In this work, an improvement of the results presented by  Abellanas et al. (Weak Equilibrium in a Spatial Model. International Journal of Game Theory, 40(3), 449-459) is discussed. Concretely, this paper investigates an abstract game of competition between two players that want to earn the maximum number of points from a finite set of points in the plane. It is assumed that the distribution of these points is not uniform, so an appropriate weight to each position is assigned. A definition of equilibrium which is weaker than the classical one is included in order to avoid the uniqueness of the equilibrium position typical of the Nash equilibrium in these kinds of games. The existence of this approximated equilibrium in the game is analyzed by means of computational geometry techniques.
Cite this paper
L. Dolores, R. Javier and S. Manuel, "Geometric Study of the Weak Equilibrium in a Weighted Case for a Two-Dimensional Competition Game," American Journal of Operations Research, Vol. 3 No. 3, 2013, pp. 337-341. doi: 10.4236/ajor.2013.33030.
 M. Abellanas, M. López, J. Rodrigo and I. Lillo, “Weak Equilibrium in a Spatial Model,” International Journal of Game Theory, Vol. 40, No. 3, 2011, pp. 449-459.
 D. Monderer and L. S. Shapley, “Potential Games,” Games and Economic Behavior, Vol. 14, No. 1, 1996, pp. 124143. doi:10.1006/game.1996.0044
 M. Abellanas, I. Lillo, M. López and J. Rodrigo, “Electoral Strategies in a Dynamical Democratic System,” European Journal of Operational Research, Vol. 175, No. 2, 2006, pp. 870-878. doi:10.1016/j.ejor.2005.05.019
 M. Abellanas, M. López and J. Rodrigo, “Searching for Equilibrium Positions in a Game of Political Competition with Restrictions,” European Journal of Operational Research, Vol. 201, No. 3, 2010, pp. 892-896.
 R. Aurenhammer and R. Klein, “Voronoi Diagrams,” In: J.-R. Sack and J. Urrutia, Eds., Handbook of Computational Geometry, Elsevier Science Publishers B.V. NorthHolland, Amsterdam, 2000.
 A. Okabe, B. Boots, K. Sugihara and S. Chiu, “Spatial Tessellations Concepts and Applications of Voronoi Diagrams,” John Wiley & Sons, Chichester, 2000.
 D. Serra and C. Revelle, “Market Capture by Two Competitors: The Preemptive Location Problem,” Journal of regional Science, Vol. 34, No. 4, 1994, pp. 549-561.
 M. Smid, “Closest Point Problems in Computational Geometry,” In: J.-R. Sack and J. Urrutia, Eds., Handbook on Computational Geometry, Elsevier Science, Amsterdam, 1997.
 M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, “Computational Geometry-Algorithms and Applacations,”2nd Edition, Springer, New York, 1997.
 I. Lillo, M. López and J. Rodrigo, “A Geometric Study of the Nash Equilibrium in a Weighted Case,” Applied Mathematical Sciences, Vol. 55, No. 1, 2007, pp. 2715-2725.