Bargaining over Strategies of Non-Cooperative Games

Transcription

1 Games 05, 6, 73-98; do:0.3390/g Artcle OPEN ACCESS games ISSN Barganng over Strateges of Non-Cooperatve Games Guseppe Attanas, *, Aurora García-Gallego, Nkolaos Georgantzís,3 and Aldo Montesano 4 BETA, Unversty of Strasbourg, 6 Av. de la Foret Nore, Strasbourg, France LEE & Department of Economcs, Unverstat Jaume I, Avda. Sos Baynat s/n, Campus Ru Sec, 07 Castellón, Span; E-Mal: mgarca@eco.u.es 3 School of Agrculture Polcy and Development, Unversty of Readng, P.O. Box 37, Readng RG6 6AR, UK; E-Mal: n.georgantzs@readng.ac.uk 4 Department of Economcs, Boccon Unversty, va Roentgen, 036 Mlan, Italy; E-Mal: aldo.montesano@unboccon.t * Author to whom correspondence should be addressed; E-Mal: attanas@unstra.fr; Tel.: +33-(0) Academc Edtor: Bahar Leventoglu Receved: June 05 / Accepted: 7 August 05 / Publshed: 3 August 05 Abstract: We propose a barganng process supergame over the strateges to play n a non-cooperatve game. The agreement reached by players at the end of the barganng process s the strategy profle that they wll play n the orgnal non-cooperatve game. We analyze the subgame perfect equlbra of ths supergame, and ts mplcatons on the orgnal game. We dscuss exstence, unqueness, and effcency of the agreement reachable through ths barganng process. We llustrate the consequences of applyng such a process to several common two-player non-cooperatve games: the Prsoner s Dlemma, the Hawk-Dove Game, the Trust Game, and the Ultmatum Game. In each of them, the proposed barganng process gves rse to Pareto-effcent agreements that are typcally dfferent from the Nash equlbrum of the orgnal games. Keywords: barganng; supergame; confrmed proposals; confrmed agreements JEL Classfcaton: C7; C73; C78

2 Games 05, Introducton In several two-player non-cooperatve games, lke the Prsoner s Dlemma or the Trust Game, Nash equlbrum s not Pareto-effcent. However, laboratory experments have shown that n these games human subects often choose strateges that are Pareto-effcent. Ths s even more obvous n real world stuatons, n whch Pareto superor outcomes are sustaned, despte ther devaton from equlbrum. In fact, whle ths type of behavor cannot be consdered to be ratonal n the game-theoretc sense, a lot has been wrtten on the ways n whch socetes and ndvduals mplement socally desrable devatons from the non-cooperatve equlbrum outcome. In real-lfe stuatons, ndvduals often bargan on how to behave n a strategc context. For example, whle frms cooperate to form a cartel or a ont venture, an ndvdually proftable devaton that would lead the agreement to collapse, can be avoded by explct commtments to the cooperatve profle. In fact, the evdence shows that cartels are more lkely to be abandoned by ndvdual defectors than ont ventures because the former, beng llegal, are restrcted to depend on tact agreements, whereas the latter can normally be formed on the bass of a fully specfed agreement regulatng ndvdual actons. Therefore, rather than a theoretcal curosty, the protocol outlned below corresponds to usual practces n contract negotaton, where the proposer contrbutes a commtment rule, developed also by the second party, fully descrbng the actons to be followed by both. It s therefore nterestng to analyze how barganng procedures over game strateges can be modeled and whch are the consequences on effcency n the context of the orgnal strategc stuaton. Whle the effcency of outcomes has been a central ssue n non-cooperatve game theory, the role of barganng as a determnant of ndvdual actons n non-cooperatve games has not been systematcally explored both from a theoretcal and from an expermental pont of vew. In ths paper, we llustrate the consequences of usng alternatng proposal protocols as the means of lettng players reach an agreement about how to behave n a non-cooperatve two-player game G of complete nformaton. The two players must play the orgnal game G, and we assume that they know whch wll be the equlbrum outcome(s) of G. Before playng G, they may bargan over whch strategy they wll play n G, accordng to a specfc barganng mechansm, whch we call Confrmed Proposal process (CP(G) henceforth). Through ths process they can reach an agreement over the strategy profles to be played n G. We call confrmed agreement the outcome of CP(G),.e., the agreed strategy profle of G. The two players have the possblty, but not the duty, to bargan through CP(G). Moreover, after CP(G), they have the possblty, but not the duty, to play G accordng to the confrmed agreement n CP(G). They commt to play G accordng to the confrmed agreement n CP(G) only f ths s proftable for both players. Therefore, a confrmed agreement n CP(G) does not mply the commtment to play t n G: f the confrmed agreement yelds each player a payoff at least equal to the one obtaned n the Nash equlbrum of G, then both players commt through thrd-party mplementaton to play G accordng to the agreed strategy profle; otherwse they play G drectly,.e., wthout any agreement over strateges. Relevant references are [ 3].

