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Orderclustered fixed point theorems on chaincomplete preordered sets and their applications to extended and generalized Nash equilibria
 Linsen Xie^{1},
 Jinlu Li^{2}Email author and
 Wenshan Yang^{3}
https://doi.org/10.1186/168718122013192
© Xie et al.; licensee Springer 2013
 Received: 27 April 2013
 Accepted: 8 July 2013
 Published: 22 July 2013
Abstract
In this paper, we introduce the concept of orderclustered fixed point of setvalued mappings on preordered sets and give several generalizations of the extension of the AbianBrown fixed point theorem provided in (MasColell et al. in Microeconomic Theory, 1995), which is from chaincomplete posets to chaincomplete preordered sets. By using these generalizations and by applying the orderincreasing upward property of setvalued mappings, we prove several existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chaincomplete preordered sets.
MSC:46B42, 47H10, 58J20, 91A06, 91A10.
Keywords
 chaincomplete preordered sets
 orderclustered fixed point
 AbianBrown fixed point theorem
 orderincreasing upward mapping
 nonmonetized noncooperative game
 generalized Nash equilibrium
 extended Nash equilibrium
1 Introduction
In traditional game theory, fixed point theory in topological spaces or metric spaces has been an essential tool for the proof of the existence of Nash equilibria of noncooperative games, in which the payoff functions of the players take real values (see [1–6]). Recently, the concept of nonmonetized noncooperative games has been introduced where the payoff functions of the players take values in ordered sets, on which the topological structure may not be equipped. The existence of generalized and extended Nash equilibria of nonmonetized noncooperative games has been studied by applying fixed point theorems to ordered sets. These games are also named generalized games by some authors (see [7–13]). Naturally, fixed point theory on ordered sets has revealed its crucial importance in this new subject in game theory.
For singlevalued mappings, Tarski provided a fixed point theorem on complete lattices; and Abian and Brown extended it to a fixed point theorem on chaincomplete posets (see [14]). In [15], Fujimoto generalized Tarski’s fixed point theorem from singlevalued mappings to setvalued mappings on complete lattices and gave some applications to vectorcomplementarity problems. Very recently, in [8] and [12], Li provided several versions of extension of both the AbianBrown fixed point theorem and the FujimotoTarski fixed point theorem to setvalued mappings on chaincomplete posets, which are applied to prove the existence of generalized and extended Nash equilibria of nonmonetized noncooperative games on posets. We list one of them below for easy reference.
Theorem 2.2 in [12] Let $(P,\succcurlyeq )$ be a chaincomplete poset and let $F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\}$ be a setvalued mapping. If F satisfies the following three conditions:

A1. F is orderincreasing upward;

A2′. $F(x)$ is inductive with a finite number of maximal elements for every $x\in P$;

A3. There is a y in P with $y\preccurlyeq u$ for some $u\in F(y)$.
Then F has a fixed point, that is, there exists ${x}^{\ast}\in P$ such that ${x}^{\ast}\in F({x}^{\ast})$.
In some nonmonetized noncooperative games, both the domains and the ranges of payoff functions may be preordered sets instead of posets; that is, some different elements in the domain or in the range may be orderindifferent or orderequivalent. This is the motivation to consider the fixed point theorems on preordered sets in this paper. This aspect can be more precisely demonstrated by the following example.
For any positive integer k, let ${\mathbb{R}}^{k}$ denote the kdimensional Euclidean space. Let ${\succcurlyeq}^{k}$ be the binary relation on ${\mathbb{R}}^{k}$, which is defined as: $x{\succcurlyeq}^{k}y$ whenever $\parallel x\parallel \ge \parallel y\parallel $ for $x,y\in {\mathbb{R}}^{k}$. It is clear that ${\succcurlyeq}^{k}$ is a preorder relation on ${\mathbb{R}}^{k}$. Hence $({\mathbb{R}}^{k},{\succcurlyeq}^{k})$ is a preordered set. Then any noncooperative game with all the strategies in $({\mathbb{R}}^{m},{\succcurlyeq}^{m})$ and the values of payoff in $({\mathbb{R}}^{n},{\succcurlyeq}^{n})$, for some positive integers m and n, is a nonmonetized noncooperative game, which will be defined in Section 3.
