A wide range no-regret theorem

In a sequential decision problem at any stage a decision maker, based on the history, takes a decision and receives a payoff which depends also on the realized state of nature. A strategy, f , is said to be as good as an alternative strategy g at a sequence of states, if in the long run f does, on average, at least as well as g does. It is shown that for any distribution, μ, over the alternative strategies there is a strategy f which is, at any sequence of states, as good as μ-almost any alternative g. 1I am grateful to Ehud Kalai, Rann Smorodinsky and Eilon Solan for fruitful discussions on this subject. I am also grateful to two anonymous referees of Games and Economic Behavior. This research was partially supported by the Israel Science Foundation, Grant no. 178/9910.0.