The Core of a Strategic Game * Parkash Chander February, 2016 Revised: September, 2016 Abstract In this paper we introduce and study the γ-core of a general strategic game and its partition function form. We show that the γ-core of an arbitrary strategic game is smaller than the conventional α- and β- cores. We then consider the partition function form of a strategic game and show that a well-known class of partition function games admit nonempty γ-cores. Finally, we show that each γ-core payoff vector (a cooperative solution concept) can be supported as an equilibrium outcome of an intuitive non-cooperative game and the grand coalition is the unique equilibrium outcome if and only if the γ-core is non-empty. JEL classification numbers: C71-73 Keywords: strategic game, core, partition function, repeated game, Nash program * This paper has benefitted from comments by Myrna Wooders, Parimal Bag, and seminar participants at UPenn and Vanderbilt University. I am also thankful to two anonymous referees of this journal for their excellent comments. Jindal School of Government and Public Policy. URL: http://www.parkashchander.com. E-mail: parchander@gmail.com. 0
1. Introduction In this paper, we introduce and study the γ-core of a general strategic game and its partition function form. 1 We show that the γ-core is a stronger concept than the classical α- and β- cores in the sense that it is generally smaller. We then consider the partition function form of a strategic game and show that a well-known class of games (Ray and Vohra, 1997 and Yi, 1997) admit nonempty γ-cores. A growing branch of the literature seeks to unify cooperative and non-cooperative approaches to game theory through underpinning cooperative game theoretic solutions with non-cooperative equilibria, the Nash Program for cooperative games. 2 In the same vein, we show that the γcore payoff vectors (a cooperative solution concept) of a general strategic game and its partition function form can be supported as equilibrium payoff vectors of a non-cooperative game and the grand coalition is the unique equilibrium outcome if and only if the γ-core of the game is nonempty. The non-cooperative game for which the γ-core payoff vectors are equilibrium outcomes is an intuitive description of how the players may agree to form a partition when they know in advance what their payoffs will be in each partition. It consists of infinitely repeated two-stages. In the first stage of the two-stages, which begins with the finest partition as the status quo, each player announces whether he wishes to stay alone or to form a nontrivial partition. In the second stage of the two-stages, the players form a partition as per their announcements. The two-stages are repeated if the outcome of the second stage is the finest partition as in the status quo from which the game began in the first place. The paper also introduces a stronger concept of γ-core, to be called the strong γ-core, which is independent of the beliefs that the deviating coalitions may have regarding the formation of coalitions by the remaining players. 1 In contrast, Chander and Tulkens (1997) introduce a similar concept for a specific game. 2 Analogous to the microfoundations of macroeconomics, which aim at bridging the gap between the two branches of economic theory, the Nash program seeks to unify the cooperative and non-cooperative approaches to game theory. Numerous papers have contributed to this program including Perry and Reny (1994), Pe rez-castrillo (1994), Compte and Jehiel (2010), and Lehrer and Scarsini (2013), for example. 1
The paper is organized as follows. Section 2 introduces the γ-core for a general strategic game and compares it with the classical α- and β- cores. Section 3 shows that a well-known class of games admit nonempty γ-cores. It then introduces an infinitely repeated game of coalition formation and shows that the γ-core payoff vectors are equilibrium payoff vectors. It also introduces the concept of strong γ-core. Section 5 draws the conclusion. 