The Core of a Strategic Game *

Transcription

1 The Core of a trategic Game * Parkash Chander January, 2014 Abstract This paper introduces and studies the γ-core of a general strategic game. It shows that a prominent class of games admit nonempty γ-cores. It also shows that the γ-core payoff vectors (a cooperative solution concept) can be supported as equilibrium outcomes of a non-cooperative game and the grand coalition is the unique equilibrium outcome if and only if the γ-core is nonempty. As an application of this result, it shows that the γ-core of an oligopoly game is nonempty and, therefore, the oligopoly may become a monopoly unless prevented by law. JEL classification numbers: C71-73, D43, L13 Keywords: strategic game, core, repeated game, Nash program, oligopoly game. * First version January I thank an anonymous referee for helpful suggestions that improved the paper significantly. I have also benefitted from comments by Claude d Aspremont, Larry amuelson, Myrna Wooders, Parimal Bag, and seminar participants at UPenn and Vanderbilt University. Jindal chool of Government and Public Policy. parchander@gmail.com. 0

2 1. Introduction This paper introduces and studies the γ-core of a general strategic game. 1 It shows that the γcore of a strategic game is a stronger concept than the classical α- and β- cores in the sense that it is generally smaller. As an illustration of its applicability, it shows that a prominent 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 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. As an application of this result, we show that the γ-core of an oligopoly game with any number of not necessarily identical firms is nonempty 3 and, therefore, the oligopoly may become a monopoly unless prevented by law. The non-cooperative game for which the γ-core payoff vectors are equilibrium outcomes is intuitive and explicitly models the process by which 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. 4 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. It is shown that if the partition function representation of the strategic game is weakly superadditive, then, as the γ- 1 A similar concept was introduced earlier in Chander and Tulkens (1997), but 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 Rubinstein (1982), Perry and Reny (1994), Pe rez-castrillo (1994), and Lehrer and carsini (2012), for example. 3 However, this result is of an independent interest. Zhao (1999) and Radner (2001) show that the α- and β- cores of an oligopoly game are equal and nonempty. ince the γ-core, as will be shown, is generally smaller, the existence of a nonempty γ-core also implies the existence of nonempty α- and β- cores. 4 If the game is limited to a single play of the two-stages, then, as shown by Ray and Vohra (1997) and Yi (1997), the grand coalition is not an equilibrium outcome. 1

3 core requires, breaking apart into singletons upon deviation by a coalition is a subgame-perfect equilibrium strategy of the remaining players. ince all games may not satisfy weak superadditivity, the paper also introduces a stronger concept of γ-core, to be called the strong γ-core, which, unlike the γ-core, does not require the remaining players to break apart into singletons upon deviation by a coalition. In fact, it is independent of the beliefs the deviating coalitions may have regarding the formation of coalitions by the outsiders. We show that the oligopoly game admits a nonempty strong γ-core (a subset of the γ-core) if all firms are identical. The paper is organized as follows. ection 2 introduces the γ-core for a general strategic game and compares it with the classical α- and β- cores. ection 3 shows that a class of prominent games admit nonempty γ-cores. It introduces the 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. ection 4 proves that the oligopoly game admits a nonempty γ-core and identifies sufficient conditions under which an oligopoly may become a monopoly. It also shows that the strong γ-core of the oligopoly game is nonempty if all firms are identical. ection 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 and its complement by N\. Given a strategy t = (t 1,, t n ) T, let t (t i ) i, t (t j ) j N\, and (t, t ) t = (t 1,, t n ). Given a coalition N, the induced strategic game Γ = (N, T, u ) is defined as follows: The player set is N = {, (j) j N\ }, i.e., coalition and all j N\ are the players (thus the game has n s + 1 players); 5 5 The small letters n and s denote the cardinality of sets N and, respectively. 2

