Game version of the Bruss’ problem

In the sequential decision making task known as the best choice problem, n items are presented in a random order one at a time. After each item, the decision maker can determine only their relative ranks. The goal of decision maker is to select the best of all n items without the possibility of recalling previously observed items. Many important business decisions such as choosing a venture partner, adopting technologyical innovation, or hiring an employee can be modeled using a sequential decison approach. The best choice problem is the name for a class of sequential decision problems where the goal is to select the best one of n items. The decision maker either to stop the search with the current item or to continue depends only on the relative ranks of the items that have been observed so far. The paper deal with the continuous-time two person non-zero sum game extension of the best choice problem. The objects appear according to the Poisson process with unknown intensity assumed to be exponentially distributed. Each player can choose only one applicant. If both players would like to select the same one, then the priority is assigned randomly. The aim of the players is to choose the best candidate. A construction of Nash equilibria for such game is given. The optimal stopping problem for such stream of option has been presented and solved by Bruss (1987). The considered game is a generalization of the discrete time finite horizon two person non-zero sum game with stopping of Markov process solved by Szajowski (1993). The extension of the game to admit the sets of the randomized stopping times is taken into account. The Nash values for the randomized Nash equilibria is obtained. Analysis of the solutions for different lotteries are given.