Metric design guides equilibrium choice in multiplayer games
Metric-Based Equilibrium Selection in Noncooperative Differentiable Games
Computer Science and Game Theory
Summary
In some games where players make decisions independently, several stable outcomes can happen. The authors study how to use special mathematical tools called metrics to steer which outcome the players settle on without changing the outcomes themselves. They show how to make one desired outcome stable and a rival outcome unstable, so learning tends to lead to the preferred result. They test this approach on classic game scenarios like the stag hunt and prisoner's dilemma and demonstrate it selects the intended equilibrium.
What this means in practice
- •For multi-agent system designers: Control which stable outcome automated agents converge to by designing metrics that guide learning dynamics.
- •For game theory software developers: Implement metric-based methods to reliably select desired equilibria in simulations of strategic interactions.
Authors
Karan Mahesh, Runyu Zhang, Michael R. Benjamin, Gioele Zardini
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.