Approximating Bimatrix Nash Equilibrium Via Trilinear Minimax

TL;DR

This paper introduces Trilinear Minimax Relaxation (TMR), a novel approach to approximate bimatrix Nash Equilibria that is computationally efficient and extendable to multiple players, addressing the PPAD-complete complexity challenge.

Contribution

The paper proposes TMR, a new relaxation method for bimatrix NE that can be computed efficiently and extended to multi-player games, providing approximate solutions.

Findings

TMR can be computed via linear programming in O(mn) time.

TMR provides bounds on players' payoffs in approximate Nash Equilibria.

Extension of TMR to multi-player scenarios is feasible.

Abstract

The Bimatrix Nash Equilibrium (NE) for $m \times n$ real matrices $R$ and $C$, denoted as the {\it Row} and {\it Column} players, is characterized as follows: Let $\Delta =S_m \times S_n$, where $S_k$ denotes the unit simplex in $\mathbb{R}^k$. For a given point $p=(x,y) \in \Delta$, define $R[p]=x^TRy$ and $C[p]=x^TCy$. Consequently, there exists a subset $\Delta_* \subset \Delta$ such that for any $p_*=(x_*,y_*) \in \Delta_*$, $\max_{p \in \Delta, y=y_*}R[p]=R[p_*]$ and $\max_{p \in \Delta, x=x_* } C[p]=C[p_*]$. The computational complexity of bimatrix NE falls within the class of {\it PPAD-complete}. Although the von Neumann Minimax Theorem is a special case of bimatrix NE, we introduce a novel extension termed {\it Trilinear Minimax Relaxation} (TMR) with the following implications: Let $\lambda^*=\min_{\alpha \in S_{2}} \max_{p \in \Delta} (\alpha_1 R[p]+ \alpha_2C[p])$ and $\lambda_*=\max_{p \in \Delta} \min_{\alpha \in S_{2}} (\alpha_1 R[p]+ \alpha_2C[p])$. $\lambda^* \geq \lambda_*$. $\lambda^*$ is computable as a linear programming in $O(mn)$ time, ensuring $\max_{p_* \in \Delta_*}\min \{R[p_*], C[p_*]\} \leq \lambda^*$, meaning that in any Nash Equilibrium it is not possible to have both players' payoffs to exceed $\lambda^*$. $\lambda^*=\lambda_*$ if and only if there exists $p^* \in \Delta$ such that $\lambda^*= \min\{R[p^*], C[p^*]\}$. Such a $p^*$ serves as an approximate Nash Equilibrium. We analyze the cases where such $p^*$ exists and is computable. Even when $\lambda^* > \lambda_*$, we derive approximate Nash Equilibria. In summary, the aforementioned properties of TMR and its efficient computational aspects underscore its significance and relevance for Nash Equilibrium, irrespective of the computational complexity associated with bimatrix Nash Equilibrium. Finally, we extend TMR to scenarios involving three or more players.