Nonlinear Analysis: Theory, Methods & Applications
Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems
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 parametrized by payoffs or by both payoffs and strategy sets, denote by the set of all Nash equilibrium points of . is said to be Hadamard well-posed (briefly, H-wp) if it has a unique Nash equilibrium and for any sequence of games converging to , and any must converge to . 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 converging to , and any must have a subsequence converging to a Nash equilibrium of . 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 . A sequence is said to be an asymptotically Nash equilibrium (briefly, a-NE) if If and are topological spaces, then the game is said to be Tykhonov well-posed (briefly, T-wp) if has a unique Nash equilibrium towards which every a-NE of converges. Furthermore, by modifying the definition for an approximate solution, we could introduce Levitin–Polyak well-posedness. To do this, let and be metric spaces and consider two subsets and . A sequence is said to be closed to if , where denotes the metric function generated by . Given a game , a sequence is said to be a Levitin–Polyak asymptotically Nash equilibrium (briefly, LPa-NE) if , and Then the game is said to be Levitin–Polyak well-posed (LP-wp) if it has a unique Nash equilibrium towards which every LPa-NE of 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 and be two Hausdorff topological spaces and be a set-valued map, where denotes all nonempty subsets of . Then is said to be upper semicontinuous at if for each open set in with , there exists an open neighborhood of such that for each ; is said to be upper semicontinuous on if is upper semicontinuous at every point ; is said to be an usco map if is
H-wp of Ky Fan’s points in a compact setting
Let be a nonempty compact subset of a Hausdorff topological space. Let be the collection of all functions such that (1) for each is lower semicontinuous and (2) .
For each , define
Clearly, is a metric space.
Write . Given , such an is called a Ky Fan’s point of ; see [9]. Denote by the set of all Ky Fan’s points of ; then,
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 and be three Hausdorff topological spaces and and be two set-valued maps. Suppose that is an usco map and there exists a continuous map such that for each . Then is an usco map. Proof For any , since and is compact, so is . Let be any open set in with
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 is
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