2. The γ-core of a strategic game We denote a strategic game with transferable utility by Γ = (N, T, u) where N = {1,, n} is the player set, T = T 1 T n is the set of strategy profiles, T i is the strategy set of player i, u = (u 1,, u n ) is the vector of payoff functions, and u i is the payoff function of player i. A strategy profile is denoted by t = (t 1,, t n ) T. We denote a coalition by S and its complement by N\S. Given a strategy t = (t 1,, t n ) T, let t S (t i ) i S, t S (t j ) j N\S, and (t S, t S ) t = (t 1,, t n ). Given a coalition S N, the induced strategic game Γ S = (N S, T S, u S ) is defined as follows: The player set is N S = {S, (j) j N\S }, i.e., coalition S and all j N\S are the players (thus the game has n s + 1 players where the small letters n and s denote the cardinality of sets N and S, respectively ); The set of strategy profiles is T S = T S j N\S T j where T S = i S T i is the strategy set of player S and T j is the strategy set of player j N\S; The vector of payoff functions is u S = (u S S, u S j ) j N\S where u S S (t S, t S ) = i S u i (t S, t S ) is the payoff function of player S and u j S (t S, t S ) = u j (t S, t S ) is the payoff function of player j N\S, for all t S T S and t S j N\S T j. Notice that if (t S, t S) = t is a Nash equilibrium of the induced game Γ S, then u S S (t S, t S) = i S u i (t S, t S) i S u i (t S, t S) for all t S T S. Thus, for each S N, a Nash equilibrium of the induced game Γ S assigns a payoff to coalition S which it can obtained without cooperation from the remaining players. If the induced game Γ S has multiple Nash equilibria, then any one with the highest payoff for S can be selected; such a payoff exists if the strategy sets are compact 2
(or finite) and the payoff functions are continuous. In this way, a unique payoff can be assigned to each coalition. The γ-characteristic function of the strategic game Γ is the function w γ (S) = i S u i (t S, t S), S N, where (t S, t S) T is a Nash equilibrium of the induced game Γ S with the highest payoff for coalition S. The pair (N, w γ ) is a characteristic function game representation of the strategic game Γ. A payoff vectors x R n is feasible if i N x i = w γ (N). Definition 1 The γ-core of the strategic game Γ, or equivalently the core of its characteristic function game representation (N, w γ ), is the set of feasible payoff vectors x R n such that i S x i w γ (S), for all S N. Thus, the γ-core is the set of payoff vectors which cannot be improved upon by any coalition by deviating irrespective of which Nash equilibrium of the resulting induced game may be played. Since w γ (N), by definition, is unique, applying any other selection procedure to the sets of Nash equilibria of the induced games Γ S, S N, will lead to a core which contains the γ-core, i.e., the γ-core is a subset of any other core that may be similarly defined by selecting among the equilibria of the induced games. 2.1 The γ-core and the α- and β- cores It is natural to compare the γ-core of a strategic game with the classical α- and β- cores (Aumann, 1961 and Scarf, 1971). Recall that the α-core of a strategic game is based on the assumption that the players outside a coalition adopt those strategies which are least favorable to the coalition. Accordingly, the worth of a coalition S N under this concept is w α (S) = max ts T S min tn\s T N\S i S u i (t S, t N\S ). By contrast, in the β-core concept, the worth of a coalition S is w β (S) = min tn\s T N\S max ts T S i S u i (t S, t N\S ). It is well-known that the α- and β- cores are large. In fact, they are often too large. Shapley and Shubik (1969) observe in this connection that in these core concepts a coalition always expects the worst so far as actions of the outside players are concerned which may be so costly that they should be discounted in determining what a coalition is worth. Chander (2007) shows 3
that such behavior may, in fact, may require the outside players to not follow even their dominant strategies. The proposition below shows that the γ-core of a strategic game is in general smaller. It is well-known that the β-core is a subset of the α-core. Proposition 1 The γ-core of a general strategic game is a subset of the β-core, which is a subset of the α-core. 3 Proof: We need to show that w β (S) w γ (S), S N. Given S N, let (t S, t S) and (t S, t S) be such that w γ (S) = i S u i (t S, t S) and w β (S) = i S u i (t S, t S). Let t S (t S ) = arg max ts T S i S u i (t S, t S ), t S T S. Then, by definition, w β (S) = i S u i (t S ( t S), t S) i S u i (t S (t S ), t S ) for all t S T S. In particular, i S u i (t S ( t S), t S) i S u i (t S ( t S), t S ) = w γ (S). Hence, w β (S) w γ (S), and, therefore, w α (S) w β (S) w γ (S), S N. Examples are easily constructed in which the inequalities established in the proposition are strict. Since the proposition holds for an arbitrary strategic game, it can be applied to a variety of economic models. For example, Proposition 2.1 concerning an oligopoly game in Lardon (2012) is a special case of this proposition. 3. The partition function and the γ-core Given a partition of the total player set into coalitions, a partition function (Thrall and Lucas, 1963) assigns a payoff to each coalition in the partition. A strategic game can be converted into a partition function if each induced strategic game in which each coalition in the partition becomes one single player admits a unique Nash equilibrium. Since in many applications (see e.g. Chander and Tulkens, 1997 and Lardon, 2012) each induced game indeed admits a unique Nash equilibrium, we henceforth restricts ourselves to strategic games which have a well-defined partition function form. 3 The proposition also implies that the γ-core solutions are not inconsistent with the α- and β- core solutions. 4
