Bilateral Bargaining with Externalities

Transcription

1 Unversty of Toronto From the SelectedWorks of Joshua S Gans October, 2007 Blateral Barganng wth Externaltes Catherne C de Fontenay, Melbourne Busness School Joshua S Gans Avalable at:

2 Blateral Barganng wth Externaltes * by Catherne C. de Fontenay and Joshua S. Gans Unversty of Melbourne Frst Draft: 12 th August, 2003 Ths Verson: 30 th October, 2007 Ths paper provdes an analyss of a non-cooperatve parwse barganng game between agents n a network. We establsh that there exsts an equlbrum that generates a coaltonal barganng dvson of the reduced surplus that arses as a result of externaltes between agents. That s, we provde a non-cooperatve justfcaton for a cooperatve dvson of a non-cooperatve surplus. The resultng dvson s akn to the Myerson- Shapley value wth propertes that are partcularly useful and tractable n applcatons. We demonstrate ths by examnng frm-worker negotatons and buyer-seller networks. Journal of Economc Lterature Classfcaton Number: C78. Keywords. barganng, Shapley value, Myerson value, networks, games n partton functon form. * We thank Roman Inderst, Stephen Kng, Roger Myerson, Arel Pakes, Arel Rubnsten, Mchael Schwarz, Jeff Zwebel, semnar partcpants at New York Unversty, Rce Unversty, the Unversty of Auckland, the Unversty of Calforna-San Dego, the Unversty of Sydney, the Unversty of Southern Calforna, the Unversty of Toronto, the Unversty of Washngton (St Lous), Wharton, Yale Unversty, partcpants at the Australan Conference of Economsts, the Australasan Meetngs of the Econometrc Socety, the Internatonal Industral Organzaton Conference, and, especally, Anne van den Nouweland and three anonymous referees for helpful comments. Responsblty for all errors les wth the authors. Correspondence to: J.Gans@unmelb.edu.au. The latest verson of ths paper s avalable at

3 1. Introducton There are many areas of economcs where market outcomes are best descrbed by an ongong sequence of nterrelated negotatons. When frms negotate over employment condtons wth ndvdual workers, patent-holders negotate wth several potental lcensors, and when competng frms negotate wth ther supplers over procurement contracts, a network of more or less blateral relatonshps determnes the allocaton of resources. To date, however, most theoretcal developments n non-cooperatve barganng have ether focused on the outcomes of ndependent blateral negotatons or on multlateral exchanges wth a sngle key agent. The goal of ths paper s to consder the general problem of the outcomes that mght arse when many agents bargan blaterally wth one another and where negotaton outcomes are nterrelated and generate external effects. Ths s an envronment where (1) surplus s not maxmzed because of the exstence of those external effects and the lack of a multlateral mechansm to control them; and (2) dstrbuton depends upon whch agents can negotate wth each other. Whle cooperatve game theory has developed to take nto account (2) by consderng payoff functons that depend on the precse poston of agents n a graph of network relatonshps, t almost axomatcally rules out (1). In contrast, non-cooperatve game theory embraces (1) but restrcts the envronment consdered symmetry, two players, small players, etc. to avod (2). Here we consder the general problem of a set of agents who negotate n pars. All agents may be lnked, or certan lnks may not be possble for other reasons (e.g., anttrust laws preventng horzontal arrangements among frms). Our envronment s such that pars of agents negotate over varables that are jontly observable. Ths mght be a jont acton such as whether trade takes place or an ndvdual acton undertaken by one agent but observed by the

4 2 other (e.g., effort or an nvestment). We specfy a non-cooperatve game whereby each par of agents n a network bargans blaterally n sequence. Parwse negotatons utlze an alternatng offer approach where offers and acceptances are made n antcpaton of deals reached later n the sequence. Moreover, those negotatons take place wth full knowledge of the network structure and how terms relate to that structure should t change. Specfcally, the network may become smaller should other pars of agents fal to reach an agreement. We consder a stuaton n whch the precse agreement terms cannot be drectly observed outsde a par. Thus, agents can observe the network of potental agreements but not the detals of agreements they are not a party to. Ths s a reasonable assumpton n a number of appled settngs. In a labor market settng, ths would be akn to frms observng the employment levels n rvals but not wages or hours. In a wholesale market, ths s akn to rval supplers observng competng product lnes beng sold downstream but not exact quanttes or supply terms. Wth some restrctons on belefs, there s a unque equlbrum outcome of the ncomplete nformaton game. That outcome nvolves agents negotatng actons that maxmze ther jont surplus (as n Nash barganng) takng all other actons as gven. Hence, wth externaltes, outcomes are what mght be termed blaterally effcent rather than socally effcent. The equlbrum set of transfers also gves rse to a precse structure; namely, a payoff that depends upon the weghted sum of values to partcular coaltons of agents. Ths has a coaltonal barganng structure but wth several mportant dfferences. Frst, the presence of externaltes means that coaltons do not maxmze ther surplus, as equlbrum actons are blaterally effcent rather than socally effcent. Second, coaltons may mpose externaltes on other coaltons; thus, the partton of the whole space s relevant. Thus, the equlbrum outcome

