Randomized Optimal Stopping Times for a Class of Stopping Games

In Dynkin's formulation [Dokl. Akad. Nauk, 185 (1969), pp. 16–19 (in Russian)] of the stopping game problem, two players observe sequential states of a homogeneous Markov chain. Each player can stop it at any stage. When the chain is stopped, the game ends and player 1 receives from player 2 the sum depending on the player who stopped the chain and on its state at the moment of stopping.
Here we consider stopping games for the chain with the state space being the set of nonnegative integers. For the state $n>0$, the only possible transitions are either into the state $n+1$ or into the absorbing state 0 with zero payoffs (the break of the chain). The payoffs are defined so that the optimality equations have no solutions with use of pure strategies only. We obtain the solutions for these games with use of randomized stopping times. The qualitative characteristics of solutions are determined with the limiting behavior of payoffs.