A note on Berge equilibrium

R. Nessah (1) , M. Larbani (2) , T. Tazdait (3)

1 UMR CNRS 8179
2 Department of Business Administration, Faculty of Economics and Management Sciences, IIUM University Jalan Gombak, 53100 Kuala Lumpur
3 CIRED - centre international de recherche sur l'environnement et le développement

This work is a contribution on the problem of the existence of Berge equilibrium. Abalo and Kostreva give an existence theorem for this equilibrium that appears in the papers [K.Y. Abalo, M.M. Kostreva, Berge equilibrium: Some recent results from fixed-point theorems, Appl. Math. Comput. 169 (2005) 624-638; K.Y. Abalo, M.M. Kostreva, Some existence theorems of Nash and Berge equilibria, Appl. Math. Lett. 17 (2004) 569-573]. We found that the assumptions of these theorems are not sufficient for the existence of Berge equilibrium. Indeed, we construct a game that verifies Abalo and Kostreva's assumptions, but has no Berge equilibrium. Then we provide a condition that overcomes the problem in these theorems. Our conclusion is also valid for Radjef's theorem, which is the basic reference for [K.Y. Abalo, M.M. Kostreva, Berge equilibrium: Some recent results from fixed-point theorems, Appl. Math. Comput. 169 (2005) 624-638; K.Y. Abalo, M.M. Kostreva, Some existence theorems of Nash and Berge equilibria, Appl. Math. Lett. 17 (2004) 569-573; K.Y. Abalo, M.M. Kostreva, Fixed points, Nash games and their organizations, Topol. Methods Nonlinear Anal. 8 (1996) 205-215; K.Y. Abalo, M.M. Kostreva, Equi-well-posed games, J. Optim. Theory Appl. 89 (1996) 89-99]. © 2007 Elsevier Ltd. All rights reserved.

Domaines: Economies et finances

R. Nessah, M. Larbani, T. Tazdait. A note on Berge equilibrium. Applied Mathematics Letters, 2007, 20 (8), pp.926. ⟨10.1016/j.aml.2006.09.005⟩. ⟨hal-00716706⟩