Solutions of Bimatrix Coalitional Games

Transcription

1 Applied Mathematical Sciences, Vol. 8, 2014, no. 169, HIKARI Ltd, Solutions of Bimatrix Coalitional Games Xeniya Grigorieva St.Petersburg State University Faculty of Applied Mathematics and Control Processes University pr. 35, St.Petersburg, , Russia Copyright c 2014 Xeniya Grigorieva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The PMS-vector is defined and computed in [1] for coalitional games with perfect information. Generalizati-on of the PMS-vector for the case of Nash equilibrium in mixed strategies is proposed in this paper. Mathematics Subject Classification: 91Axx Keywords: bimatrix games, coalitional partition, Nash equilibrium, Shapley value, PMS-vector, games with perfect information 1 Introduction The new approach to the solution of bimatrix coalitional games is proposed. Suppose N-person game Γ with finite sets of strategies is given. The set of players N is divided on two subsets (coalitions) S, N\S each acting as one player. The payoff of player S (N\S) is equal to the sum of payoffs of players from S (N\S). The Nash equilibrium (NE) in mixed strategies is calculated (in the case of multiple NE [2] the solution of the correspondly coalitional game will be not unique). Mathematical expectation of the payoffs coalition S (N\S) in the NE in mixed strategies is allocated according to the Shapley value [3]. The resulting payoffs vector will be a generalization of the PMS-vector defined and computed in [1] for coalitional games with perfect information. Then the payoff of coalitions S, N\S which appears with positive probability in the NE is allocated proportionally to the PMS-vector.

2 8436 Xeniya Grigorieva 2 Statement of the problem Suppose n-person game Γ = {N, X 1,..., X n, K 1,...,K n } in normal form with the coalitional partition Σ = {S 1,..., S l }, l n, S i S j =, i j, be given. Consider the game in normal form Γ Σ = {N, X 1,..., X l, H 1,...,H l } between l players, where the players are coalitions from partition Σ. Consider coalition S i, consisting of s i players. Notions: m j is the number of strategies of player j; X j = { } x k j is the set of the strategies of player j; k=1,m j X Si = X j is the set of the strategies of coalition S i, i. e. Cartesian product of the sets of players strategies, which are included into coalition S i ; vector x i X Si of dimension s i = S i is the strategy of player i in the game Γ Σ ; H Si = K j, i. e. payoff of player S i is a sum of payoffs of the players from coalition S i. l Si = X Si = l j is the number of pure strategies of coalition S i ; l Σ = l Si is the number of l-tuples in pure strategies in the game Γ Σ. i=1,l Consider the case, when the set of players N is divided into two separated coalitions S and N\S each acting as one player. Then we get the coalitional bimatrix game Γ ( Ã, B) = { N, XS,X N\S, H S, H N\S }, where à = {H S (x i, x j )} xi X S, x j X N\S, B = { HN\S (x i, x j ) } x i X S, x j X N\S. It s required to find the optimal imputation rule for each coalition and in some sense optimal strategies for the coalitions. 3 Solution of the problem 1. We shall consider the case when the matrixes à and B can be reduced to square matrixes A and B: det A 0, det B 0. In other cases the iterational methods can be used.

3 Solutions of bimatrix coalitional games 8437 Solve the bimatrix game Γ (A, B), i. e. find a NE in the mixed strategies by formulas x = v 2 ub 1 ; y = v 1 A 1 u, where v 1 = 1/(uA 1 u), v 2 = 1/(uB 1 u), u = (1,..., 1), using the theorem about a completely mixed equilibrium ([4], p. 135). In the case of multiple NE [2] the solution of the corresponding coalitional game will be not unique. 2. Calculate the NE value in mixed strategies: where v (S) = i X S E ( x, ȳ) = [v (S), v (N\S)], j X N\S a ij ξ i η j, v (N\S) = i X S j X N\S b ij ξ i η j, x = {ξ i } i XS, y = {η j } j XN\S. One can show that v (S) v i, where v i maximal guaranteed payoff i S of i-th player, i S, under condition that the players from coalition N\S use mixed strategy from NE. This follows from the superadditivity of the characteristic function defined as maximal guaranteed payoff of the coalition S. 3. In [1] the definition of PMS-vector in pure strategies for coalitional games with perfect information has been given. Find PMS-vector (imputation) in mixed strategies as the mathematical expectation over the 2-tuples of strategies generated by NE. In the beginning define PMS-vector in mixed strategies for the game Γ Σ. Let µ = ( µ 1,..., µ l ) be l-tuple NE in mixed strategies in the game Γ Σ, where each mixed strategy of coalition S i is a vector µ i = ( µ 1 i,..., µ l ) S i i, µ j i 0, j = 1, l Si, Denote a payoff of coalition S i in NE by l Si j=1 µ j i = 1. where p k = i=1,l v (S i ) = l Σ k=1 p k H k (S i ), i = 1, l, µ j i i, j i = 1, l Si, k = 1, l Σ, probability of the payoff s realization H k (S i ) of coalition S i, when players choose their pure strategies x i in l-tuple NE in mixed strategies µ. The value H k (S i ) is random variable. There could be many l-tuple NE in the game, therefore, v (S 1 ),..., v (S l ), are not uniquely defined. Consider for each coalition S i Σ, i = 1, l, a cooperative game G Si supposing that the players which are not included into the coalition S i, use NE strategies from l-tuple µ.