3 Games 05, 6 75 The barganng process CP(G) over the strategy profle can be llustrated by the followng dalogue between the two players and : s Player (): If I play strategy, whch strategy would you play?. Player (): If you play s, I would play s. Player (): Ether Ok, I confrm that I ll play strategy s, and so let us play ( s, s ), or No, f 3 you play, I would play. s In the former case, the barganng process ends. In the latter case, s 3 Player (): Ether Ok, I confrm that I ll play strategy s, and so let us play ( s, s ), or No, f 3 4 you play, I would play. s s If player confrms, the barganng process ends. Otherwse the barganng process contnues wth player ether acceptng the proposed strategy profle or proposng another strategy. And so on and so forth. Therefore, there s an orgnal game G whose playng leads to the two players fnal payoffs, and a supergame CP(G) whose playng may lead n case an agreement s reached to the two players chosen strateges n the orgnal game G. Indeed, CP(G) s an nteractve strategc stuaton where a player, n order to gve offcal acceptance of a contract, must confrm her proposed strategy combned wth the strategy counterproposed by her opponent after havng heard the former player s proposal. We call equlbrum confrmed agreement the correspondng equlbrum contract between players n the barganng supergame CP(G), leadng to a strategy profle to be played n the orgnal game G f t yelds both players a payoff that s not smaller than the one obtaned by playng G drectly. Notce that our confrmed proposal process s a smple barganng process that does not make a player ntrude upon her opponent s strategc choce. In fact, each player only ndcates her strategy n G, wth the other one ndcatng her own strategy after havng heard ths proposal. Ths makes our work dfferent from [4], where Rubnsten s [5] model s extended ntroducng barganng wthout commtment: n [4], Rubnsten s [5] dea of lettng the frst player proposng the whole strategy profle thereby ntrudng upon her opponent s strategc choce s mantaned. A smlar remark can be made about the Proposer Commtment Procedure n [6]: the randomly selected proposer suggests an agreement for the whole set of actve players. Smlarly to our process, the agreement concerns players strateges. Dfferently from ours, the proposer also suggests the strategy of other players. For ths reason, the procedure n [6] cannot be seen as a generalzaton of our process. In fact, we wll show n the followng that n our case a subgame perfect equlbrum may not exst, whle n ther case t always exsts. Fnally, our approach s smlar to Brams [7] theory of moves n the fact that he mposes a dynamc process over players pars of strateges whose fnal state determnes players payoffs. However, [7] conceves a sequence of players actual strateges so as to reach a par of strateges that would not be further modfed, whle n our approach there s a sequence of proposals of possble strateges to be performed once confrmed. The concept of confrmed proposals has been frst examned n the game-theoretcal lterature by [8], focusng on the Prsoner s Dlemma as the orgnal game G. They let the two players bargan

4 Games 05, 6 76 over the strateges to play n the Prsoner s Dlemma: the barganng supergame CP(G) ends when one of the two players confrms her proposal gven the proposal of her opponent. At that pont, the orgnal Prsoner s Dlemma s played accordng to the proposed and confrmed strategy profle. It s shown that when players alternate n exertng the power to end the barganng supergame CP(G) played over the strateges of a Prsoner s Dlemma, the unque equlbrum confrmed agreement s the cooperatve (Pareto-effcent) outcome. The authors test ther theory n the lab: the expermental results provde support for the predcton of cooperaton n socal dlemma games wth confrmed proposals. In ths paper, we provde a general analyss of the confrmed proposal process over a complete-nformaton game G wth two players and fnte strategy spaces. We dscuss exstence, unqueness, and effcency of an equlbrum agreement n CP(G). Furthermore, we llustrate the consequences, n terms of equlbrum behavor, of applyng such barganng process to some non-cooperatve games very common n the expermental lterature: the Prsoner s Dlemma, the Hawk-Dove Game, the Trust Game, and the Ultmatum Game. The remanng part of the paper s structured as follows. In Secton, we descrbe the barganng process wth confrmed proposals ntroduced n the paper. In Secton 3, we dscuss exstence, unqueness, and effcency of the equlbrum agreement obtaned through ths process. In Secton 4 we apply the confrmed proposal process to some two-player non-cooperatve games extensvely used n the expermental research. Secton 5 concludes.. The Barganng Supergame Throughout the paper, we consder only non-cooperatve games G wth complete nformaton and we restrct the analyss to the two-player case. We assume that players are ratonal (.e., they have complete and transtve preferences over the set of payoffs), G has at least one equlbrum n pure or mxed strateges, and players know the equlbrum/a. Before playng G, they can bargan over whch strategy to play n G, eventually not the equlbrum one. However, the barganng process CP(G) starts only f both players go along wth enterng ths procedure and whch one of them wll be the frst mover n the supergame CP(G). Therefore, whle the standard descrpton of G assumes that any communcaton between players s forbdden, our barganng process CP(G) mplctly leads players to tactly communcate and bargan before playng G, eventually mplementng bndng agreements on how to play G. CP(G) s an nfnte-horzon dynamc game n whch the two players alternate proposals. Any proposal by a player s one of the possble strateges that she can adopt n G. The supergame CP(G) ends when a player confrms the proposal she made the prevous perod n whch she was actve: at ths pont, f t s worth t for both of them (compared to the Nash equlbrum of G), the two players can commt to play G accordng to the confrmed strateges n CP(G). In the case of no confrmaton, the player ndcates a dfferent strategy, whch becomes the counter-proposal to the last strategy proposed by the opponent. Then, the opponent may confrm or not the latter strategy. Ths structure of confrmaton proposng twce consecutvely the same strategy means confrmng t can be nterpreted as a chan between proposals. Attanas et al. [8] examne the non-chaned case wth alternatng proposals: the frst mover starts proposng her strategy, then the second mover counterproposes her strategy, fnally the frst mover confrms or not

5 Games 05, 6 77 Except for the selecton of the frst mover at the begnnng of the supergame, 3 the rules of the game are symmetrc. Let us denote by Sk the fnte strategy space for player k (wth k =, ) n the orgnal noncooperatve game G. Player k s set of possble proposals n the supergame wth confrmed proposals CP(G) concdes wth Sk. As a consequence, the set of possble agreements n CP(G) concdes wth the set of strateges of G,.e., the product set S S contans all the possble agreements of CP(G). Denote by s t k the strategy proposed by player k n perod t. Suppose that player starts the supergame CP(G). The sequence of alternatng proposals s as follows: Perod. Player proposes a strategy s S to player. Player would actually play s f (and only f) she would confrm ths strategy after the counter-proposal of player. Perod. Player proposes a strategy s S to player. Ths strategy would actually be played f (and only f) ether wll confrm her prevous strategy or wll confrm her proposal after the counter-proposal of player. Perod 3. Player chooses whether or not to confrm her prevous strategy s. If she confrms s, 3.e., s = s, then the barganng process ends, through the sequence ( s ), wth the, s, s confrmed agreement ( s ), and the two players receve the payoffs correspondng to the, s strategy profle ( s ) n the orgnal game G. If she does not confrm,.e., she proposes a, s 3 new strategy s s, the barganng process contnues wth as player s proposal and as player s counter-proposal to s proposal. Perod 4. Player chooses whether or not to confrm her prevous strategy s. If she confrms s, 4 3.e., s, then the barganng process ends, through the sequence ( ), wth the = s s, s, s 3 confrmed agreement ( s ), and the two players receve the payoffs correspondng to the, s 3 strategy profle ( s ) n the orgnal game G. If she does not confrm,.e., she proposes a, s new strategy s s, the barganng process contnues wth s as player s proposal and as player s counter-proposal to s proposal. And so on and so forth. t T Therefore, CP(G) s characterzed by sequences of proposals ( s k) t=, one for each perod t and for the actve player k at perod t, wth k = n odd perods and k = n even perods, and T = 3, 4,..., +. t T 3 4 A sequence ( s k) t= s a feasble hstory h of CP(G) f s, s,... S and s, s,... S, where the set of possble proposals n each perod t of CP(G) s the fnte strategy space Sk for player k (wth k =, ) n the orgnal non-cooperatve game G. 4 0 Let H be the set of all feasble hstores h, where h ndcates the ntal, empty hstory of CP(G),.e., before perod, and h t for t =,,... ndcates a feasble hstory before perod t +. s s s 3 s s the strategy profle. In the former case, the barganng process ends and the confrmed strateges are played n the orgnal game. In the latter case, the barganng process restarts wthout any constrant due to the proposals made before. 3 The frst mover n CP(G) ether can be selected at random or the players should agree over her dentty. However, for many orgnal games, the dentty of the frst mover n CP(G) s rrelevant for the equlbrum confrmed agreement. In partcular, ths s rrelevant for all orgnal games consdered n ths paper. An orgnal game where the dentty of the frst mover s relevant for the equlbrum confrmed agreement obtaned n CP(G) s the Battle of Sexes (see footnote 7). 4 Beng S k ndependent of t, we omt the superscrpt t when ndcatng the set of possble proposals n perod t.

