Regret in the On-line Decision Problem

At each point in time a decision maker must make a decision. The payoff in a period from the decision made depends on the decision as well as on the state of the world that obtains at that time. The difficulty is that the decision must be made in advance of any knowledge, even probabilistic, about which state of the world will obtain. A range of problems from a variety of disciplines can be framed in this way. In this paper we survey the main results obtained, as well as some of their applications.