arXiv:2605.14879v3 Announce Type: replace Abstract: Many multi-agent games and simulated environments involve several independent, often learning, agents repeatedly competing for a shared resource. Conventional fairness measures assess whether resources are shared equally overall, but cannot distinguish orderly turn-taking from irregular access patterns that produce long, unpredictable waiting times despite similar cumulative outcomes. We introduce Rotational Periodicity, a family of computationally efficient metrics combining two complementary measurements, the regularity of waiting times between successful accesses and the balance of access frequencies across agents. We evaluate them against a more detailed family of alternation metrics in a repeated threshold-congestion game, the Honey-Jar Game, in which two to ten reinforcement-learning agents compete for exclusive access to a shared high-value resource. Our experiments show that independently trained Q-learning agents often coordinate substantially worse than random-policy agents, even though conventional fairness metrics consistently report favourable outcomes. Every member of the family closely reproduces the rankings of the more computationally expensive alternation metrics, while computing six to twenty times faster as the number of agents grows. A direct timing study, which separates the two quantities that drive that cost, measures the advantage growing in proportion to the number of agents out to two thousand agents. These findings show that evaluating multi-agent learning systems requires temporally aware measures of coordination, not only aggregate outcomes, and that efficient proxies make this practical for larger multi-a