6 Games 05, 6 78 t T T T A hstory z= ( sk) t= H of CP(G) s termnal f sk = s k,.e., player k actve at perod T confrms her prevous proposal, made two perods before. Denote wth Z the set of termnal hstores of CP(G) and let H : = H \ Z denote the complementary set of non-termnal (or partal) hstores. For nstance, 0 h, h and h always ndcate non-termnal hstores by constructon, snce no confrmaton s possble before perod 3, whle h T for T = 3, 4,... can ndcate ether a non-termnal or (n the case of confrmaton n perod T ) a termnal hstory. 3 4 A strategy for player k n CP(G) s a functon σ k : H Sk such that s, s,... S and s, s,... S for all h H. Notce that, snce n each perod the only actve player k can always choose among all of t t her possble strateges s k S k of G, the set of possble proposals sk ( h t ) at each hstory h does not t depend on the specfc hstory h, but only on the player s dentty ( or ): t s S f the actve player k =, and S otherwse. If no strategy profle of G s ever confrmed by ether player n any t = 3, 4, n CP(G), then no agreement s reached through barganng,.e., CP(G) has no equlbrum, and G s played drectly (.e., wthout any agreement over strateges). The orgnal game G s played drectly also f there s a confrmed agreement n CP(G), but t does not lead each player a payoff at least equal to the one she would get n equlbrum by playng G drectly. In ths case, a commtment to play G accordng the agreed strategy profle n CP(G) s not possble. Let us now ntroduce assumptons about players preferences over agreements n CP(G). t t t t Denote wth f ( sk, s k ) the outcome of G n the case the agreement ( sk, s k ) over the strategy profle to be played n G would be confrmed n the supergame CP(G) n perod t, wth t = 3, 4,... We assume that each player k s preference relaton k satsfes statonarty,.e., the preference t t' t t' between two agreements does not depend on tme: f sk = sk, sk = sk, and, t t t t t' t' t' t' then f ( sk, s k ) k f ( s k, s k ) f and only f f ( sk, s k ) k f ( s k, s k ) for all t t, wth tt, ' = 3,4,... The assumpton of statonarty of preferences means that a player s preferences do not depend on perod t of an agreement n CP(G), but only on the outcome of G due to ths agreement. A player k s mpatent f the tme of the agreement s relevant and she prefers to reach the same t t' t t' agreement n an earler than later perod,.e., f sk = sk and sk = sk, then t t t' t' f ( sk, s k ) k f( sk, s k ) for all t< t, wth tt, ' = 3,4,... ; she s patent f the tme of the agreement t t' t t' t t t' t' s rrelevant,.e., f sk = sk and sk = sk, then relaton f ( sk, s k ) ~k f ( sk, s k ) for all t< t, wth tt, ' = 3,4,... The assumpton of mpatence helps n selectng, among several payoff-equvalent strategy profles of CP(G), those leadng to the earlest confrmed agreement. In the followng analyss, we wll show that the number of statonary equlbra of CP(G), whch generate the same equlbrum n G, shrnks f players are mpatent. 3. General Results about the Equlbrum of the Barganng Supergame In ths secton we dscuss exstence, unqueness and effcency of the equlbrum confrmed agreement of CP(G) through several examples of orgnal games G.

7 Games 05, 6 79 A Subgame Perfect Nash Equlbrum of CP(G) s a par of strateges ( σk*, σ k*) n the supergame t such that one of the two players k at perod t makes a proposal s k that s the same as n perod t,.e., t t sk s t t = k. Ths leads to the equlbrum confrmed agreement ( sk, s k ), whch leads to the agreed strategy profle ( sk*, s k*) to be played n G. We look for equlbra of CP(G) by applyng the followng reasonng. Notce that CP(G) s a dynamc (super)game, whch we represent below through a game tree. In every decson node after perod, the actve player can confrm her prevous proposal. We apply the followng weakdomnance argument: we assume that the actve player confrms at perod t her prevous proposal at perod t f confrmaton gves her an outcome that s not worse than the best outcome she can get n the subgame of CP(G) that she enters n the case of no confrmaton at perod t. Then, n those subgames that are fnte because of confrmaton, we apply backward nducton. Example shows an orgnal game G wth several subgame perfect equlbra of CP(G), all leadng to the same equlbrum confrmed agreement. The exstence of a subgame perfect equlbrum of CP(G), and therefore of an equlbrum confrmed agreement, s not guaranteed. Ths s shown n Example. There s no equlbrum f, n each perod t of CP(G), no player has an ncentve to confrm the proposal she made n perod t. A player does not confrm her proposal because she beleves she can obtan ether a better agreement n the contnuaton game of CP(G), or a better equlbrum outcome by playng G drectly. Furthermore, t can happen that, although there exsts a subgame perfect equlbrum of CP(G), for one player the correspondng equlbrum confrmed agreement s worse off than the equlbrum outcome obtanable by playng G drectly. In ths case players are not able to commt on playng G accordng to the equlbrum confrmed agreement of CP(G), hence G s played drectly. Example 3 shows such a stuaton. Example : One equlbrum confrmed agreement. Consder the two-player smultaneous game G n Fgure. The set of strateges for player and player s, respectvely, S = {Superor, Inferor}, henceforth S = {S, I}, and S = {Left, Rght}, henceforth S = {L, R}. Fgure, wth a > b > c > d, shows, besdes the smultaneous-move orgnal game G, also all the possble agreements of CP(G), the barganng supergame wth confrmed proposals bult on t. Fgure. Orgnal game G wth one equlbrum confrmed agreement. The orgnal game G has the profle (S, R) as Nash equlbrum. Let us now calculate the subgame perfect equlbrum outcome of the barganng supergame CP(G). Observe Fgure. The set of feasble payoffs of CP(G) s the same as the orgnal game G n Fgure. The frst of the two payoffs always

