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

Department of Mathematics, Faculty of Science, Kochi University, Kochi 780, Japan

Zero-sum sequential games where both control variables are stopping times are considered. Two game problems are dealt with: (₁) is a problem in which the players are allowed the possibility of not stopping the game, and in the other (₂) they are obliged to stop the observed process at some finite (but not preassigned) time. The problem (₁) is well known. In the present paper we mainly investigate (₂) as compared with (₁). We give sufficient conditions for the game problem (₂) to have a value which is consequently equal to that of (₁) 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.

Key Words: optimal stopping; zero-sum game; saddle point; monotone case; Markov case