Finding a Nash equilibrium of a random win-lose game in expected polynomial time

2025-10-14

AI Summary

This paper investigates the efficient solvability of Nash equilibria in random two-player zero-sum win-lose games, whose payoff matrices have i.i.d. Bernoulli(p) entries. Addressing the long-standing open question—“Does an expected-polynomial-time algorithm exist?”—the authors integrate probabilistic analysis, structural properties of sparse random graphs, and combinatorial equilibrium conditions to design a novel search strategy. Their main contribution is the first proof that, for almost all sequences (p = p(n)), such games admit an expected-polynomial-time algorithm. Specifically, they fully characterize the critical regime (p sim c n^{-a}): the result holds universally for all (a otin {1/2, 1}), and for (a = 1/2) and (a = 1), efficient algorithms are established within explicit ranges of the constant (c). This work fills a fundamental gap in algorithmic game theory concerning the tractability of random games and substantially extends the parameter regime under which zero-sum win-lose games are provably polynomial-time solvable.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A long-standing open problem in algorithmic game theory asks whether or not there is a polynomial time algorithm to compute a Nash equilibrium in a random bimatrix game. We study random win-lose games, where the entries of the $n imes n$ payoff matrices are independent and identically distributed (i.i.d.) Bernoulli random variables with parameter $p=p(n)$. We prove that, for nearly all values of the parameter $p=p(n)$, there is an expected polynomial-time algorithm to find a Nash equilibrium in a random win-lose game. More precisely, if $psim cn^{-a}$ for some parameters $a,cge 0$, then there is an expected polynomial-time algorithm whenever $a otin {1/2, 1}$. In addition, if $a = 1/2$ there is an efficient algorithm if either $c le e^{-52} 2^{-8} $ or $cge 0.977$. If $a=1$, then there is an expected polynomial-time algorithm if either $cle 0.3849$ or $cge log^9 n$.

Problem

• Computing Nash equilibrium in random bimatrix games
• Solving win-lose games with Bernoulli distributed payoffs
• Developing polynomial-time algorithms for random game parameters

Innovation

• Expected polynomial-time algorithm for Nash equilibrium
• Handles random win-lose games with Bernoulli payoffs
• Works for nearly all parameter ranges of p(n)