Extending the theory of conflict equilibria

Abstract

E. R. Smol'yakovUDC 517.9New symmetric coalition conflict equilibria are proposed. Together with already known equilibria, they allow one to find the strongest equilibrium in the majority of static and dynamic conflict problems.We propose new symmetric coalition conflict equilibria without which it is often impossible to find the strongest conflict equilibrium in static and dynamic conflict problems.The theory of conflict equilibria [1] is mainly based on equilibria that take into account individual preferences of the participants of a conflict; a system of such equilibria was called the base one. However, the author mentioned in [1, p. 118] that the base system of equilibria may appear insufficient in problems with three and more participants. A more complete set of equilibria may be needed. It takes into account competitive relations between any coalitions of the participants and is called in [1, p. 118] an extended base system. The experience of solving conflict problems shows that the need for such "coalition" equilibria arises quite often, when it is impossible (even using an iterative scheme to generate new equilibria, formulated in Theorem 1.1.9 in [1], and applying nonsymmetric base equilibria [1, p. 34-36]) to select the strongest equilibrium among several most strong equilibria indistinguishable within the framework of the base system of equilibria.Though the idea of forming "coalition" equilibria is demonstrated in [1, p. 118] by formulating an A¢-equilibrium, it is obviously insufficient, even for an expert, to use this idea to formulate correctly the whole set of "coalition" equilibria.The present paper proposes a statement of "coalition" equilibria and uses a conflict problem with three participants (which is considered as a noncooperative game on the one hand and as a cooperative game on the other hand) as an example to show that a final decision that is quite suitable for all the participants and fully satisfies practical requirements can be obtained only in aggregate with the new equilibria being proposed.On the one hand, the paper proposes new concepts of equilibria, never considered before, and on the other hand, it gives a complete presentation of hierarchical orders of well-known base conflict equilibria [1,2]. 225 1060-0396/07/4302-0225