Communications in Mathematical Analysis

Nash Equilibrium and Saddle Points for Multifunctions

Pando G. Georgiev, Tamaki Tanaka, Daishi Kuroiwa, and Panos M. Pardalos

Full-text: Open access


We introduce new definitions of semi-continuity for multifunctions, combining the topological and the ordered structure of a Banach space induced by a closed convex cone. We prove two types Nash equilibrium theorems for multifunctions using scalarization and the Ky Fan’s inequality. As corollaries we obtain saddle point theorems for convex-concave multifunctions, which can be considered as generalization to the vector-valued set-valued case of the Von Neumann minimax theorem.

Article information

Commun. Math. Anal., Volume 10, Number 2 (2011), 118-127.

First available in Project Euclid: 13 February 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46020

saddle points set-valued mappings semi-continuity minimax Nash equilibrium


Georgiev, Pando G.; Tanaka, Tamaki; Kuroiwa, Daishi; Pardalos, Panos M. Nash Equilibrium and Saddle Points for Multifunctions. Commun. Math. Anal. 10 (2011), no. 2, 118--127.

Export citation


  • J.-P. Aubin, I. Ekeland, Applied Nonlinear Analysis, J. Willey and Sons, 1984.
  • Georgiev, Pando Gr.; Tanaka, Tamaki, Minimax theorems for vector-valued multifunctions, Proc. Workshop on Nonlinear Analysis and Convex Analysis, Research Institute for Mathematical Sciences, Kyoto, Japan, Aug.28-30, 2001, RIMS Kokyuroku 1187, (2001), pp.155–164.
  • I.-S. Kim and Y.-T. Kim, saddle points of set-valued maps in topological vector spaces, Appl. Math. Letters 12 (1999), 21-26.
  • D. T. Luc and C. Vargas, A saddle point theorem for set-valued maps, Nonlinear Anal. 18, 1-7 (1992).
  • Pareto Optimality, Game Theory and Equilibria, co-editors: Altannar Chinchuluun, Panos Pardalos, Athanasios Migdalas and Leonidas Pitsoulis, Edward Elgar Publishing, (2008).
  • Chr. Tammer, A generalization of Ekeland's variational principle, Optimization 25, 1992, 129-141.
  • Chr. Tammer, A variational principle and applications for vectorial control approximation problems, Reports of the Inst. of Optimization and Stohastics, Martin-Luther Universität Halle-Wittenberg, 1996.
  • T. Tanaka, Generalized semicontinuity and existence theorems for cone saddle points, Appl. Math. Optim., 36 (1997), 313-322.
  • T. Tanaka, Generalized quasiconvexitues, cone saddle points and minimax theorem for vector-valued functions, J. Optim. Theory Appl. 81, 2(1994), 355-377.
  • T. Tanaka, Two types of minimax theorems for vector-valued functions, J. Optim. Theory Appl., 68, 2 (1991), 321-334.
  • T. Tanaka, Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces, J. Optim. Theory Appl., 62 (1989), 127-138.
  • T. Tanaka, Some minimax problems of vector-valued functions, J. Optim. Theory Appl., 59 (1988), 505-524.