4 The set of strategy profiles is T = T j N\ T j where T = i T i is the strategy set of player and T j is the strategy set of player j N\; The vector of payoff functions is u = (u, u j ) j N\ where u (t, t ) = i u i (t, t ) is the payoff function of player and u j (t, t ) = u j (t, t ) is the payoff function of player j N\, for all t T and t j N\ T j. Notice that if (t, t ) = t is a Nash equilibrium of the induced game Γ, then u (t, t ) = i u i (t, t ) i u i (t, t ) for all t T. 6 Thus, for each N, a Nash equilibrium of the induced game Γ assigns a payoff to coalition which it can obtained without cooperation from the remaining players. If the induced game Γ has multiple Nash equilibria, then any one with the highest payoff for can be selected. 7 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 () = i u i (t, t ), N, where (t, t ) T is a Nash equilibrium of the induced game Γ with the highest payoff for coalition. 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). 8 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 x i w γ (), for all 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. 9 6 Note that a Nash equilibrium of an induced game is not an equilibrium of the original game. It is exactly the same as a quasi-hybrid solution corresponding to a partition in Zhao (1991). This equilibrium concept has appeared in the previous literature also under other names such as partial agreement equilibrium in Chander and Tulkens (1997) and Lardon (2012). 7 uch a payoff exists if the strategy sets are compact (or finite) and the payoff functions are continuous. 8 Note that the grand coalition is an efficient partition, since it can choose at least the same strategies that the players may choose in any partition. 9 Applying any other refinement procedure to the sets of Nash equilibria of the induced games Γ, 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. 3

5 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. 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 N under this concept is w α () = max t T min tn\ T N\ contrast, in the β-core concept, the worth of a coalition is w β () = min tn\ T N\ max t T i u i (t, t N\ ). i u i (t, t N\ ). By It is well-known that the α- and β- cores are large. In fact, they are often too large. hapley and hubik (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 that such behavior may, in fact, 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 (Aumann, 1961 and carf, 1971). 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. 10 Proof: We need to show that w β () w γ (), N. Given N, let (t, t ) and (t, t ) be such that w γ () = i u i (t, t ) and w β () = i u i (t, t ). Let t (t ) = arg max t T i u i (t, t ), t T. Then, by definition, w β () = i u i (t ( t ), t ) i u i (t (t ), t ) for all t T. In particular, i u i (t ( t ), t ) i u i (t ( t ), t ) = w γ (). Hence, w β () w γ (), and, therefore, w α () w β () w γ (), N. Examples are easily constructed in which the inequalities established in the proposition are strict. ince the proposition holds for a general strategic game, it can be applied to a variety of economic models. E.g., Proposition 2.1 concerning an oligopoly game in Lardon (2012) is a special case of this proposition 10 The proposition also implies that the γ-core solutions are not inconsistent with the α- and β- core solutions. 4

6 3. The γ-core and the partition function 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 by assigning to each coalition in a partition the Nash equilibrium payoff of the coalition in the induced strategic game in which each coalition in the partition becomes one single player (Zhao, 1991 and Ray and Vohra, 1997). More formally, a set P = { 1,, m } is a partition of N if i j = for all i, j {1,, m}, i j, and m i=1 i = N. Let v( i ; P) 0 denote the Nash equilibrium payoff of coalition i in the induced strategic game in which each coalition j, j = 1,, m, becomes one single player. 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 ection, we shall treat a partition function as the primitive, but recall when necessary that the partition function is generated from a strategic game. One implication of that 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 a 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 whatever strategies the members of the coalition can choose individually. 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 all singletons, given the strategies of the remaining players. Accordingly, the non-cooperative 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 payoff of the grand coalition. Let [N] and [N\] denote the finest partitions of N and N\, 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 x i v(; {, [N\]}) for all N. 5

7 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 x i v(; P) for all partitions P such that P. In fact, the definition implies that the γ-core of a partition function game (N, v) is equal to the core of the characteristic function game (N, w γ ) 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. In particular, the feasible payoff vector with equal shares is a γ-core payoff vector. In addition to illustrating the applicability of the γ-core, this result also enables us to contrast Theorem 3 below with the related results in the previous literature. We need some additional notation. A partition function (N, v) is symmetric if for every partition P = { 1,, m }, s i = s j v( i ; P) = v j ; P. Proposition 2 Let (N, v) be a symmetric partition function game such that for every partition P = { 1,, m }, v( i ; P)/s i v( 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 x i v(; {, [N\]}) for all N. ince the grand coalition is an efficient partition, v(; {, [N\]}) + i N\ v(i; {, [N\]}) v(n; {N}) = i N x i. ince v(; {, [N\]})/s v(i; {, [N\]}), i N\, and x i = x j, i, j N, this inequality can hold only if v(; {, [N\]})/s x i. Hence, v(; {, [N\]}) i x i for all N. For symmetric games, equal sharing of 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 11 ee Hafalir (2007) for alternative core concepts for partition function games. Applications and properties of these alternative core concepts, however, remain largely unexplored. 6