5 3 s a Shapley allocaton generalzed to partton functon spaces (as n Myerson 1977b) and further to networks n those partton spaces, but over a surplus that s characterzed by blateral rather than socal effcency. 1 Thrd, the restrcted communcaton space may gve rse to further neffcences, f certan agents are mssng lnks between them and cannot negotate, but nstead choose ndvdually optmal actons (see Jackson and Wolnsky, 1996). In sum, we have a non-cooperatve equlbrum that s a generalzed Shapley dvson of a non-cooperatve surplus, whch s easy to use n appled settngs. To our knowledge, no smlar smple characterzaton exsts n the lterature for a mult-agent barganng envronment wth externaltes. The usefulness of ths soluton to appled research seems clear. The semnal paper n the theory of the frm, Hart and Moore (1990), assumes that agents receve the Shapley value n negotatons; capturng the mpact of substtutablty wthout the extreme solutons of other concepts such as the core. However, there s an nherent dscomfort to applyng Shapley values n non-cooperatve settngs, because Shapley values assume that groups always agree to maxmze ther surplus, even n the presence of externaltes. As a result, the theory of the frm has lmted the types of strategc nteractons that can be studed. 2 1 In the absence of externaltes, t reduces to the Myerson value, and f, n addton, the network s complete, t reduces to the Shapley value. 2 Stole and Zwebel (1996) examned an envronment where a frm bargans blaterally wth a gven set of workers. Whle ther treatment s for the most part axomatc, focusng on a natural noton of stable agreements, they do post an extensve form game for ther envronment. In ther extensve form game, there s a fxed order n whch each worker bargans wth the frm over the wage for a unt of labor (.e. there s no acton space). Any gven negotaton has the worker and frm takng turns n makng offers to the other party that can be accepted or rejected. Rejected offers brng wth them an nfntesmal probablty of an rreversble breakdown where the worker leaves employment forever. Otherwse, a counter-offer s possble. If the worker and frm agree to a wage (n exchange for a unt of labour), the negotatons move on to the next worker. The twst s that agreements are not bndng n the sense that, f there s a breakdown n any blateral negotaton, ths automatcally trggers a replayng of the sequence of negotatons between the frm and each remanng worker. Ths new subgame takes place as f no prevous wage agreements had been made (reflectng a key assumpton n Stole and Zwebel s axomatc treatment that wage agreements are not bndng and can be renegotated by any party at any tme). Stole and Zwebel (1996, Theorem 2) clam that ths extensve form game gves rse to the Shapley value as the unque subgame perfect equlbrum outcome (somethng they also derve n ther axomatc treatment).

6 4 Ths game also allows us to contrbute to the modelng of buyer-seller networks. Up untl now, the papers addressng ths ssue have needed to restrct ther attenton to envronments wth a restrctve network structure, such as common agency, or to an envronment wth no competton n downstream markets. 3 Our soluton combnes the ntutveness and computablty of Shapley values wth the consequences of externaltes for effcency. As such, t s capable of general applcaton n these envronments. 4 The paper proceeds as follows. In the next secton, we ntroduce our extensve form game. The equlbrum outcomes of that game are charactersed n Sectons 3 and 4; frst wth the equlbrum outcomes as they pertan to actons and then to dstrbuton. Secton 5 then consders partcular economc applcatons ncludng wage barganng wth competng employers and buyer seller networks. A fnal secton concludes. 2. Barganng Game There are N agents and a graph, L (the network), of connectons between them. Each lnked par, L, has assocated wth t a jont acton, x X, 5 where X s a compact nterval of the reals. We normalze x so that f a par s not lnked, L, then x Each However, we demonstrate below that the nformatonal structure between dfferent blateral negotatons must be more precsely specfed (Stole and Zwebel mplctly assume that the precse wage that s pad to a worker s not observed by other workers), and certan specfc out of equlbrum belefs specfed, for ther result to hold. As wll be apparent below, our extensve form barganng game s a natural extenson of thers to more general economc envronments. 3 For example, Cremer and Rordan, 1987; Kranton and Mnehart, 2001; Inderst and Wey, 2003; and Prat and Rustchn, There s also a lterature on neffcences that arse n non-cooperatve games wth externaltes (see, for example, Jehel and Moldovanu, 1995). The structure of our non-cooperatve game s of a form that elmnates these and we focus, n partcular, on equlbra wthout such neffcences. As such, that lterature can be seen as complementary to the model here. 5 The acton here s lsted here as a scalar but could easly be consdered to be a vector. 6 For example, f x s an acton that s taken only by, chooses the optmal level for ther own payoff, whch we normalze to zero. The extenson to acton spaces n whch the optmal level depends on the actons of others s trval, as wll be seen n Theorems 1 and 2: wll choose ts best response, holdng as gven all other actons.

7 5 agent,, has a payoff functon, u ( x ) j N t, where the frst term s a utlty functon and t s a transfer (postve or negatve) from to j. The utlty functon s a strctly concave and contnuously dfferentable functon of the vector of all jont actons nvolvng ; but we mpose no structure on the utlty to of actons not nvolvng (that s, externaltes) except to assume that N u (x) s globally concave n x. Fx an exogenous orderng of lnked pars. 7 When ts turn n the order comes, each par,, negotates over ( x, t ). The parwse barganng game s descrbed below. Importantly, t s assumed that, f there s an agreement n that game, only and j can observe the agreed ( x, t ), however, t s assumed that breakdowns between pars s common knowledge. As a breakdown wll sever a par s lnk, a new network state wll arse (e.g., f s negotatons break down, the new network s K L ). Formally, t s ths network state that s common knowledge. We follow Stole and Zwebel (1996) and assume that agreements are non-bndng wth respect to a change n network state. Thus, n the event of a breakdown, any agreement between a par stll lnked on the new network state can be unlaterally re-opened. In the model, we presume that the negotaton game s smply repeated for the new network state, because one party wll always fnd t attractve to renegotate. Crtcally, however, t s the antcpaton of equlbrum outcomes n renegotaton subgames that plays a crtcal role n negotated outcomes n the ntal network state. Ths modelng choce effectvely assumes some contractual ncompleteness wth respect to a change n the network state. 8 An alternatve approach would be to assume, followng Inderst 7 In the equlbrum we focus on, the precse orderng wll not matter. 8 Many contracts contan clauses that allow for renegotaton when a materal change n crcumstances arses.

8 and Wey (2003), 9 that ntal negotatons are not just over x ( L), t ( L ) that would arse n the ntal network state but also over each x ( K), t ( K ) for all possble network states, K L, where K. That s, agreements would be network contngent and bndng. It turns out that the equlbrum of nterest that we analyze below arses n both the non-bndng and bndng cases. For expostonal ease, we focus on the non-bndng case and demonstrate the extenson to the bndng case n the appendx. Barganng for each par proceeds accordng to the Bnmore, Rubnsten and Wolnsky (1986) protocol. Frst, or j are randomly selected to be the proposer and makes an offer, based on the current network state K, x ( K), t ( K ) whch the recever can accept or reject. Acceptance closes the negotatons and the next parwse negotaton n the order begns. We assume that pror to any offer beng made, there s an exogenous probablty, 1, that negotatons between a par ceases and no agreement can be made, otherwse the negotaton 6 game between and j begns agan. 10 Thus, rejecton may also trgger a breakdown n negotatons n whch case ths becomes common knowledge and, as past agreements are nonbndng, a new order and round of negotatons between all pars n K on results where s arbtrarly close to 1. begns. We wll focus 9 Inderst and Wey (2003) model multlateral negotatons as occurrng smultaneously; any agent nvolved n more than one negotaton delegates one agent to bargan on ther behalf n each negotaton. Ths alternatve specfcaton may be approprate for stuatons where negotatons take place between frms. Agents could not observe the outcomes of negotatons they were not a party to. Ths would avod the need to specfy belefs precsely n any equlbrum. As our model apples more generally than just between frms, we chose not to rely on a smlar specfcaton here. Note, also, that Inderst and Wey s treatment of ndvdual negotatons s axomatc rather than a full extensve-form, as they merely post that agents splt the surplus from negotatons n each dfferent contngency. In an extensve form game, one would also have to model how and why pars choose to negotate over contngences that are very unlkely to arse. 10 Note that usually n such games, rejecton trggers a breakdown possblty. Here, for techncal rather than substantve reasons, we adopt a symmetrc conventon that exogenous breakdowns that sever future agreement possbltes between the par can occur pror to an offer beng made. Ths has the same mpact as the alternatve assumpton but for the possblty of a breakdown pror to any offer beng made.

