Online Learning with Variable Stage Duration

We consider online learning in repeated decision problems, within the framework of a repeated game against an arbitrary opponent. For repeated matrix games, well known results establish the existence of no-regret strategies; such strategies secure a long-term average payoff that comes close to the maximal payoff that could be obtained, in hindsight, by playing any fixed action against the observed actions of the opponent. In the present paper we consider the extended model where the duration of each stage of the game may depend on the actions of both players, while the performance measure of interest is the average payoff per unit time. We start the analysis of online learning in repeated games with variable stage duration 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 forecasts, and examine their performance guarantees. In either case we show that the long-term average payoff is higher than a certain function of the empirical distribution of the opponent’s actions, and in particular is strictly higher than the minimax value of the repeated game whenever that empirical distribution deviates from a minimax strategy in the stage game.