Weak equilibrium in a spatial model

Abstract

Spatial models of two-player competition in spaces with more than one dimensión almost never have pure-strategy Nash equilibria, and the study of the equilibrium positions, if they exist, yields a disappointing result: the two players must choose the same position to achieve equilibrium. In this work, a discrete game is proposed in which the existence of Nash equilibria is studied using a geometric argument. This includes a definition of equilibrium which is weaker than the classical one to avoid the uniqueness of the equilibrium position. As a result, a "región of equilibrium" appears, which can be located by geometric methods. In this área, the players can move around in an "almost-equilibrium" situation and do not necessarily have to adopt the same position.

Searching for optimal positions through directional data in a political competition model

“…An important contribution of this model is that some of the determined positions of equilibrium do not fulfil the requirement that the two parties must choose the same strategy. This is a problem that usually arises in competition models between two players and for which different solutions have typically been found: restricting the positions of the players, studying mixed-strategy Nash equilibrium, weakening the definition of equilibrium, studying uncovered sets or considering valence issues, among others (see, for example, Abellanas et al, 2006Abellanas et al, , 2010Abellanas et al, , 2011Roemer, 2001;McKelvey, 1976;Stokes, 1963;Díaz-Báñez et al, 2011). The use of arcs of tolerance is a new solution to this problem, that has plausible interpretations in various areas aside from political competition.…”

“…Political competition models developed in the plane in which equilibrium positions are sought (Nash, 1951) establish that in situations of economic and social stability, such balance, if fact existing, is unique and in a central position with respect to voters' preferences (see Plott, 1967;Kramer, 1973;McKelvey, 1976). Thus political parties must present similar moderate stances in order to achieve the greatest number of voters (Abellanas et al, 2010(Abellanas et al, , 2011Roemer, 2001;López and Rodrigo, 2009).…”

Geometric Study of the Weak Equilibrium in a Weighted Case for a Two-Dimensional Competition Game

“…Now it is stated that the condition is sufficient: If is not the empty set, then any player can locate in 1 to ensure that the other player cannot get an earning bigger than in any location, because the first player controls all the subsets of H with weight bigger than .…”

“…Works exist that generalize the study of the Nash equilibrium in a competitive game with two players [1,2]. This generalization is usually performed by means of new definitions of equilibrium that are weaker than the classical one.…”

“…In this paper, the definition of weak equilibrium and the game stated in [1] is adapted to a more general game where an appropriate weight is assigned to each position the players aim for. The results included in said paper have been then generalized to be adapted to these new considerations.…”