9 7 The game concludes when all pars on a gven network state have reached an agreement or there are no lnked pars left. In ths case, all agents receved ther agreed payments and choose ther contracted actons (f any) wth unlnked pars choosng actons and transfers of 0. Coaltonal value and effcency As antcpated n the ntroducton, the equlbrum we focus on from ths barganng game gves rse to payoffs that reflect those found n coaltonal game theory. For that reason, t s useful to provde addtonal notaton to reflect coaltonal value. For a gven network, K, the resultng equlbrum set of actons, x ˆ( K), leads to agent payoffs whch sum to a coaltonal value, ( ˆ x ( K )). When a subset of agents, S N, are lnked only to each other we wll u N also consder the sub-coalton value, ( ˆ x ( K )). u S An mportant concept n ths paper s blateral effcency, defned as follows: Defnton (Blateral Effcency). For a gven graph, K, a vector of actons, x ˆ( K) xˆ ( K), xˆ ( K),..., xˆ ( K), xˆ ( K),... xˆ ( K) satsfes blateral effcency f and only f: n 23 n 1, n xˆ ˆ xˆ ˆ arg max u x, ( K) \ x ( K) u x, ( K) \ x ( K) K xˆ ( ) x j K f. 0 K Under our concavty and contnuty assumptons, xˆ ( K) exsts and s unque for every K. Consstent wth ths defnton, we defne vˆ( S, K) u ( ˆ( K)) x as the coaltonal value to a set S of players (lnked only to each other) when actons are blaterally effcent. 11 Note that the values v ˆ(.) are unque gven our concavty assumptons on u (.). It s useful to note that, n some stuatons, blateral effcency wll concde wth the effcent outcome normally presumed n coaltonal game theory. Specfcally, t s easy to see 11 Note that there s a dstncton between these coaltonal values and those normally analyzed n coaltonal game theory. In coaltonal game theory, the sum of utltes n a coalton would descrbe a characterstc functon where the actons were chosen to maxmze coaltonal value. Here, we defne coaltonal value wth respect to an equlbrum set of actons arsng from our non-cooperatve barganng game. S

10 8 that f there were no externaltes so that for each, u ( x ) was ndependent of x jk for all jk j, k, and a complete network, then maxmzng parwse utltes would result n a maxmzaton of the sum of all utltes of agents lnked n the network. Feasblty Dependng on the nature of the externaltes, and the structure of barganng, an agent may be better off wthout one of ther lnks, and therefore mght unlaterally trgger a breakdown. 12 To make our analyss tractable, we need to restrct the underlyng envronment to rule ths out, n any state of the network (N, L). Stole and Zwebel (1996) term ths feasblty: Defnton (Feasble Payoffs). An equlbrum set of payoffs u ( ˆ( )) ˆ x L k ( ) k t K N s N feasble f and only f, for any K L and any K, u ( x ˆ( K )) t ˆ ( K ) u ( x ˆ( K )) t ˆ ( K ). kn k kn In what follows, we smply assume that the prmtves of the envronment are such that feasblty s assured; after characterzng the equlbrum, we provde a smple suffcent condton for feasblty to hold. 13 However, for any gven applcaton, feasblty s somethng that would have to be confrmed n order to drectly apply our equlbrum characterzaton below. If t dd not hold, then our barganng game wll have an equlbrum where not all lnks would be mantaned; resultng n nterestng predctons n some envronments. Belef structure Gven that our proposed game nvolves ncomplete nformaton, the game potentally allows for many equlbrum outcomes. We need to mpose some structure on out of 12 For nstance, as Maskn (2003) demonstrates, when an agent may be able to free rde upon the contrbutons and choces of other agents, that agent may have an ncentve to force breakdowns n all ther negotatons so as to avod ther own contrbuton. Maskn shows that ths s the case for stuatons where there are postve externaltes between groups of agents (as n the case of publc goods). 13 Although t s always satsfed n envronments where there are no externaltes.

11 9 equlbrum belefs that allows us to characterze a unque equlbrum for a gven underlyng envronment. Ths s an ssue that has drawn consderable attenton n the contractng wth externaltes lterature (McAfee and Schwartz, 1994; Rey and Vergé, 2004). It s not our ntenton to revst that lterature here. Suffce t to say that the most common assumpton made about what players beleve about actons that they do not observe s the smple noton of passve belefs. We wll utlze t below. To defne t, let {( xˆ ( K), tˆ ( K)) } be a K L L set of equlbrum agreements between all negotatng pars. Defnton (Passve Belefs). When receves an offer from j of x ( K) xˆ ( K) or t ( K) tˆ ( K), does not revse ts belefs regardng any other unobserved acton n the game. At one level, ths s a natural belef structure that mmcs Nash equlbrum reasonng. 14 That s, f s belefs are consstent wth equlbrum outcomes as they would be n a perfect Bayesan equlbrum then under passve belefs, t holds those belefs constant off the equlbrum path. At another level, ths s precsely why passve belefs are not appealng from a game-theoretc standpont. Specfcally, f receves an unexpected offer from an agent t knows to be ratonal, a restrcton of passve belefs s tantamount to assumng that makes no nference from the unexpected acton (e.g., by sgnalng). Nonetheless, as we demonstrate here, passve belefs play an mportant role n generatng tractable and nterpretable results from our extensve form barganng game; smplfyng the nteractons between dfferent blateral negotatons. 3. Equlbrum Outcomes: Actons In explorng the outcomes of ths non-cooperatve barganng game, t s useful to focus 14 McAfee and Schwartz (1995, p.252) noted that: one justfcaton for passve belefs s that each frm nterprets a devaton by the suppler as a tremble and assumes trembles to be uncorrelated (say, because the suppler apponts a dfferent agent to deal wth each frm). Smlarly, the passve belefs equlbrum n ths paper s tremblng hand perfect n the agent perfect form. A proof of ths s avalable from the authors.

