Algorithms of optimal control methods for solving game theory problems

Abstract

Purpose

The aim of the paper is to present the theory and algorithms based on the methods of systems optimal control for a numerical solution of a defined mathematical model of a system as well as that of a mathematical model of game theory.

Design/methodology/approach

The paper brings a formulation of the mathematical model of a problem of systems optimal control with distributed parameters in Hilbert space. The mathematical model of the optimal control problem includes equations that also occur in the defined mathematical model of the theory of a two player zero‐sum game. Optimization problems of game theory have been defined for the purpose of finding a saddle point of a functional satisfying task constraints ε>0.

Findings

In order to find a saddle point of a functional and that one of a functional with a limitation, a designed algorithm of an iterative gradient method is presented. Furthermore, the paper contains a concept of algorithms designing that can be applied to a numerical solution of the defined problem of game theory. These algorithms can be realized on the basis of the methods of systems optimal control. After an adjoint state of the system is defined, a saddle point of a functional will be characterized by equations and inequalities.

Originality/value

The contribution of the paper lies in the formulation of the theorems which express the necessary and sufficient conditions of optimality for saddle points of a functional. Furthermore, it has been proved that algorithms of methods of systems optimal control with distributed parameters can be used for the solution of a mathematical model of game theory. The paper contains original results achieved by the authors within scientific projects.

Citation

Jadlovská, A. and Hrubina, K. (2011), "Algorithms of optimal control methods for solving game theory problems", Kybernetes, Vol. 40 No. 1/2, pp. 290-299. https://doi.org/10.1108/03684921111118059