The Concentration Game: A Repeated Zero-Sum Game Unifying Bayesian Updating, Regret, and Concentration

Summary

This paper (arXiv:2608.18061) by Akshay Balsubramani introduces a two-player zero-sum repeated game between a learner and nature whose value identity simultaneously generates Bayesian updating and an exact accounting of exponential-weights regret. The terminal payoff equals the maximum gain a comparator can achieve at fixed relative entropy from the prior, while each one-step constraint is an information budget on nature's move under the learner's mixed action. When the learner's move is otherwise unrestricted, Gibbs/Bayes weights emerge as the unique Bellman equalizer, with log-partition functions serving as value functions. Regret decomposes exactly into three parts: per-round information loss, an additive re-tempering drift accounting for changes in measurement scale across rounds, and the information the comparator carries relative to the prior. Standard variance and bounded-range proxies are looser relaxations of this decomposition, which holds universally. The same comparator-class geometry explains classical large-deviation bounds, and methods in bandits, posterior sampling, aggregation, and boosting appear as special cases of a single regret decomposition.