12 10 frst on the equlbrum actons that emerge before turnng to the transfers and ultmate payoffs. Of course, the equlbrum descrbed s one n whch actons and transfers are jontly determned. It s for expostonal reasons that we focus on each n turn. Theorem 1. Suppose that agents hold passve belefs, and that feasblty holds for each K L. Gven ( NL,, ) as 1, any perfect Bayesan equlbrum nvolves each L takng the blaterally effcent actons, xˆ ( L ). Ths result says that actons are chosen to maxmze parwse utlty holdng those of others as gven. It s easy to see that, n general, the outcome wll not be effcent. 15 The ntuton behnd the result s subtle. Consder a par, and j, negotatng n an envronment where all other pars have agreed to the equlbrum choces n any past negotaton, there s one more addtonal negotaton stll to come and that negotaton nvolves and another agent, k. Gven the agreements already fxed n past negotatons, the fnal negotaton between and k s smply a blateral Bnmore, Rubnsten, Wolnsky barganng game. That game would ordnarly yeld the Nash barganng soluton f and k had symmetrc nformaton regardng the mpact of ther choces on ther jont utlty, u ( x, x,.) u ( x, x,.). Ths wll be the case f k k k and j agree to the equlbrum x ˆ. However, f and j agree to x xˆ, and k wll have dfferent nformaton. Specfcally, whle under passve belefs, k wll contnue to base ts offers and acceptance decsons on an assumpton that x ˆ has occurred, s offers and acceptances wll be based on x. That s, wll make an offer, ( t k, x k ( x )), that maxmzes u ( x, xk,.) tk rather than u ( xˆ, x,.) t subject to k acceptng that offer. k k In ths case, the queston becomes: wll and j agree to some x xˆ? If they do, ths wll alter the equlbrum n subsequent negotatons. wll antcpate ths, however, the assumpton of 15 As noted earler, t wll be effcent f there are no externaltes and the network s complete. Consequently, ths can be vewed as a generalsaton of the effcency results of Segal (1999, Proposton 3).

13 11 passve belefs means that j wll not. That s, even f they agreed to x xˆ, j would contnue to beleve that x ˆk wll occur. For ths reason, j wll contnue to make offers consstent wth the proposed equlbrum. On the other hand, wll make an offer, ( t, x ), that maxmses u ( x, x ( x ),.) t ( x ) t ( x ( x )) rather than u ( x, xˆ,.) t tˆ subject to j acceptng that k k k offer. We demonstrate that ths s equvalent to choosng: k k x arg max u ( x, x ( x ),.) u ( x, xˆ,.) u ( xˆ, x ( x ),.) x k j k k k whch, by the envelope theorem appled to x k, has x xˆ, the blaterally effcent acton. By a smlar argument, agents do not fnd t worthwhle to devate n a seres of several negotatons. 4. Equlbrum Outcomes: Transfers and Payoffs Turnng now to consder equlbrum transfers and payoffs, we demonstrate here that whle surplus s determned n a non-cooperatve manner (from Theorem 1), under the same passve belefs assumpton, surplus dvson takes on a form attractvely smlar to coaltonal barganng outcomes. In partcular, dependng upon the nature of externaltes and the network of blateral negotatons, the dvson of whatever surplus s created gves agents a generalzaton of ther Myerson value on that reduced surplus. As such, the dvson of surplus between players has an appealng coaltonal structure even f the surplus s non-cooperatvely determned. It s useful frst to consder the Myerson value and related concepts n coaltonal game theory. Shapley s famous soluton assumed that all agents were lnked to each other. Myerson (1977a) generalsed that noton by allowng for the possblty that cooperaton may be restrcted ntally to an (exogenously gven) graph (N, L), even before any coaltons have broken lnks wth other players. Jackson and Wolnsky (1996) further extended the restrctons mposed by

14 12 graphs by allowng the structure of the graph wthn a coalton tself (e.g., whether agents are lnked drectly or ndrectly) to affect the payoff to a coalton; somethng that we have permtted here. The Myerson value s somewhat restrctve n that t s not defned n stuatons where dfferent groups of agents mpose externaltes upon one another. In another paper, Myerson (1977b) generalsed the Shapley value to consder externaltes by defnng t for games n partton functon form. In ths paper, below we defne a further generalzaton of the Myerson value to allow for a partton functon space as well as a graph of potental communcatons (as n Navarro 2007). The characterstc functon (.e. the total payoff, v, to any gven coalton of players) n such an envronment depends on the structure of the entre graph, both ntra- and nter-coalton. In order to state the equlbrum payoffs, we need to ntroduce notaton to express parttons of agents. P { P1,..., P p } s a partton of the set N f and only f () p 1 P N; () P ; and () for all j k, Pj Pk. We defne p as the cardnalty of P. The set of all parttons of N s P N. For a gven network (N, K), we can now defne a graph, (N, K P ), parttoned P by, P. That s, K { jk K s.t. j, k P }. In other words, (N, K P ) s a graph parttoned by P. We are now n a poston to state our man result. Theorem 2. Suppose that agents hold passve belefs, and that feasblty holds for each K L. Gven ( NL,, ) as 1, there exsts a unque perfect Bayesan equlbrum wth each agent recevng: P 1 ˆ 1 1 P u ( ˆ( )) ( 1) ( 1)! ˆ x L t P v( S, L ) j N N. PP SP n S P ( P 1)( n S ) SS The rght hand sde s, n fact, a generalzed Myerson value or Myerson value n partton

