Applied Mathematical ciences, Vol. 8, 2014, no. 170, 8451-8457 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ams.2014.410889 tatic Model of Decision-maing over the et of Coalitional Partitions Xeniya Grigorieva t.petersburg tate University Faculty of Applied Mathematics and Control Processes University pr. 35, t.petersburg, 198504, 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 wor is properly cited. Abstract Let be N the set of players and M the set of proects. The coalitional model of decision-maing over the set of proects is formalized as family of games with different fixed coalitional partitions for each proect that required the adoption of a positive or negative decision by each of the players. The players strategies are decisions about each of the proect. Players can form coalitions in order to obtain higher income. Thus, for each proect a coalitional game is defined. In each coalitional game it is required to find in some sense optimal solution. olving successively each of the coalitional games, we get the set of optimal n-tuples for all coalitional games. It is required to find a compromise solution for the choice of a proect, i. e. it is required to find a compromise coalitional partition. As an optimality principles are accepted generalized PMvector [1], [2] and its modifications, and compromise solution. Matthematics ubect Classification: 90Axx Keywords: coalitional game, PM-vector, compromise solution 1 Introduction The set of agents N and the set of proects M are given. Each agent fixed his participation or not participation in the proect by one or zero choice. The participation in the proect is connected with incomes or losses which the
8452 Xeniya Grigorieva agents wants to maximize or minimize. Agents may form coalitions. This gives us an optimization problem which can be modeled as game. This problem we call as static coalitional model of decision-maing. Denote the players by i N and the proects by M. The family M of different games are considered. In each game G, M the player i has two strategies accept or reect the proect. The payoff of the player in each game is determined by the strategies chosen by all players in this game G. As it was mentioned before the players can form coalitions to increase the payoffs. In each game G coalitional partition is formed and the problem is to find the optimal strategies for coalitions and the imputation of the coalitional payoff between the members of the coalition. The games G 1,..., G m are solved by using the PM-vector [1], [2] and its modifications. Then having the solutions of games G, = 1, m the new optimality principle - the compromise solution is proposed to select the best proects M. 2 tate of the problem Consider the following problem. uppose: N = {1,..., n} is the set of players; X i = {0 ; 1} is the set of pure strategies x i of player i, i = 1, n. The strategy x i can tae the following values: x i = 0 as a negative decision for the some proect and x i = 1 as a positive decision; l i = 2 is the number of pure strategies of player i; x is the n-tuple of pure strategies chosen by the players; X = X i is the set of n-tuples; i=1, n µ i = (ξi 0, ξi 1 ) is the mixed strategy of player i, where ξi 0 is the probability of maing negative decision by the player i for some proect, and ξi 1 is the probability of maing positive decision correspondingly; M i is the set of mixed strategies of the i-th player; µ is the n-tuple of mixed strategies chosen by players for some proect; M = M i is the set of n-tuples in mixed strategies for some proect; i=1, n K i (x) : X R 1 is the payoff function defined on the set X for each player i, i = 1, n, and for some proect. Thus, for some proect we have noncooperative n-person game G ( x): G (x) = N, {X i } i=1, n, {K i (x)} i=1, n, x X. (1) Now suppose M = {1,..., m} is the set of proects, which require maing positive or negative decision by n players.
tatic model of decision-maing over the set of coalitional partitions 8453 A coalitional partitions Σ of the set N is defined for all = 1, m: Σ = { 1,..., l }, l n, n = N, q = q, l = N. =1 Then we have m simultaneous l-person coalitional games G (x Σ ), = 1, m, in a normal form associated with the respective game G (x): G (x Σ ) = N, { X } =1, l, Σ, { H (x Σ ) }, = 1, m. =1, l, Σ (2) Here for all = 1, m: x = {x i } i is the l-tuple of strategies of players from coalition, = 1, l; X = i x Σ = ( x 1 of all coalitions; X i is the set of strategies x of coalition, = 1, l, i. e. Cartesian product of the sets of players strategies, which are included into coalition ; ),..., x X, x X l, = 1, l is the l-tuple of strategies X = l =1, l = X l Σ = =1,l X = l is the set of l-tuples in the game G (x Σ ); i l i is the number of pure strategies of coalition ; is the number of l-tuples in pure strategies in the game G (x Σ ). M is the set of mixed strategies µ ( ) l of the coalition, = 1, l; µ = µ 1..., µ, µ, ξ 0, ξ = 1, l, µ ξ = 1, is the mixed ξ=1 strategy, that is the set of mixed strategies of players from coalition, = 1, l; µ Σ = ( µ ),..., µ 1 M, µ M l, = 1, l, is the l-tuple of mixed strategies; M = =1, l M l is the set of l-tuples in mixed strategies. From the definition of strategy x of coalition it follows that x Σ = ) ( x,..., x 1 and x = (x1,..., x n ) are the same n-tuples in the games l G(x) and G (x Σ ). However it does not mean that µ = µ Σ. Payoff function H : X R 1 of coalition for the fixed proects, = 1, m, and for the coalitional partition Σ is defined under condition
