Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave

Two Remarks on the Infinite Approximation of a Finite World in Economic Models

“…e fixed points theorems used in the game theory literature rely heavily on the machinery of the topological structure of either the payoff function and/or strategy space. On the other hand, existing results that can be applied to finite strategy sets assume other mathematical structures on the strategy space (e.g., supermodularity [6,13], potential games [5], or symmetry and integer sets [7]). Unfortunately, these results do not help distinguish the type (pure of mixed) of equilibria in the Prisoners Dilemma, the Battle of Sexes, and the game of Matching Pennies.…”

“…In particular, Nash theorem for finite games only guarantees that an equilibrium (pure or mixed) exists, and therefore, the problem of an existence of equilibrium points in pure strategies for finite games remains open. Although many subsequent studies have provided sufficient conditions for the existence of pure strategies equilibrium (for example, see Rosenthal [4], Monderer and Shapley [5], Topkis [6], and Limura and Wanatabe [7]), to my knowledge, there are no known characterizations of pure strategy NE for finite games. ere are also several generalizations of the Nash theorem that do not work if the strategy space is restricted to be a finite set [8,9].…”

Discrete hotelling pure location games: potentials and equilibria

“…Various classes of finite games in strategic form that have a (pure) equilibrium are identified. We mention here: potential games, supermodular games, symmetric games with integrally concave payoffs ( [9]) and games with increasing best-response correspondences.…”

“…Below we quickly compare results for the dHG and the cHg. As for the dHg general results for the structure of the equilibrium set only are available for the demand function f (d) = w d (0 < w ≤ 1), we do this for this demand function. 9 A new notion: the function σ : E → R defined by σ(e 1 , e 2 ) = e 1 + e 2 is referred to as Nash sum.…”

A Convex Hull’ Characterization

“…In spite of this, it is well known that an arbitrary convex set has not a "convex basis", that is, it is impossible to find an unique sequence, with finite number of elements of the set, which span, by linear convex combinations, any other. In Algebra it is proved that n ℜ is free over the standard basis, this implies some properties, one of these is that an arbitrary sequence of a module A may be written as a linear combination of the vectors It is known, see [7] and [8], that two persons, infinite, symmetric games, with a compact, convex set as strategy space and a linear convex function as payoff, have solutions. In the last section we obtain the form of the payoff of these games.…”