More formally, a set P = {S 1,, S m } is a partition of N if S i S j = for all i, j {1,, m}, i j, and m i=1 S i = N. Let v(s i ; P) 0 denote the Nash equilibrium payoff of coalition S i in the induced strategic game in which each coalition S j, j = 1,, m, becomes one single player, i.e., within the coalition the individual strategies are selected so as to maximize the payoff of the coalition: the sum of the payoffs of its members. Then, (N, v) denotes the partition function form of the strategic game (N, T, u). Notice that the γ-characteristic function w γ is a specific restriction of the partition function v. For brevity, we shall sometimes refer to the partition function game (N, v) simply as partition function (N, v). In this Section, we shall treat a partition function as the primitive, but recall as and when necessary that the partition function is generated from a strategic game. One implication of this is that the grand coalition is an efficient partition, since the coalition of all players can choose at least the same strategies as the players in any partition. Accordingly, we assume below that the grand coalition is an efficient partition. Another implication is that the members of a coalition in a partition may unanimously decide to dissolve the coalition, i.e., to not give effect to the coalition. This possibility arises from the fact that a coalition can choose the same strategies that its members would choose individually in a Nash equilibrium of the strategic form of the game. In terms of the strategic game underlying the partition function, dissolving a coalition is equivalent to the players in the coalition choosing the same strategies that they will choose if they were singletons, given the strategies of the remaining players. Accordingly, the noncooperative game introduced below allows for this possibility. Given a partition function game (N, v), a payoff vector (x 1,, x n ) is feasible if i N x i = v(n; {N}). In words, a feasible payoff vector represents a division of the worth of the grand coalition. Let [N] and [N\S] denote the finest partitions of N and N\S, respectively. Definition 2 The γ-core of a partition function (N, v) is the set of feasible payoff vectors (x 1,, x n ) such that i S x i v(s; {S, [N\S]}) for all S N. Notice that the γ-core of a partition function (N, v) does not require a feasible payoff vector (x 1,, x n ) to satisfy i S x i v(s; P) for all partitions P such that S P. In fact, the definition implies that the γ-core of a partition function game (N, v) is equal to the core of the 5
characteristic function game (N, w γ ). Hafalir (2007) notes that many alternative core concepts for partition function games are possible, but focuses on the γ-core, referred to as the s-core by him, and establishes several properties of the γ-core. We now establish below some additional properties of the γ-core. 3.1 A class of games with nonempty γ-cores A number of studies have focused on symmetric games in which the grand coalition is the efficient partition and larger coalitions in each partition have lower per-member payoffs (see e.g. Ray and Vohra, 1997 and Yi, 1997). We show that the γ-cores of games with this property are nonempty. 4 In particular, the feasible payoff vector with equal shares is a γ-core payoff vector. We need some additional notation. A partition function (N, v) is symmetric if for every partition P = {S 1,, S m }, s i = s j v(s i ; P) = v S j ; P, where the small letters s i and s j denote the cardinality of the sets S i and S j, respectively. Proposition 2 Let (N, v) be a symmetric partition function game such that for every partition P = {S 1,, S m }, v(s i ; P)/s i v(s j ; P)/s j if s i s j, i, j {1,, m}. If the grand coalition is an efficient partition, then (N, v) admits a nonempty γ-core. Proof: Let (x 1,, x n ) be the feasible payoff vector with equal shares, i.e., and x i = x j, i, j N. We claim that (x 1,, x n ) belongs to the γ-core of (N, v). i N x i = v(n; {N}) We need to show that i S x i v(s; {S, [N\S]}) for all S N. Since the grand coalition is an efficient partition, v(s; {S, [N\S]}) + i N\S v(i; {S, [N\S]}) v(n; {N}) = i N x i. Since v(s; {S, [N\S]})/s v(i; {S, [N\S]}), i N\S, and x i = x j, i, j N, this inequality can hold only if v(s; {S, [N\S]})/s x i. Hence, v(s; {S, [N\S]}) i S x i for all S N. Chander (2007) assumes symmetric players and investigates an infinitely repeated game of coalition formation for a specific partition function game. In what follows we consider a general 4 See Helm (2001) and Stamatopoulos (2015) for sufficient conditions for the existence of a nonempty γ-core for games which are not necessarily symmetric. 6
partition function game and assume that the players are not necessarily symmetric. This raises new challenges. One of these new challenges is how the worth/payoff of a coalition in a partition may be distributed among its members. For symmetric games, equal sharing of the payoffs among the members of each coalition in a partition is natural, but not if the game is not symmetric. To extend our analysis to games which the players are not necessarily symmetric, we introduce now a more general class of payoff sharing rules of which the equal payoff sharing rule is a special case. Given a partition function game (N, v), a payoff sharing rule is a mapping x: R n which associates to each partition P = {S 1,, S m } a vector of individual payoffs x(p) R n such that j S i x j (P) = v(s i ; P), S i P. 5 A mapping x: R n is the equal payoff sharing rule if for each partition P = {S 1,, S m }, x i (P) = x j (P) for each i, j S k, k = 1,, m. The equal payoff sharing rule, as in the class of symmetric games in Proposition 2, is a special case of the following general class of payoff sharing rules. A payoff sharing rule x: R n is monotonic if for each partition P = {S 1,, S m } and each coalition S i P, x j (P) > ( )x j ({N}) for all j S i if and only if v(s i ; P) > ( ) x j ({N}) j S i. A monotonic payoff sharing rule assigns higher (resp. lower) payoffs to each member of a coalition in a partition if the total payoff of the coalition is lower (resp. higher) in the grand coalition. In other words, a monotonic rule assigns payoffs to members of each coalition in a partition such that their individual payoffs are either all higher or all lower compared to their individual payoffs in the grand coalition. Thus, monotonic payoff sharing rules ensure unanimity among the members of a coalition on coalition s decision to leave or not leave the grand coalition if they are farsighted and can foresee the resulting partition that will form subsequent to departure of their coalition. 5 In contrast, Hart and Kurz (1983) do not require the sum of payoffs of the members of a coalition in a partition to be equal to the payoff/worth of the coalition in the partition. This comes about from their efficiency assumption in which the worth of the grand coalition is assumed to accrue as the total payoff of all players in any coalition structure. 7
Definition 3 A payoff sharing rule x: R n is proportional if for each partition P = {S 1,, S m } and each coalition S i P, x j (P) = x j ({N}) [v(s i ; P)/ k Si x k ({N})], j S i. Proportional sharing rules are clearly monotonic and the equal payoff sharing rule is clearly proportional and, therefore, monotonic. It will be convenient to denote a proportional sharing rule x: R n simply by a feasible payoff vector (x 1,, x n ) meaning that for each partition P = {S 1,, S m } and each coalition S i P, x j (P) = x j [v(s i ; P)/ k Si x k ]. Theorem 3 below holds for all monotonic payoff sharing rules, but to be concrete we shall restrict ourselves to a proportional payoff sharing rule. 3.2 The non-cooperative foundations of the γ-core We show that each γ-core payoff vector can be supported as an equilibrium payoff vector of a non-cooperative game. This means that the γ-core as a solution concept can be arrived at from a very different point of view. The non-cooperative game, to be called the payoff sharing game, consists of infinitely repeated two-stages. The first stage of the two-stages begins from the finest partition [N] as the status quo and each player announces some nonnegative integer from 0 to n. In the second stage of the two-stages, all those players who announced the same positive integer in the first stage form a coalition and may either give effect to the coalition or dissolve it. All those players who announced 0 remain singletons. Thus no coalition with two or more players can be formed without the consent of all players as any player can announce the same positive number and no player can be forced to form a coalition with another player as a player can announce 0. If the outcome of the second stage is not the finest partition, the game ends and the partition formed remains formed forever. 