agreement when matched to bargan. Snce 2, 3 or 2, 4 are matched frst wth probablty /2, the expected total surplus generated by the MPE approaches /2 + /2 2 = 3/2 < 2 = μg tr+pont as δ. Note that usng Proposton 2, we can mmedately evaluate the lmt MPE payoffs to be /56.96 for player, 5/8 =.625 for player 2, and 9/ for players 3 and 4. One nterestng feature of ths example s that G tr+pont s not unlaterally stable wth respect to the lmt MPE payoffs. 9 Indeed, f player 4 severed hs lnk wth player 2, the lne of 4 network would ensue, and player 4 s lmt MPE payoff would ncrease from 9/56 to /2. 20 Thus player 4 would be better off f he could credbly commt to never trade wth player 2. Proposton 4. For every δ 0,, the game Γ δ G tr+pont has a unque MPE. In the MPE agreement occurs wth probablty across every lnk selected for barganng n the frst perod. Proof. We show that for every δ 0,, all MPEs of Γ δ G tr+pont nvolve agreement wth condtonal probablty for every par of players matched to bargan n the frst perod. Fx a dscount factor δ 0, and an MPE σ of Γ δ G tr+pont. Denote by v the expected payoff of player under σ. We frst argue that agreement occurs under σ wth probablty n the frst perod f the lnk, 2 or 3, 4 s selected for barganng. We only treat the case of the former lnk, as the latter s smlar. The strategy profle σ determnes a dstrbuton over ont outcomes for players and 2, where an outcome for a gven player specfes the tme of an agreement nvolvng that player and the share he receves. For every realzaton of agreements under σ, the sum of the correspondng dscounted payoffs for players and 2 s not greater than. Indeed, when 2 reaches an agreement wth, the sum of the undscounted payoffs of the two players s ; when 2 reaches an agreement wth 3 or 4, player 2 s undscounted payoff cannot exceed and player s s 0 an agreement between 2 and 3 or 4 solates ; f player 2 never reaches an agreement, then both and 2 receve 0 payoffs; thus the expected dscounted payoffs of and 2 satsfy v + v 2. Therefore, δv + v 2 <, and hence players and 2 reach an agreement under σ f matched to bargan n the frst perod of the game. Let p and q denote the probabltes of frst perod agreement across the lnks 2, 3 and 2, 4, respectvely condtonal on the respectve lnk beng selected for barganng. We next show by contradcton that p = q. Wthout loss of generalty, assume that p > q. It must be that p > 0, q <. Hence δv 2 + v 3 δv 2 + v 4,sov 3 v 4. It can be easly seen that the payoffs satsfy δ δv 3 + δv δv 2 + δv 3 + qδv 3 v 3 = 4 v 4 = 4 δ δv 4 + δv 3 + δv 4 + pδv In each of the two sums, the frst term represents the contnuaton payoffs of /2 receved by players 3 and 4 condtonal on the lnk, 2 beng selected for barganng n the frst perod. The second term corresponds to an agreement between 6. 9 See Jackson and Wolnsky 996 and Manea 20 for defntons of stablty. 20 In general, to apply the concept of stablty consstently we would need to use an equlbrum selecton crteron for networks wth multple payoff non-equvalent MPEs as n Secton 5. However, ths ssue s nconsequental for the current argument, snce both Γ δ G lne 4 and Γ δ G tr+pont have unque MPE payoffs for every δ.

12 2 D. Abreu, M. Manea / Games and Economc Behavor players 3 and 4 when matched to bargan. Here we use the fact that the selecton of the lnks, 2 and 3, 4 leads to trade under σ.theterm δv 2 + δv 3 /2 appears n the evaluaton of the payoff of player 3 because under σ, f δv 2 + v 3 <, then player 3 offers δv 2 when selected to make an offer to 2 and player 2 accepts wth condtonal probablty ; f δv 2 + v 3 =, then player 3 obtans a contnuaton payoff of δv 2 = δv 3 when selected to make an offer to 2 regardless of whether the offer s accepted or reected. Smlarly, the thrd term n the expresson for v 4 can be explaned by the nequalty δv 2 + v 4. The last terms n the two equatons reflect the probabltes of agreements that player 2 reaches wth 4 3, leavng player 3 4 solated. Snce δv 2 + v 3, p > q, v 3 > 0, we have that δ δv 3 + δv δv 2 + δv 3 + qδv 3 v 3 = 4 > 4 δ δv 3 + δv 4 + δv 3 + pδv Puttng together 6.2 and 6.3, we obtan that v 4 v 3 < 3 p 4 δv 4 v 3. Ths leads to a contradcton, as δ 3 p 4 < and v 4 v 3 0. We have establshed that p = q. It s easy to check that f p = q = 0thenv = v 2 = v 3 = v 4 < /2, and hence δv 2 + v 3 <, contradctng p = 0. Therefore, p = q > 0. Assume, by contradcton, that p = q <. Usng arguments smlar to those above, t can be argued that the payoffs solve v = 4 2 δv + δv pδv + δ 2 v 2 = 4 2 δv 2 + δv + 2δv 2 + δ 2 v 3 = δ δv 3 + pδv v 4 = v 3. For every p [0, ], the unque soluton of the system of lnear equatons above s gven by 4 + δ 3δ 2 v p = 24 δ4 δ + 2δ 2 + δp8 5δ 4 + δ 3δ 2 + 2δp + δ v 2 p = 24 δ4 δ + 2δ 2 + δp8 5δ + δ v 3 p = v 4 p = 24 δ2 p. Note that the expresson v 2 p + v 3 p v 2 0 v 3 0 can be smplfed to δ + δp4 δ2 δ + δp4 3δ 22 δ4 δ2 p4 δ4 δ + 2δ 2 + δp8 5δ, whch s non-postve. Hence δv 2 p + v 3 p δv v 3 0 < forallp [0, ]. Therefore, the equlbrum payoffs satsfy δv 2 + v 3 <, whch contradcts p <. We thus need p = q =. It can be mmedately verfed that the strateges n whch player offers δv when chosen to make an offer to and player accepts offers greater than or equal to δv and reects smaller offers defne an MPE. The arguments above establsh that ths consttutes the unque MPE. Proposton 4 leaves open the possblty that effcency mght be attanable as δ when non-markovan strateges are consdered. A companon paper Abreu and Manea, forthcomng shows that ths s ndeed the case. The canoncal specfcaton of asymptotcally effcent equlbra for arbtrary networks s delcate. Interestngly, a key smplfcaton s

