Subgame Perfect Farsighted Stability

Transcription

1 Subgame Perfect Farsighted Stability Daniel Granot Eran Hanany June 2016 Abstract Farsighted game theoretic solution concepts have been shown to provide insightful results in many applications. However, we provide simple examples demonstrating that the existing solution concepts may fail to incorporate some important aspects of farsighted reasoning. This failure stems apparently from the reliance of these concepts on constructs such as indirect dominance which only provide limited foresight. Accordingly, we explore in this paper the implications of replacing indirect dominance with subgame perfection. That is, we propose a new farsighted solution concept, the Subgame Perfect Consistent Set (SPCS), based on consistency in the spirit of the von Neumann Morgenstern solution and on subgame perfect equilibrium. Surprisingly, the SPCS is shown to always lead to Pareto efficiency in farsighted normal form games. This result is demonstrated in various oligopolistic settings, and is shown to imply, for example, that players who follow the SPCS reasoning are always able to share the monopolistic profit in farsighted settings based on Bertrand and Cournot competition, and are always able to achieve coordination and Pareto efficiency in decentralized supply chain contracting, even when they are unable to form coalitions. Key Words: Dynamic games, normal form games, farsighted stability, largest consistent set, oligopolistic competition 1 Introduction The introduction of farsighted game theoretic solution concepts was motivated by criticism of myopia raised against both cooperative and noncooperative solution concepts. For example, the Cournot equilibrium model was already criticized by Chamberlin (1933) on the grounds that firms in the model ignore reactions by rival firms to their actions, the von Neumann and Morgenstern solution (vnm 1944) was criticized by Harsanyi (1974) for failing to incorporate foresight, and the strong Nash equilibrium was criticized (see Chwe 1994) for not being consistent in the sense that it may reject outcomes in view of the existence of a strictly profitable deviation to another outcome even though this latter outcome is not a strong Nash equilibrium. This research was supported by the Natural Sciences and Engineering Research Council of Canada. Operations and Logistics Division, Sauder School of Business, University of British Columbia, Vancouver BC, Canada V6T 1Z2. daniel.granot@sauder.ubc.ca Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel. hananye@post.tau.ac.il 1

2 As compared to myopic solution concepts, such as the vnm solution, core, or Nash equilibrium, farsighted solution concepts characterize stable outcomes that are immune to deviations by farsighted players who recognize that their own deviations may induce multiple deviations by other players. Indeed, motivated by the above shortcoming of myopic solution concepts, a variety of farsighted solutions were proposed, starting with Harsanyi (1974) who pioneered the approach that coalitional behavior should be analyzed using subgame perfect equilibrium strategies of an extensive form game. His aim was to provide a non-cooperative foundation to the vnm solution in cooperative game theory. Greenberg (1990) introduced the conservative and optimistic stable standards of behavior, which possess vnm type consistency. Konishi and Ray (2003) introduced a dynamic Markov process of coalition formation, and characterized history independent equilibria based on coalition value functions representing infinite horizon discounted payoffs. Differently from the above approaches to farsightedness, the vnm farsighted stable set (vnm FSS, Chwe 1994) and the largest consistent set (LCS, Chwe 1994) take a cooperative view. The vnm FSS was shown by Chwe (1994) to be contained in the LCS, and farsightedness in both concepts 1 is achieved via the indirect dominance relation, introduced by Harsanyi (1974) and formalized by Chwe (1994). The LCS is (internally) consistent in the sense that a defection by coalition from a state 1 in the LCS to another state 2 is deterred, if subsequent deviations by other coalitions from 2 could lead to a state 3 in the LCS which indirectly dominates 2 and at which not all members of are strictly better off with respect to 1. Moreover, such defection from a state outside the LCS is not deterred, making the LCS (externally) consistent. When the set of states in the game is finite or countably infinite, it was shown by Chwe (1994, Proposition 2) that the LCS is also externally stable in the sense that every state not in the LCS is indirectly dominated by another state in the LCS. The above farsighted solution concepts have led to various refinements and modifications, and wereappliedinavarietyofsettings. Forexample,theywereemployedinvariousstudieson coalition formation (e.g., Mauleon and Vannetelbosch (M&V) 2004, Granot and Sošić 2005, and Nagarajan and Sošić 2007), and network stability (e.g., Page et al. 2005, and Herings et al. 2009). Not surprisingly, farsighted stable solution concepts could yield quite different insights than those derived from myopic stable solution concepts. For example, in coalition formation studies (e.g., M&V 2004, and Granot and Yin 2008), it is shown that under certain conditions, the unique farsighted stable coalition structure is the grand coalition while the unique myopic stable coalition is the stand alone structure. We will elaborate in Section 2 on the above as well as other related solution concepts and the various settings at which they were applied. We provide simple examples (see Section 3) demonstrating that the existing solution concepts may fail to incorporate some important aspects of farsighted reasoning. Since this failure stems from relying on concepts such as indirect dominance, we ask what are the implications of assuming instead that players foresight is modelled via subgame perfection. To this end, in this paper we introduce a new farsighted solution concept, referred to as the Subgame Perfect Consistent Set 1 For precise definitions of both concepts see Section 2. 2

