Journal of Mathematical Sciences, Vol. 105, No. 6, 2001 THREE-PERSON STOPPING GAME WITH PLAYERS HAVING PRIVILEGES D. Ramsey and K. Szajowski (Wroc law, Poland) UDC 519.2 1. Introduction In the paper, a construction of Nash equilibria for a random-priority, finite-horizon, three-person, nonzero-sum stopping game is given. Let (X n , F n , P x ) N n=0 be a homogeneous Markov process defined on a probability space (Ω, F , P) with state space (E, B). At each moment n =1, 2,...,N , the decision-makers (henceforth called Player 1, Player 2, and Player 3) are able to observe the Markov chain sequentially. Each player has his own utility function g i : E →ℜ, i =1, 2, 3, and at moment n each decides separately whether to accept or reject the realization x n of X n . We assume that the functions g i are measurable and bounded. If it happens that two or three players have selected the same moment n to accept x n , then a lottery decides which player gets the right (priority) of acceptance. Let 0 ≤ γ n ≤ δ n ≤ 1 for n =1, 2,...,N . According to the lottery, at moment τ , if three players would like to accept x τ , then Player 1 is chosen with probability γ τ , Player 2 with probability δ τ − γ τ , and Player 3 with probability 1 − δ τ . If only two players compete for the observation x τ , then the priority of Player 1 is proportional to γ τ , Player 2’s priority is proportional to δ τ − γ τ , and Player 3’s priority is proportional to 1 − δ τ . The players rejected by the lottery may select any other realization x n at a later moment n, τ<n ≤ N . Once accepted, a realization cannot be rejected; once rejected, it cannot be reconsidered. If a player has not chosen any realization of the Markov process, he gets g ∗ i = inf x∈E g i (x). The aim of each player is to choose a realization that maximizes his expected utility. The problem will be formulated as a three-person nonzero-sum game with the concept of Nash equilibrium as the solution. The two-person nonzero- sum stopping game with permanent priority for Player 1 has been solved by Ferenstein [2]. Random priority for such games has been considered by Szajowski [14]. The n-person stopping-game model has been investigated by Enns and Ferenstein [1]. Such games are also strictly connected with the optimal stopping of stochastic processes. The ideas of Kuhn [5] and Rieder [9] as well as Yasuda [15] and Ohtsubo [7] are adopted to this random-priority game model. The inspiration for these game models is the secretary problem. For the original secretary problem and its extension, the reader is referred to Gilbert and Mosteller [4], Freeman [3] or Rose [10]. Related games can be found in [6, 8, 11–13], where nonzero-sum versions of the games have been investigated. A review of these problems can be found in [8]. In noncooperative nonzero-sum games, one of the possible definitions of a solution is the Nash equilibrium. This approach is adopted here. The mathematical model of the problem formulated above will be presented in Sec. 2, and equilibria for each lottery will be derived in Sec. 3. 2. The Game with Random Priority In problems of optimal stopping, the basic class of strategies T N are Markov times with respect to σ-fields {F n } N n=1 . We permit P(τ ≤ N ) < 1 for some τ ∈T N . This class of strategies is not sufficient in the stopping game(see [15]). So we consider a class of randomized stopping times. It is assumed that the probability space is rich enough to admit the following constructions. Definition 1 (see [15]). A simple strategy can be described by a random sequence p =(p n ) ∈P N such that for each n: (i) p n , q n , and r n are adapted to F n ; (ii) 0 ≤ p n , q n ,r n ≤ 1 a.s. If each random variable equals either 0 or 1, we call such a strategy a pure strategy. Let A i 1 ,A i 2 ,...,A i N be independent identically distributed random variables (i.i.d.r.v.) from the uniform distri- bution on [0, 1] and independent of the Markov process (X n , F n , P x ) N n=0 . Let H n be the σ-field generated by F n , {A i 1 ,A i 2 ,...,A i n }. A randomized Markov time τ (p i ) for strategy p i =(p i n ) ∈P N,i is defined by τ (p i ) = inf {N ≥ n ≥ 1: A n ≤ p i n }. We denote by M N i , i =1, 2, 3, the sets of all randomized strategies of the ith player. Clearly, if each p i n is either zero or one, then the strategy is pure and τ (p i ) is, in fact, an {F n }-Markov time. In particular, an {F n }-Markov time τ i corresponds to the strategy p i =(p i n ) with p i n = I {τ i =n} , where I A is the indicator function for the set A. Proceedings of the Seminar on Stability Problems for Stochastic Models, Nal¸ eczow, Poland, 1999, Part I. 1072-3374/01/1056-2599$25.00 c  2001 Plenum Publishing Corporation 2599