13 D. Abreu, M. Manea / Games and Economc Behavor acheved by defnng MPEs of a modfed barganng game n whch agreements that would lead to neffcency are prohbted by fat. The overall strateges nvolve non-markovan threats and rewards to sustan the artfcal prohbtons, wthn a completely non-cooperatve and subgame perfect equlbrum constructon. Thus the current analyss of MPEs plays an unexpectedly crtcal role n our constructon of asymptotcally effcent equlbra. 7. Concluson Networks are ubqutous and have been the subect of much scholarly attenton n recent years Jackson, 2008 offers an excellent overvew. However, there has been lmted analyss of decentralzed trade n a network settng. Models of decentralzed barganng n networks provde a natural framework to nvestgate the connecton between network structure, feasble agreements, and the dvson of the gans from trade. In the model ntroduced here we establsh the exstence of MPEs and show that MPEs are not necessarly unque. We relate the propertes of MPEs to features of the underlyng network and provde a method to construct MPEs. Fnally, we demonstrate that n some networks MPEs are ncompatble wth effcent trade even asymptotcally as players become patent or the tme between matchngs goes to zero. Ths robust fndng motvates our companon paper Abreu and Manea, forthcomng, whch focuses on the constructon of asymptotcally effcent hence, n general, non-markovan equlbra. Nevertheless, that equlbrum constructon has a strong Markovan flavor as t reles on MPEs of a modfed barganng game, the exstence of whch s premsed on arguments developed here. Many open questons reman, ncludng the analyss of network structures whch lead to multplcty or neffcency of MPEs. It s unclear at ths stage whether useful characterzatons are attanable. Another nterestng drecton s to endogenze the matchng process. 2 The latter undertakng entals qualtatve changes n the model structure. These are ntrgung topcs for future research. Appendx A Proof of Lemma. We establshed the only f part after the statement of Proposton. To prove the f part, suppose that the subgame consstent collecton of MPEs σ δ G G G 0 of the games Γ δ G G G 0 wth respectve payoffs v δ G G G 0 defnes the correspondence f by , and that v f v. It follows that v = { G} 2 p Gz, where z f v. Then, there exsts q such that z = q δv δ G {, }, δv,δv + q δv, }{{}}{{}}{{}, wth q = 0 f δv + v <> and q [0, ] f δv + v =. The strategy profle σ δ G defned below consttutes an MPE wth payoffs v. We frst defne the strateges for hstores h t along whch at least one agreement occurred. Recall that Gh t denotes the network nduced by the players remanng n the subgame h t. Construct the tme t strategy of each player accordng to the date 0 behavor specfed by σ δ Gh t. 22 For hstores along whch no agreement has occurred, σ δ G specfes that when s chosen to propose to he offers mn δv,δv, and when has to respond to an offer from he accepts wth probablty any offer greater than δv, accepts wth probablty q an offer of δv, and reects wth probablty any smaller offers. 23 The subgame consstency of the collecton σ δ G G G 0 guarantees that under the constructed σ δ G the expected payoffs n any subgame of Γ δ G wth nduced network G G 0 are v δ G, and that σ δ G s an MPE wth expected payoffs v. Contnuaton of the proof of Proposton. We use Lemma to show the exstence of MPEs. We prove more generally that there exsts a subgame consstent collecton of MPEs for the games Γ δ G G Gn, where Gn denotes the subset 2 For nstance players mght be able to expend resources to ncrease the lkelhood of barganng encounters and perhaps to drect the search at specfc partners. 22 Formally, σ δ Gh t ; = σ δ Gh t h 0 ; and σ δ Gh t ; ; x = σ δ Gh t h 0 ; ; x for all Gh t and x [0, ], whereh 0 denotes an empty hstory. 23 Payoff rrelevant MPE multplcty may arse for two reasons. Frst, f δv + v >, when s selected to propose to, n the constructon above offers mn δv,δv = δv to and the offer s reected. The strateges may be modfed so that offers any mxed offer x <δv, as reecton obtans regardless f we specfy that reect offers of δv wth probablty the constrant becomes x δv. Second, when δv + v =, we stpulated that s offer to s accepted wth probablty q,and s offer to s accepted wth probablty q.ifq + q 0, 2 then the equlbrum constructon may be modfed so that the two agreement probabltes become q + ε and q ε, respectvely, for a range of values of ε.