In Section 2 of this paper, we generalize the extensions of the AbianBrown fixed point theorem provided in [12] from chaincomplete posets to chaincomplete preordered sets for setvalued mappings. Evidently, they are also generalizations of the FujimotoTarski fixed point theorem from complete lattices to chaincomplete preordered sets. In Sections 3 and 4, we apply these generalizations to prove some existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chaincomplete preordered sets.
2 Orderclustered fixed point
In this section, we recall and provide some concepts and properties of preordered sets, and we introduce the concept of orderclustered fixed point of setvalued mapping on preordered sets. Then we generalize the AbianBrown fixed point theorem from singlevalued mappings to setvalued mappings, which is also from complete lattices to preordered sets. Here we closely follow the notations from [7, 16, 17], and [14].
 1.
(reflexive) $x\succcurlyeq x$ for every $x\in P$;
 2.
(transitive) $x\succcurlyeq y$ and $y\succcurlyeq z$ imply $x\succcurlyeq z$ for all $x,y,z\in P$.
 3.
(antisymmetric) $x\succcurlyeq y$ and $y\succcurlyeq x$ imply $x=y$ for every $x,y\in P$.
In this case, $(P,\succcurlyeq )$ is simply called a poset.
Remark 2.1 It is worth mentioning for clarification that a preordered set $(P,\succcurlyeq )$ equipped with the preorder ≽ on P is defined as a partially ordered system (p.o.s.) in [17].
Let $(P,\succcurlyeq )$ be a preordered set. If A is a subset of P, then an element u of P is called an upper bound of A if $x\preccurlyeq u$ for each $x\in A$; if $u\in A$, then u is called a greatest element (or a maximum element) of A. The collection of all greatest elements (maximum elements) of A is denoted by maxA. A lower bound of A and a smallest element (or a minimum element) of A can be defined similarly. The collection of all smallest elements (minimum elements) of A is denoted by minA. If the set of all upper bounds of A has a smallest element, we call it a supremum of A; and the collection of all supremum elements of A is denoted by supA or ∨A. An infimum of A is similarly defined as a greatest element of the set of all lower bounds of A, provided that it exists; and the collection of all infimum elements of A is denoted by infA or ∧A.
It is important to note that, from the above definitions, if A is a subset of a preordered set $(P,\succcurlyeq )$, then maxA, minA, supA and infA are the subsets of P. In particular, if A is a subset of a poset $(P,\succcurlyeq )$, then maxA, minA, supA and infA are singletons, provided that they are nonempty. In this case, maxA, minA, supA and infA are simply written as the contained elements, respectively.
An element $y\in A$ is said to be a maximal element of the subset A of the preordered set $(P,\succcurlyeq )$ if for any $z\in A$ with $y\preccurlyeq z$ implies $z\preccurlyeq y$. Similar to the definition of maximal element, a minimal element of A can be defined. Then every greatest element (smallest element) of the subset A of the preordered set $(P,\succcurlyeq )$ is a maximal element (minimum element) of A; the converse does not hold.
A subset C of the preordered set $(P,\succcurlyeq )$ is said to be totally ordered (or a chain in P) whenever, for every pair $x,y\in C$, either $x\preccurlyeq y$ or $y\preccurlyeq x$. Following [17] and [4], we have the following definition.
 (i)
inductive if every totally ordered subset of (chain in) P has an upper bound in P;
 (ii)
chaincomplete whenever, for every totally ordered subset C of (a chain in) P, the set of all supremum elements of C is a nonempty subset of P; that is, $\vee C\ne \mathrm{\varnothing}$.
In game theory and decision theory, the decision makers may consider having indifference (same) utilities at some orderequivalent elements in a preordered set. It is the motivation to introduce the following concepts of orderequivalent elements and orderequivalent classes in a preordered set.
