Abstract. The optimal choice problem (also known as “the secretary problem”) is one of the classical in stochastic optimization. Initially, this problem was proposed by Martin Gardner as a puzzle. In this paper, we consider a modification of the classical secretary problem by adding the second player, who can either help the first player to find the best element by a prompt or precludes him by imposing some restrictions on the search. Nash equilibrium has been found in the explicit form of mixed strategies for three different types of the game. The asymptotic behavior of diverse numerical quantities associated with the optimal strategies for both players, as the number of objects tends to infinity, has been studied. Tabl.: 1. Refs: 9 titles.

Keywords: best choice problem, Markov chain stopping, matrix game, Nash equilibrium, mixed strategy, threshold strategy.