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Equilibrium stacks for a non-cooperative game defined on a product of staircase-function continuous and finite strategy spaces

Romanuke Vadim

Journal of Mathematics and Applications

2023

Vol. 46

79--114

A method of solving a non-cooperative game defined on a product of staircase-function strategy spaces is presented. The spaces can be finite and continuous as well. The method is based on stacking equilibria of “short” non-cooperative games, each defined on an interval where the pure strategy value is constant. In the case of finite non-cooperative games, which factually are multidimensional-matrix games, the equilibria are considered in general terms, so they can be in mixed strategies as well. The stack is any combination (succession) of the respective equilibria of the “short” multidimensional-matrix games. Apart from the stack, there are no other equilibria in this “long” (staircase-function) multidimensional-matrix game. An example of staircase-function quadmatrix game is presented to show how the stacking is fulfilled for a case of when every “short” quadmatrix game has a single pure-strategy equilibrium. The presented method, further “breaking” the initial staircase-function game into a succession of “short” games, is far more tractable than a straightforward approach to solving directly the “long” non-cooperative game would be.