6 But if the outcome of the second stage is the finest partition as in the status quo from which the game began in the first place, the two-stages are repeated, possibly ad 6 This is analogous to the rule in the infinite bargaining game of alternating offers (Rubinstein,1982) in which the game ends if the players agree to a split of the pie, but continues, possibly ad infinitum, if no agreement is reached. It is also similar to the rule that coalition formation is irreversible (e.g. Compte and Jehiel, 2010). 8
infinitum, until some nontrivial partition is formed in a future round. 7 In either case, the players receive payoffs in each period in proportion to a pre-specified feasible payoff vector (x 1,, x n ). If no nontrivial partition is formed and the game continues forever, then the players are said to agree to disagree perpetually. Notice that the payoff sharing game depends on a pre-specified feasible payoff vector and allows the players to form any partition and end the game; it does not rule out a priori any partition as a possible equilibrium outcome. The finest partition [N] is an outcome of the second stage of the two-stages if all players announce 0 in the first stage of the two-stages or some players announce the same positive integers in the first stage, but decide to dissolve the coalition(s) in the second stage. Since a nontrivial partition can be formed only with the agreement of all players, formation of a nontrivial partition is to be interpreted as an agreement among all players. Since formation of a nontrivial partition depends on the strategies of each and every player, a nontrivial partition cannot be formed without the consent of all players. Also notice that the conditional repetition of the two-stages does not imply stronger incentives to form a nontrivial partition. On the contrary, it may weaken them, since then the players stand to lose nothing by not forming a nontrivial partition in the current round as there will be opportunities to form it in a future round instead. To describe the repeated game in more concrete terms, visualize the following story: All players meet in a negotiating room to decide on formation of coalitions knowing in advance what their payoffs will be in each resulting partition. They may form any nontrivial partition or they may all decide to stay alone, i.e., form the finest/trivial partition. If the players agree to form a nontrivial partition, the meeting ends, the players receive per-period payoffs according to the pre-specified rule, and all leave the room. But if the players do not agree to form a nontrivial partition the meeting and negotiations continue and nobody leaves the room until the players agree to form a nontrivial partition. 7 Since the game starts from the finest partition, not allowing repetition of the two-stages if the outcome of the second stage is again the finest partition as in the status quo, from which the game begins in the first place, would be inconsistent. 9
Since the structure of the continuation game is exactly the same as the original game, we restrict ourselves to equilibria in stationary strategies of the repeated game. In fact, only the equilibria in stationary strategies are relevant. Accordingly, we characterize the equilibria of the repeated game by comparing only per-period payoffs of the players. We need the following definition. Definition 4 A partition function (N, v) is partially superadditive if for any partition P = k {S, [N\S]} and {S 1,, S k } such that i=1 where P = P\S {S 1,, S k }. S i = S, s i > 1, i = 1,, k, k i=1 v(s i ; P ) v(s; P) Partial superadditivity, as the term suggests, is weaker than the familiar notion of superadditivity which requires that combining any arbitrary coalitions increases their total worth. 8 In contrast, partial superadditivity requires that combining only all non-singleton coalitions increases their total worth. Unlike the familiar notion of superaditivity, it is trivially satisfied by all partition functions with three players, since each partition then has at most one non-singleton coalition. It is also satisfied by all those games with four players, since the grand coalition is an efficient partition and each partition has at most two non-singleton coalitions whose union is the grand coalition. Theorem 3 Let (N, v) be a partially superadditive partition function game with a nonempty γcore. Then, each γ-core payoff vector (x 1,, x n ) is an equilibrium payoff vector of the payoff sharing game. Proof: In order to obtain a sharper proof, we will assume all inequalities to be strict. It will be clear below that the proof holds also if the inequalities are weak. We show that in the payoff sharing game (i) to dissolve a coalition if it does not include all players is an equilibrium strategy of each player, and 8 See e.g. Hafalir (2007) for a formal definition of superadditivity. However, Hafalir uses the term full cohesiveness in place of superadditivity. 10
