Effects of Players' Random Participation to the Stability in LQ Games

We consider a linear quadratic (LQ) game with randomly arriving players, staying in the game for a random period of time. The Nash equilibrium of the game is characterized by a set of coupled Riccati-type equations for Markovian jump linear systems (MJLS), and the existence of a Nash equilibrium is proved using Brouwer's fixed point theorem. We then consider the game, in the limit as the number of players becomes large, assuming a partially Kantian behavior. We then focus on the effects of the random entrance, random exit, and partial Kantian cooperation to the stability of the overall system. Some numerical results are also presented. It turns out that in the noncooperative case, the overall system tends to become more unstable as the number of players increases and tends to stabilize as the expected time horizon increases. In the partially cooperative case, an explicit relation of the expected time horizon of each player with the minimum amount of cooperation, sufficient to stabilize the closed loop system, is derived.