Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions

Abstract

This paper characterizes the existence of equilibria in minimax inequalities without assuming any form of quasiconcavity of functions and convexity or compactness of choice sets. A new condition, called "local dominatedness property", is shown to be necessary and further, under some mild continuity condition, sufficient for the existence of equilibrium. We then apply the basic result obtained in the paper to generalize the existing theorems on the existence of saddle points, fixed points, and coincidence points without convexity or compactness assumptions. As an application, we also characterize the existence of pure strategy Nash equilibrium in games with discontinuous and non-quasiconcave payoff functions and nonconvex and/or noncompact strategy spaces.

On the Existence of Nash Equilibrium in Bayesian Games

“…Combining this observation with Proposition 2.1 and Remark 2.3 in Nessah and Tian [53] and applying Theorem 3.1 in their paper, we conclude that G b has a Nash equilibrium,…”

“…satisfies the definition of 0-transfer lower semicontinuity in σ (see [58] or Nessah and Tian [53]). Combining this observation with Proposition 2.1 and Remark 2.3 in Nessah and Tian [53] and applying Theorem 3.1 in their paper, we conclude that G b has a Nash equilibrium,…”

Berge–zhukovskii Equilibria: Existence and Characterization

“…The function x → Υ f (x) ∈ X 0 is continuous on the compact and convex X 0 . Then by Theorem 5.3 of Nessah and Tian [2013], there exists a strategy profile x ∈ X 0 so that Υ f (x) = x and by construction of Υ f , we obtain that x is a solution of the system Ψ i (x, ϕ i (x)) = 0, for each i ∈ I. According to Theorem 3.1, the strategy profile x is a BZ-equilibrium of the game (2.1).…”

“…In the following theorems, using Brouwer Theorem [Aliprantis and Border, 2006] and Theorem 5.3 of Nessah and Tian [2013], we provide some other sufficient conditions of the existence of BZ-equilibrium. Assume that X i ⊂ R ni .…”

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

“…Dasgupta and Maskin [5], Reny [6], Nessah [7], Nessah and Tian [8], and others established the existence of pure strategy Nash equilibrium for discontinuous, compact, and quasi-concave games. Baye et al [4], Yu [9], Tan et al [10], Zhang [11], Lignola [12], Nessah and Tian [13,14], Kim and Lee [15], Hou [16], Chang [17], and Tian [10] and others investigated the existence of pure strategy Nash equilibrium for discontinuous and/or non-quasi-concave games with finite or countable players by using the approach to consider a mapping of individual payoffs into an aggregator function (the aggregator function : × → is defined by ( , ) = ∑ ∈ ( , − ) for each ( , ) ∈ × . ), which is pioneered by Nikaido and Isoda [18].…”