(ii) the grand coalition N is an equilibrium outcome resulting in per-period equilibrium payoffs equal to (x 1,, x n ). The theorem is clearly true for n = 2. It will be useful to prove the theorem separately for n = 3 and n > 3. Case n = 3: We first show that (i) implies (ii) and then prove that the strategies in (i) are indeed equilibrium strategies as they imply (ii). Given strategies in (i) and players' responses to them, we derive a reduced form of the payoff sharing game as follows: Given (i), let (w 1,, w n ) be the per-period equilibrium payoff vector of the repeated game. (a) If in some period, all players do not announce the same positive integer or some player announces i = 0, then as the strategies in (i) require any non-trivial coalition is dissolved and the outcome is the finest partition implying per period payoffs of (w 1,, w n ), since the continuation game is identical to the original game. (b) If in some period, all players announce the same positive integer, then the outcome is the grand coalition, the game ends, and the per-period payoffs are equal to (x 1, x 2, x 3 ). Note that if the grand coalition is indeed an equilibrium outcome of the game, then it may be realized in the first period itself. That is because the per-period payoffs of the players would be otherwise lower in the periods preceding the period in which the grand coalition is formed, since x i > v(i; {1,2,3}), i = 1, 2, 3, by definition of (x 1, x 2, x 3 ). 9 We solve the game first with discounting, and then take the limit as the discount rate goes to zero. Let δ < 1 be the discount factor. There is no loss of generality in assuming that one of the players, say 3, chooses only between strategies i = 1 and i = 0. The same analysis holds if player 3 chooses instead between strategies i = 2 or 3 and i = 0. Given the equilibrium strategies in (i), the payoff matrix of the reduced form of the repeated game is as below: 9 To economize on commas and brackets we shall often denote partitions {{1}, {2}, {3}} by {1,2,3}, {{1,2,3}} by {123}, and {{i}, {j, k}} by {i, jj}, whenever no confusion is possible. 11
Player 3 ---------------------------------------------------------------------- i = 1 i = 0 --------------------- ---------------------- Player 2 Player 2 -------------------- ---------------------- i = 1 i 1 i = 1 i 1 Player 1 i = 1 i 1 x 1, x 2, x 3 δδ 1, δδ 2, δδ 3 δδ 1, δw 2, δδ 3 δδ 1, δδ 2, δw 3 δδ 1, δw 2, δδ 3 δδ 1, δδ 2, δδ 3 δδ 1, δδ 2, δδ 3 δδ 1, δδ 2, δδ 3 Since (x 1, x 2, x 3 ) is a γ-core payoff vector, x i > v(i; {1,2,3}) 0, i = 1, 2, 3. To find a solution of the reduced game, consider first a mixed strategy Nash equilibrium. Let p 1, p 2, p 3 be the probabilities assigned by the three players to the strategy i = 1. Then, in equilibrium each player, say 1, should be indifferent between strategies i 1 and i = 1. Therefore, w 1 = p 2 p 3 δδ 1 + (1 p 2 p 3 )δw 1 = p 2 p 3 x 1 + (1 p 2 p 3 )δw 1. If x 1 > δw 1, then i = 1 is the dominant strategy and the resulting payoff is w 1 = x 1 (> 0), confirming the inequality x 1 > δw 1. Thus, i = 1 is indeed the dominant strategy of each player for each δ 1, the grand coalition N is an equilibrium outcome, and (x 1, x 2, x 3 ) are the per-period equilibrium payoffs. 10 We now prove that the strategies in (i) are indeed equilibrium strategies, since they imply (ii). Suppose in some period, two players, say 2 and 3, announce i = 1, but player 1 announces i 1. Suppose further that in Stage 2, players 2 and 3 give effect to the coalition 23 and do not dissolve it. Such a deviation from the strategies in (i) would lead to payoffs of x 2 x 2 +x 3 v(23; {23,1}) < x 2 and x 3 x 2 +x 3 v(23; {23,1}) < x 3 for players 2 and 3 (resp.), since (x 1,, x n ) is a γ-core payoff vector and, therefore, x 2 + x 3 > v(23; {23,1}). However, if 10 Since the equalities characterizing the mixed strategy equilibrium also hold for δ = 1, (x 1, x 2, x 3 ) is indeed the per-period equilibrium payoff vector in the limit. 12
