The Coordination Gap: Multi-Agent Alternation Metrics for Temporal Fairness in Repeated Games

TL;DR

This paper introduces novel temporally sensitive metrics for evaluating coordination in multi-agent repeated games, revealing that traditional outcome-based measures can be misleading about actual coordination quality.

Contribution

It proposes six new alternation metrics and demonstrates their effectiveness in diagnosing coordination failures overlooked by conventional measures.

Findings

Traditional metrics can overestimate coordination quality.

Learned policies perform significantly worse than random baselines under new metrics.

High rewards do not necessarily indicate good temporal coordination.

Abstract

Multi-agent coordination dilemmas expose a fundamental tension between individual optimization and collective welfare, yet characterizing such coordination requires metrics sensitive to temporal structure and collective dynamics. As a diagnostic testbed, we study a BoE-derived multi-agent variant of the Battle of the Exes, formalizing it as a Markov game in which turn-taking emerges as a periodic coordination regime. Conventional outcome-based metrics (e.g., efficiency and min/max fairness) are temporally blind (they cannot distinguish structured alternation from monopolistic or random access patterns) and fairness ratios lose discriminative power as n grows, obscuring inequities. To address this limitation, we introduce Perfect Alternation (PA) as a reference coordination regime and propose six novel Alternation (ALT) metrics designed as temporally sensitive observables of coordination quality. Using Q-learning agents as a minimal adaptive diagnostic baseline, and comparing against random-policy null processes, we uncover a clear measurement failure: despite exhibiting deceptively high traditional metrics (e.g., reward fairness often exceeding 0.9), learned policies perform up to 81% below random baselines under ALT-variant evaluation, a deficit already present in the two-agent case and intensifying as n grows. These results demonstrate, in this setting, that high aggregate payoffs can coexist with poor temporal coordination, and that conventional metrics may severely mischaracterize emergent dynamics. Our findings underscore the necessity of temporally aware observables for analyzing coordination in multi-agent games and highlight random-policy baselines as essential null processes for interpreting coordination outcomes relative to chance-level behavior.