General Properties of a Payoff Function

Abstract

In this chapter we study general properties of a payoff function—the properties that a payoff function possesses under minimal assumptions on the system. Our attention is focused on proving the continuity property of a payoff function as well as proving different versions of Bellman’s principle, and, furthermore, proving the fact that strategies which are close to optimal ones can be found among natural strategies. In Chapter 5 we shall discuss the feasibility of further reduction of a set of strategies to Markov strategies without decreasing the payoff. Considering the problem of optimal stopping of a controlled process, we describe in this chapter a subclass of stopping rules, which yield the same payoff performance as the class of all possible stopping rules.