Dynamic equilibrium in games with randomly arriving players

This note follows our previous works on games with randomly arriving players [3] and [5]. Contrary to these two articles, here we seek a dynamic equilibrium, using the tools of piecewise deterministic control systems The resulting discrete Isaacs equation obtained is rather involved. As usual, it yields an explicit algorithm in the finite horizon, linear-quadratic case via a kind of discrete Riccati equation. The infinite horizon problem is briefly considered. It seems to be manageable only if one limits the number of players present in the game. In that case, the linear quadratic problem seems solvable via essentially the same algorithm, although we have no convergence proof, but only very convincing numerical evidence. We extend the solution to more general entry processes, and more importantly , to cases where the players may leave the game, investigating several stochastic exit mechanisms. We then consider the continuous time case, with a Poisson arrival process. While the general Isaacs equation is as involved as in the discrete time case, the linear quadratic case is simpler, and, provided again that we bound the maximum number of players allowed in the game, it yields an explicit algorithm with a convergence proof to the solution of the infinite horizon case, subject to a condition reminiscent of that found in [20]. As in the discrete time case, we examine the case where players may leave the game, investigating several possible stochastic exit mechanisms. MSC: 91A25, 91A06, 91A20, 91A23, 91A50, 91A60, 49N10, 93E03. Foreword This report is a version of the article [2] where players minimize instead of maximizing, and the linear-quadratic examples are somewhat different.