# Nash Equilibrium Sets in Dyadic Trimatrix Mixed-Strategy Games

## Abstract

The approach exposed in Chap. 3 is applied to solve three-matrix \(2\times 2\times 2\) mixed-strategy games. Such games, alike the games considered in the precedent chapter and the result presented in the following chapter, admit a simple graphic representation of Nash equilibrium sets in a Cartesian system of coordinates. The results of this chapter were successfully applied by Y.M. Zhukov to investigate *“An Epidemic Model of Violence and Public Support in Civil War”* in the Department of Government at Harvard University (Zhukov, Conflict Management and Peace Science, 30(1):24–52, 2013, [1]). Evidently, a full graphical representation of the Nash equilibrium sets in the Cartesian system of coordinates is not possible in the case of four and more players. We can only suppose the possibility of such representation for the four-matrix \(2\times 2\times 2\times 2\) mixed-strategy games in the Barycentric coordinate system or other coordinate systems. Sure, in the context of results presented in Chap. 6 for dyadic two-matrix games, an important problem for dyadic three-matrix games is that of defining explicitly a Nash equilibrium set function as a piecewise set-valued function. In this chapter, we solve the problem of NES function definition algorithmically as the number of components/pieces of the NES piecewise set-valued function is substantially larger in the case of dyadic three-matrix mixed-strategy games because of more pieces/components of the piecewise set-valued function that defines best response mapping graph of each player.

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