Computing the Lp-strong nash equilibrium looking for cooperative stability in multiple agents markov games

Abstract

The notion of collaboration implies that related agents interact with each other looking for cooperative stability. This notion consents agents to select optimal strategies and to condition their own behavior on the behavior of others in a strategic forward looking manner. In game theory the collective stability is a special case of the Nash equilibrium called strong Nash equilibrium. In this paper we present a novel method for computing the Strong Lp-Nash equilibrium in case of a metric state space for a class of time-discrete ergodic controllable Markov chains games. We first present a general solution for the Lp-norm for computing the Strong Lp-Nash equilibrium and then, we suggest an explicit solution involving the norms L1 and L2. For solving the problem we use the extraproximal method. We employ the Tikhonov's regularization method to ensure the convergence of the cost-functions to a unique equilibrium point. The method converges in exponential time to a unique Strong Lp-Nash equilibrium. A game theory example illustrates the main results.