Regret minimization in repeated matrix games with variable stage duration

Regret minimization in repeated matrix games has been extensively studied ever since Hannan's seminal paper [Hannan, J., 1957. Approximation to Bayes risk in repeated play. In: Dresher, M., Tucker, A.W., Wolfe, P. (Eds.), Contributions to the Theory of Games, vol. III. Ann. of Math. Stud., vol. 39, Princeton Univ. Press, Princeton, NJ, pp. 97-193]. Several classes of no-regret strategies now exist; such strategies secure a long-term average payoff as high as could be obtained by the fixed action that is best, in hindsight, against the observed action sequence of the opponent. We consider an extension of this framework to repeated games with variable stage duration, where the duration of each stage may depend on actions of both players, and the performance measure of interest is the average payoff per unit time. We start by showing that no-regret strategies, in the above sense, do not exist in general. Consequently, we consider two classes of adaptive strategies, one based on Blackwell's approachability theorem and the other on calibrated play, and examine their performance guarantees. We further provide sufficient conditions for existence of no-regret strategies in this model.