3 (SPCS), which allows players or coalitions to look arbitrarily far ahead when they consider the consequences of their deviations from an existing state. Indeed, from a given initial state, we model the (non-stationary) evolution of play resulting from players/coalitions deviations as an extensive form game, infinite in size, and define 2 the SPCS to consist of all states which can be supported by a subgame perfect equilibrium, refined to additionally satisfy both internal and external consistency in the spirit of the vnm stable sets. Thus, the SPCS shares with existing farsighted solution concepts the use of vnm type of consistency, but differently, it uses the reasoning of subgame perfection, which endows players with unlimited farsightedness, instead of alternative farsighted constructs such as indirect dominance. The SPCS also does not necessarily impose the assumption of pessimism prevalent in the existing literature. As in the case of almost all other farsighted solution concepts, the SPCS is a natural solution concept in settings in which actions are public, transient states have no payoff consequences, and payoffs are realized only when players reach a final agreement. We analyze and characterize in this paper the SPCS for the general case when coalition deviations are possible. However, we are particularly interested to study situations in which only deviations by individual players are allowed. This would correspond to farsighted settings based on Cournot and Bertrand competition, where players are not allowed to collude or coordinate their actions. We investigate the structure of the SPCS for the class of farsighted normal form games. In particular, we show that even when only single players can deviate, quite surprisingly, the SPCS solution leads to (weak) Pareto efficiency in any normal form game having a pure Nash equilibrium. Thus, the new farsighted concept leads to efficiency in farsighted versions of, e.g., Bertrand games, Cournot games, some instances of Decentralized Supply Chains coordination problems and in the Prisoner s Dilemma. For all these settings, a SPCS is a singleton set consisting of a Pareto efficient outcome weakly dominating the myopic (pure Nash) equilibrium outcome. This is achieved using strategies that are similar to grim-trigger strategies commonly used in folk theorems within the repeated games literature. However, contrary to repeated games in which efficiency is achieved as a possibility, the SPCS achieves efficiency as a necessary implication. Moreover, we show that thespcssolutionleadstoefficiency also for normal form games having no myopic equilibrium, given that one is willing to accept as a possible outcome of the game a swinging behavior in which some of the players keep on moving between states. We show that all the above results continue to hold when coalition deviations are permitted, where in this case a myopic equilibrium is a state from which no coalition strictly prefers a onestep deviation. An important contribution of our approach is that, contrary to many cooperative game models which assume Pareto efficiency from the start, we derive such efficiency within a noncooperative framework. This is undertaken using strategies that could be thought of as modelling tacit negotiation. The plan of this paper is as follows. In Section 2 we provide the formal definitions of relevant, related solution concepts in the literature. In Section 3 we provide examples motivating our SPCS. 2 Precise definition is given in Section 4. 3

4 In Section 4 we introduce the farsighted game, formally define the SPCS and investigate some of its basic properties. In Section 5 we introduce farsighted normal form games, derived from standard one-shot normal form games by allowing players, or coalitions, to publicly and repeatedly, in their turn, change their previous actions. For this class of games we prove that any set of outcomes weakly dominating a myopic equilibrium is a SPCS if, and only if, it consists of a single Pareto efficient outcome (up to equivalence to all players). The analysis is extended to farsighted normal form games without a myopic equilibrium. We also illustrate the significance of our findings by demonstrating, for example, that by contrast with existing farsighted solutions such as the LCS and the Largest Cautious Consistent Set (LCCS), introduced by M&V (2004), but similarly to the vnm FSS, farsighted players who follow the SPCS reasoning are always able to share the monopolistic profit infarsightedsettingsbasedonbertrandandcournotcompetition. Wealso show that the SPCS always leads to full coordination and Pareto efficiency in decentralized supply chain contracting even when they cannot form coalitions. Section 6 concludes with a summary and suggestions for future research. Proofs are collected in Appendix A. Appendix B provides some further comparison of the SPCS and LCS. 2 Examination of the Related Literature We briefly discuss in this section the relevant literature on farsighted solution concepts. One of the most prominent farsighted consistent solution concepts is the largest consistent set (LCS), due to Chwe (1994), wherein farsightedness is exclusively embodied in the notion of indirect dominance. Before introducing the relevant definitions, let denote the set of all possible outcomes, and for 1 and 2 in, let 1 2 mean that coalition can effect a change from state 1 to state 2. Definition 2.1 [Chwe 1994, Harsanyi 1974]. We say that a state is indirectly dominated by a state, or, ifthereexiststates 0 1, where 0 = and =, and coalitions 0 1 1, such that +1 and ( ) ( ) for all =0 1 1 and. Formally, consistent sets and the LCS are defined as follows: Definition 2.2 [Chwe 1994] We say that a set is consistent if if,andonlyif,for every coalition and state such that, there exists a state, such that = or, for which it is not the case that ( ) ( ) for all. The largest consistent set (LCS) is the union of all consistent sets. Thus, the LCS is the union of all consistent sets satisfying: (i) for any state in, no subset of players that can deviate from would elect to do so, since such a deviation could lead, via indirect dominance, to another state in in which not all members of are strictly better off (internal consistency), and (ii) for any state not in, there always exists a subset of players that can deviate from and would elect to do so, since in all states in that such a deviation may 4

