Weak equilibrium in a spatial model

Spatial models of two‐player competition in spaces with more than one dimension 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 ‘region of equilibrium’ appears, which can be located by geometric methods. In this area, the players can move around in an ‘almost‐equilibrium’ situation and do not necessarily have to adopt the same position.