Alliance in Three Person Games

In the present paper, we suggest a new concept of an optimal solution (that we call "coalitional equilibrium") based on the concepts of Nash and Berge equilibria. We apply the concept of an optimal solution where the outcome of a deviant coalition cannot increase. Then we determine sufficient conditions of existence of a coalitional equilibrium using the Germeier convolution. The convolution transforms the problem of determining a coalitional equilibrium into finding a saddle point of a special antagonistic game that can be effectively constructed based on the mathematical model of the initiall game. As an example of application, we suggest the proof of existence of a coalitional equilibrium in mixed strategies under "regular" mathematical programming limitations: continuity of players' outcome functions and compactness of sets of strategies. This work is intentionally limited to three persons to avoid cumbersome notations and calculations, even though application of the suggested method to games with more than three players is promising for solving problems of creating stable coalitions.