5 lead to via indirect dominance, all members of are strictly better off (external consistency). In the context of stability, the interpretation for indirect dominance provided by Chwe is that if and is presumed to be stable, then it is possible, not certain, that the coalitions 0 1 will move from to. Note the pessimistic criterion embedded in the definition of the LCS, which is somewhat mollified in M&V s (2004) definition of the Largest Cautious Consistent set (LCCS), described below. Chwe was aware of a possible criticism of the LCS for being too inclusive and that it may contain possibly non-intuitive outcomes. Explicitly, as he argues, If is consistent and, the interpretation is not that will be stable, but that it is possible for to be stable. If an outcome is not contained in any consistent, the interpretation is that cannot possibly be stable: there is no consistent storyinwhich is stable (Chwe 1994; p. 303). Nevertheless, as we show in Example 3.1 in Section 3, there are outcomes not contained in the LCS and for which there are very consistent stories of stability. It should be noted that, prior to Chwe, Greenberg (1990) has developed the theory of social situations, wherein he has introduced the concept of a stable standard of behavior (SB), which is claimed to capture perfect foresight by individuals or coalitions. Some precise relationships between a stable SB and the LCS were investigated by Chwe (1994). Xue (1998) has argued that the LCS captures only partial foresight, and by employing Greenberg s (1990) framework, characterized a set of paths which constitutes a stable SB. See also Mariotti (1997), wherein a farsighted solution concept similar to a stable SB is shown to achieve partial efficiency in normal form games with coalitions. Aside for the LCS, the stream of literature more directly related to this paper is concerned with various refinements of the LCS: M&V s LCCS (2004), Chwe s vnm FSS (1994), and, to some extent, Konishi and Ray s EPCF (2003) and the path dominance core (Page and Wooders, 2009). Let us first consider the Largest Cautious Consistent Set (LCCS), introduced by M&V (2004). The LCCS is a proper refinement of the LCS, which could possibly be empty, wherein players are less conservative than in the LCS. Specifically, as explained by M&V, once a coalition deviates from to 0, this coalition should contemplate the possibility of ending up with positive probability at any state 00 in the stable set via indirect dominance, i.e. 0 = 00 or A state is never stable if a coalition can deviate from to 0, and by doing so there is no risk that some coalition members will end up worse off. ThustheLCCSresultsfromamodification of the LCS pessimistic criterion. The LCCS does provide new insights in some instances, as shown, e.g., by M&V in their study of coalition formation, and by Granot and Yin (2008) in their analysis of a single-period two-stage supply chain problem. The vnm FSS is derived from the classical vnm solution by replacing direct dominance with indirect dominance. It was introduced by Chwe (1994), who showed that it is a refinement of the LCS. As such, the vnm FSS was shown to produce sharper results than the LCS in several instances, such as symmetric Cournot and Bertrand oligopoly markets (Suzuki and Muto (2006) and Masuda et al. (2000)), as will be further elaborated on in the sequel. Similarly, it was shown 5

6 by Mauleon et al. (2011) that, for one-to-one matching games, while the LCS may contain more matchings than those in the core (see also Diamantoudi and Xue, 2003), a set of matchings is a vnm FSS if and only if it is a singleton set whose unique member is a core element. See also Herings et al. (2009), who use a modification of the vnm FSS in their investigation of farsighted stable networks. Notwithstanding these sharper results, the vnm FSS could yield not very insightful results. For example, for a game based on the provision of a perfectly lumpy public good, without coalitions, it was shown by Kawasaki and Muto (2009) that the vnm FSS includes almost all individually rational outcomes. Another approach to farsightedness, which was employed in the analysis of supply chain models, is due to Konishi and Ray (2003). Therein, the authors have introduced a dynamic process of coalition formation (PCF), given by a transition probability : [0 1] such that P ( ) =1 for each. APCF induces a value function for each player, which captures the infinite horizon discounted payoff to each player starting from any initial point under the Markov process. A PCF is an equilibrium PCF (EPCF) if a coalitional move to some other state can be supported by the expectation of a higher future value. Evidently, the EPCF approach to farsightedness and coalition formation is quite different from the reasoning leading to the LCS, LCCS, and the SPCS, the solution concept introduced in this paper. By contrast with these three concepts, in which payoffs to players are realized only at the final state and wherein players may entertain different conjectures about other players behavior, in the EPCF players get discounted state-dependent payoffs at each stage, and all players have common beliefs about future moves. Note that in the EPCF the strategies are Markovian, i.e., they depend only on the current state and not on the history of arrival to that state. By contrast, asitwillbeshowninthesequel,thestrategies in the SPCS could be history-dependent. One of the attractions of the EPCF is that the set of all absorbing states under all deterministic (i.e., ( ) {0 1} ) absorbing EPCFs is a refinement, typically strict, of the LCS when the discount factor is large enough. This result was used by e.g., Sošić (2006) and Nagarajan and Sošić (2007, 2009) in their analysis of several Operations Management models. We note that the EPCF approach to coalition formation may lead to non-efficient outcomes in normal form games, as is evident from Observation 2 and the ensuing discussion in Konishi and Ray (2003). Page and Wooders (2009) have introduced a model of network formation whose primitives consist of a set of networks, players preferences, rules of network formation, and a dominance relation on feasible networks. As noted by the authors, their network formation game can be viewed as an abstract game with a finite set of states,, and a dominance relation among states which could be either the direct dominance, (Chwe s) indirect dominance, or path dominance. The main concept introduced by Page and Wooders is the path dominance core, which consists of all states which are not path dominated by any other distinct state. The path dominance core could be empty. However, if it is not empty, and if the game is such that for any two states, and, there exists a coalition that can move from to, then all states in the path dominance core are Pareto efficient. 6

