On Lyapunov Game Theory Equilibrium: Static and Dynamic Approaches

Abstract

This paper suggests a game theory problem in which any feasible solution is based on the Lyapunov theory. The problem is analyzed in the static and dynamic cases. Some properties of Nash equilibria such as existence and stability are derived naturally from the Lyapunov theory. Remarkable is that every asymptotically stable equilibrium point (Nash equilibrium point) admits a Lyapunov-like function and if a Lyapunov-like function exists it converges to a Nash/Lyapunov equilibrium point. We define a Lyapunov-like function as an Lp-norm from the multiplayer objective function to the utopia minimum as a cost function. We propose multiple metrics to find the Nash/Lyapunov equilibrium and the strong Nash/Lyapunov equilibrium. Finding a Nash/Lyapunov equilibrium is reduced to the minimization problem of the Lyapunov-like function. We prove that the equilibrium point properties of Nash and Lyapunov meet in game theory. In order to validate the contributions of the paper, we present a numerical example.