Bahar Kaynar. Optimal Stopping in a Stochastic Game

Abstract. In the popular TV game show “The Price is Right”, the “Showcase Showdown” game of chance is played on every show. In the game each of three players in turn spins the wheel once or twice attaining some total score, and then waits for the results of the succeeding players’ spins. The object of the game is to have the highest score, from one or two spins, without going over a given upper limit. This game of
chance has been analyzed in Coe and Butterworth and Tijms.

This paper considers a variant of this stochastic game, where each player has no information about the results and actions of the other players. Each player chooses one or two random numbers between 0 and 1. The player can decide to stop after the first draw or to continue for a second draw. The decision must be made without knowing what the other players have done. The object of the game is to have the highest total score without going over 1. In case the total scores of all players exceed 1, the winner is the player whose score is closest to 1. In the paper, we will characterize the optimal stopping rule a player should use in order to maximize its probability of winning.