7 (5, 1, 1) A (2, 10, 1000) {1} B (10, 20, 10) {2} C (1, 1, 100) {3} D Figure 3.1: Transition tree in Example 3.1 Brams (1994) theory of moves (TOM) (see also Brams and Wittman 1981) offers a dynamic theory approach to normal form games that bears some resemblance to our farsighted game approach to normal form games. Indeed, TOM attempts to characterize non-myopic equilibria, or stable outcomes, when players in a normal form game think ahead, and the predictions TOM provides depend, as in our case, on the starting point or status quo state. However, TOM is only developed for two-person normal form games in which each player has only two strategies, the players according to TOM play in a strict alternating order, and, more importantly, TOM makes simplifying assumptions to ensure a finite game tree representation of the dynamic normal form game. Related extensions to TOM s non-myopic equilibria include Kilgour (1984). 3 Motivating Examples Farsightedness in the LCS, LCCS, and the vnm FSS, stems exclusively from indirect dominance (Definition 2.1) which, as demonstrated in Examples 3.1 and 3.2, provide players with limited foresight. The lack of complete farsightedness in these examples is used to motivate the introduction of our new solution concept, the SPCS, wherein players are endowed with complete farsightedness stemming from subgame perfection. Example 3.1 Consider the game associated with Figure 3.1 that has four states with transitions between states as illustrated in the figure. Each node represents a possible state of the game, and theoutgoingarrowsrepresentapossiblemovebysomecoalitiontoadifferent state. If the final state is, the utilities to the three players are, respectively, (5 1 1). Player1 can move from to state, in which the utilities to the players, if it is the final state, are ( ). Then, from state, player2 can move to state, in which the utilities to the three players are ( ). Also at state, player3 can move to state, in which the utilities are ( ). No other moves are possible. Let us first characterize the LCS and LCCS. Since states and are terminal states, no deviation therefrom are possible, and both belong to the LCS and the LCCS. State cannot be a final state, since player 2 can move from to and strictly increase his utility. Thus, state is neither in the LCS nor in the LCCS. We further note that according to the farsighted reasoning of both the 7

8 LCS and the LCCS, player 3 will not move from state to state, since at state his utility is 100, which is strictly lower than his utility at. Next, consider state. Since, as clarified above, at state player 2 will move to state and player 3 will not move to state, and since the utility to player 1 at state, 10, is strictly larger than his utility at state, player1 will move to state. Thus, LCS = LCCS = { }. Next, let us consider the vnm FSS. State is directly dominated by, andstate is indirectly dominated by. Further, no deviation from or is possible, and thus, neither state indirectly dominates the other. So, { } is both internally and externally stable and forms the unique vnm FSS. Thus, all three farsighted solution concepts do not view state as possibly stable. In particular, it follows from Chwe that since state is not in the LCS, there is no consistent story in which state is stable. However, the instability of state according to the LCS, and, in fact, according also to the LCCS and the vnm FSS, is predicated on the reasoning that player 3 will not deviate from state to state becauseheisstrictlybetteroff at state than at state. But according to these three solution concepts, state is also not stable and cannot be a final state. Thus, this comparison is irrelevant and, in fact, is not consistent with the spirit of the LCS according to which only final outcomes matter. Indeed, while under indirect dominance, a deviation from state can only lead to final state, it is quite compelling to conclude that upon realizing that state cannot possibly be a final state, a farsighted player 3, given an opportunity, will deviate from state to state, just to preempt a future deviation by player 2 from state to state at which player 3 is strictly worse off. Thus, by contrast with the implications of indirect dominance, under more complete farsightedness, a deviation from state can end up in the final state. Therefore,even though state in not a member of the LCS, there is evidently a consistent and compelling story for state to be stable: player 1 will prefer not to move from state sincehemayendupatstate at which he is strictly worse off. Indeed, according to the farsighted solution concept, SPCS, developed in this paper, it will be verified in the sequel that SPCS = { } if is viewed as sufficiently likely to occur, or if players use a pessimistic criterion. It can be shown that the path dominated core with indirect dominance relation coincides with the LCS, LCCS and vnm FSS. With direct dominance relation, since state for player 1 does not directly dominate state, the path dominance core is { }. In the absence of state, LCS= LCCS = vnm FSS = path dominance core with indirect dominance relation = SPCS = { }, while the path dominance core with direct dominance relation, unfortunately, is { }. Next, consider Example 3.2, which illustrates that even in very simple settings one may need to analyze an infinite game tree in order to determine the farsighted outcomes. Example 3.2 ConsiderthegamedepictedinFigure3.2. Sincethereisnospecification which coalition/player is moving at a state in which more than one coalition/player can move, in principle it could happen that player 1 gets a turn to move at state for any finite number of times before player 2 gets a turn to move at this state. Thus, the game tree associated with Example 3.2 is infinite, wherein, player 1 will move from state to state and then for any finite number of times 8

