[Paper Review] On the Verification and Computation of Strong Nash Equilibrium

Nicola Gatti, Marco Rocco+1 more|arXiv (Cornell University)|Nov 16, 2017

Game Theory and Applications19 references4 citations

TL;DR

This paper presents a polynomial-time algorithm to verify whether a strategy profile is a strong Nash equilibrium (SNE) in n-agent games, and introduces a spatial branch-and-bound algorithm that iteratively queries a Nash equilibrium oracle to compute SNEs. The key contribution is proving SNE verification is in P, making SNE computation in NP, and demonstrating the method's superiority over NE enumeration on mixed-strategy SNE instances.

ABSTRACT

Computing equilibria of games is a central task in computer science. A large number of results are known for \emph{Nash equilibrium} (NE). However, these can be adopted only when coalitions are not an issue. When instead agents can form coalitions, NE is inadequate and an appropriate solution concept is \emph{strong Nash equilibrium} (SNE). Few computational results are known about SNE. In this paper, we first study the problem of verifying whether a strategy profile is an SNE, showing that the problem is in $\mathcal{P}$. We then design a spatial branch--and--bound algorithm to find an SNE, and we experimentally evaluate the algorithm.

Motivation & Objective

• Address the lack of computational methods for strong Nash equilibria (SNE) in general games, especially when coalitions are possible.
• Investigate the complexity of verifying whether a given strategy profile is an SNE, particularly in the presence of multilateral and mixed-strategy deviations.
• Design and evaluate a practical algorithm for computing SNEs, focusing on two-agent games with mixed strategies.
• Demonstrate that standard NE benchmark testbeds like GAMUT are unsuitable for SNE evaluation due to their bias toward pure-strategy SNEs.
• Develop a new instance generator that produces games admitting only mixed-strategy SNEs for meaningful experimental evaluation.

Proposed method

• Reduce the SNE verification problem to a multi-agent minmax problem with bounded action sets, proving it is solvable in polynomial time.
• Design a spatial branch-and-bound algorithm that alternates between computing a Nash equilibrium via an oracle and verifying whether it is an SNE.
• Integrate the SNE verification subroutine into the branch-and-bound framework to prune non-SNE solutions efficiently.
• Use mixed-integer programming (MIP) as the NE-finding oracle, enabling efficient NE computation and integration with the SNE verification step.
• Implement and compare multiple configurations: Iterated MIP 2StrongNash, Algorithm 3 with random state selection, and PNS-based NE enumeration.
• Generate new test instances using a custom generator that ensures only mixed-strategy SNEs exist, avoiding the bias of existing GAMUT benchmarks.

Experimental results

Research questions

• RQ1 Is the problem of verifying whether a strategy profile is a strong Nash equilibrium in P for general n-agent games?
• RQ2 Can a spatial branch-and-bound algorithm effectively compute SNEs by combining NE oracles and SNE verification?
• RQ3 What is the worst-case number of calls to the NE-finding oracle required by the proposed algorithm in two-agent games?
• RQ4 Is it possible to formulate the conditions for an SNE using a finite set of constraints, including mixed-strategy multilateral deviations?
• RQ5 What is the smoothed complexity of computing an SNE in general games?

Key findings

• SNE verification is in P for any n-agent game, as it reduces to a bounded multi-agent minmax problem.
• The problem of finding an SNE is in NP, since verification is in P and a solution can be guessed and verified efficiently.
• The proposed spatial branch-and-bound algorithm with Iterated MIP 2StrongNash outperforms NE enumeration by a factor of over 100x on 100x100 games.
• The algorithm terminates within 150 seconds on 100x100 games, while NE enumeration does not complete in one day.
• The standard GAMUT benchmark is unsuitable for SNE evaluation because all instances either admit pure SNEs or no SNE at all.
• Mixed-strategy multilateral deviations must be explicitly considered in SNE conditions, as they cannot be captured by finite constraints over pure deviations alone.