15 13 functon space defned on characterstc functons where agents take ther blaterally effcent actons. Thus, n equlbrum, we have a generalzed Myerson value type dvson of a reduced surplus. That surplus s generated by a blaterally effcent outcome n whch each blateral negotaton maxmses the negotators own sum of utltes whle gnorng the external mpact of ther choces on other negotatons (as n Theorem 1). 16 As n Theorem 1, the proof reles upon the agents holdng passve belefs n equlbrum. Wthout passve belefs, the equlbrum outcomes are more complex and do not reduce to ths smple structure. That smplcty s, of course, the mportant outcome here. What we have s a barganng soluton that marres the smple lnear structure of cooperatve barganng outcomes wth easly determned actons based on blateral effcency. It s that smplcty that allows t to be of practcal value n appled work. Suffcent Condton for Feasblty Now that we have derved the payoffs, we can provde suffcent condtons on the structure of externaltes for feasblty to hold. Theorem 3. Suppose that v( N, L ) s such that for any set of agents, h, who are connected to each other by L but otherwse not connected to any agents n N/h, v( h, K) v( h, K ), for any K L. Then the payoffs defned by Theorem 2 wll be feasble. The proof s n the Appendx. The condton n the proposton mples that any negatve externaltes wthn a coalton are counterbalanced by benefts to beng part of the coalton, but does not rule out the possblty that other coaltons mght experence negatve externaltes resultng from the actons of the coalton 16 It s easy to demonstrate that when there are no externaltes (.e., u ( x ( L)) s ndependent of x kl for any k and l not connected to ), ths value s equvalent to the Myerson value and, n addton, f t s defned over a complete graph, t s equvalent to the Shapley value.

16 14 5. Applcatons We now consder how our basc theorems apply n several of specfc contexts where mult-agent blateral barganng has played an mportant role. Stole and Zwebel s Wage Barganng Game Stole and Zwebel (1996; hereafter SZ) develop a model of wage barganng between a number of workers and a sngle frm. The workers cannot negotate wth one another or as a group. Thus, the relevant network has an underlyng star graph wth lnks between the frm and each ndvdual worker. A key feature of SZ s model s that barganng over wages s nonbndng; that s, followng the departure of any gven worker (that s, a breakdown), ether the frm or an ndvdual worker can elect to renegotate wage payments. Nonetheless, what s sgnfcant here s that, when a frm cannot easly expand the set of workers t can employ ex post, there wll be a wage barganng outcome wth workers and the frm recevng ther Myerson values. Ths happens even f workers dffer n ther productvty, outsde employment wages, and f work hours are varable. Moreover, f there were many frms, each of whom could bargan wth any avalable worker ex post, each frm and each worker would receve ther Myerson value over the broader network. As such, our results demonstrate that a Myerson value outcome can be employed n sgnfcantly more general envronments than those consdered by SZ. It s nstructve to expand on ths latter pont as t represents a sgnfcant generalsaton of the SZ model and yelds mportant nsghts nto the nature of wage determnaton n labor markets. Suppose that there are two dentcal frms, 1 and 2, each of whom can employ workers from a common pool wth a total sze of n. All workers are dentcal wth reservaton wages

17 15 normalzed here to 0, and supply a unt of value. If, say, frm 1 employs n 1 of them, t produces profts of F(n 1 ); where F(.) s non-decreasng and concave. The frms only compete n the labor and not the product market. 17 In ths nstance, as Fn ( 1) does not depend on n 2 and vce versa, the actons agreed upon wll maxmze ndustry value, defned as 1 frm receves n v( n) max F( n ) F( n n ). By Theorem 2, each n1 1 1 n ( n1)( n2) 1 1 ( n) ( 1) v( ) 2 F( ) ( n 2) F( ) and each worker receves 1 2 n w( n) v( n) (2 n) F( ) ( 1) v( ). 18 It s straghtforward to n n( n1)( n2) 0 demonstrate that wn ( ) s decreasng n n as n the SZ model. It s nterestng to examne the effect of frm competton n the labor market by consderng wage outcomes when the two frms above merge. 19 In ths case, the barganed wage, ( ) 1 1 w M n, becomes w n ( n ) v ( n ) v ( ). One would normally expect that wm ( n) w ( n) M n n( n 1) 0 as there s a reducton n competton for workers wth a merged frm; pushng wages down. However, ths s only the case f: n 1 n( n1)( n2) 0 ( n 2 ) v( ) 2 F( ) 0 (1) whch does not always hold. For example, suppose that workers can work part tme for each 1 frm, then v( ) 2 F( ). In ths case, the LHS of (1) becomes 2 n 1 n( n1)( n2) 2 2 ( n 2 ) F ( ) F ( ). The terms wthn the summaton move from negatve to 0 1 postve and so f F( ) F( ) s decreasng n then the entre expresson may be negatve so 2 17 It should be readly apparent that our model here wll allow for competng, non-dentcal frms as well as a heterogeneous workforce. 18 The complete dervaton of these values can be provded by the authors on request. 19 Stole and Zwebel (1998) also consdered a smlar ssue but wth a small number of heterogeneous workers.

18 16 that w ( n) w ( n). Thus, the smple ntuton may not hold. M The model reveals why workers may be able to approprate more surplus facng a merged frm than two competng ones: f the producton functon s very flat between 0.5n and n, a worker has relatvely poor outsde optons. Even f there s another frm to negotate wth, by movng ther, ther labor adds very lttle value there, and hence ther wage s low. General Buyer-Seller Networks Perhaps the most mportant applcaton of the model presented here s to the analyss of buyer-seller networks. These are networks where downstream frms purchase goods from upstream frms and engage n a seres of supply agreements; the jont acton between buyer and seller beng the amount of nput that wll be suppled from the seller to the buyer. 20 Sgnfcantly, t s often assumed for practcal and anttrust reasons that the buyers and sellers do not negotate wth others on the same sde of the market. Hence, the analyss takes place on a graph wth restrcted communcaton and negotaton optons. In ths lterature, models essentally fall nto two types. The frst assumes that there are externaltes between buyers (as mght happen f they are frms competng n the same market) but that there s only a sngle seller (e.g., McAfee and Schwartz, 1994; Segal, 1999; de Fontenay and Gans, 2004). The second lterature assumes that there are multple buyers and sellers, but assumes that each buyer s n a separate market, so there s no competton n the fnal-good market (Cremer and Rordan, 1987; Kranton and Mnehart, 2001; Inderst and Wey, 2003; Prat and Rustchn, 2003). Our envronment here encompasses both of these model types permttng externaltes between buyers (and ndeed sellers) as well as not restrctng the 20 The transfer payment can be thought of as a lump-sum payment or a per-unt payment. The two are equvalent f quanttes are agreed-upon at the same tme as prce. (But ths model excludes envronments n whch a per-unt prce s negotated, and the downstream frm subsequently orders quanttes at that prce.)