8 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 = { 1,, m } a vector of individual payoffs x(p) R n such that j i x j (P) = v( i ; P), i P. 12 A mapping x: R n is the equal payoff sharing rule if for each partition P = { 1,, m }, x i (P) = x j (P) for each i, j 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. A payoff sharing rule x: R n is monotonic if for each partition P = { 1,, m } and each coalition i P, x j (P) > ( )x j ({N}) for all j i if and only if v( i ; P) > ( ) x j ({N}) j 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 sharing rules can ensure unanimity among the members of a coalition on whether to deviate or not from the grand coalition if they are farsighted and can foresee the resulting partition that will form subsequent to their deviation. Definition 3 A payoff sharing rule x: R n is proportional if for each partition P = { 1,, m } and each coalition i P, x j (P) = x j ({N}) [v( i ; P)/ k i x k ({N})], j 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 = { 1,, m } and each coalition i P, x j (P) = x j [v( i ; P)/ k i x k ]. Theorem 3 below holds for all monotonic payoff sharing rules, but to obtain a sharper proof we shall restrict ourselves to a proportional rule. 12 Thus, a payoff sharing rule enables the players to compare their individual payoffs in different partitions. In contrast, Hart and Kurz (1983) assume that the individual payoffs of the players are the same irrespective of which partition is formed and equal to the hapley value allocations that the players would receive in the grand coalition. 7

9 3.2 Non-cooperative foundations of the γ-core We show that the γ-core payoff vectors can be supported as equilibrium payoff vectors 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 and indicates that it may be relevant in other contexts too. 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. 13 If the outcome of the second stage is not the finest partition, the game ends and the partition formed remains formed forever. 14 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 infinitum, until some nontrivial partition is formed in a future round. 15 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, i.e., the players agree to disagree perpetually. Note that the payoff sharing game 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. ince 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. 16 Also notice that the conditional repetition of the two-stages 13 Thus no coalition with two or more players can be formed without the consent of all players and no player can be forced to form a coalition with another player. 14 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 in Compte and Jehiel (2010) that coalition formation is irreversible. 15 ince 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 began in the first place, would be inconsistent. 16 ince formation of a nontrivial partition depends on the strategies of each and every player, a nontrivial partition cannot be formed without the agreement of all players. 8

10 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. ince 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, the game admits equilibria only in stationary strategies. 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 weakly superadditive if for any partition P = k {, [N\]} and { 1,, k } such that i=1 where P = P\ { 1,, k }. i =, s i > 1, i = 1,, k, k i=1 v( i ; P ) v(; P) Weak superadditivity, as the term suggests, is weaker than the familiar notion of superadditivity which requires that combining any arbitrary coalitions increases the total worth of the coalitions. 17 In contrast, weak superadditivity requires that combining only all nonsingleton coalitions increases the total worth of coalitions. Unlike the familiar notion of superaditivity, it is trivially satisfied by all partition functions with three players and also by all those with four players if, as is in the present case, the grand coalition is an efficient partition. 17 ee e.g. Hafalir (2007) for definition who uses the term full cohesiveness for superadditivity. 9

11 Theorem 3 Let (N, v) be a weakly superadditive partition function game with a nonempty γcore. Then, any 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 to the reader that the proof holds also if the inequalities are weak. We show that in the payoff sharing game (i) (ii) to dissolve a coalition if it does not include all players is an equilibrium strategy of each player, and 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 must be an equilibrium outcome 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 10

