Games of stopping with infinite horizon

The paper considers two-person zero-sum games of stopping: two players sequentially observe a stochastic process with infinite time horizon. Player I selects a stopping time and player II picks the distribution of the process. The pay-off is given by the expected value of the stopped process. Results of Irle on existence of value and equivalence of randomization for such games with finite time horizon, where the set of strategies for player II is dominated in the measure-theoretical sense, are extended to the infinite time case. Furthermore the paper treats such games when the set of strategies for player II is not dominated. A counterexample shows that even in the finite time case such games may not have a value. Then a sufficient condition for the existence of value is given which applies to prophet-type games.