Let $(P,{\succcurlyeq}^{P})$ be a preordered set. For any $x,y\in P$, we say that x, y are ${\succcurlyeq}^{P}$order equivalent (${\succcurlyeq}^{P}$order indifferent), which is denoted by $x{\sim}^{P}y$, whenever both $x{\succcurlyeq}^{P}y$ and $y{\succcurlyeq}^{P}x$ hold. It is clear that ${\sim}^{P}$ is an equivalence relation on P. For any $x\in P$, let $[x]$ denote the orderequivalent class (orderindifferent class) containing x, which is called a ${\succcurlyeq}^{P}$cluster (or simply an order cluster, or a cluster, if there is no confusion caused). Let $P/{\sim}^{P}$ or $\tilde{P}$ denote the collection of all order clusters in the preordered set $(P,{\succcurlyeq}^{P})$. So, $x\in [x]\in \tilde{P}$ for every $x\in P$.
Given two preordered sets $(X,{\succcurlyeq}^{X})$ and $(U,{\succcurlyeq}^{U})$, we say that a singlevalued mapping $F:X\to U$ is orderincreasing (or orderpreserving) if $x{\preccurlyeq}^{X}y$ in X implies $F(x){\preccurlyeq}^{U}F(y)$ in U. An orderincreasing singlevalued mapping $F:X\to U$ is said to be strictly orderincreasing whenever $x{\prec}^{X}y$ implies $F(x){\prec}^{U}F(x)$. We say that a setvalued mapping $F:X\to {2}^{U}\mathrm{\setminus}\{\mathrm{\varnothing}\}$ is orderincreasing upward if $x{\preccurlyeq}^{X}y$ in X and $u\in F(x)$ imply that there is $w\in F(y)$ such that $u{\preccurlyeq}^{U}w$. F is said to be orderincreasing downward if $x{\preccurlyeq}^{X}y$ in X and $w\in F(y)$ imply that there is $u\in F(x)$ such that $u{\preccurlyeq}^{U}w$. A setvalued mapping F is said to be orderincreasing whenever F is orderincreasing both upward and downward.
Orderincreasing mappings from a preordered set to a preordered set have the following equivalent classes preserving property.
Lemma 2.3 Let $(X,{\succcurlyeq}^{X})$ and $(U,{\succcurlyeq}^{U})$ be two preordered sets and let $F:X\to U$ be an orderincreasing singlevalued mapping. Then, for every x in X, $y\in [x]$ implies $F(y)\in [F(x)]$ in U.
Proof Without any confusion, $[x]$ is understood to be an orderequivalence class in $(X,{\succcurlyeq}^{X})$ with respect to the preorder ${\succcurlyeq}^{X}$ and $[F(x)]$ is understood to be an orderequivalence class in $(U,{\succcurlyeq}^{U})$ with respect to the preorder ${\succcurlyeq}^{U}$. For every x in X, $y\in [x]$ if and only if both of $x{\preccurlyeq}^{X}y$ and $y{\preccurlyeq}^{X}x$ hold in X. Then the orderincreasing property of F implies that both of $F(x){\preccurlyeq}^{U}F(y)$ and $F(y){\preccurlyeq}^{U}F(x)$ hold in U. That is, $F(y)\in [F(x)]$. □
In game theory and decision theory, there are useful mappings $F:X\to U$ between two preordered sets which map orderindifferent elements in X to orderindifferent elements in U. That is, if the inputs of the mapping F are ${\succcurlyeq}^{X}$order equivalent elements in X, then the outputs of F are ${\succcurlyeq}^{U}$order equivalent elements in U. This leads us to the following definition.
Lemma 2.5 Every orderpreserving singlevalued mapping from a preordered set to a preordered set is clusterpreserving.
Proof The order clusterpreserving property (2.1) immediately follows from Lemma 2.3. □
 1.
a fixed point of F, whenever $x\in F(x)$;
 2.
an ${\succcurlyeq}^{X}$clustered fixed point (an orderclustered fixed point, or simply a clustered fixed point) of F, whenever there is a $w\in [x]$ such that $w\in F(x)$.