8 Games 05, 6 80 refers to player, as n G. In Fgure a (left-hand sde) t s assumed that the frst mover n CP(G) s player. In Fgure b (rght-hand sde) t s assumed that the frst mover n CP(G) s player. (a) (b) Fgure. CP(G), G beng the orgnal game n Fgure, wth (a) or (b) as frst mover. The above mentoned weak-domnance argument apples to each of the two CP(G) n Fgure as follows: In every decson node ξ n perod t, the actve player weakly prefers the proposal whch leads, n the subgame wth root ξ, to an outcome whch s better or ndfferent for her than the best outcome that s obtaned by choosng another proposal at ξ. We call ths proposal weakly domnant, and we mark the correspondng branch from perod t to t+ wth a bold lne. Whenever there are two or more weakly-domnant proposals at a gven node n t, the correspondng branches from t to t+ are marked wth dotted bold lnes. For nstance, n Fgure a, player, after the hstory (S, L, I), proposes L and so she confrms the agreement (I, L), because n ths way she obtans the hghest possble payoff a. Usng backward nducton, we fnd that player, after hstory (S, L), proposes S and so she confrms the agreement (S, L), because n ths way she obtans the payoff c rather than the payoff she would obtan by ndcatng I (her payoff n ths case would be d). Gong backward, player, after s ntal proposal S, counter-proposes R, snce ths leads to obtan the payoff c rather than the payoff d, whch she would obtan by counter-proposng L. Usng the same reasonng throughout CP(G), we fnd that there are three subgame perfect equlbra σ *, σ *) n pure (supergame) strateges of CP(G), the ( correspondng hstores beng (S, R, S) for the frst equlbrum, and (I, L, S, R, S) and (I, R, S, R) for the other two. In each of them, the equlbrum confrmed agreement s (S, R). Ths s true f patence s assumed. If, nstead, both players are mpatent, then the unque subgame perfect equlbrum leads to

9 Games 05, 6 8 the hstory (S, R, S), where the agreement (S, R) s confrmed by player n perod 3, the earlest possble perod of confrmaton. 5 The same equlbrum confrmed agreement s obtaned f the frst mover n CP(G) s player (see Fgure b). Also n ths case, we fnd that (S, R) s confrmed n two subgame perfect equlbra f patence s assumed the correspondng hstores beng (L, S, R, S) and (R, S, R), and n only one equlbrum leadng to hstory (R, S, R) f players are mpatent. Therefore, ndependently of players level of (m)patence, and of whoever s the frst mover n CP(G), after the end of CP(G), the two players commt to play (S, R) n G. In ths specfc example, ths s also the Nash equlbrum of G: n the followng, we wll show that a necessary (but not suffcent) condton for a Nash equlbrum of G to be an equlbrum confrmed agreement of CP(G) s weak Pareto effcency (see Proposton ). Example : No equlbrum confrmed agreement. Our soluton procedure does not always allow for an equlbrum of CP(G). Consder Fgure 3. The orgnal game G has only one Nash equlbrum n mxed strateges, wth player one choosng strategy S wth probablty ps ( ) = 0.5 and player two choosng strategy L wth probablty ql ( ) = 0.5. Ths leads to expected payoffs V = V =.5. Both expected payoffs are larger than, the second-lowest possble payoff of G. Fgure 3. Orgnal game G wth no equlbrum confrmed agreement. In Fgure 4 we show the barganng supergame CP(G) wth player as frst mover. Assumng patence, the weak-domnance argument mples that none of the two players k makes a proposal whch leads to obtan the payoff of ether 0 or, snce she can obtan a hgher expected payoff Vk n the mxed-strategy Nash equlbrum of G by not barganng through CP(G). 5 A possble par of strateges leadng to the unque equlbrum agreement (S, R) n CP(G) of Fgure a s the followng: S f h 0 f f R h S S h ( S, L) = = f f L h I S h ( S, R) = = f 3 L h = ( S, L, I) σ* = S f h = ( I, L), σ * = f 3 R h = ( I, L, S) S f h = ( I, R) f 3 4 R h = ( I, R, S) S f h = ( I, L, S, R) f 5 4 L h = ( I, R, S, L, I) S f h = ( I, R, S, L). Ths par of strateges s one of the three subgame perfect equlbra when both players are patent, and the unque subgame perfect equlbrum when they are mpatent.

