Approximating Bimatrix Nash Equilibrium Via Trilinear Minimax
Bahman Kalantari
TL;DR
The paper addresses the computational challenge of bimatrix Nash Equilibria by introducing a Trilinear Minimax Relaxation (TMR) that recasts payoffs as a weighted blend of the two players' values, enabling a computable upper bound λ^* via a primal-dual pair of linear programs. By analyzing M[α,p] = α_1 R[p] + α_2 C[p] and constructing a derived solution Q^*, the authors derive tight relations to NE payoffs and provide conditions under which approximate NE can be computed, including cases where λ^* = λ_* and where equalized payoffs are achievable through scaling. They further characterize optimal TMR solutions, reveal structural properties such as at most two positive entries in Q^*, and introduce constructive procedures (square-root and trivial solutions) to obtain practical approximations with potentially higher payoffs than NE for one or both players. The work extends to Multilinear Minimax Relaxation (MMR) for more players, offering a general, LP-based framework for approximating NE in multilinear settings, with computational advantages over traditional algorithms like Lemke-Howson or Govindan-Wilson. Overall, the approach provides a computable, theory-backed pathway to approximate Nash Equilibria and balanced outcomes, while highlighting scaling strategies and extensions to broader multilinear scenarios.
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 $Δ=S_m \times S_n$, where $S_k$ denotes the unit simplex in $\mathbb{R}^k$. For a given point $p=(x,y) \in Δ$, define $R[p]=x^TRy$ and $C[p]=x^TCy$. Consequently, there exists a subset $Δ_* \subset Δ$ such that for any $p_*=(x_*,y_*) \in Δ_*$, $\max_{p \in Δ, y=y_*}R[p]=R[p_*]$ and $\max_{p \in Δ, x=x_* } C[p]=C[p_*]$. 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 $λ^*=\min_{α\in S_{2}} \max_{p \in Δ} (α_1 R[p]+ α_2C[p])$ and $λ_*=\max_{p \in Δ} \min_{α\in S_{2}} (α_1 R[p]+ α_2C[p])$. $λ^* \geq λ_*$. $λ^*$ is computable as a linear programming in $O(mn)$ time, ensuring $\max_{p_* \in Δ_*}\min \{R[p_*], C[p_*]\} \leq λ^*$, meaning that in any Nash Equilibrium it is not possible to have both players' payoffs to exceed $λ^*$. $λ^*=λ_*$ if and only if there exists $p^* \in Δ$ such that $λ^*= \min\{R[p^*], C[p^*]\}$. Such a $p^*$ serves as an approximate Nash Equilibrium. We analyze the cases where such $p^*$ exists and is computable. Even when $λ^* > λ_*$, 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.