It is clear that the inverse of the above implication does not hold. Hence, orderclustered fixed points are generalizations of the fixed points. It is important to notice that if $(X,{\succcurlyeq}^{X})$ is a poset, then an orderclustered fixed point coincides with a fixed point.
 A1.
F is orderincreasing upward;
 A2.
$SF(x)$ is an inductive subset of P for each $x\in P$;
 A3.
There is a y in P with $y\preccurlyeq u$ for some $u\in F(y)$.
Then F has an orderclustered fixed point, that is, there exists ${x}^{\ast}\in P$ with $w\in [{x}^{\ast}]$ such that $w\in F({x}^{\ast})$.
From assumption A2, $SF(b)$ is inductive; and therefore the totally ordered subset (chain) C of $SF(b)$ has an upper bound in $SF(b)$, say c, with $c\in SF(b)$. From (2.2), it yields that there is a $u\in F(b)$ such that $c\preccurlyeq u$.
which implies that $b\in A$. Hence b is an upper bound of the given chain C in A; and therefore A is inductive.
Then applying Zorn’s lemma (Theorem I.2.7 in [17]), the inductive set A has a maximal element, say ${x}^{\ast}\in A$. Equation (2.3) implies that there is $w\in F({x}^{\ast})$ such that ${x}^{\ast}\preccurlyeq w$. For this element $w\in F({x}^{\ast})$ with ${x}^{\ast}\preccurlyeq w$, from assumption A1 in this theorem, there is $z\in F(w)$ with $w\preccurlyeq z$, which implies that $w\in A$. Since ${x}^{\ast}$ is a maximal element of A, from ${x}^{\ast}\preccurlyeq w$, we must have ${x}^{\ast}\sim w\in F({x}^{\ast})$. Hence, ${x}^{\ast}$ is an orderclustered fixed point of F. This theorem is proved. □
Theorem 2.8 Let $(P,\succcurlyeq )$ be a chaincomplete preordered set and let $F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\}$ be a setvalued mapping. If F satisfies conditions A1 and A3 given in Theorem 2.7 and
A2′. $F(x)$ is inductive with a finite number of maximal element clusters for every $x\in P$,
then F has an orderclustered fixed point.
By applying (2.6), the proof of (2.5) is similar to the proof of Theorem 2.2 in [12]. Then, from the claim (2.5), we have $b\preccurlyeq {u}_{j}\in F(b)$ for some j with $1\le j\le m$; and hence $b\in A$. This implies that C has an upper bound b in A; and therefore, A is inductive. The rest of the proof is very similar to the proof of Theorem 2.7. □
As a consequence of Theorem 2.8, we obtain the following special cases of the extensions of the AbianBrown fixed point theorem in preordered sets.
Corollary 2.9 Let $(P,\succcurlyeq )$ be a chaincomplete preordered set and let $F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\}$ be a setvalued mapping. If F satisfies conditions A1 and A3 given in Theorem 2.7 and
A2″. $F(x)$ has a maximum element for every $x\in P$,
then F has an orderclustered fixed point.
If we consider some special cases of the chaincomplete preordered set $(P,\succcurlyeq )$ for which condition A3 can be automatically satisfied, then we obtain the following corollaries that can also be considered as extensions of both the FujimotoTarski fixed point theorem from complete lattices to a chaincomplete preordered set∖and the AbianBrown fixed point theorem from singlevalued mappings to setvalued mappings on chaincomplete preordered sets.
Corollary 2.10 Let $(P,\succcurlyeq )$ be a chaincomplete preordered set with $\wedge P\ne \mathrm{\varnothing}$ in P. Let $F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\}$ be a setvalued mapping. If F satisfies condition A1 given in Theorem 2.7 and one of conditions A2, A2′, or A2″ given in Theorems 2.7, 2.8 and Corollary 2.9, respectively, then F has an orderclustered fixed point.