9 (6, 6) A {1} (0, 12) B {2} (8, 8) C {1} Figure 3.2: Transition tree in Example 3.2 could stay at state or move from state to and back to. Notwithstanding the difficulty that may arise from an infinite game tree, the analysis of the game is straightforward. Let us first characterize the LCS and LCCS. Clearly, since there is no possible deviation from state, itisafinal state, belonging to both the LCS and LCCS. State cannot be a final state, since player 1 can move from it to state, at which he is strictly better off. Next, consider state. Since a move by player 1 to state cansubsequentlyleadtoamovebyplayer1 back to state, state is contained in the LCS. Furthermore, since player 2 is strictly better off at state than at state, according to the farsighted reasoning of the LCS (and the LCCS), player 2 will not move from state to state. Thus,amovebyplayer1 from state to state can only be followed by areturnmovebyplayer1 from state to state. Therefore, state is also in the LCCS, and, we have, LCS = LCCS = { }. Further,sincestate does not indirectly dominate state, and state directly dominates state, the vnm FSS and the path dominance core (both with direct and indirect dominance) coincide with the LCS and the LCCS. Since the game tree of Example 3.2 is infinite, the formal analysis of the SPCS associated with Example 3.2, carried out in Section 4, is slightly more involved. However, intuitively, state is not a final state, and while player 2 is strictly better off at state than at state, the former is just a pie in the sky for him. Indeed, notwithstanding player 2 reluctance to move from state to state, he is strictly better off at state than at state, whichwillbethefinal state if he refuses to deviate to state. Thus, it is optimal for player 2 to effect a deviation from state to state once he has such an opportunity. Therefore, player 1 will deviate from state to state having full confidence that player 2 will eventually deviate from state to state, and therefore, state cannot be a final state. The above analysis suggests that in Example 3.2 the only farsighted final state is and, indeed, as we will demonstrate in the sequel, the SPCS for Example 3.2 consists of the unique state. 4 Model and Solution: The Subgame Perfect Consistent Set Consider a dynamic game in which players can act repeatedly and publicly by moving between states they care about. For example, consider a group of firms engaged in contract negotiation, in which a state is a vector of contract parameters set by the firms, and each firm is responsible for adifferent part of the vector. Players only care about the final state reached, irrespective of the sequence of actions that lead to it. Thus actions have no significant costs, and the time frame in which they are taken is short with no relevant discounting of the final state utility. Despite the 9

10 dynamic nature of the game, it can be thought of as a one period interaction between the players. This setting can be described as a game in effectiveness form (Chwe 1994, see also Greenberg 1990), denoted by h ( ) ( ) i,where: is a finite set of players; is a measurable set of possible states; the utility function : R for each determines player s utility from a state (when it is a final state); and the binary relation for each describes the players ability to change the current state, where 1 2 for 1 2 means that coalition of players can alter the current state 1 by moving to a new state 2 (with the empty coalition always having null effectiveness, i.e. = ). A state issaidtobeterminalifthereis no 0, 0 6=, and, such that 0. For example, in the contract negotiation setting, can be the set of all contract parameter vectors possibly set by the firms, and the effectiveness of a singleton coalition = { }, consisting of a single player, allows the player to alter the current state by changing their own contract part without changing the parameters set by the other players. Furthermore, in this example, the effectiveness of a coalition consisting of two or more players can describe agreements between members of the coalition that allow particular simultaneous changes in the parameters set by all members of the coalition (again without changing the parameters set by players outside the coalition). As mentioned above, we envision an extensive form game with perfect information, in which all actions are public, which we call the Farsighted Game (FG). The complete description of the game is given by the following sequence of events. First, an initial state in is publicly chosen by nature. Then, an infinite sequence of stages initiates, each stage consisting of the following two steps: (Step 1) some coalition is publicly selected by nature; and (Step 2) the selected coalition can publicly keep the current state unchanged or move to a new state in according to its effectiveness. Formally, define the set of all possible histories in the game, H, by the set of all possibly infinite sequences =( ) =0, including the empty sequence when = 1, such that 0 is some initial state in, for all odd numbers 1, is some coalition, and for all even numbers 2,, such that = 2 or 2 1. For two finite histories 0 H with even, positive cardinality such that 0 0 = 1 or 1 0 0,denoteby( 0 ) the history in H obtained when is followed by 0. Denotethesetofinfinite histories by H. We say that a history H converges for player if there exist ( ) and an even, positive number such that ( )= [ ( )] for every even such that.wedefine 0 to be the minimal such.ahistory converges when it converges for all players, in which case we omit the player index and just write ( ) and 0. Convergence occurs, in particular, when 0 is a terminal state, or when is finite. Define the player function : H\H { }, where denotes a nature move, by ( ) =, ( ) = for every finite history with odd cardinality, and ( ) = for every finite history with even, positive cardinality. Nature s move at history for which ( ) = is determined by a full support probability measure,,defined over when = and over 2 otherwise. We assume that this probability can depend only on the last state in the history, i.e., = 0 when = 0 for any non-empty 0 with ( ) = All the results in the paper would continue to hold if this assumption was omitted, but the pessimistic criterion 10