19 17 numbers or set of lnks between ether sde of the market. In so dong, we have demonstrated that when there are no spllovers between dfferent agent pars then ndustry profts are maxmsed. Thus, t provdes a general statement of the broad concluson of the buyer-seller network lterature. Smlarly, we have a farly precse characterzaton of outcomes when there are externaltes: frms wll produce Cournot quanttes, n the sense that the contracts of upstream frm A wth downstream frm 1 wll take the quanttes sold by A to downstream frms 2, m as gven; and the quanttes sold by B to downstream frms 1, m as gven. Ultmately, the framework here allows one to characterze fully the equlbrum outcome n a buyer-seller network where buyers compete wth one another n downstream market. The key advantage s that the cooperatve structure of ndvdual frm payoffs makes ther computablty relatvely straghtforward. For example, consder a stuaton wth m dentcal downstream frms each of who can negotate wth two (possbly heterogeneous) supplers, A and B. In ths stuaton, applyng Theorem 2, A s payoff s: m x x m ( 1) x ˆ A va, B ( m x) x0 x 0 m2 mxaxb mxaxb ( 2) m xa xb vˆ ˆ A xa xb vb xb xa m mx A m m xa 0 m2 m xa xb xa 0 xb 0 xa x B ( 1) vˆ A xa xb mxb 1 (2) where vˆ AB, ( m x) s the blaterally effcent (.e., Cournot) surplus that can be acheved when both supplers can both supply m x downstream frms and vˆ A( xa x B ) s the blaterally effcent surplus generated by A and x A downstream frms when those x A downstream frms can only be suppled by A and there are x B downstream frms that can only be suppled by B (wth no downstream frms able to be suppled by both). Thus, wth knowledge of vˆ AB, ( m s), vˆ A( xa x B )

20 18 and vˆ B ( xa x B ), usng demand and cost assumptons to calculate Cournot outcomes, t s a relatvely straghtforward matter to compute each frms payoff. One mplcaton of Theorem 2 s the parsmony of the structure: relatvely few terms mpact on the fnal payoff. These payoffs do not nclude the surplus created by ndustry envronments n whch some frms are lnked to both upstream frms and some frms are lnked to only one upstream frm, even though such envronments are possble, and are consdered by the players n ther barganng. Sgnfcantly, ths soluton can be used to analyze the effects of changes n the network structure of a market. The lnear structure makes comparsons relatvely smple. For example, Kranton and Mnehart (2001) explore the formaton of lnks between buyers and sellers whle de Fontenay and Gans (2005) explore changes n those lnks as a result of changes n the ownershp of assets. The cooperatve game structure of payoffs n partcular ts lnear structure makes the analyss of changes relatvely straghtforward. It s also convenent for analyzng the effect of non-contractble nvestments (e.g., Inderst and Wey, 2003). 6. Concluson and Future Drectons Ths paper has analyzed a non-cooperatve blateral barganng game that nvolves agreements that may mpose externaltes on others. In so dong, we have demonstrated that the generaton of overall surplus s lkely to be neffcent, as a result of these externaltes, but surplus dvson results n payoffs that are the weghted sums of surplus generated by dfferent coaltons. As such, there exsts an equlbrum barganng outcome that nvolves a cooperatve dvson of a non-cooperatve surplus. Ths s both an ntutve outcome but also one that provdes a tractable foundaton for appled work nvolvng nterrelated blateral exchanges.

21 19 7. Appendx Proof of Theorem 1 As there s a probablty that there s an exogenous breakdown between a par pror to any offer beng made by them, wth < 1, there s a non-zero chance that no agreements wll be reached. In a perfect Bayesan equlbrum, agents hold consstent belefs along the equlbrum path; note that, every sub-network wll appear on the equlbrum path, albet, ultmately, wth arbtrarly small probablty. Because agents hold passve belefs, when they observe a breakdown between other agents and consequently a new subgame, they assume that the breakdown was due to ths (1 )-mprobable event rather than due to a devaton from equlbrum, and play ther equlbrum strateges n the subgame. The agents nvolved n the breakdown never play aganst each other agan, and are forward-lookng n ther dealngs wth other agents. Hence behavor n each sub-network s ndependent of how that sub-network was reached. We focus attenton frst on stuatons where there s an arbtrary and on offers that are made f there s an ntal chance to make them. We note here that as goes to 1, the probablty that an offer s not made falls to zero and the equlbrum outcome wll reflect that. Wthout loss n generalty, therefore, let the current state of the network be L, and let { xˆ ( L), tˆ ( L)} be the conjectured equlbrum outcome and also agents passve belefs L regardng unobserved actons. We need only consder the ncentves for one player,, to devate. Suppose s nvolved n k negotatons, and re-name the agents that negotates wth as 1 to k. Suppose that s consderng devatng n the negotaton wth k. If makes the frst offer, solves the followng problem: subject to max u ( x, x ˆ( L) \{ xˆ ( L)}) t tˆ ( L) xk, tk k k k s sn \{, k} u ( x, x ˆ( L) \{ xˆ ( L)}) t tˆ ( L) ( L) (1 ) ( L k) k k k k sk k k sn \{, k} Here, x ˆ( L) s the vector of conjectured equlbrum actons, ( L) s k s expectaton of ther payoff f t makes a counter-offer, and k ( L k) s k s equlbrum payoff f there s a breakdown n negotatons between and k and a renegotaton subgame s trggered. As dscussed above, both agents have consstent expectatons about equlbrum actons and transfers n the sub-network; thus, n ths negotaton, they both take k ( L k) as gven. The ncentve constrant reflects the passve belefs of both players: Player mplctly assumes that f k were to reject an offer and make a counter-offer, k would make the equlbrum counter-offer. And assumes that k wll not change behavor n subsequent negotatons (although such devatons wll make the offer even more proftable for k). k beleves that has not devated n pror negotatons, and that f ths out-of-equlbrum offer s refused, wll stll k

