Strengthened Pareto Equilibrium for Games on Intersecting Sets *
- PDF / 160,990 Bytes
- 11 Pages / 594 x 792 pts Page_size
- 66 Downloads / 205 Views
STRENGTHENED PARETO EQUILIBRIUM FOR GAMES ON INTERSECTING SETS*
E. R. Smol’yakov
UDC 517.9
Abstract. A new concept of equilibrium is offered. It can be used to define a fair distribution in cooperative games on intersecting game sets and to refine the hierarchical dependence between known equilibria. Keywords: conflict problem on intersecting sets, game equilibrium.
Problems with side interests of participants (with partially overlapping game sets) form a new section of the theory of games and conflicts. Such problems are more natural models of conflicts in real life than the classical problems considered on a game set common to all participants [1–9]. The first work on game theory on intersecting game sets was published in 2003 [10]. The new theory [10–15] stipulated the development of specific concepts of equilibrium and the concept of a strong threat. In order not to complicate this presentation by cumbersome constructions generated by the dynamics and possibilities of formation of any coalitions of participants, we will restrict ourselves to conflict problems in static statement, in particular, coalitions consisting of only one and N -1 participants (P1 and PN -1 coalitions) without considering arbitrary coalitions Pk of any number k of participants (k = 1, K , N ). In [14], a rather strong concept of equilibrium is proposed for conflict problems with side interests of participants that exists not always and, in [15], vice versa, a too weak one. This work proposes a compromise equilibrium between them, which is efficient when defining a fair distribution in cooperative games.
FORMULATIONS OF CONFLICT EQUILIBRIA Problems with side interests of participants are considered under the following natural assumptions. Assumption 1. Let Q i , i = 1, K , N , be metric spaces, let Q = Q1 ´ K ´ Q N , and let G i , i = 1, K , N , be the compact game set of the ith participant in Q; let also a set G = G1 Ç K Ç G N ¹ Æ (although this is not a matter of principle), and let G ¢ = G1 È K È G N . Next, let a continuous function (functional) J i ( q ) whose maximization is of interest for the ith participant be defined on a set G i , let q i be the strategy of the ith participant, and let q = ( q1 , K , q N ) be the vector of strategies of all participants. It is natural to assume that the ith participant can choose his strategy q i from the projection PrQi G¢ of the set G ¢ onto the space Q i or the section G ¢ ( q i ) and that admissible strategies q i = ( q i , K , q i -1 , q i + 1 , ..., q N ) of the coalition
PN -1 competing with him are similarly defined. *
This work was supported by the Russian Foundation for Basic Research, No. 15-01-08838-a and No. 18-01-00842-a.
M. V. Lomonosov Moscow State University, Moscow, Russia, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2018, pp. 45–55. Original article submitted December 19, 2016. 552
1060-0396/18/5404-0552 ©2018 Springer Science+Business Media, LLC
The interests of all players are explicitly conflicting only on the set G,
Data Loading...
