Double stopping by two decision-makers

A problem of optimal stopping of the discrete-time Markov process by two decision-makers (Player 1 and Player 2) in a competitive situation is considered. The zero-sum game structure is adopted. The gain function depends on states chosen by both decision-makers. When both players want to accept the realization of the Markov process at the same moment, the priority is given to Player 1. The construction of the value function and the optimal strategies for the players are given. The Markov chain case is considered in detail. An example related to the generalized secretary problem is solved.