11 Note that the choice of the initial state by nature is made only to reflectthefactthatnone of the players can affect this choice one could alternatively define the game with a prespecified initial state instead of random selection. Note also that in a setting where players can repeatedly observe the actions made by others and adjust their own actions accordingly, full farsightedness necessitates unbounded play: actions are made taking into account the fact that any move by a coalition may be counteracted with a further move by another coalition, without limit. Such an approach is in the spirit of modelling farsighted negotiation. It is natural therefore that players only care about the final outcome of the negotiation process, with no intermediate payoffs. A strategy of a coalition is a function : { H\H ( ) = } such that ( ) { = 1 or 1 }, specifying the action taken by this coalition after any history in the game where this coalition is selected to make a choice (a complete contingent plan). A strategy profile is a vector =( ) specifying the strategy of each coalition. Denote by the vector of strategies for all coalitions except for. We consider subgame perfect equilibria of the FG. For the purpose of defining such equilibria, it is sufficient to consider subgames beginning with coalitions decisions, i.e. following finite histories with even, positive cardinality. Following such, astrategyprofile generates a probability measure over the infinite histories 0 beginning with that are dictated by. LetΣ denote the set of strategy profiles which, following, generate a probability measure having support consisting only of infinite converging histories. Whenever Σ, this strategy profile generates for player a distribution over ( 0 ). This distribution has countable support because there are finitely many coalitions that may be selected and convergence occurs after a finite history. Elements of this support are called final states. To define,let ( Φ = ( 0 ) 0 0 =0 0 H 0 = ) 0 1 and 0 +1 = 0 [( 0 ) =0 ] =. +2 } The set of final states according to strategy profile following history, i.e. the support of,is Ψ = { ( 0 ) 0 Φ }, and then ( ) = P Q { 0 Φ ( 0 )= } = ( ( 0 0 ) 1 ) for. =0 In a subgame that follows a finite history with ( ) =, coalition has a strict preference relation  over strategy profiles, where weak preference % and indifference are defined from the strict preference  in the usual way. When the coalition is a singleton player, we assume the following. Assumption 4.1 For each player, the preference relation Â, when restricted to Σ,iscontinuous and satisfies  0 whenever strictly first order stochastically dominates 0, where is ordered according to. in Definition 4.4 was additionaly assumed. 11

12 For our analysis, the important implication of this assumption arises when considering a strategy profile which leads to a degenerate distribution over final states, i.e. some 0 for sure (with probability 1). In this case, the strategy profile is strictly worse than any strategy profile leading to a non-degenerate distribution over final states which assigns positive probability only to states in at least as good as 0. For general coalitions we assume the following. Assumption 4.2 For each coalition, the preference relation Â, when restricted to Σ,satisfies  0 whenever % 0 for all members, with strict preference for at least one member of. Givencoalitionpreferences,wecandefine a strategy profile as a subgame perfect equilibrium if % (ˆ ) for every coalition, everyfinite history with ( ) =, and every strategy ˆ for this coalition. In analyzing the farsighted game described above, we seek a solution concept that produces a set of states considered farsighted stable. As mentioned in the Introduction, our proposed solution is based on subgame perfection and a consistency notion in the spirit of vnm stable sets (von Neumann and Morgenstern 1944). Roughly speaking, this consistency notion requires the solution set of states to satisfy the following two properties: (i) farsighted internal consistency: for each state, no coalition prefers to move away from, anticipating that such a move would eventually endupin ; and (ii) farsighted external consistency: for each state, therealwaysexistsa coalition that prefers to move away from, again anticipating that such a move would eventually endupin. To formalize where a move might potentially lead to under subgame perfection, we say that a state 2 is reachable from a state 1 if for a subgame in which 1 is the initial state, there exists a subgame perfect equilibrium strategy profile such that 2 is a final state (i.e., the play converges to it with positive probability). Denote by ( 1 ) the set of states 2 reachable from 1. Clearly for a state to be a final state according to a subgame perfect equilibrium, it must be reachable from itself. This leads to the following definition, which relies on the logic of subgame perfection to reflect farsightedness with respect to some set of states. Definition 4.1 Given a strategy profile and a set of states, say that is -farsighted, denoted,if is a subgame perfect equilibrium that satisfies the following two requirement: (a) for any history =( 1 2 ) such that ( ) and { } =1 2 \, ( ) = ; and (b) for any history =( ) such that 1 6= 2,thesetoffinal states is Ψ = ( 2 ). Requirement (a) says that after choices by all coalitions to stay at an initial state reachable from itself, is the final state for sure. According to requirement (b), after an initial move from an initial state by some coalition, all reachable states in from 2, and only them, are final states. Both requirements hold on or off equilibrium path. Our solution concept relies on this 12

13 formulation of farsightedness, and involves an additional sense of a fixed point apart from the usual one delivered by equilibrium: The set is exactly the set of states from which, on equilibrium path of a -farsighted strategy profile, no coalition moves when selected as initial states. Formally: Definition 4.2 Asetofstates is a Subgame Perfect Consistent Set (SPCS) of a Farsighted Game (FG) if there exists such that if, and only if, ( ) = for any coalition and any finite history =( 1 2 ). Whenever satisfies definition 4.2 with respect to,wesaythat supports as a SPCS. It will be also useful to consider the final states that a SPCS generates after the selection of initial states outside the consistent set: Definition 4.3 For a SPCS and \, define the attraction set, ( ), asthesetofall states in that, according to some subgame perfect equilibrium supporting as a SPCS, are final states following the selection of as an initial state. We now investigate some basic properties of our solution concept. First, the following example demonstrates that a SPCS may fail to exist. In this example the failure is due to the lack of a subgame perfect equilibrium. Example 4.1 There is one player, is the set of natural numbers, ( ) = for each, and the player can only move from =1to any 1 (with no other possible moves included in the effectiveness relation). In this example there is no SPCS because there is no subgame perfect equilibrium: when =1is selected as an initial state, for any choice the player can make there exists a strictly better choice. Despite such examples, the following proposition provides sufficient conditions for existence of anon-emptyspcs. Proposition 4.1 Whenever is finite and the effectiveness relation is acyclic, there exists a nonempty SPCS. When analyzing farsighted normal form games in Section 5, we will show that the assumptions that is finite and the effectiveness relation is acyclic are not necessary for existence. Indeed, a non-empty SPCS can exist also for games with an infinite or when the effectiveness relation is cyclic. In general, the SPCS may potentially depend on the probability measures governing the coalition selection process, e.g., when players use expected utility maximization. Nonetheless, it is possible for our solution to be insensitive to the particular probabilities assumed. To allow such invariance, we can assume, similarly to Chwe (1994) and (particularly to) M&V (2004), that players use a pessimistic criterion to make their choices. 13