players 2 and 3 adhere to the strategies and thus dissolve the coalition, then the game will be repeated and their payoffs from that, as shown above, will be x 2 and x 3, which are higher. Thus, it is ex post optimal for both players 2 and 3 to dissolve the coalition, which player 1 must take into account when deciding his strategy. 11 This proves (i) as well. Case n > 3: The extension to n > 3 follows from the fact that if the partition function game is partially superadditive, then any partition other than the finest partition has at least one nonsingleton coalition whose payoff is lower compared to its payoff in the grand coalition. More formally, let S 1,, S k be the non-singleton coalitions in the partition P = {S 1,, S m }. Let k S = i=1 S i and P = P\{S 1,, S k } {S}. Then, i=1 v(s i ; P) v(s; P ) < k i=1 j S i ), i S x i (= x j since (N, v) is partially superadditive and (x 1,, x n ) is a γ-core x j payoff vector. This implies v(s i ; P) < j S i for at least one non-singleton coalition S i of partition P. If the members of such a non-singleton coalition dissolve the coalition, then another coalition among the remaining non-singleton coalitions will similarly have a lower payoff then its payoff in the grand coalition, and so on. It is therefore ex post optimal for the members of each non-singleton coalition to dissolve their coalition so that the game is repeated and the grand coalition is formed. k Since the grand coalition, by assumption, is an efficient partition, it follows that the equilibrium outcome of the payoff sharing game is efficient. In contrast, Ray and Vohra (1997) and Yi (1997) show that the grand coalition is not an equilibrium outcome, even though, as Proposition 2 shows, the γ-cores of their games are nonempty and the players' payoffs in each partition are proportional to a pre-specified γ-core payoff vector. 12 The intuition for their contrasting result is the following: If the two-stages in a three-player game are to be played only once, then for a player i considering a unilateral deviation from the grand coalition, the coalition 11 The argument here is not that players 2 and 3 can force player 1 to merge with them by threatening to dissolve the coalition (and thus deny player 1 the opportunity to free ride), but rather that given the strategies in (i) and the players responses to them, such an action is ex post optimal for players 2 and 3, i.e., a subgame-perfect equilibrium strategy. 12 Though these games are not partially superadditive, Theorem 3 still applies since in these games each partition other than the finest includes at least one non-singleton coalition (namely, the largest coalition) which is worse-off. 13
structure {i, jj} rather than the finest partition {i, j, k} is the strategically relevant partition as the strategies of the other two players will not be oriented towards the finest partition if the twostages are not to be repeated and the payoffs of the other two players are higher in the partition {i, jj} than in the finest partition {i, j, k}, which is true especially in the case of superadditive partition function games. Therefore, if the payoff of player i in the partition {i, jj} is higher than its γ-core payoff in the grand coalition, player i can benefit by deviating from the grand coalition as that would lead to formation of the partition {i, jj} and not the finest partition {i, j, k}. Hence, in three-player symmetric games the three coalition structures with a pair and a singleton and not the grand coalition are the equilibrium outcomes if the payoff sharing game is limited to a single play of the two-stages. Theorem 3 does not show that the grand coalition is the unique equilibrium outcome of the payoff sharing game. E.g., in the case of three-player games the finest partition {1,2,3} is not only the status quo, but also an equilibrium outcome. However, this equilibrium is Pareto dominated, since (x 1, x 2, x 3 ), by hypothesis, is a γ-core payoff vector and thus x i > v(i; {1,2,3}), i = 1, 2, 3. Therefore, this equilibrium will never be played, since the players know the structure of the game and know that their rivals know the structure of the game and so on. Hence, if the players believe that perpetual disagreement is not a strategically relevant equilibrium outcome, their equilibrium strategies will be oriented towards formation of the grand coalition and no partition other than the grand coalition can be sustained as an equilibrium outcome, since a unilateral deviation by a player from the grand coalition will lead to the finest partition and repetition of the two-stages as the other two players strategies would remain focused on formation of the grand coalition in which their payoffs are higher. Therefore, the finest partition resulting in repetition of the two-stages, rather than any of the nontrivial partitions consisting of a pair and a singleton, is the strategically relevant partition for a player considering unilateral deviation from the grand coalition. Hence, the grand coalition is the only equilibrium outcome if the players believe that perpetual disagreement is not a strategically 14