10 Games 05, 6 8 Fgure 4. CP(G), G beng the orgnal game n Fgure 3, wth as frst mover. Ths mples that no player k s able to confrm n CP(G) an agreement where she gets a payoff hgher than Vk. In fact, f player s frst proposal n perod s S, the hstory that results by takng nto account weak domnance and backward nducton s the ntal hstory (S, R, I, L) repeated nfnte tmes. 6 If, nstead, player s frst proposal n perod s I, the hstory that results by takng nto account weak domnance and backward nducton s the ntal hstory (I, L, S, R) repeated nfnte tmes. Thus, no equlbrum confrmed agreement s obtaned n CP(G). Ths s the case also when the frst mover would be player. Snce the reasonng s analogous, we omt the graphcal representaton of ths supergame n Fgure 4. Gven that there s no equlbrum confrmed agreement n CP(G), players have to drectly play the orgnal game G, thereby gettng the expected payoffs V = V =.5. Example 3: One equlbrum confrmed agreement that s not played. It can be the case that, although CP(G) has an equlbrum confrmed agreement, no commtment to play G accordng to the equlbrum confrmed agreement of CP(G) s possble, snce one of the two players would get a hgher payoff by drectly playng G,.e., n the Nash equlbrum of G. Ths happens, for example, when the orgnal game G s the Entry Game. In ths two-stage game, player (the potental entrant) chooses whether to Enter (E) or to Stay Out (S) of the market, wth (the ncumbent) decdng whether to Accommodate (A) or to Fght (F) f the entrant decdes to enter. The strategc form of the game n Fgure 5, where x := x f E, wth x = A, F, and a > b > c > d, represents all the possble agreements of CP(G). Notce that the hghest possble payoff for player s b. 6 Indeed, the sub-tree n perods 3 7, after the sequence of proposals (S, R), concdes wth the sub-tree n perods 7 after s proposal R n perod 6. The same holds for perods 5, 5 9,

11 Games 05, 6 83 Fgure 5. Orgnal game G wth one equlbrum confrmed agreement that s not played. In the unque (subgame perfect) Nash equlbrum of G, s entry takes place, wth accommodatng t. Hence, both players get a payoff equal to b. Conversely, n all subgame perfect equlbra of CP(G), the entrant stays out. The two possble versons of CP(G) when G s the Entry Game are n Fgure 6. The frst verson, n Fgure 6a, represents the case n whch player, the potental entrant n the orgnal game, moves frst n CP(G). In the second verson, Fgure 6b, player, the ncumbent n the orgnal game, s the frst mover. For both CP(G) n Fgure 6, there are two payoff-equvalent equlbrum confrmed agreements, whch nvolve the entrant to stay out. When the frst mover s player (Fgure 6a), there are three equlbrum termnal hstores: ( E, FSF,, ), ( S, A, E, F, S, F ), and ( S, A, S). When the frst mover s player (Fgure 6b), there are two equlbrum termnal hstores: ( A, EFSF,,, ) and ( A, S, A). (a) (b) Fgure 6. CP(G), G beng the orgnal game n Fgure 5, wth (a) or (b) as frst mover. In the two equlbrum confrmed agreements, ( S, A ) and ( SF, ), player gets a payoff equal to c. Conversely, by playng drectly G, she obtans a payoff of b. Consequently, she wll not commt to play G accordng to the strategy profle agreed n CP(G), and the players must play G drectly. Ths result also holds n the case both players would be mpatent. In fact, the unque equlbrum confrmed agreement n both CP(G) n Fgure 6a and CP(G) n Fgure 6b would be ( S, A), confrmed n perod 3: player would get c by barganng through CP(G) and b by playng drectly G.

12 Games 05, 6 84 Unqueness of the equlbrum confrmed agreement. If an equlbrum exsts, we can ntroduce Proposton, concernng the unqueness of the equlbrum confrmed agreement. Proposton. If the equlbrum for CP(G) exsts for a gven frst mover, and G s generc, then the equlbrum confrmed agreement s unque, hence players agree on a unque behavor n G. If G s not generc, then multple confrmed agreements are possble, although beng payoff-equvalent for at least one player. The ntuton behnd Proposton s as follows. Consder an orgnal game G. It s generc f each player s not ndfferent between two outcomes stemmng from two dfferent strategy profles of G,.e., t cannot be f ( s, s ) ~k f ( s', s ') f s s' and/or s s ' for k =,. Suppose that CP(G) has two equlbrum confrmed agreements, ( s*, s *) and ( s '*, s '*). An equlbrum confrmed agreement n CP(G) s assocated to one or more termnal hstores. A termnal hstory s a branch of the game tree of CP(G) that starts at perod and ends wth a confrmaton. Let us now reason by contradcton. If G s generc, then one of the two equlbrum confrmed agreements should be better than the other for player, and player could have the same preference or the opposte preference. Supposng that player s the frst mover n CP(G) and that f( s*, s *) f( s '*, s '*), one could have ether f( s*, s*) f( s'*, s'*) or f( s*, s*) f( s'*, s'*). In the former case, t s f( s*, s *) f( s '*, s '*) for k =,. The supergame CP(G) s wth k complete nformaton, hence players know the game tree of the barganng process. The equlbrum confrmed agreements are assocated to two dfferent termnal hstores. There are, n general, several termnal hstores assocated to the same equlbrum confrmed agreement: let us consder all pars of termnal hstores where each hstory leads to a dfferent equlbrum confrmed agreement. For any par, there s a perod t of CP(G) where the two termnal hstores dverge, thereby ncludng each one after t a dfferent sub-branch of the game. Ths sub-branch, and hence the consequent equlbrum confrmed agreement, s chosen by the player actve at t : she chooses the sub-branch that wll lead to the equlbrum confrmed agreement that s better for her. Snce both players have the same preferences over the two supposed equlbrum confrmed agreements, no equlbrum termnal hstory wll lead to ( s '*, s '*). Thus, ( s '*, s '*) cannot be an equlbrum confrmed agreement. For nstance, n Fgure b, the agreement (S, L) cannot be confrmed n equlbrum (the same s true n Fgure a). In fact, both players prefer the agreement (S, R) to (S, L). The termnal hstores leadng to (S, L) are: (L, S, L), (R, S, L, S), and (R, I, L, S, L). The termnal hstores leadng to (S, R) are: (L, S, R, S), (R, S, R), and (R, I, L, S, R, S). Compare parwse termnal hstores leadng to (S, L) wth termnal hstores leadng to (S, R): the player actve at the perod where the two termnal hstores dverge chooses the sub-branch leadng to (S, R). For nstance, consderng the par (L, S, L) and (L, S, R, S), the two hstores dverge at perod 3, where player s actve: she prefers proposng R nstead of L. In the latter case, t s f( s*, s*) f( s'*, s'*) and f( s*, s*) f( s'*, s'*). Consder all termnal hstores leadng to ether ( s*, s *) or ( s '*, s '*). Take a par of termnal hstores where one leads to ( s*, s *) and the other one leads to ( s '*, s '*). For ths par, there s a perod t of CP(G)