12 coalition is formed, since x i > v(i; {1,2,3}), i = 1, 2, 3, by definition of (x 1, x 2, x 3 ). 18 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 strategies in (i), the payoff matrix of the reduced form of the repeated game is as below. Player i = 1 i = Player 2 Player i = 1 i 1 i = 1 i 1 Player 1 i = 1 i 1 x 1, x 2, x 3 δw 1, δw 2, δw 3 δw 1, δw 2, δw 3 δw 1, δw 2, δw 3 δw 1, δw 2, δw 3 δw 1, δw 2, δw 3 δw 1, δw 2, δw 3 δw 1, δw 2, δw 3 ince (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 δw 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 > 18 To economize on commas and brackets whenever possible, we shall denote partitions {{1}, {2}, {3}} by {1,2,3}, {{123}} by {123}, and {i, {jk}} by {i, jk}. 11

13 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. 19 We now prove that the strategies in (i) are indeed equilibrium strategies, since they imply (ii). uppose in some period, two players, say 2 and 3, announce i = 1, but player 1 announces i 1. uppose further that in tage 2, players 2 and 3 give effect to the coalition 23 and do not dissolve it. uch 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 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. 20 This proves (i) as well. Case n > 3: The extension to n > 3 follows from the fact that if the partition function game is weakly 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 1,, k be the non-singleton coalitions in the partition P = { 1,, m }. Let k = i=1 i and P = P\{ 1,, k } {}. Then, i=1 v( i ; P) v(; P ) < k i=1 j i ), i x i (= x j since (N, v) is weakly superadditive and (x 1,, x n ) is a γ-core x j payoff vector. This implies v( i ; P) < j i for at least one non-singleton coalition 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. k 19 ince 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. 20 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 their strategies in (i) and the players responses to it, such an action is ex post optimal for players 2 and 3, i.e., a subgame-perfect equilibrium strategy. 12

14 ince the grand coalition, by assumption, is an efficient partition, it follows that the payoff sharing game admits an efficient equilibrium. 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. 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 structure {i, jk} 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 two-stages are not to be repeated and the payoffs of the other two players are higher in the partition {i, jk} 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, jk} 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, jk} and not the finest partition {i, j, k}. Hence, in threeplayer 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 13

15 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 relevant equilibrium. 21 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 which satisfy weak superadditivity. Though, as already noted, weak 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 = { 1,, m } [N], j i x j > v( i ; P) for at least one coalition i P such that s i > In contrast, the finest partition is a strategically relevant equilibrium in Ray and Vohra (1997) since they assume that coalitions can never merge, they can only become finer. 14

16 The strong γ-core is clearly a subset of the γ-core. Unlike many other core concepts in the extant literature (see e.g. Lekeas, 2013 and Yong, 2004), the strong γ-core is independent of any beliefs the deviating coalitions may have regarding the formation of coalitions by the outsiders. 22 We now identify a class of games for which the strong γ-core is nonempty. A symmetric partition function admits a nonempty strong γ-core if the inequalities in Proposition 2 are strict, i.e., v( i ; P)/s i < (=) v( j ; P)/s j if s i > (=)s j and the feasible payoff vector with equal shares belongs to the strong γ-core. Thus the class of symmetric games in Ray and Vohra (1997), Yi (1997), and Chander (2007) among others admit nonempty strong γ-cores, since, as can be easily seen, the relevant inequalities in these games are indeed strict. Theorem 3 * 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: ame as the proof of Theorem The γ-core of an oligopolistic market The literature on oligopolistic markets has focused mostly on non-cooperative models. Formation of coalitions among oligopolistic firms engaging in cooperative behavior has been studied only recently. This has been done (see e.g. Lekeas, 2013) by first converting the oligopoly game to a partition function game by computing the quasi-hybrid solution for each partition in Zhao (1991). We shall follow the same approach and use the Nash equilibrium of an induced game which, as noted above, is exactly the same as the quasi-hybrid solution for a partition, to convert the oligopoly game into a partition function game. ince theorems 3 and 3 * show that the grand coalition is an equilibrium outcome of a non-cooperative game of coalition formation, they suggest promising avenues for applications to an oligopoly. However, for either of these theorems to apply, the oligopoly game must admit at least a nonempty γ-core. Thus, we first need to show that an oligopoly with any number of not necessarily identical firms admits a nonempty γ-core and a nonempty strong γ-core if the firms are identical. 22 Unlike the other core concepts, the beliefs of deviating coalitions regarding the outsiders in the case of γ-core are not really exogenous, since, as the proof of Theorem 3 shows, breaking apart into singletons upon deviation by a coalition is a subgame-perfect equilibrium strategy of the outsiders. 15