14 Definition 4.4 Pessimistic criterion for player : the preference relation Â,whenrestrictedto Σ, is continuous and satisfies  0 whenever a worst state in the support of is strictly better than a worst state in the support of 0, orif strictly first order stochastically dominates 0 where is ordered according to. This pessimistic criterion implies that a sure outcome is strictly preferred to any nondegenerate distribution that gives positive probability to some state strictly worse than. Let us now formally investigate the SPCS of the motivating examples of Section 3. In Example 3.1, the attraction set ( ) ={ }. This is supported by the strategy profile in which player 1 does not move from, player2 moves from to and player 3 moves from to. Clearly properties (a) and(b) indefinition 4.1 and the additional property in Definition 4.2 are satisfied for this strategy profile. It forms a subgame perfect equilibrium because player 2 prefers to and player 3 prefers, thefinal state after a move to, to, thefinal state after not moving from and anticipating a later move of player 2 from to ; player 1 uses the pessimistic criterion in Definition 4.4, thus prefers the degenerate distribution of utilities resulting from to the distribution of utilities from and that gives positive probability to, whichisaworse outcome for player 1 than. Notethatstate is reachable from itself, but state is not. Note also that in this example the SPCS implies that player 3 will move from to despite the fact that is worse than, because player 3 perceives rather than as a final state. Next we apply the SPCS to Example 3.2. Intuitively, it is not a subgame perfect equilibrium strategy for player 1 to remain forever at state. Thus, state can never be a final outcome of the game. Since transient states have no payoffs, there is no meaning for averaging the payoffs of transient states, thus no player would be content with player 1 repeatedly moving forever between states and. More specifically, the unique SPCS is { } with attraction sets ( ) ={ } for each \. This is supported by the strategy profile in which player 1 moves from to (unless previously stayed at, in which case player 1 stays at forever), stays at until player 2 plays,andthenmovesbackfrom to ; player2 moves from to (unless previously stayed at, in which case player 2 stays at forever). This strategy profile forms a subgame perfect equilibrium because player 2 prefers, thefinal state after a move to, to, which occurs after not moving from and anticipating a later move of player 1 from to and staying there forever. Furthermore, player 1 prefers a move from, thefinal state after not moving from forever, to, thefinal state after moving from to and anticipating a later move of player 2 from to ; inthecasethatplayer2 does not move from, player1 prefers, thefinal state after returning to, to, thefinal state after remaining at, which leads to remaining there forever. Note that state is reachable from itself, but state is not, thus player 1 is not restricted by property (a) in Definition 4.1 to stay at after both of them did not move from. This example shows the importance of non-stationary strategies in supporting the desirable efficient final state. Player 1 s choice depends not only on the current state but also on the history of play by player 2: ifplayer 2 has not yet been selected to make a choice, player 1 gives player 2 the opportunity to move to the efficient state ; but after player 2 was selected to make a choice and chose not to move, player 1 14

15 givesupandreturnsto. Note that the fact that this strategy does not allow player 2 more than one opportunity to move is not crucial: one could imagine player 1 giving player 2 a finite number of such opportunities to move until giving up and returning to ; in any such strategy, player 2 will eventually move to on the equilibrium path. Finally, in the last section of his paper, Chwe (1994) noted that many details are left out of the definition of the LCS, leaving important issues blurred and resulting in a limited farsightedness. We now revisit the examples that Chwe introduced to demonstrate the shortcomings of the LCS, and show that, in contrast, the SPCS solution overcomes these problems. In Examples , even though the LCS and the SPCS coincide, they lead to different final states if the game starts from an initial state outside these sets. Example 4.2 Optimal rather than better choice. = {1} (i.e., there is a single player) with = { }, {1}, {1} and 1 ( ) 1 ( ) 1 ( ). LCS = { }, implying that when is the initial state, player 1 will move to either or, even though is strictly preferred to. As Chwe has noted: This example illustrates how the largest consistent set does not incorporate any idea of best response : coalitions will move to any, not just the best, of the outcomes which are better than the status quo (Chwe, 1994: page 322). In this example, the unique SPCS = LCS, since and are terminal states according to the effectiveness relation. However, the attraction set { } ( ) ={ }, since,when is the initial state, is the unique outcome that can be supported as a final state according to a subgame perfect equilibrium in the game tree corresponding to this example. Thus the unique SPCS implies a move from to. Indeed, this demonstrates that in contrast to the LCS, the SPCS is predicated on the notion of best response. Example 4.3 Not joining a coalition if it is better otherwise. = {1 2} with = { }, {1 2}, {2}, 1 ( ) 1 ( ) 1 ( ) and 2 ( ) 2 ( ) 2 ( ). Again, since and are terminal states, and both are preferred by both players to state, the unique SPCS = LCS = { }. Thus from an initial state, the LCS implies that either coalition {1 2} will move to state or player 2 will move to state. The difficulty with this conclusion, as Chwe noted, is that player 2 prefers state to state, thus should prefer not to join player 1 to move to, as he could do better by himself moving to. Indeed, there exists a subgame perfect equilibrium in the game tree corresponding to this example, in which the coalition {1 2} never moves following the selection of state as an initial state. This choice can be interpreted as resulting from player 2 s preference not to join player 1 in such a move, and instead to wait until an opportunity arises to move alone from state to state. Consequently the implication of the SPCS in this case is that player 2 will not join player 1 in a move to state. Note, however, that the attraction set { } ( ) ={ }, as there also exists a subgame perfect equilibrium in which the coalition {1 2} sometimes moves from to, in which case there is positive probability for to be a final state when the initial state is. Example 4.4 Waiting and not moving until a better opportunity arises. = {1 2} with = { }, {1}, {2}, 1 ( ) 1 ( ) 1 ( ), and 2 ( ) 2 ( ) 2 ( ). We have LCS = { }, implying that from an initial state, eitherplayer1 will move to state or player 2 will move 15

