Unilaterally competitive games with more than two players

On a Fixed Point Theorem for General Multivalued Mappings on Finite Sets with Applications in Game Theory

“…erefore, our result not only sharpens the Nash theorem (as it can pin down the exact class of finite games have pure strategy equilibria) but is also able to successfully separate the completely mixed strategy equilibria from the pure ones. ird, our results readily generalize the existing results by the authors of [27,28,30,37] who provided sufficient conditions for the existence of pure strategy Nash equilibria. Furthermore, in the context of finite games, we are able to generalize the existence theorems of potential games ( [31]) and supermodular games [38][39][40][41].…”

“…Hence, s ∈ UNBR(s) is a NE. (11 Our notion of UNBR is in similar veins as the concepts of better (or best) improvement paths studied by [30,34]).…”

“…In fact, although the Nash theorem is one of the most important results in game theory, it only guarantees that an equilibrium (pure or mixed) exists and therefore a part of the problem of existence of equilibrium points in pure strategies remains unsolved. While most of the subsequent research on the problem of existence of equilibrium points in games have generalized Nash's theorem in more general and abstract in nite spaces, (for example, see [5][6][7][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23][24][25][26]), fewer studies have retained the nite strategy space assumption (3 for example, see [27][28][29][30][31][32]). (It is important to note that the generalization of the problem to in nite spaces with a richer topological structure does not solve the pure strategy equilibrium problem when the space is nite and mixed strategies are not allowed).…”

“…e cyclicity of best response maps in games was also considered by the authors of [34] who show that some weak acyclicity condition is necessary for the existence of a unique Nash equilibrium in every subgame. More recently, the authors of [30] gave a related sufficient condition for the existence of pure strategy Nash equilibria for the class of finite games (with n > 2 players) that satisfy some unilaterally competitive (UC) property which was first developed by the authors of [35]. (4 While the authors of [35] gave some interesting properties of the UC class of games, the authors of [28] show in the first place that all quasi-concave, symmetric UC games have a symmetric pure strategy equilibrium and more recently (in [30]) that all UC games with at least three players have a Nash equilibrium).…”

“…More recently, the authors of [30] gave a related sufficient condition for the existence of pure strategy Nash equilibria for the class of finite games (with n > 2 players) that satisfy some unilaterally competitive (UC) property which was first developed by the authors of [35]. (4 While the authors of [35] gave some interesting properties of the UC class of games, the authors of [28] show in the first place that all quasi-concave, symmetric UC games have a symmetric pure strategy equilibrium and more recently (in [30]) that all UC games with at least three players have a Nash equilibrium). Another class of games that have the pure strategy Nash equilibrium property was given by the authors of [36,37] who proved the existence of equilibrium for all symmetric quasi-concave finite games two-player zero-sum games.…”