17 4.1 The model The set of oligopolistic firms is N = {1,, n}. Let p(q) denote the inverse demand function faced by these firms, where q is the total demand. We assume that the inverse demand function is differentiable and strictly decreasing, i.e., p (q) < 0. Let c i (q i ) denote the cost function of firm i with c i (0) = 0. We assume that the cost function of each firm is differentiable, strictly increasing and strictly convex, i.e., c i (q i ) > 0, q i > 0, and c i (q i ) > 0, q i 0. Assuming strict convexity of the cost functions enables us to avoid the problem of multiple solutions, but it should be clear to the reader that the results below also hold if the cost functions are linear. The profit function of each firm i is π i (q 1,, q n ) = p(q)q i c i (q i ), where q = j N q j. In order to avoid corner solutions, we assume that there exists a q 0 such that p(q 0 ) + p (q 0 )q 0 c i (q 0 ) < 0, p(nq 0 ) > 0, and c i (0) = 0, i N.These assumptions imply that a profit maximizing firm will never produce an output larger than q 0 even if it has the capacity to do so and will always produce a positive amount irrespective of the output of other profit maximizing firms. 23 Many quadratic cost functions and linear demand functions satisfy the above assumptions. We assume further that the revenue function p(q)q i of each firm i is concave in q 1,, q n. Thus, the marginal revenue p(q) + p (q)q i of each firm i is non-increasing in the output of other firms, i.e., p (q) + p (q)q i 0 for each fixed q i 0. Note that this condition is satisfied, if the inverse demand function p(q) is decreasing and concave. 4.2 The oligopoly game Let T i = [0, q 0 ], T = T 1 T n, and π = (π 1,, π n ). We shall refer to the strategic game (N, T, π) as the oligopoly game. Clearly, each strategy set T i is compact and convex and each π i is concave and continuous in q 1,, q n. Lemma 4 The oligopoly game (N, T, π) admits a unique Nash equilibrium (q 1,, q n ). Proof: ee the appendix to the paper. 23 The output level q 0 is not to be interpreted as the capacity constraint that is often assumed in models of oligopoly (see e.g. Radner, 2001 and Lardon, 2012). Our formulation allows the firms to expand capacity when necessary leading to a more general analysis and a notion of core that is independent of any exogenously fixed capacity constraints. 16

18 Let (N, T, π ) denote the induced game when coalition forms. ince each π i (q 1,, q n ) is concave and continuous in q 1,, q n, the payoff function π (q 1,, q n ) (= π i (q 1,, q n )) i of coalition is also concave and continuous in q 1,, q n. Moreover, the strategy set j T j of is compact and convex. Therefore, the induced game (N, T, π ) admits a unique Nash equilibrium. Proposition 5 For each coalition N, let (q 1,, q n ) denote the unique Nash equilibrium of the induced game (N, T, π ).Then, (i) q = i N i N q i = q, and (ii) q j q j for each j N\. That is if a cartel forms the total industry output is lower, but the output of each independent firm is higher. Proof: (i) uppose contrary to the assertion that q > q. Then, either q j > q j for some j N\ or i > i q i. In the former case, c j q j = p(q ) + q j p (q ) > p(q ) + q j p (q ) q i p(q ) + q j p (q ) = c j q j q j > q j, which is a contradiction. In the latter case, for each i, c i q i = j p (q ) + p(q ) < i q i p (q ) + p(q ) i q i p (q ) + p(q ) q j q i p (q ) + p(q ) = c i (q i ) q i > q i, which contradicts our supposition that i > q i Therefore, q q. q i q i i. (ii) uppose contrary to the assertion that q j < q j for some j N\. Then, since q q as show n, c j q j = p(q ) + q j p (q ) < p(q ) + q j p (q ) p(q ) + q j p (q ) = c j q j q j < q j, which contradicts our supposition that q j < q j for each j N\. Therefore, q j q j for all j N\. 4.3 The γ-core of an oligopoly game Let P = { 1,, m } be some partition. Then, the induced game in which each coalition i becomes one single player admits a unique Nash equilibrium. That is because the payoff function π j (q 1,, q n ) j i of each coalition i is concave and continuous. Moreover, the strategy set j i T j of i is compact and convex. Therefore, as in Lemma 4, each induced game admits a unique Nash equilibrium. Let v( i ; P) be equal to the Nash equilibrium payoff of coalition i in 17

