Technical Note Alternative Theorem: Equilibrium in Abstract Economies

We prove a new theorem for the existence of equilibrium in discontinuous games in which the players’ preferences are neither complete nor transitive. Our result is an alternative version of Shafer and Sonnenschein ([1975] J. Math. Econ. 2, 345–348), He and Yannelis ([2016] Econ. Theory 61, 497–513), Reny ([2016] Econ. Theory Bull.) and Carmona and Prodczeck ([2016] Econ. Theory 61, 457–478).

Nash Equilibrium in Discontinuous Games

We review the discontinuous games literature, with a sharp focus on conditions that ensure the existence of pure and mixed strategy Nash equilibria in strategic form games and of Bayes-Nash equilibria in Bayesian games.

Quasi-Transfer Continuity and Nash Equilibrium

We introduce a new notion of continuity, called quasi-transfer continuity, and show that it is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. This holds true in a large class of discontinuous games. We show that our result strictly generalizes the pure strategy existence theorem of Carmona [Carmona, G. [2009] An existence result for discontinuous games, J. Econ. Theory 144, 1333–1340]. We also show that our result is neither implied by nor does it imply the existence theorems of Reny [Reny, J. P. [1999] On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056] and Baye et al. [Baye, M. R., Tian, G. and Zhou, J. [1993] Characterizations of the existence of equilibria in games with discontinuous and nonquasiconcave payoffs, Rev. Econ. Studies 60, 935–948].

Existence of Equilibria for Discontinuous Games in General Topological Spaces with Binary Relations

We provide several results on the existence of equilibria for discontinuous games in general topological spaces without any convexity structure. All of the theorems yielding existence of equilibria here are stated in terms of the player’s preference relations over joint strategies.