Teaching the applications of optimisation in game theory's zero sum and non-zero sum games

Abstract

We apply linear and non-linear programming to find the solutions for Nash equilibriums and Nash arbitration in game theory problems. Linear programming was shown as a viable method for solving mixed strategy zero-sum games. We review this methodology and suggest a class of zero-sum game theory problems that are well suited for linear programming. We applied this theory of linear programming to non-zero sum games. We suggest and apply a separate formulation for a maximising linear programming problem for each player. We move on the Nash arbitration method and remodel this problem as a non-linear optimisation problem. We take the game's payoff matrix and we form a convex polygon. Having found the status quo point (x*, y*), we maximise the product (x-x*)(y-y*) over the convex polygon using KTC non-linear optimisation techniques. The results give additional insights into game theory analysis.