Reducing the complexity of computing the values of a Nash equilibrium

TL;DR

This paper introduces a novel algorithm for computing the value of Nash equilibria in two-player zero-sum games, including Colonel Blotto, with reduced complexity and broader applicability beyond traditional fixed-point methods.

Contribution

The paper presents a new algorithm that simplifies Nash equilibrium computation by avoiding the need to determine both players' strategies, applicable to all two-player zero-sum games.

Findings

The algorithm reduces computational complexity compared to existing LP and fixed-point methods.

It can be extended to all two-player zero-sum games for equilibrium value computation.

Theoretical analysis shows superiority over traditional LP solvers and random assignment methods.

Abstract

The Colonel Blotto game, formulated by Emile Borel, involves players allocating limited resources to multiple battlefields simultaneously, with the winner being the one who allocates more resources to each battlefield. Computation of the Nash equilibrium, including of two person, zero sum, mixed strategy Colonel Blotto games have encountered issues of scalability and complexity owing to their PPAD completeness. This paper proposes an algorithm that computes the same value as the Nash equilibrium but cannot be characterized by the Fixed point Theorems of Tarski, Kakutani and Brouwer. The reduced complexity of the proposed algorithm is based on dispensing with the need for computing both players Nash strategies in Colonel Blotto games. The same algorithm can, therefore, be extended to all two person, zero sum games to compute the value of the Nash equilibrium. The theoretical superiority of the proposed algorithm over both LP solvers and another method that computes the same value of the game as its Nash equilibrium by a random assignment of probabilities to the active strategy set of the defending player, is also proposed.