Nash Equilibrium in Games with Ordered Outcomes

We study Nash equilibrium in games with ordered outcomes. Given game with ordered outcomes, we can construct its mixed extension. For it the preference relations of players are to be extended to the set of probability measures. In this work we use the canonical extension of an order to the set of probability measures.
It is shown that a finding of Nash equilibrium points in mixed extension of a game with ordered outcomes can be reduced to search so called balanced matrices, which was introduced by the author. The necessary condition for existence of Nash equilibrium points in mixed extension of a game with ordered outcomes is a presence of balanced submatrices for the matrix of its realization function. We construct a certain method for searching of all balanced submatrices of given matrix using the concept of extreme balanced matrix. Necessary and sufficient conditions for Nash equilibrium point in mixed extension of a game with ordered outcomes are given also.