Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems

doi:10.1016/j.na.2005.10.018

Nonlinear Analysis: Theory, Methods & Applications

Volume 66, Issue 4, 15 February 2007, Pages 777-790

https://doi.org/10.1016/j.na.2005.10.018 Get rights and content

Abstract

We give a unified approach to Hadamard well-posedness for some nonlinear problems such as those of Ky Fan’s point and quasi-variational inequality. As applications, we obtain some well-posed theorems for Nash equilibrium points.

Introduction

For well-posed optimization problems, there are concepts of two main types: Hadamard and Tykhonov well-posedness [1]. There are several surveys in [4] focused on the ideas of well-posedness and stability of problems in vector optimization, calculus of variations and other related fields. In 1995, Revalski gave a survey [7] on well-posedness of optimization problems, in which he presented various notions of well-posedness and some results concerning the relation between them. Similarly, there are two types of well-posedness in game theory, namely, Hadamard and Tykhonov well-posedness. Contrary to the case for optimization problems, there are few results on well-posedness of Nash equilibrium points (for short, NE). In 1995, a survey by Patrone [5] reviewed the ideas and some results of Tykhonov well-posedness for Nash equilibrium points. Of course, like for optimization problems, in addition to Tykhonov well-posedness, we may introduce other types of well-posedness for Nash equilibrium points. Recently, Pusillo Chicco [6] presented several open problems on well-posedness in the context of Nash equilibrium points, containing the problem of exploring the relationship between Tykhonov well-posedness and Hadamard well-posedness. In Section 7 of [6] Pusillo Chicco wrote: “Another problem that can be considered as ‘classical’ in the context of well-posedness in optimization is the connection between Tykhonov well-posedness and Hadamard well-posedness. It would be interesting to explore this relationship in the context of NE: given an approximate equilibrium, how to build a game ‘close’ to the original one for which it will become an equilibrium?”

Generally speaking, Hadamard types of well-posedness of a problem mean the continuous dependence of the solution on the data of such a problem. Tykhonov types of well-posedness such as Tykhonov and Levitin–Polyak well-posedness deal with the behavior of a prescribed class of sequences of approximate solutions. In this paper, we focus on various types of well-posedness of Nash equilibrium points of a noncooperative game in strategic form (briefly, a game). To be precise, for Hadamard well-posedness, one considers a family of payoff functions and/or a family of strategy sets, both of which are topologized. Given a game $Γ$ parametrized by payoffs or by both payoffs and strategy sets, denote by $F (Γ)$ the set of all Nash equilibrium points of $Γ$ . $Γ$ is said to be Hadamard well-posed (briefly, H-wp) if it has a unique Nash equilibrium $x^{*}$ and for any sequence of games ${Γ_{n}}$ converging to $Γ$ , and any $x_{n} \in F (Γ_{n}), {x_{n}}$ must converge to $x^{*}$ . $Γ$ is said to be generalized Hadamard well-posed (briefly, gH-wp) if it has at least one Nash equilibrium and for any sequence of games ${Γ_{n}}$ converging to $Γ$ , and any $x_{n} \in F (Γ_{n}), {x_{n}}$ must have a subsequence converging to a Nash equilibrium of $Γ$ . When one considers the Tykhonov type of well-posedness, one must give notions of approximate solutions for a game, i.e., approximate equilibrium points. For the sake of simplicity, consider a two-person game $Γ = (X, Y, f, g)$ . A sequence ${(x_{n}, y_{n})} \subset X \times Y$ is said to be an asymptotically Nash equilibrium (briefly, a-NE) if $sup_{x \in X} f (x, y_{n}) - f (x_{n}, y_{n}) \to 0,$ $sup_{y \in Y} g (x_{n}, y) - g (x_{n}, y_{n}) \to 0 .$ If $X$ and $Y$ are topological spaces, then the game $Γ$ is said to be Tykhonov well-posed (briefly, T-wp) if $Γ$ has a unique Nash equilibrium towards which every a-NE of $Γ$ converges. Furthermore, by modifying the definition for an approximate solution, we could introduce Levitin–Polyak well-posedness. To do this, let $X$ and $Y$ be metric spaces and consider two subsets $A \subset X$ and $B \subset Y$ . A sequence ${(x_{n}, y_{n})} \subset X \times Y$ is said to be closed to $A \times B$ if $d ((x_{n}, y_{n}), A \times B) \to 0$ , where $d ((x, y), A \times B) ≔ inf {d ((x, y), (x^{'}, y^{'})) : (x^{'}, y^{'}) \in A \times B}$ denotes the metric function generated by $A \times B$ . Given a game $(A, B, f, g)$ , a sequence ${(x_{n}, y_{n})} \subset X \times Y$ is said to be a Levitin–Polyak asymptotically Nash equilibrium (briefly, LPa-NE) if $d ((x_{n}, y_{n}), A \times B) \to 0$ , and $sup_{x \in A} f (x, y_{n}) - f (x_{n}, y_{n}) \to 0,$ $sup_{y \in B} g (x_{n}, y) - g (x_{n}, y_{n}) \to 0 .$ Then the game $Γ$ is said to be Levitin–Polyak well-posed (LP-wp) if it has a unique Nash equilibrium towards which every LPa-NE of $Γ$ converges. Also, if we consider generalizations of Tykhonov types of well-posedness, we only need the existence of Nash equilibrium points and require that every corresponding approximate sequence has a subsequence converging to a Nash equilibrium.

