A note on Berge equilibrium

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)...

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Veröffentlicht in:Applied mathematics letters 2007-08, Vol.20 (8), p.926-932
Hauptverfasser: Nessah, Rabia, Larbani, Moussa, Tazdait, Tarik
Format: Artikel
Sprache:eng
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Zusammenfassung: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].
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2006.09.005