16 to state. The difficulty with this conclusion, as pointed out by Chwe, is that both players prefer state to state, so that if player 1 is provided with the opportunity to be the first to move from the initial state, he should rather forfeit this opportunity and wait until player 2 is able to move. Although for this example the unique SPCS = LCS, according to the SPCS, if provided with the opportunity to move, Player 1 would not move to state. Indeed, the attraction set { } ( ) ={ } because state is the unique outcome that can be supported by a subgame perfect equilibrium in the game tree corresponding to this example following the selection of state as an initial state. Example 4.5 Moving to preempt other moves that may lead to worse outcomes. = {1 2} with = { }, {1}, {2}, 1 ( ) 1 ( ) 1 ( ), and 2 ( ) 2 ( ) 2 ( ). Here, LCS = { }, implying that from an initial state, player1 will move to state. However, as noted by Chwe, even though player 2 prefers state to state, he should deviate from to given an opportunity to do so; otherwise player 1 will deviate to state, whichforplayer2 is even less preferred than state. Although the unique SPCS = LCS, the attraction set { } ( ) ={ }, as both these states can be supported as final states according to a subgame perfect equilibrium in the game tree corresponding to this example following the selection of state as an initial state. Thus the SPCS generates a move from state to either to or. Note that this example is somewhat similar to Example Farsighted Normal Form Games In this section we analyze normal form games when they are viewed as farsighted games. Definition 5.1 Agameineffectiveness form h ( ) ( ) i is a -normal form game if is integer with 1, =,where is the set of alternatives available to player, andforeach 1 2 and each coalition, 6=, theeffectiveness relation 1 2 holds if, and only if, and 1 = 2 for each. A normal form game is a -normal form game for some. We view the concept of a pure Nash equilibrium as myopic, as it does not involve farsighted reasoning. Formally, a myopic equilibrium is defined as follows. Definition 5.2 In a normal form game, is a myopic equilibrium if ( ) ( ) for each and such that. 4 We show that the SPCS solution provides a surprisingandstrikingconclusioninthefarsighted analysis of normal form games: whenever the game possesses a myopic equilibrium, Pareto efficiency 4 Our analysis also applies when considering Stackelberg games (and extensive form games with perfect information in general) in their reduced normal form. This is justified by the view that in the farsighted perspective, the leader and the follower are indistinguishable. Thus, in this case, the results concerning farsighted normal form games apply when replacing pure Nash equilibria with pure Stackelberg equilibria. 16

17 is always and necessarily achieved. Moreover, Pareto efficiency is achieved even when coalition deviations are not permitted. When analyzing farsighted normal form games we assume that each player s utility function is continuous and that a SPCS is required to be a compact set. To state the main result we need some additional pieces of terminology/notation. We say that a state is Pareto efficient if there does not exist a state 0 such that ( 0 ) ( ) for all. Further, we write that { } whenever a subset is countable and consists of states that are all equivalent for allplayers,i.e.thereexists such that ( 0 )= ( ) for each player and 0. We can now state our main result. Theorem 5.1 (1) For any normal form game, if is a non-empty SPCS, then { } for some Pareto efficient ; and (2) For any normal form game, if { } for some Pareto efficient such that ( ) ( ) for some myopic equilibrium and all, then is a SPCS. For a 1-normal form game, the strategies supporting a SPCS in Theorem 5.1 are similar to grimtrigger strategies commonly used in folk theorems within the repeated games literature (see, e.g., Osborne and Rubinstein 1994), adapted to our setting where a repeated stage game is not required. The strategies involve a threat of reaching an undesirable outcome off equilibrium path in order to create incentives to reach a good outcome on equilibrium path. These strategies could be thought of as social norms that are publicly known to all players in the game. Since no player has an incentive to deviate from acting according to these social norms, they are indeed implemented, leading to the outcome of the corresponding SPCS. But note that contrary to folk theorems in repeated games, which do not claim to achieve efficiency, we achieve efficiency as a necessary implication. Thus the contribution of Theorem 5.1 is the conclusion that whenever the normal form game possesses a pure Nash equilibrium, such social norms must lead to Pareto efficiency, which is achieved rather than assumed. Theorem 5.1 implies that a game with no Pareto efficient states cannot have a SPCS. Existence of Pareto efficient states is a mild condition satisfied in many interesting settings as demonstrated in the following Example. Example 5.1 Prisoner s Dilemma Negotiation. In this 1-normal form game there are two players, player R (the row player) and player C (the column player), each having two available alternatives and utilities as in the following matrix For this game there is a unique SPCS = { } with utilities (3 3), i.e., ( )=3for both, and the attraction sets are ( ) = for each \. ThisSPCSissupportedbythefollowing strategy profile : after a selection of any initial state, each player moves to the alternative 17