Metric-Based Equilibrium Selection in Noncooperative Differentiable Games

2026-10-05

AI Summary

This study addresses the problem of selectively attracting specific equilibria in non-cooperative differentiable games without altering their equilibrium locations. To this end, it proposes a conditioning framework based on state-dependent symmetric positive-definite metrics, which achieves targeted equilibrium control by smoothly interpolating between stabilizing and destabilizing strategies. The authors demonstrate that player-independent metrics fail to preserve the stability of differential Nash equilibria, and accordingly construct a unified single metric field that retains all equilibria while prescribing their local stability types. The effectiveness of this approach is successfully validated through experiments on a continuous-commitment Stag Hunt game and an entropy-regularized Iterated Prisoner’s Dilemma.

Abstract

Metric conditioning can change which equilibrium attracts learning dynamics without moving the equilibria of a differentiable game. We study how to use state-dependent symmetric positive-definite (SPD) metrics to select among known equilibria by keeping a designated equilibrium attracting while making a rival unstable. Building on classical results on matrix stability under positive-definite multiplication, we characterize when such destabilization is possible and show how restricting the metric to act independently on each player limits the stability changes that can be achieved at differential Nash equilibria. We then combine a stabilizing metric at the selected equilibrium with a destabilizing metric at the rival through smooth interpolation, yielding a single metric field that preserves every equilibrium of the game while assigning the desired local stability types. We also derive a step-size condition for implementing the method in discrete time. The approach is validated on a continuous-commitment stag hunt and an entropy-regularized iterated prisoner's dilemma, where it selects the desired equilibrium in both settings.

Problem

• differentiable games
• equilibrium selection
• metric conditioning
• learning dynamics
• Nash equilibria

Innovation

• differentiable games
• equilibrium selection
• symmetric positive-definite metrics
• Nash equilibria
• smooth interpolation