22 20 accept the equlbrum counter-offer. 21 The transfer payment provdes a degree of freedom that allows to make the constrant bnd; therefore: t ( L) (1 ) ( L k) tˆ ( L) u ( x, x ˆ( L) \{ xˆ ( L)}) k k k k k k k sn \{, k} and solves (omttng terms that do not depend on x k ): max u ( x, xˆ ( L) \{ xˆ ( L)}) u ( x, x ˆ( L) \{ xˆ ( L)}) xk k k k k k Hence, unless xˆ k ( L) s blaterally effcent relatve to all other equlbrum actons, a proftable devaton exsts. Now we consder what happens f consders devatng n offers to both k and k 1. Let us assume, for smplcty, that always gets to make the frst offer, notng that, f ths were not the case, as approaches 1, player would smply reject any dfferent offer. Havng concluded agreements wth 1 through k 2, s offers to k 1 and k solve: max u ( x, x, x ˆ( L) \{ xˆ ( L), xˆ ( L)}) t t t ( L) x, k1, xk, t, k1, tk, k 1 k, k 1 k, k 1 k s sn \{, j, k} subject to: u ( x, x ˆ( L) \{ xˆ ( L)}) t t ( L) ( L) (1 ) ( L (, k 1)) (3) k 1, k 1, k 1, k 1 s, k 1 k 1 k 1 sn \{, k 1} u ( x, x ˆ( L) \{ xˆ ( L)}) t t ( L) ( L) (1 ) ( L k) (4) k k k k sk k k sn \{, k} Note that, because of passve belefs, k 1 does not nfer that a devaton wll change s preferred x k offer to k; nstead k 1 expects the equlbrum xˆ k ( L). When the transfers t, k1 and t k are chosen to make constrants (3) and (4) bnd, the choce of x, k1 and x k s equvalent to solvng: max u ( x, x, xˆ ( L) \{ xˆ ( L), xˆ ( L)}) u ( x, xˆ ( L) \{ xˆ ( L)}) u ( x, xˆ ( L) \{ xˆ ( L)}) x, k1, xk, k 1 k, k 1 k k 1, k 1, k 1 k k k k By teraton on agents from k 2 to 1, s optmal offers of actons { x} j 1 { x 1, x 2,... xk} to agents 1 to k wll satsfy: k { xk } j1 k k k ˆ ˆ ˆ ˆ j1 x j1 j x j1 max u ({ x }, ( L) \{ x ( L)} ) u ( x, ( L) \{ x ( L)}) If there s an nteror soluton to ths problem, t s clear that the frst-order condtons for each acton the frst-order condton for blateral effcency and to ths problem are the same. 22 Because of assumed compactness, contnuty and dfferentablty, applyng Mlgrom and Segal 21 k mantans these belefs even f refuses the equlbrum counter-offers for many rounds. Thus, there s no possblty of credble (costly) sgnallng. 22 If each u s concave n ts jont actons, as we have assumed, then ths maxmzaton problem s concave. The proof s avalable from the authors on request.

23 21 (2002, Corollary 4), the results hold even f the optmal value of x s a corner soluton. 23 Thus, we can conclude that the equlbrum values of x ˆ( L) are blaterally effcent, otherwse a proftable devaton exsts. Proof of Theorem 2 The proof of ths theorem has two parts. Frst, we consder the set of condtons that characterze the unque coaltonal barganng allocaton n a partton functon envronment when the communcaton structure s restrcted to a graph. Second, we wll demonstrate that the equlbrum of our non-cooperatve barganng game consdered n Theorem 1 satsfes these condtons. Part 1: Condtons Charactersng the Generalzed Myerson Value Begnnng wth Myerson (1977a), a way of demonstratng a coaltonal barganng allocaton was to state characterstcs of that allocaton that themselves determne that an allocaton satsfyng them was unque. Then one would demonstrate that a partcular allocaton satsfed those characterstcs. Hence, t could be concluded that that allocaton was the unque outcome of the coaltonal barganng game. Myerson (1977a) used ths approach and Jackson and Wolnsky (1996) extended t to demonstrate that the Myerson value was the outcome of a graph-restrcted coaltonal game. Stole and Zwebel (1996) used ths to prove Shapley value equvalence for ther wage barganng game. Myerson (1977b) defnes a cooperatve value for a game n partton functon space but does not consder the possblty of a restrcted communcaton structure nor does he provde a charactersaton of that outcome based on condtons such as far allocaton and component balance. Let v(s, K P ) be the underlyng coaltonal value of a game n partton functon form wth total number of agents (S) and graph of communcaton (K). Here are some defntons mportant for the results that follow. Some defntons: Defnton (Connectedness). Agents and j are connected n network L f there exsts a sequence of agents ( 1, 2,..., t ) such that 1 and t j and l, l 1 L for all l{1,2,..., t 1}. s drectly connected to j f L. Defnton (Component). A set of agents h N s a component of N n L f () all h, j h, j are connected n (N,L); and () for any h, j h, and j are not connected. The set of all components of (N,L) s C(L). Defnton (Allocaton Rule). An allocaton rule s a functon that assgns a payoff vector, Y ( N, v, L) R N, for a gven (N, v, L). 23 Applyng the teratve process n the proof, f s optmal offer was x k, 1 takng nto account an offer to k of x k, the total dervatve of the objectve functon wth respect to xk, 1 s equal to the partal dervatve of the objectve functon holdng x k constant at x k, even f that functon s not dfferentable n x k. Iteratng back to 1, ths verson of the envelope theorem can accommodate optmal values of actons that may be corner solutons.

