Guoqiang Tian

Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity

This paper generalizes the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem of Ky Fan (“Game Theory and Related Topics,” pp. 151–156, North-Holland, Amsterdam, 1979; and Math. Ann.266, 1984, 519–537) and the Ky Fan minimax inequality by introducing a class of the generalized closedness and continuity conditions, which are called the transfer closedness and transfer continuities. We then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions. We also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions. Thus our results generalize many of the existence theorems in the literature.

Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs

This paper characterizes pure-strategy and dominant-strategy Nash equilibrium in non-cooperative games which may have discontinuous and/or non-quasiconcave payoffs. Conditions called diagonal transfer quasiconcavity and uniform transfer quasiconcavity are shown to be necessary and, with conditions called diagonal transfer continuity and transfer upper semicontinuity, sufficient for the existence of pure-strategy and dominant-strategy Nash equilibrium, respectively. The results are used to examine the existence or non-existence of equilibrium in some well-known economic games with discontinuous and/or non-quasiconcave payoffs. For example, we show that the failure of the existence of a pure-strategy Nash equilibrium in the Hotelling model is due to the failure of an aggregator function to be diagonal transfer quasiconcave—not the failure of payoffs to be quasiconcave, as has been elsewhere conjectured.

Implementation of the Lindahl Correspondence by a Single-Valued, Feasible, and Continuous Mechanism

This paper considers the problem of designing mechanisms whose Nash allocations coincide with the Lindahl allocations for public goods economies with more than one private good. Unlike previous mechanisms, the mechanism presented here has a single-valued, feasible, and continuous outcome function. Furthermore, when there are no public goods in economies, feasible and continuous implementation of the (constrained) Walrasian correspondence can be obtained as a corollary of our Theorem 1.

Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization

Abstract This paper gives necessary and sufficient conditions for 1. (1) a function to attain its maximum on a compact set 2. (2) the set of maximum points of a function on a compact set to be non-empty and compact, and 3. (3) the maximum (marginal) correspondence to be closed. We do so by introducing a class of transfer continuities which characterize the essence of topological structures of functions and correspondences for extreme points and significantly weaken the conventional continuities. Thus our results generalize the classical Weierstrass theorem and the Maximum Theorem of Berge ( Espaces topologiques et fonctions multivoques , Donod, Paris, 1959; Topological Spaces , Macmillan, New York, 1963, p. 116), by giving necessary and sufficient conditions. Furthermore, we generalize the maximum theorem of Walker, (International Economic Review, 1979, 20, 267–270) by relaxing the openness of the graph of preference correspondences and the lower semi-continuity of the feasible action correspondence. By applying our maximum theorems to game theory and economics, we can generalize many of the existence theorems on Nash equilibrium of games and equilibrium of the generalized games (the so-called abstract economies) in the literature.

Completely feasible and continuous implementation of the Lindahl correspondence with a message space of minimal dimension

Abstract This paper deals mainly with the problem of designing mechanisms whose Nash allocations coincide with the Lindahl allocations. It goes beyond the previously existing literature in that it uses outcome functions that are individually feasible, balanced, and continuous, and further, has a message space of minimal dimension and thus is informationally efficient.

Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations

This paper provides necessary and sufficient conditions for the existence of greatest and maximal elements of weak and strict preferences, and unifies two very different approaches used in the related literature (the convexity and acyclicity approaches). Conditions called transfer FS-convexity and transfer SS-convexity are shown to be necessary and, in conjunction with transfer closedness and transfer openness, sufficient for the existence of greatest and maximal elements of weak and strict preferences, respectively. The results require neither the continuity nor convexity of preferences, and are valid for both ordered and unordered binary relations. Thus, the results generalize almost all of the theorems on existence of maximal elements of preferences that appear in the literature.

The maximum theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities

In this paper we generalize Berges Maximum Theorem to the case where the payoff (utility) functions and the feasible action correspondences are not lowersemicontinuous. The condition we introduced is called the Feasible Path Transfer Lower Semicontinuity (in short, FPT l.s.c.). By applying our Maximum Theorem to game theory and economics, we are able to prove the existence of equilibrium for the generalized games (the so-called abstract economics) and Nash equilibrium for games where the payoff functions and the feasible strategy correspondences are not lowersemicontinuous. Thus the existence theorems given in this paper generalize many existence theorems on Nash equilibrium and equilibrium for the generalized games in the literature.

Quasi-Variational Inequalities without the Concavity Assumption

Abstract This paper generalizes a foundational quasi-variational inequality by relaxing the (0-diagonal) concavity condition. The approach adopted in this paper is based on continuous selection-type arguments and hence it is quite different from the approach used in the literature. Thus it enables us to prose the existence of equilibrium of the constrained noncooperative games without assuming the (quasi) convexity of loss functions.

Fixed Points Theorems for Mappings with Non-compact and Non-Convex Domains

This note gives some fixed point theorems for lower and upper semi-continuous mappings and mappings with open lower sections defined on non-compact and non-convex sets. It will be noted that the conditions of our theorems are not only sufficient but also necessary. Also our theorems generalize some well-known fixed point theorems such as the Kakutani fixed point theorem and the Brouwer-Schauder fixed point theorem by relaxing the compactness and convexity conditions.

EQUILIBRIUM IN ABSTRACT ECONOMIES WITH A NON-COMPACT INFINITE DIMENSIONAL STRATEGY SPACE, AN INFINITE NUMBER OF AGENTS AND WITHOUT ORDERED PREFERENCES

Abstract The purpose of this note is to prove the existence of an equilibrium for abstract economies with a non-compact infinite-dimensional strategy space, a countably infinite number of agents and without ordered preferences. We do so by generalizing the result of Yannelis and Prabhakar to a non-compact strategy space.