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.

翻译：多智能体协调困境揭示了个体优化与集体福利之间的根本张力，然而表征此类协调需要能够感知时间结构与集体动态的度量指标。作为诊断性测试基准，我们研究了一种源于“异性博弈”的多智能体变体——改编自“异性博弈”（Battle of the Exes, BoE），并将其形式化为一个马尔可夫博弈，其中轮流交替作为周期性协调机制出现。传统基于结果的度量指标（如效率、最小/最大公平性）对时间不敏感（无法区分结构化交替与垄断性或随机访问模式），且随着智能体数量n增长，公平率丧失了区分能力，掩盖了不平等现象。为解决此局限，我们引入完美交替（Perfect Alternation, PA）作为参考协调机制，并提出六种新型交替（Alternation, ALT）度量指标，将其设计为协调质量的时间敏感可观测变量。以Q学习智能体作为最小自适应诊断基线，并与随机策略空过程对比，我们揭示了一个显著的测量失效现象：尽管学习策略展现出看似较高的传统指标（如奖励公平性常超过0.9），但在ALT变体评估下其表现比随机基线低达81%，该缺陷在双智能体场景中已然存在，并随n增长而加剧。这些结果表明，在此设定下，高聚合收益可能与恶劣的时间协调共存，而传统度量可能严重误判涌现动态。我们的发现强调了分析多智能体博弈中协调行为时需采用时间感知可观测变量，并突出了随机策略基线作为解释协调结果相对于随机水平行为之基础性空过程的重要性。