Proof Take $y\in \wedge P$ in P, then for any $u\in F(y)$, we obviously have $y\preccurlyeq u$. So, $y\in \wedge P$ satisfies assumption A3 in Theorem 2.7. □
As consequences, when posets are considered as special preordered sets, the extensions of the AbianBrown fixed point theorem in posets provided in [12] immediately follow from Corollary 2.10. As a matter of fact, we have the following result on a chaincomplete poset, which is more general than the extension of Tarski’s fixed point theorem on complete lattice by Fujimoto [15].
Corollary 2.11 Let $(P,\succcurlyeq )$ be a chaincomplete poset with ∧P exists in P. Let $F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\}$ be a setvalued map. If F satisfies condition A1 given in Theorem 2.7 and one of conditions A2, A2′, or A2″ given in Theorems 2.7, 2.8, and Corollary 2.9, respectively, then F has a fixed point.
3 Applications to the extended Nash equilibria of nonmonetized noncooperative games on chaincomplete preordered sets
In this section, we recall the concepts of the nonmonetized noncooperative games and the generalized and extended Nash equilibria of these games on preordered sets, which were studied in [8–13]. Then we apply the extensions of the AbianBrown fixed point theorem provided in the last section to prove some existence theorems for the extended Nash equilibrium of nonmonetized noncooperative games on chaincomplete preordered sets, which can be considered as extensions of the results proved by Li in [12], which are on chaincomplete posets.
 1.
the set of n players, which is denoted by $N=\{1,2,3,\dots ,n\}$;
 2.
the collection of n strategy sets $\{{S}_{1},{S}_{2},\dots ,{S}_{n}\}$, for the n players respectively, such that $({S}_{i},{\succcurlyeq}_{i})$ is a chaincomplete preordered set for player $i=1,2,3,\dots ,n$, with notation $S={S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$;
 3.
the outcome space $(U;{\succcurlyeq}^{U})$ that is a preordered set;
 4.
the n payoff functions ${f}_{1},{f}_{2},\dots ,{f}_{n}$, where ${f}_{i}$ is the payoff function for the player i that is a mapping from ${S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ to the preordered set $(U;{\succcurlyeq}^{U})$, for $i=1,2,3,\dots ,n$. We define $f=\{{f}_{1},{f}_{2},\dots ,{f}_{n}\}$.
This game is denoted by $\mathrm{\Gamma}=(N,S,f,U)$.
An nperson nonmonetized noncooperative game defined in this section is also called a generalized game (see [7]). Now we recall the extensions of the concept of Nash equilibrium of noncooperative games to the generalized Nash equilibrium and the extended Nash equilibrium of nonmonetized noncooperative games.
 1.a generalized Nash equilibrium of this game if, for every $i=1,2,3,\dots ,n$, the following order inequality holds:${f}_{i}({x}_{i},{\stackrel{{}_{\u2322}}{x}}_{i}){\preccurlyeq}^{U}{f}_{i}({\stackrel{{}_{\u2322}}{x}}_{i},{\stackrel{{}_{\u2322}}{x}}_{i})\phantom{\rule{1em}{0ex}}\text{for all}{x}_{i}\in {S}_{i};$
 2.an extended Nash equilibrium of this game if, for every $i=1,2,3,\dots ,n$, the following order inequality holds:${f}_{i}({x}_{i},{\stackrel{{}_{\u2322}}{x}}_{i}){\nsucc}^{U}{f}_{i}({\stackrel{{}_{\u2322}}{x}}_{i},{\stackrel{{}_{\u2322}}{x}}_{i})\phantom{\rule{1em}{0ex}}\text{for all}{x}_{i}\in {S}_{i}.$
It is clear that any generalized Nash equilibrium of an nperson nonmonetized noncooperative game is an extended Nash equilibrium of this game; and the converse may not be true.
Furthermore, for any $x=({x}_{1},{x}_{2},\dots ,{x}_{n})\in S$, we have $[x]=([{x}_{1}],[{x}_{2}],\dots ,[{x}_{n}])$, where $[x]$ stands for an ${\succcurlyeq}^{S}$cluster in $\tilde{S}$ and $[{x}_{1}],[{x}_{2}],\dots ,[{x}_{n}]$ are order clusters in ${\tilde{S}}_{1},{\tilde{S}}_{2},\dots ,{\tilde{S}}_{n}$, respectively.
