Optimal Stopping in Sequential Games With or Without a Constraint of Always Terminating

Abstract

Zero-sum sequential games where both control variables are stopping times are considered. Two game problems are dealt with: (𝒢1) is a problem in which the players are allowed the possibility of not stopping the game, and in the other (𝒢2) they are obliged to stop the observed process at some finite (but not preassigned) time. The problem (𝒢1) is well known. In the present paper we mainly investigate (𝒢2) as compared with (𝒢1). We give sufficient conditions for the game problem (𝒢2) to have a value which is consequently equal to that of (𝒢1) and, in parallel with it, present the constructive algorithm of the value in the natural form. The saddle point in each problem is found under a certain condition. The monotone case and the Markov case are finally investigated as the special cases.

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