13 Games 05, 6 85 where the two termnal hstores dverge, thereby ncludng each one after t a dfferent sub-branch of the game. Ths sub-branch, and hence the equlbrum confrmed agreement, s chosen by the player actve at t : she chooses the sub-branch that wll lead to the equlbrum confrmed agreement that s better for her. If the actve player makng ths choce s, then n the par of hstores the one leadng to ( s '*, s '*) s elmnated; f ths actve player s, the termnal hstory leadng to ( s*, s*) s elmnated. Then, the number of termnal hstores leadng to ether ( s*, s *) or ( s '*, s '*) reduces by. Iteratng ths procedure, only termnal hstores leadng to the same equlbrum confrmed agreement would survve: only one equlbrum confrmed agreement exsts. For nstance, n Fgure b, the agreement (I, R) cannot be confrmed n equlbrum (the same s true n Fgure a). In fact, consder the two agreements (I, R) and (S, R): player prefers (I, R), whle player prefers (S, R). The only termnal hstory leadng to (I, R) s (R, I, R). The termnal hstores leadng to (S, R) are: (L, S, R, S), (R, S, R) and (R, I, L, S, R, S). Frst, compare (R, I, R) wth (L, S, R, S). The two hstores dverge at perod, where player s actve: she prefers proposng R nstead of L, thereby elmnatng (L, S, R, S). Then, compare (R, I, R) wth (R, I, L, S, R, S): the two hstores dverge at perod 3, where player s actve; she prefers proposng R nstead of L, thereby elmnatng (R, I, L, S, R, S). Fnally, compare (R, I, R) wth (R, S, R): the two hstores dverge at perod, where player s actve; she prefers proposng S nstead of I, thereby elmnatng (R, I, R). Consequently, the only equlbrum confrmed agreement s (S, R), whch can be obtaned also through the other two termnal hstores (L, S, R, S) and (R, I, L, S, R, S). In fact, although they have been elmnated n the comparson wth (R, I, R), they are stll equlbrum termnal hstores of CP(G) n Fgure b. Fnally, notce that for dfferent frst movers n CP(G), a dfferent equlbrum confrmed agreement may emerge. 7 If G s not generc, t can be f ( s, s ) ~k f ( s, s ) f s s and/or s s for k =,. Suppose that f( s*, s*) f( s'*, s'*) and f ( s*, s *) ~ f( s '*, s '*). If the player confrmng the agreement s player, then ether ( s*, s *) or ( s '*, s '*) can be confrmed. Hence, both agreements can be confrmed n equlbrum, wth player beng ndfferent between the two. If the player confrmng the agreement s player, f she s gven the possblty to confrm ( s*, s*), she certanly does t. If she s gven the possblty to confrm ( s '*, s '*), she does t f, by not confrmng, player would confrm an agreement yeldng player a lower payoff than n ( s '*, s '*). All ths s shown n next example. Example 4: Multple equlbrum confrmed agreements. As stated n Proposton, a non-generc game may have multple equlbrum confrmed agreements. Fgure 7, wth a > b > c > d, provdes an example of a non-generc orgnal game G wth two equlbrum confrmed agreements n CP(G). G s non-generc snce player gets the same payoff for L and R f player plays S. 7 An example s gven by the Battle of Sexes Game, whch s not analyzed here. The equlbrum confrmed agreement for CP(G) when the frst mover s player concdes wth the Nash equlbrum for G that s more convenent for player, and vce versa: n ths example a second-mover advantage emerges.

14 Games 05, 6 86 Fgure 7. Orgnal game G wth multple confrmed agreements. Fgure 8 shows CP(G) wth player (Fgure 8a) and player (Fgure 8b) as frst mover. In both CP(G) n Fgure 8, the two equlbrum agreements are (S, L) and (S, R). In fact, nether of the two players s able to confrm an agreement allowng one player to get the hghest possble payoff a. If such an agreement would be confrmed, one of the two players would get a, and the other one would receve d (the lowest possble payoff). Hence, the player gettng d would never confrm ths contract. Further, ths player would not counter-propose a strategy of G that would allow the other player to confrm such a contract. Ths means that, n any perod of CP(G), player does not reply wth proposal I to s proposal L, and player does not reply wth proposal R to s proposal I. Thus, for each player the hghest reachable payoff n a confrmed agreement s b. Thus, when a player has the possblty to get b by confrmaton, she confrms the prevous proposal. Gven that G s non-generc, when t s player confrmng an agreement n equlbrum, player gets ether b agreement (S, R) or c agreement (S, L). However, player can confrm an agreement that gves her c when, by not confrmng at t, she would allow to confrm at t + an agreement gvng a payoff equal to d. (a) (b) Fgure 8. CP(G), G beng the orgnal game n Fgure 7, wth (a) or (b) as frst mover. Multplcty of confrmed agreements holds also f both players are mpatent. As expected, mpatence reduces the number of subgame perfect equlbra of CP(G): f s the frst mover, the two equlbrum hstores are (L, S, L) and (R, S, R); f s the frst mover, the two equlbrum hstores are

