N person stopping game with players given priority randomly (Mathematical Decision Making under uncertainty and ambiguity)
David Ramsey, Krzysztof Szajowski · Kyoto University Research Information Repository (Kyoto University) · 2000
In the paper a construction of Nash equilibria for a random priority finite horizon $N$ -person stopping game is given.The normal form of the game is formulated.The random priority scheme for the players is modeled by division of the unit interval and a sequence of random variables with uniform distribution on it.The strategies of the players are triples of randomized stopping times.A recursive procedure is developed to calculate the Nash value and the equilibrium strategies. IntroductionIn the paper the $\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}N$ person stopping game is considered.At each moment $t=$ $1,2,$ $\ldots$ , $T$ the decision makers (he,nceforth called players) are able to observe sequentially the homogeneous Markov process $(X_{t}, F_{t}, \mathrm{P}_{x})_{t=0}^{T}$ defined on $(\Omega, F, \mathrm{P})$ with state space $(\mathrm{E}, B)$ .The players have utility functions $g_{i}$ : $\mathrm{E}\mapsto\Re,$ $i=1,2,$ $\ldots$ , $N$ and at each moment $t$ each decides separately whether to accept the realization $x_{t}$ of $X_{t}$ or not.If it happens that more than one player has selected the same moment $t$ to accept the state, then a lottery decides which player gets the right (priority) of acceptance.According to the lottery, at moment $\tau$ , if players $\{i_{1}, i_{2}, \ldots, i_{i}\}$ would like to accept $x_{\tau}$ then Player $r$ is chosen with probability proportional to $p_{r,\mathcal{T}},$ $r\in\{i_{1}, i_{2}, \ldots, i_{l}\}$ .The players rejected