Julio Clempner, Jesus Medel, Alin Cârsteanu
Centro de Investigación en Computación (CIC)
Instituto Politécnico Nacional
Apartado Postal 75-476, C.P. 07738, Mexico City, MEXICO
e-mail: julio@k-itech.com
Centro de Investigación e Innovación Tecnológica (CIITEC)
Instituto Politécnico Nacional
Apartado Postal 75-476, C.P. 07738, Mexico City, MEXICO
e-mail: jmedel@ipn.mx
Centro de Investigación y de Estudios Avanzados del
Instituto Politécnico Nacional (Cinvestav)
Apartado Postal 14-740, C.P. 07360, Mexico City, MEXICO
e-mail: alin@math.cinvestav.mx

Abstract.In this paper we extend the non-cooperative game theory including the Lyapunov's equilibrium and stability criteria, locally and robustly. Equilibrium and stability conditions are obtained by defining the utility function as a Lyapunov function. In this sense, we present some properties about utility functions to show that an isomorphism can be determined between any utility function and the Lyapunov utility function. We introduce the Lyapunov equilibrium point as an alternative definition to the Nash equilibrium point for games. We prove that the concept of Lyapunov equilibrium coincides in this case with the concept of Nash equilibrium. The advantage of this approach is that fixed-point conditions for games are given by the definition of the Lypunov function. We show that the game is asymptotically stable in the Lyapunov sense. A formal treatment leading to interesting mathematical results, and open problems in game theory are presented.

Received: January 12, 2005

AMS Subject Classification: 91A10, 91A30, 91A40, 93D30, 93D09

Key Words and Phrases: noncooperative games, utility theory for games, game-theoretic models, scalar and vector Lyapunov functions, robust stability

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 19
Issue: 4