15 Games 05, 6 87 (S, L, S) and (S, R, S). In partcular, termnal hstory (S, L, S) wth confrmng an agreement yeldng her a payoff equal to c emerges n equlbrum because f player would not confrm (S, L) n perod 3, player would confrm (I, L) n perod 4. Pareto effcency of the equlbrum confrmed agreement. If an equlbrum exsts, we can ntroduce Proposton, concernng the Pareto effcency of the equlbrum confrmed agreement. Proposton. Every equlbrum confrmed agreement n CP(G) s weakly Pareto-effcent. The ntuton behnd Proposton s the followng. Proposton states that a termnal hstory leads to an equlbrum confrmed agreement of CP(G) only f no other termnal hstory leads to an agreement whch strongly Pareto domnates the equlbrum one. Consequently, the equlbrum confrmed agreement s weakly Pareto-effcent, snce the set of agreements that can be confrmed n CP(G) concdes wth the set of strategy profles of G. Let us now reason by contradcton. Suppose that there exsts a strategy profle of G and thus a termnal hstory of CP(G) that leads to an agreement whch strongly Pareto domnates the equlbrum confrmed agreement (wth regard to the case where there s only one equlbrum confrmed agreement). Then, there s a perod t of CP(G) where ths termnal hstory dverges from the equlbrum one, thereby ncludng, after t, a dfferent sub-branch of the game. The player actve at t chooses the sub-branch leadng to the Pareto domnatng agreement. Consequently, the canddate neffcent equlbrum agreement s not reached: an neffcent equlbrum cannot exst. Let us now consder the case where there s more than one equlbrum confrmed agreement. By Proposton, all these agreements are payoff-equvalent for at least one player. Then, none of them s strongly Pareto superor to the other one: both are weakly Pareto-effcent (see Example 4 n the prevous paragraph, Fgures 7 and 8). 4. Confrmed Agreements n Standard Two-Player Games In ths secton we apply our barganng process CP(G) to some well-known G extensvely analyzed n the expermental lterature. In partcular, we focus on those games where subects n the lab often choose strateges leadng to Pareto-effcent outcomes that do not concde wth the Nash equlbra of the game. We wll show that n all these games G our barganng process CP(G) gves rse to Pareto-effcent agreements that dffer from the Nash equlbrum of the orgnal games G. Frst, we analyze two examples n whch the orgnal game G s a smultaneous game. Then, by mantanng the assumpton of two players only, we concentrate on two examples where the orgnal game G s a two-stage dynamc game wth perfect nformaton. Notce that the fact that G s dynamc does not matter for the scope of barganng. Indeed, n CP(G) players bargan over strateges of G. Hence, we drectly represent these dynamc games through ther strategc form. Ths shows how our barganng process can be appled to every two-player dynamc game wth fnte strategy spaces. Prsoner s Dlemma. The orgnal game G s a standard smultaneous-move Prsoner s Dlemma. The sets of players feasble proposals CP(G) concde wth ther sets of actons n the orgnal game: S = S = {Defect, Cooperate}, henceforth {D, C}. Fgure 9, wth a > b > c > d, shows the smultaneous-move orgnal game and all the possble agreements of CP(G). 8 8 Notce that the Prsoner s Dlemma has been ntroduced n the game-theoretcal lterature by explctly excludng the possblty of barganng. Therefore, by allowng the two prsoners to play CP(G) before playng G, we end up examnng a

16 Games 05, 6 88 The orgnal game G has the profle Fgure 9. Prsoner s Dlemma as orgnal game G. ( D, D) as equlbrum n domnant actons. Let us now fnd the subgame perfect equlbrum outcome of CP(G). Observe Fgure 0, where CP(G) s represented wth player as frst mover. Gven that the orgnal game s symmetrc, CP(G) wth player as frst mover s totally analogous to the one n Fgure 0. The theoretcal predcton for CP(G) where G s the Prsoner s Dlemma s the followng: Proposton 3. The subgame perfect equlbrum of CP(G) where G s the Prsoner s Dlemma s unque, and leads, n perod 3, to an equlbrum confrmed agreement where both players cooperate (C, C). The ntuton behnd Proposton 3 s as follows. Fgure 0. CP(G), G beng the Prsoner s Dlemma, wth player as frst mover. Consder Fgure 0 (player s the frst mover n CP(G)). After hstory (D, D, C), player can confrm D n perod 4, thereby gettng the hghest payoff a. Thus, hstory (D, D, C, C) s weakly domnated, and so we elmnate the subgame startng wth C after (D, D, C). By backward nducton, player confrms D after hstory (D, D). Reasonng n the same way, player confrms D after hstory (D, C), thereby gettng the hghest payoff a. Thus, hstory (D, C, C) s weakly domnated, and so we elmnate the subgame startng wth C after (D, C). By backward nducton, player counter-proposes D to s ntal proposal D. Analogously, player confrms D after hstory (C, D, D, C), and player dfferent strategc stuaton. However, expermental studes seem to suggest that players behave as f an mplct barganng occurs (see footnote 0).

17 Games 05, 6 89 confrms D after hstory (C, C, D, D, C). By backward nducton, the equlbrum termnal hstory (C, C, C) emerges. The same subgame perfect equlbrum s found when player s the frst mover. Thus, n the unque subgame perfect equlbrum of CP(G) n Fgure 0, player () starts by proposng strategy C to player (), who counter-proposes strategy C. Then, player () confrms her strategy C, such that the strategy profle ( CC, ) s the (unque) equlbrum confrmed agreement. Ths s reached already n perod t = 3, after the frst nteracton among players takes place. The equlbrum confrmed agreement of CP(G), ( CC, ), Pareto-domnates ( D, D), the Nash equlbrum of G. Therefore, both players commts to play G accordng to the agreement confrmed through CP(G). Hawk-Dove Game. The orgnal game G s the Hawk-Dove smultaneous-move game (see [9]). The set of players feasble proposals, whch concdes wth the set of players strateges n the orgnal game G, s S = S = {Hawk, Dove}, henceforth {H, D}. Fgure shows the smultaneous-move orgnal game and, also, all the possble agreements n CP(G). Parameters are such that a> b> c> d. Fgure. Hawk-Dove Game as orgnal game G. The orgnal game G has two Nash equlbra n pure strateges: ( H, D ) and ( D, H ). CP(G) s represented n Fgure, wth player as frst mover. Gven that the orgnal game G s symmetrc, CP(G) wth player as frst mover s totally analogous to the one n Fgure. Fgure. CP(G), G beng the Hawk-Dove Game, wth player as frst mover.