19 the induced game. Then, (N, v) is a partition function game representation of the oligopoly game (N, T, π). A familiar method to prove the existence of the core of a characteristic function game (see e.g. Helm (2001)) comes from the well-known Bondareva-hapley theorem (Bondareva, 1963 and hapley, 1967). It uses the following concept of a balanced collection of coalitions. Given the set of players N, let C denote the set of all coalitions that can be formed and for each i N, let C i = { C: i } denote the subset of all coalitions of which i is a member. Then, C is a balanced collection of coalitions, if for each coalition C, there exists a δ [0,1] such that δ = 1 C i for every i N. According to Bondareva-hapley theorem, a characteristic function game (N, w) has a nonempty core if and only if every balanced collection of coalitions C. C δ w() w(n) for Given a balanced collection of coalitions C, we derive an important inequality concerning the equilibrium outputs (q 1,, q n ) of the firms in the family of induced games (N, T, π ), C. This uses the inequalities i N\ q i i N\ q i and i q i i q i, N, established in Proposition 5 in which (q 1,, q n ) is the Nash equilibrium of the induced game (N, T, π ) and (q 1,, q n ) is the Nash equilibrium of the oligopoly game (N, T, π). Lemma 6 If C is a balanced collection of coalitions, then for each i N, δ q j j N C i i N C i δ q i. That is, the average total output of all firms is not smaller than the total average output of all firms, where the average is taken over the coalitions which include firm i. Proof: ee the appendix to the paper. Let w γ (. ) denote the γ-characteristic function of the oligopoly game (N, T, π), i.e., w γ () = i π i ( q 1,, q n ), N, where (q 1,, q n ) is the Nash equilibrium of the induced game (N, T, π ). Theorem 7 The oligopoly game (N, T, π) has a nonempty γ-core. Proof: In view of the Bondareva-hapley theorem, we need to show that w γ (N) for every balanced collection of coalitions C. By definition C δ w γ () 18

20 C δ w γ () = C δ i π i (q 1,, q n ) = i N Ci δ π i (q 1,, q n ) i N π i ( C δ (q 1,, q i n )) (since C i δ = 1 and each π i (. ) is concave) = i N [p( δ j N q j ) C i δ q i c i ( δ q i C i C i )] p i N C i δ q i i N C i δ q i i N c i ( C i δ q i ) (by Lemma 6) w γ (N), since w γ (N) i N [p(q)q i c i (q i )] for all (q 1,, q n ) and q = i N q i. It is well-known that the monopoly game may not be weakly superadditive for some demand and cost functions. Thus, we look for conditions under which Theorem 3 * applies instead. Proposition 8 The oligopoly game admits a nonempty strong γ-core if all firms are identical. Proof: We show that the payoff vector with equal shares belongs to the strong γ-core. Given a partition P = { 1,, m }, let (q 1,, q n ) be the outputs of the firms in the Nash equilibrium of the induced game in which each coalition i, i = 1,, m, becomes one single player. The first order conditions imply c j q j = p( k N q k ) + k i q k p ( k N q k ), j i, i = 1,, m. ince the firms are identical, the cost function c j is convex and the inverse demand function p is strictly decreasing, these conditions imply q j < q k if firm j is a member of a larger coalition in partition P than firm k. Thus, π j (q 1,, q n ) < π k (q 1,, q n ) if firm j is a member of a larger coalition in the partition P than firm k. ince the grand coalition is the efficient partition, i N π i (q 1,, q n ) < π i (q 1,, q n ) where q 1,, q n are the outputs when all firms merge. i N ince the firms are identical, π j (q 1,, q n ) < π j (q 1,, q n ) if firm j is a member of the largest coalition in partition P. 24 Furthermore, the payoffs of all coalitions in the finest partition are equal and lower than their payoffs π j (q 1,, q n ), j N, in the grand coalition. These inequalities imply that the payoff of the largest coalition is lower and the vector with equal payoffs belongs to the strong γ-core. ince the strong γ-core is nonempty if the cost functions are identical and the coalitional payoffs depend continuously on marginal costs, it must also be nonempty for cost functions which are nearly identical, i.e., for an oligopoly consisting of firms which are not "too" 24 If no coalition is the largest, then the inequality holds for all coalitions which are not smaller than any other coalition. 19