4 8438 Xeniya Grigorieva Definition 3.1 Let w (S i : K) = v (K) be characteristic function in the cooperati-ve game G Si, where K S i. Divide payoff w (S i ) = v (S i ) between the players of coalition S i, according to Shapley value [3] Sh = (Sh 1,..., Sh Si ): Sh i = S S S i (s 1)! (s s )! s! [w (S ) w (S \ {i})] i = 1, s, (1) where s = S ( s = S ) is the number of elements of set S (S ) and w (S ) is a maximal guaranteed payoff of the subcoalition S S. Denote Sh (S k ) = (Sh (S k : 1),..., Sh (S k : s k )), where s k is the number of elements of set S k. Moreover w (S i ) = s i j=1 Sh (S i : j). Then PMS-vector for the NE in mixed strategies in the game Γ Σ is defined as PMS (Γ Σ ) = (PMS 1 (Γ Σ ),..., PMS N (Γ Σ )), where PMS j (Γ Σ ) = Sh (S i : j), j S i, i = 1, l. 4. Divide payoffs H k (S i ) of coalition S i for every i = 1, l occurring with positive probability when l-netuple in the game Γ Σ is played, proportionally to the components of Shapley value: λ j (S i ) = Sh (S i : j) Sh (S i : j), j S i, i = 1, l. (2) One can show that is Sh (S i : j) = l Σ p k H jk (S i ), where H jk (S i ) = λ j H k (S i ), k=1 j S i i = 1, l. Then in bimatrix coalitional game matrixes of players payoffs j = 1, N, are defind as follows A j = λ j A, j S; B j = λ j B, j N\S. 4 Example Let there be 3 players in the game. Each of them has two strategies (see table 1). The payoffs of each player are defined for all three-tuples. 1. Compose and solve the coalitional game, i. e. find NE in mixed strategies in the game: η = 3/7 1 η = 4/7 0 0 ξ = 1/3 1 ξ = 2/3 1 2 (1, 1) [6, 1] [3, 2] (2, 2) [4, 3] [4, 2] (1, 2) [4, 5] [6, 3] (2, 1) [8, 1] [3, 2].

5 Solutions of bimatrix coalitional games 8439 Table 1: The strategies The payoffs I II III I II III Figure 1: It s clear, that first matrix row is dominated by the last one and the second is dominated by third. One can easily calculate NE and we have y = (3/7, 4/7), x = (0, 0, 1/3, 2/3). Then the Nash value of the game in mixed strategies is calculated as E (x, y) = 1 7 [4, 5] [8, 1] [6, 3] + 8 [ [3, 2] = 7, 7 ] [ = 5 1 ] 3 7, Divide the game s Nash value in mixed strategies according to Shapley s value (1). For this purpose find the maximal guaranteed payoffs v {I} and v {II} of players I and II (see fig.1). At the fig. 1 pure strategies of coalition N\S and its mixed strategy y are given horizontally at the right side. Pure strategies of coalition S and its mixed strategy x are given vertically. Inside the table players payoffs from the coalition S and the total payoff of the coalition S are given at the right side.

6 8440 Xeniya Grigorieva Fix a NE strategy of a third player as ȳ = (3/7, 4/7). At the fig. 1 the mathematical expectations of the players payoffs from coalition S when mixed NE strategies are used by coalition N\S are located at the left. Then, look at the fig. 1: min H 1 (x 1 = 1, x 2, ȳ) = min { 2 2; 4 } = 2 2 ; 7 min H 1 (x 1 = 2, x 2, ȳ) = min { 2 5; 7 0} = 0; v 1 = max { 2 2; 7 0} = 2 2; 7 min H 2 (x 1, x 2 = 1, ȳ) = min { 2; 2 7} 3 = 2 ; min H 2 (x 1, x 2 = 2, ȳ) = min {1; 4} = 1; v 2 = max {2; 1} = 2. Thus, maxmin payoff for player 1 is v {I} = 2 2 and for player 2 is v {II} = 2. 7 Hence, Sh 1 (ȳ) = 2 5, Sh 7 2 (ȳ) = Thus, PMS-vector is equal: PMS 1 = ; PMS 2 = ; PMS 3 = Now dividing the payoffs of coalition S in pure strategies proportional to the Shapley vector (see (2)) we get λ 1 = 19, λ 36 2 = Hence, the newly defind payoffs of players I and II from coalition S are: A I = λ 1 A = ( Conclusion ) = ( ), A II = λ 2 A = ( In this paper the algorithm of getting imputation proportional to the PMSvalue in the NE in mixed strategies, and the example which show realization of the proposed approach are given. ). References [1] L. Petrosjan, S. Mamkina, Dynamic Games with Coalitional Structures, International Game Theory Review, 8(2) (2006), [2] J. Nash, Non-cooperative Games, Ann. Mathematics 54 (1951), [3] L. S. Shapley, A Value for n-person Games. In: Contributions to the Theory of Games (Kuhn, H. W. and A. W. Tucker, eds.) (1953), Princeton University Press.

7 Solutions of bimatrix coalitional games 8441 [4] L. Petrosjan, N. Zenkevich, E. Semina, The Game Theory. - M.: High School, Received: November 9, 2014; Published November 27, 2014