18 Games 05, 6 90 The theoretcal predcton for CP(G) where G s the Hawk-Dove game s the followng: Proposton 4. The subgame perfect equlbrum of CP(G) where G s the Hawk-Dove Game s unque, and leads, n perod 3, to an equlbrum confrmed agreement where both players cooperate (D, D). The statement n Proposton 4 can be obtaned followng the same ntutve reasonng as the one we have provded for Proposton 3 (n each subgame from perod 3 onward, each actve player confrms the agreement gvng her the hghest possble payoff, and by backward nducton the equlbrum termnal hstory emerges). The equlbrum s the same f ether player or player s the frst mover n CP(G). Thus, n the unque subgame perfect equlbrum of CP(G) n Fgure, player () starts by proposng strategy D to player (), who counter-proposes strategy D. Then, player () confrms her strategy D, such that the strategy profle (D, D) s the (unque) equlbrum confrmed agreement, reached already n perod t = 3, after the frst nteracton among players takes place. Notce that the equlbrum confrmed agreement of CP(G) does not Pareto-domnate any of the two Nash equlbra of G. If the Nash equlbrum s (D, H), then player prefers to play drectly G rather than partcpatng n CP(G). If the Nash equlbrum s (H, D), then player prefers to play drectly G rather than partcpatng n CP(G). In these two cases, a commtment over playng G accordng to (D, D) s not possble. If nstead players do not know whch of the two Nash equlbra wll be played, they could commt over playng G accordng the strateges agreed through CP(G). For nstance, ths happens f they attrbute a probablty of 50% to each of the two Nash equlbra of G, and b > 0.5(a + c). Trust Game. The orgnal game G s the Trust Mngame, a two-stage game wth both the trustor and the trustee havng only two possble actons (see [0]). Player (the trustor) decdes whether to Trust (T) or to Not trust (N) player (the trustee). In case trusts, total profts are hgher. In that case, would decde whether to Grab (G) or to Share (S) the hgher profts. The strategc form of the Trust Mngame s depcted n Fgure 3, where x x f T, wth x = G, S, and b > c > d, := a > b > c, a+ d = b + b. Ths fgure also represents all the possble agreements of CP(G). Fgure 3. Trust Mngame as orgnal game G. In the unque subgame perfect equlbrum of the orgnal game G, does not trust, whle the latter would choose to grab f had trusted her n the frst place,.e., ( NG, ). Fgure 4 represents the two possble versons of CP(G): In Fgure 4a the trustor () n the orgnal game G s the frst mover n CP(G), whle n Fgure 4b the trustee () n the orgnal game G s the frst mover n CP(G). In the latter case, s ntal proposal n CP(G) s her ntenton to grab or to share the hgher total profts n the case would trust her.

19 Games 05, 6 9 (a) (b) Fgure 4. CP(G), G beng the Trust Mngame, wth (a) or (b) as frst mover. Notce that, snce G s a dynamc game, players bargan n CP(G) over strateges that eventually nduce the same termnal hstory n G. However, n order for to confrm an agreement n CP(G), a player has to re-propose the same strategy of G n two subsequent perods of CP(G) where she s actve. Accordng to ths rule, ( SNG,, ) s not a termnal hstory of CP(G) n Fgure 4b, even though both strategy profles ( NS, ) and ( NG, ) nduce the same termnal hstory n the orgnal game G. The theoretcal predcton for CP(G) where G s the Trust Mngame s the followng: Proposton 5. The equlbrum confrmed agreement of CP(G) where G s the Trust Mngame s unque, and leads player to trust player and player to share the hgher profts ( TS, ). The ntuton behnd Proposton 5 s as follows. Independently from the dentty of the frst mover n CP(G), the agreement ( TG, ) allowng to get the hghest possble payoff a would never be confrmed n equlbrum. If such an agreement would be confrmed, would get a, and would receve d (the lowest possble payoff). Hence, player would never confrm ths contract. Further, ths player would not counter-propose strategy T, when ths strategy would allow to confrm such a contract: n any perod of CP(G) where she s actve, player does not reply wth proposal T to s proposal G. Thus, the hghest reachable payoff n an agreement s b for player : when player has the possblty to get b by confrmaton, she does t. Therefore, n Fgure 4a, where the frst mover s player, player would confrm S both after hstory ( NST,, ) and after hstory ( TGNST,,,, ). In both cases, she would get b. Moreover, player would confrm T after hstory ( TS, ). Consequently, the equlbrum termnal hstores are ( NSTS,,, ), ( TGNSTS,,,,, ) and ( T, S, T ). All lead to the same equlbrum confrmed agreement ( TS, ). In Fgure 4b, where the frst mover s player, player would confrm S both after the hstory ( GNST,,, ) and after hstory ( ST, ), and she would get b. Consequently, the equlbrum termnal hstores are ( GNSTS,,,, ) and ( STS,, ). All lead to the same equlbrum confrmed agreement ( TS, ). If both players are mpatent, there s ust one subgame perfect equlbrum and therefore one equlbrum termnal hstory n both CP(G) n Fgure 4. When the frst mover s player, ths hstory s ( T, S, T ); when the frst mover s player, ths hstory s ( STS,, ). Therefore, n both cases, the agreement ( TS, ) s confrmed n perod 3. The equlbrum confrmed agreement of CP(G) Pareto-domnates the Nash equlbrum of G. Therefore, players commt to play G accordng to the agreement confrmed through CP(G).

20 Games 05, 6 9 Ultmatum Game. The orgnal game G s the Ultmatum Mngame, a two-stage game wth both the proposer and the respondent havng only two possble actons (see []). In the orgnal game G, (proposer) can offer a far (F) or unfar (U) dvson to (respondent); the latter, after havng receved s offer, may ether accept (A) or reect (R). The set of s possble strateges concdes wth the set of her possble actons, whle the set of s possble strateges s S = AA, ARRARR,,, wth x y := x f F { } and y f U, wth x = A, R and y = A, R. The strategc form of the Ultmatum Mngame n Fgure 5 (wth a > b > c > d) also represents all the possble agreements of CP(G). 9 Fgure 5. Ultmatum Mngame as orgnal game G. In the unque subgame perfect equlbrum of the orgnal game G, unfar dvson takes place, wth acceptng both s offers,.e., ( U, AA). Fgure 6 represents the two possble versons of CP(G): n Fgure 6a the proposer () n the orgnal game G s the frst mover n CP(G), whle n Fgure 6b the respondent () n the orgnal game G s the frst mover n CP(G). In ths latter case, s ntal proposal n CP(G) s her ntenton to accept or to reect for each of the two possble strateges (far or unfar) of player. (a) (b) Fgure 6. CP(G), G beng the Ultmatum Mngame, wth (a) or (b) as frst mover. 9 Recall that confrmaton s acheved through re-proposal of the same strategy of G. Thus a hstory lke ( AR, F, AA) s not a termnal hstory for CP(G) when s the frst mover, even though both strategy profles ( F, AR ) and ( F, AA) nduce the same termnal hstory n the orgnal game G.