So far, results on well-posedness of Nash equilibrium have been obtained mainly for Tykhonov well-posedness. In this paper, we develop a new approach to well-posedness of Nash equilibrium in game theory. More precisely, we will establish the relation between different types of well-posedness of the Nash equilibrium. By applying set-valued analysis, we obtain some results on Hadamard well-posedness and then we can derive Tykhonov types of well-posedness from the relation. In particular, with our approach, it is effective to study Levitin–Polyak well-posedness of Nash equilibrium points. With our approach, it is also much easier to study generalizations of well-posedness of Nash equilibrium points.

This paper is organized as follows. In Section 2, a unified theorem of generalized Hadamard and Hadamard well-posedness for some nonlinear problems will be given. From this unified theorem, in Section 3, we derive the gH-wp and H-wp theorems for Ky Fan’s points and the solutions of quasi-variational inequalities. In Section 4, as applications of the results in Section 3, we derive gH-wp and H-wp theorems of equilibrium points of noncooperative games for three models:

(A)
games parametrized by payoffs;
(B)
games parametrized by payoffs and by strategy spaces; and
(C)
games parametrized by payoffs and by feasible strategy correspondences.

In Section 5, we give some results concerning the relation between different types of well-posedness of Nash equilibrium points and some results on the Tykhonov type of well-posedness. To be precise, given an approximate Nash equilibrium for models (A), (B) and (C), we will construct a game close to the original game such that the approximate Nash equilibrium becomes a Nash equilibrium of the approximate game. And under some stronger conditions for strategy spaces we obtain generic well-posedness of saddle point problems. Finally, in Section 6, we state briefly some results on generalized well-posedness of Nash equilibrium points.

Section snippets

Unified theorems of Hadamard well-posedness

First we recall some notions stated in the following definition; see [3].

Definition 2.1

Let $X$ and $Y$ be two Hausdorff topological spaces and $F : Y \to 2^{X}$ be a set-valued map, where $2^{X}$ denotes all nonempty subsets of $X$ . Then $F$ is said to be upper semicontinuous at $y \in Y$ if for each open set $U$ in $X$ with $U \supset F (y)$ , there exists an open neighborhood $O (y)$ of $y$ such that $U \supset F (y^{'})$ for each $y^{'} \in O (y)$ ; $F$ is said to be upper semicontinuous on $Y$ if $F$ is upper semicontinuous at every point $y \in Y$ ; $F$ is said to be an usco map if $F$ is

H-wp of Ky Fan’s points in a compact setting

Let $X$ be a nonempty compact subset of a Hausdorff topological space. Let $M_{1}$ be the collection of all functions $φ : X \times X \to R$ such that (1) for each $y \in X, x \mapsto φ (x, y)$ is lower semicontinuous and (2) ${sup}_{(x, y) \in X \times X} | φ (x, y) | < + \infty$ .

For each $φ, ψ \in M_{1}$ , define $ρ (φ, ψ) = sup_{(x, y) \in X \times X} | φ (x, y) - ψ (x, y) | .$

Clearly, $(M_{1}, ρ)$ is a metric space.

Write $M = {φ \in M_{1} : there exists x^{*} \in X such that φ (x^{*}, y) \leq 0 for all y \in X}$ . Given $φ \in M$ , such an $x^{*}$ is called a Ky Fan’s point of $φ$ ; see [9]. Denote by $F (φ)$ the set of all Ky Fan’s points of $φ$ ; then, $φ \mapsto F (φ)$

Hadamard well-posedness of Nash equilibrium points

In this section, we study Hadamard well-posedness of Nash equilibrium points for three models: Model (A), Model (B) and Model (C); see [10].

We need the following lemma:

Lemma 4.1

Let $X, Y$ and $Z$ be three Hausdorff topological spaces and $F : Y \to 2^{X}$ and $G : Z \to 2^{X}$ be two set-valued maps. Suppose that $F$ is an usco map and there exists a continuous map $T : Z \to Y$ such that $G (z) = F (T (z))$ for each $z \in Z$ . Then $G$ is an usco map.

Proof

For any $z \in Z, y = T (z) \in Y$ , since $G (z) = F (y)$ and $F (y)$ is compact, so is $G (z)$ .

Let $U$ be any open set in $X$ with $U$

Relation between different types of well-posedness

In this section, we introduce concepts of other types of well-posedness of Nash equilibrium points, namely, Tykhonov and Levitin–Polyak well-posedness. We will study the relation between Hadamard well-posedness and these types of well-posedness. Also, by applying Theorem 2.1, we can derive from the relation some Levitin–Polyak well-posed and Tykhonov well-posed theorems for Nash equilibrium points. Speaking precisely, for Models (A), (B) and (C), we introduce notions of asymptotically Nash

Generalized Tykhonov and Levitin–Polyak well-posedness

In [7], Revalski gave the definitions of generalized well-posedness for minimization problems. Similarly, we may give generalizations of various types of well-posedness for Nash equilibrium points and relations similar to those in the previous section. Generalized well-posedness for Nash equilibrium points is more interesting since the uniqueness of Nash equilibrium is not a generic property, while it is true for minimization problems and saddle point problems (cf. [2] and [8]).

A game $f \in L$ is

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Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems

Abstract

Introduction

Section snippets

Unified theorems of Hadamard well-posedness

H-wp of Ky Fan’s points in a compact setting

Hadamard well-posedness of Nash equilibrium points

Relation between different types of well-posedness

Generalized Tykhonov and Levitin–Polyak well-posedness

Journal of Mathematical Economics

Journal of Mathematical Analysis and Applications

Nonlinear Analysis, Theory, Methods and Applications

Well-Posed Optimization Problems

Generic well-posedness of optimization problems and the Banach–Mazur game

Theory of Correspondences