21 heterogeneous. This means that an oligopoly with homogenous or nearly homogeneous firms is likely to become a monopoly. 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 prominent class of symmetric games admit nonempty γ-cores. We showed that the γ-core payoff vectors can be supported as equilibrium outcomes of a non-cooperative game which describes an intuitive process by which 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 the γ-core is nonempty. We also introduced the concept of strong γ-core, which unlike other core concepts that have been proposed in the literature, is independent of the beliefs that deviating coalitions may have regarding the coalitions that will be formed by the outsiders. In order to apply our concepts and results for a general strategic game or a partition function representation thereof, we demonstrated that the γ-core of an oligopoly game with any number of not necessarily identical firms is nonempty, and the strong γ-core of the oligopoly game is nonempty if the firms are identical. This means that an oligopoly with nearly homogenous firms is likely to become a monopoly -- unless prevented by law. Appendix Proof of Lemma 4: ince each π i (. ) is concave and continuous in q 1,, q n, the game (N, T, π) has a Nash equilibrium (q 1,, q n ). uppose contrary to the assertion that the game has another Nash equilibrium, say (q 1,, q n ), and (q 1,, q n ) (q 1,, q n ). Without loss of generality, let 20

22 q = i N q i i N q i = q. ince (q 1,, q n ) (q 1,, q n ), q i > q i for at least one i. Furthermore, p (q )q i + p(q ) > p (q )q i + p(q ) p (q )q i + p(q ), since q q and by assumption the marginal revenue of each firm is non-increasing with total demand. From the first order conditions for a Nash equilibrium c i (q i ) = p (q )q i + p(q ) > p (q )q i + p(q ) = c i (q i ) implying q i < q i, which is a contradiction. Proof of Lemma 6: We first note a useful accounting identity: for each i N, Ci δ j q j = Ci δ ( j N q j j q j ) = j N q j Ci δ j q j = C δ i q i Ci δ j q j = C\Ci δ j q j. For any arbitrary i, C i δ j N q j = δ j q j + δ j q j C i C i C i δ j q j + Ci δ j q j (by Proposition 5) = C i δ j q j + C\Ci δ j q j (by the accounting identity) C i δ j q j + δ j q j (by Proposition5) C\C i = C δ j q j = i N δ q i. C i The following numerical example may help in understanding the proof of Lemma 6. To economize on commas and brackets we shall denote coalitions {ijk} and {ij} simply by ijk and ij, respectively. Let n = 4 and C = {12, 13, 14, 234} be the balanced collection of coalitions with weights given by 1/3, 1/3, 1/3 and 2/3, respectively. We first verify the accounting identity used in the proof of Lemma 6 for = 1. By definition, Ci δ j q j = 1 (q q 4 ) + 1 (q q 4 ) + 1 (q q 3 ) = 2 (q q 3 + q 4 ) = C\Ci δ j q j. The proof for i = 2, 3, 4 is analogous. We next verify the inequality in Lemma 6. C i δ j N q j = 1 (q q q q 12 4 ) + 1 (q q q q 13 4 ) + 1 (q q q q 4 14 ) 21

23 = 1 3 (q q 2 12 )+ 1 3 (q q 3 13 ) (q q 4 14 ) (q q 4 12 ) (q q 4 13 ) (q q 3 14 ) 1 3 (q q 2 12 )+ 1 3 (q q 3 13 ) (q q 4 14 ) (q 3 + q 4 ) (q 2 + q 4 ) (q 2 + q 3 ) (by Proposition 5) = 1 3 (q q 2 12 )+ 1 3 (q q 3 13 ) (q q 4 14 ) (q 2 + q 3 + q 4 ) 1 3 (q q 2 12 )+ 1 3 (q q 3 13 ) (q q 4 14 ) (q q q ) (by Proposition 5) = 1 3 q q q q q q q q q = i N C i δ q i. 22