8454 Xeniya Grigorieva that: H (x Σ ) H (x Σ ) = K i (x), = 1, l, = 1, m, Σ, (3) i where K i (x), i, is the payoff function of player i in the n-tuple x Σ. Definition 1. A set of m coalitional l-person games defined by (2) is called static coalitional model of decision-maing. Definition 2. olution of the static coalitional model of decision-maing in pure strategies is x Σ, that is Nash equilibrium (NE) in a pure strategies in l-person game G (x Σ ), with the coalitional partition Σ, where coalitional partition Σ is the compromise coalitional partition (see 2.2). Definition 3. olution of the static coalitional model of decision-maing in mixed strategies is µ Σ, that is Nash equilibrium (NE) in a mixed strategies in l-person game G (µ Σ ), with the coalitional partition Σ, where coalitional partition Σ is the compromise coalitional partition (see 2.2). Generalized PM-vector is used as the coalitional imputation [1], [2]. 3 Algorithm for solving the problem 3.1 Algorithm of constructing the generalized PM-vector in a coalitional game. Remind the algorithm of constructing the generalized PM-vector in a coalitional game [1], [2]. 1. Calculate the values of payoff H (x Σ ) for all coalitions Σ, = 1, l, for coalitional game G (x Σ ) by using formula (3). 2. Find NE [3] x Σ or µ Σ (one or more) in the game G (x Σ ). The payoffs vector of coalitions in NE in mixed strategies E (µ Σ ) = { v ( )} Denote a payoff of coalition in NE in mixed strategies by =1, l. where v ( ) l Σ = τ=1 p τ, Hτ, (x Σ ), = 1, l Σ, H τ, (x Σ ) is the payoff of coalition, when coalitions choose their pure strategies x in NE in mixed strategies µ Σ. p τ, = =1,l µ ξ, ξ = 1, l, τ = 1, l Σ, is probability of the payoff s realization H τ, (x Σ ) of coalition.
tatic model of decision-maing over the set of coalitional partitions 8455 The value H τ, NE in the game, therefore, v ( 1 (x Σ ) is random variable. There could be many l-tuple of ) ( ),..., v l, are not uniquely defined. The payoff of each coalition in NE E (µ Σ ) is divided according to hapley s value [4] h ( ) = ( h ( : 1),..., h ( : s)) : h ( : i) = i (s 1)! (s s )! s! [v ( ) v ( \ {i})] i = 1, s, (4) where s = (s = ) is the number of elements of sets ( ), and v ( ) are the total maximal guaranteed payoffs all over the. Moreover v ( ) s = h ( : i). i=1 Then PM-vector in the NE in mixed strategies µ Σ in the game G (x Σ ) is defined as where PM (µ Σ ) = ( PM 1 (µ Σ ),..., PM n (µ Σ )), PM i (µ Σ ) = h ( : i), i, = 1, l. 3.2 Algorithm for finding a set of compromise solutions. We also remind the algorithm for finding a set of compromise solutions ([5]; p.18). { } C PM (M) = arg min max max PM i PM i. i tep 1. Construct the ideal vector R = (R 1,..., R n ), where R i = PM i = max PM i is the maximal value of payoff s function of player i in NE on the set M, and is the number of proect M: PM 1 1... PM 1 n......... PM m 1... PM m n... PM 1 1... PM n n tep 2. For each find deviation of payoff function values for other players from the maximal value, that is i = R i PM i, i = 1, n: R 1 PM 1 1... R n PM 1 n =.......... R 1 PM m 1... R n PM m n
8456 Xeniya Grigorieva tep 3. From the found deviations i for each select the maximal deviation i = max i among all players i: i R 1 PM 1 1... R n PM 1 n......... R 1 PM m 1... R n PM m n = 1 1... 1 n......... m 1... m n 1 i 1.... m i m tep 4. Choose the minimal deviation for all from all the maximal deviations among all players i i = min i = min max i. i The proect C PM (M), on which the minimum is reached is a compromise solution of the game G (x Σ ) for all players. 3.3 Algorithm for solving the static coalitional model of decision-maing. Thus, we have an algorithm for solving the problem. 1. Fix a, = 1, m. 2. Find the NE µ Σ in the coalitional game G (x Σ ) associated with the noncooperative game G(x) and find imputation in NE, that is PM (µ Σ ). 3. Repeat iterations 1-2 for all other, = 1, m. 4. Find compromise solution, that is C PM (M). 4 Conclusion A static coalitional model of decision-maing and algorithm for finding optimal solution are constructed in this paper. References [1] X. Grigorieva, olutions of Bimatrix Coalitional Games, Applied Mathematical ciences, vol. 8, 2014, no. 169, 8435-8441. http://dx.doi.org/10.12988/ams.2014.410880 [2] L. Petrosan,. Mamina, Dynamic Games with Coalitional tructures, International Game Theory Review, 8(2) (2006), 295 307. http://dx.doi.org/10.1142/s0219198906000904 [3] J. Nash, Non-cooperative Games, Ann. Mathematics 54 (1951), 286 295. http://dx.doi.org/10.2307/1969529
tatic model of decision-maing over the set of coalitional partitions 8457 [4] L.. hapley, A Value for n-person Games. In: Contributions to the Theory of Games (Kuhn, H. W. and A. W. Tucer, eds.) (1953), 307 317. Princeton University Press. [5] O.A. Malafeev, Managed conflict system, PbU, Pb., 2001. Received: November 15, 2014; Published November 27, 2014