relevant equilibrium. 13 Clearly, this argument can be extended, as in the proof of Theorem 3, to the payoff sharing game with more than three players. The converse of Theorem 3 is also true, i.e., if the grand coalition is the unique equilibrium outcome of the payoff sharing game, then the equilibrium payoffs must be equal to a γ-core payoff vector. This too is more easily seen by focusing on the case of three-player games. If the grand coalition is the only equilibrium outcome, then since the players, despite having the opportunity, do not form a partition consisting of a pair and a singleton and end the game, payoffs of the players in no pair must be higher than in the grand coalition. Furthermore, since the players do not form a partition consisting of a pair and a singleton, each player i can induce the finest partition by not agreeing to form the grand coalition and, therefore, obtain at least the payoff v(i; {1,2,3}). These conditions are exactly the same as those required of a γ-core payoff vector. Clearly, the argument can be extended to games with more than three players. To conclude, if the players believe and know that their rivals believe that perpetual disagreement is not a strategically relevant equilibrium and so on..., then the grand coalition is the unique equilibrium outcome of the payoff sharing game if and only if the γ-core is nonempty. 3.3 A stronger concept of γ-core It may seem that Theorem 3 applies only to those partition function games that satisfy partial superadditivity. Though, as already noted, partial superadditivity is weaker than the familiar notion of suparadditivity and satisfied by all partition function games with 3 or 4 players, it may not be satisfied in some applications of interest. However, even in such cases Theorem 3 holds for a subset of the γ-core payoff vectors. We need the following definition to identify that subset. Definition 5 The strong γ-core of the partition function game (N, v)is the set of feasible payoff vectors (x 1,, x n ) such that x i v({i}; [N]) for all i N and for every partition P = {S 1,, S m } [N], j Si x j > v(s i ; P) for at least one coalition S i P such that s i > 1. 13 In Ray and Vohra (1997) the finest partition is a strategically relevant equilibrium, since in their formulation coalitions can never merge; they can only become finer. 15
A forthcoming paper studies the strong γ-core in more details and provides additional justification for this concept. For the present, it is sufficient to note that the strong γ-core is a subset of the γ-core and independent of any beliefs the deviating coalitions may have regarding the formation of coalitions by outsiders. Theorem 4 Let (N, v) be a partition function with a nonempty strong γ-core. Then, any strong γ-core payoff vector (x 1,, x n ) is an equilibrium payoff vector of the payoff sharing game. Proof: Same as the proof of Theorem 3. 6. Conclusion We have introduced a concept of core for a general strategic game which is nicely related to the classical concepts of α- and β- cores and established its many properties. As an illustration of its applicability, it was shown that a well-known class of symmetric games admit nonempty γcores. We showed that the γ-core payoff vectors can be supported as equilibrium outcomes of a non-cooperative game which is an intuitive description how the players may reach an agreement on which partition to form when they know what their payoffs will be in each partition. This non-cooperative game consists of infinitely repeated two-stages. The fact that the two-stages may be repeated infinitely many times plays a crucial role in obtaining the result that the grand coalition is the unique equilibrium outcome if and only if the γ-core of the partition function game is nonempty. In contrast, if the game is limited to a single play of the two-stages, then as in the previous literature (see e.g. Ray and Vohra,1997, and Yi, 1997) the grand coalition is not an equilibrium outcome even if, as shown in Proposition 2, the γ-core is nonempty. 16
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