24 22 Defnton (Component Balance). An allocaton rule, Y, s component balanced f ( N, v, L) ( h, L) for every h C( L), where ( h, L) h u. h Defnton (Far Allocaton). An allocaton rule, Y, s far f ( N, v, L) ( N, v, L ) ( N, v, L) ( N, v, L ) for every L. j j The method of proof wll be the followng. Frst, Lemma 1 establshes that under component balance and far allocaton, there s a unque allocaton rule. Second, we show that the generalzed Myerson value satsfes far allocaton and component balance. Thus, usng Lemma 1, ths mples that the generalzed Myerson value s the unque allocaton rule for ths type of cooperatve game. Frst, we note the followng result from Navarro (2007): Lemma 1 (Navarro, 2007). For a gven cooperatve game (N, v, L), under component balance and far allocaton, there exsts a unque allocaton rule. Next, n both an earler verson of ths paper and the workng paper verson of Navarro (2007), there s a proof that the generalzed Myerson value where agent receves: p1 1 1 P ( N, L) ( 1) ( p 1)! v( S, L ) N PP SP N S P ( p 1)( N S ) SS satsfes component balance and far allocaton. Navarro (2007) states a theorem to that effect: Lemma 2 (Navarro, 2007). The generalzed Myerson value for the game (N, v, L) n partton functon form satsfes component balance and far allocaton. Part 2: The non-cooperatve barganng game satsfes these condtons We want to show that the non-cooperatve barganng game satsfes far allocaton and component balance over a cooperatve game wth value functon vˆ( N, L ) as determned by blateral effcency. Note that Theorem 1 demonstrates that the unque equlbrum of the barganng game under passve belefs nvolves achevng blateral effcency. Ths defnes an mputed value functon. We now want to show that, for ths equlbrum, the two condtons are satsfed for the game wth ths value functon. When and j negotate, the current state of the network s L. When and j bargan together, let t be the transfer that offers, whch would gve a payoff vˆ and v ˆ to and j respectvely; j s offer t would, f accepted, lead to payoffs vˆ j the transfers are chosen to make the ncentve constrant bnd, the offers satsfy: vˆ vˆ vˆ (1 ) ( N, L ) j 1 j vˆ vˆ vˆ (1 ) ( N, L ) 1 j 1 j 2 j 2 j j j j and v ˆ respectvely. Gven that where ( N, L ) s the payoff to after a breakdown wth j. (Recall that f an offer s rejected, the order of offers s randomzed agan; so ether or j may make the next offer, wth 0.5 j j (5)

25 23 probablty each). The payoff of a player, v ˆ, s smply ther utlty from the actons taken plus equlbrum transfers t ˆk from other players (whch may be negatve): vˆ ˆ ˆ ˆ u t tk and j j j kn \ j k kn \ j vˆ uˆ tˆ tˆ (where transfer t k s zero f and k do not have a barganng lnk). Also, the total amount that and j have to dvde s gven by the other barganng relatonshps: f t ˆk s the equlbrum transfer from k to : vˆ vˆ vˆ vˆ uˆ tˆ uˆ tˆ j j j j k j kj kn \{, j} kn \{, j} j Usng (5) to substtute out vˆ and v ˆ n the frst part of (6): j (6) vˆ vˆ ( N, L ) vˆ ( N, L ) vˆ j j 2 j 2 j 2 2 j vˆ ( N, L ) vˆ ( N, L ) j j 2 j 2 j vˆ ( N, L ) vˆ ( N, L ) j j j Note from (5) that n the lmt, as tends towards 1, payoffs vˆ and ˆ v become the same payoff v, and therefore, vˆ ( N, L ) vˆ ( N, L ) whch s the far allocaton condton. ˆ j j Now consder condton (6) and ts analogue for every barganng lnk n the component that ncludes and j. In the lmt, as tends towards zero, the condton becomes v u tˆ for each, where transfer t s zero f and j do not have a barganng lnk. ˆ kn \ k Therefore, for a gven component, h: v u tˆ u tˆ u ˆ k k h h kn \ h kh\ h because there are no transfers to agents that you do not bargan wth. The non-zero transfers n ths summaton term are all between agents n h, and, therefore, the summaton ncludes both t ˆ and ( tˆ ), whch cancel out. Ths demonstrates component balance. Bndng Contngent Contracts We now extend Theorems 1 and 2 by demonstratng that they apply for the game wth bndng contngent contracts. Let an arbtrary order of negotatons be fxed, and suppose the order of negotatons s known to all players. In the negotaton between and j, and j negotate over all possble contngences that may stll occur. The proof wll show that the equlbrum actons and transfers consstent wth Theorems 1 and 2 for the nonbndng contracts barganng game also form a unque equlbrum of the contngent contract barganng game. Suppose that, when any makes an offer to any j, ther equlbrum offer s composed of:

26 24 an offer of the blaterally effcent actons xˆ ( K ) for each contngency K n whch and j are stll lnked; an offer of the transfers tˆ ( K ) that satsfy (5) for each contngency K n whch and j are stll lnked. Suppose that and j are the frst par to negotate, n network L. They expect all other pars to negotate the agreements descrbed above. Therefore, actons xˆ ( L ) and tˆ ( L ) satsfy (5), and hence, are the outcome of blateral barganng between and j. As approaches 1, and j are ndfferent as to the actons and transfers negotated n other contngences. Notce, however, that f and j assgn any postve probablty to any contngency other than L (or a number of other contngences), these contngent offers automatcally satsfy condtons (5) and (6). Suppose, for nstance, that they assgn probablty to one other contngency K. To satsfy the above condtons, s offer must satsfy max ˆ ˆ ˆ x ( ), ( ) ( ( ), ( ) \{ ( )}) ( ) ( ) K t K u x K x K x K tk K t K x ( L), t ( L) kn \{, j} (1 ) u ( ( ), ˆ( ) \{ ˆ ( )}) ˆ x L x L x L tk ( L) t ( K) kn \{, j} u ˆ ˆ ˆ j ( x ( K), x( K) \{ x ( K)}) t ( K) t jk ( K) j \{, } subject to ˆ kn j v j( K) (1 ) j( K ) j ˆ (1 ) ( ( ), ˆ( ) \{ ˆ ( )}) ( ) ˆ j j u j x L x L x L t L t jk ( L) kn \{, j} (1 ) v ( L) (1 ) ( L ) Clearly the equlbrum offers from the non-bndng game satsfy these condtons. Would and j have an ncentve to devate and negotate dfferent contngent contracts, n order to nfluence the other negotatons? For example, would and j want to negotate a contract that s very favorable to n the event of a breakdown between and k, n order to mprove s barganng power n negotatons wth k? Gven that we are assumng passve belefs by all agents, a devaton by these two would not change k s equlbrum belefs, and, therefore, would not mprove s barganng poston wth k. Now let us consder a negotaton that s further down n the lne of negotatons. Suppose that and j negotate after players a and b. Then, f a and b have not had a breakdown n negotatons, and j do not negotate over the contngences n whch a and b have a breakdown, for nstance, as that wll clearly not occur. However, f a and b have ndeed had a breakdown, they negotate over those contngences, and not over any contngences n whch a and b are stll lnked. From a and b s pont of vew, therefore, and j behave n the same way as n the nonbndng contract game: they negotate a contract n whatever contngency they fnd themselves n, not constraned by any earler agreement. Therefore, they expect them to reach the agreements descrbed n the non-bndng contract game.