Average Case Performance of Replicator Dynamics in Potential Games via Computing Regions of Attraction

TL;DR

This paper investigates the average case behavior of replicator dynamics in potential games by analyzing regions of attraction, invariant functions, and equilibrium stability, providing new insights into game efficiency and equilibrium selection.

Contribution

It introduces a geometric framework for analyzing the stability and efficiency of potential games with multiple equilibria, including computing regions of attraction and invariant functions.

Findings

Computed regions of attraction for equilibria.

Identified invariant functions in potential games.

Provided average case performance metrics for game efficiency.

Abstract

What does it mean to fully understand the behavior of a network of adaptive agents? The golden standard typically is the behavior of learning dynamics in potential games, where many evolutionary dynamics, e.g., replicator, are known to converge to sets of equilibria. Even in such classic settings many critical questions remain unanswered. We examine issues such as: Point-wise convergence: Does the system actually equilibrate even in the presence of continuums of equilibria? Computing regions of attraction: Given point-wise convergence can we compute the region of asymptotic stability of each equilibrium (e.g., estimate its volume, geometry)? System invariants: Invariant functions remain constant along every system trajectory. This notion is orthogonal to the game theoretic concept of a potential function, which always strictly increases/decreases along system trajectories. Do dynamics in potential games exhibit invariant functions? If so, how many? How do these functions look like? Based on these geometric characterizations, we propose a novel quantitative framework for analyzing the efficiency of potential games with many equilibria. The predictions of different equilibria are weighted by their probability to arise under evolutionary dynamics given uniformly random initial conditions. This average case analysis is shown to offer novel insights in classic game theoretic challenges, including quantifying the risk dominance in stag-hunt games and allowing for more nuanced performance analysis in networked coordination and congestion games with large gaps between price of stability and price of anarchy.