Proof The proof is straightforward and is omitted here. □
The following theorem provides a result for the existence of an extended Nash equilibrium of nonmonetized noncooperative games on preordered sets.
 G1.
${f}_{i}({S}_{i},{x}_{i})$ is an inductive subset of the preordered set $(U,{\succcurlyeq}^{U})$;
 G2.
The ${\succcurlyeq}_{i}$downward set of the inverse image $\{{z}_{i}\in {S}_{i}:{f}_{i}({z}_{i},{x}_{i})\phantom{\rule{0.25em}{0ex}}\mathit{\text{is a maximal element of}}\phantom{\rule{0.25em}{0ex}}{f}_{i}({S}_{i},{x}_{i})\}$ is an inductive subset of ${S}_{i}$;
 G3.
For $x{\preccurlyeq}^{S}y$ in S, if ${z}_{i}\in {S}_{i}$ with ${f}_{i}({z}_{i},{x}_{i})$ is a maximal element of ${f}_{i}({S}_{i},{x}_{i})$, then there is ${w}_{i}\in {S}_{i}$ with ${z}_{i}{\preccurlyeq}_{i}{w}_{i}$ such that ${f}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${f}_{i}({S}_{i},{y}_{i})$.
then this game Γ has an extended Nash equilibrium $\stackrel{{}_{\u2322}}{x}=({\stackrel{{}_{\u2322}}{x}}_{1},{\stackrel{{}_{\u2322}}{x}}_{2},\dots ,{\stackrel{{}_{\u2322}}{x}}_{n})$. Moreover, every element in $[\stackrel{{}_{\u2322}}{x}]$ is an extended Nash equilibrium of Γ.
Proof The first part of the proof is similar to the proof of Theorem 3.4 in [12]. Since $({S}_{i},{\succcurlyeq}_{i})$ is a chaincomplete preordered set for every $i=1,2,\dots ,n$, then from Lemma 3.3, $(S,{\succcurlyeq}^{S})$ is also a chaincomplete preordered set equipped with the product order ${\succcurlyeq}^{S}$.
Now we prove that the mapping T satisfies all conditions A1, A2 and A3 in the first extension of the AbianBrown fixed point theorem in a chaincomplete preordered set provided in the preceding section.
At first, we show that T satisfies assumption A1: T is ${\succcurlyeq}^{S}$increasing upward. For any given $x{\preccurlyeq}^{S}y$ in S and for any $z=({z}_{1},{z}_{2},\dots ,{z}_{n})\in T(x)$, for every $i=1,2,\dots ,n$, we have ${z}_{i}\in {T}_{i}(x)$, that is, ${f}_{i}({z}_{i},{x}_{i})$ is a maximal element of ${f}_{i}({S}_{i},{x}_{i})$. Then from hypothesis G3 of this theorem, there is ${w}_{i}\in {S}_{i}$ with ${z}_{i}{\preccurlyeq}_{i}{w}_{i}$ such that ${f}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${f}_{i}({S}_{i},{y}_{i})$, that is, ${w}_{i}\in {T}_{i}(y)$. Take $w=({w}_{1},{w}_{2},\dots ,{w}_{n})$. We obtain that $z{\preccurlyeq}^{S}w$ and $w\in T(y)$. Hence, T is isotone.
From assumption G2 of this theorem, the ${\succcurlyeq}_{i}$downward set of {${z}_{i}\in {S}_{i}$: ${P}_{i}({z}_{i},{x}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{x}_{i})$} is inductive, which implies that $S{T}_{i}(x)$ is an inductive subset of S, so is $ST(x)$.
Finally, for the given elements $p,q\in S$ in this theorem, from condition (3.1), we have $p{\preccurlyeq}^{S}q$ and ${f}_{i}({q}_{i},{p}_{i})$ is a maximal element of ${f}_{i}({S}_{i},{p}_{i})$. This implies that $q\in T(p)$ with $p{\preccurlyeq}^{S}q$; and hence the mapping T satisfies assumption A3.
This shows that $\stackrel{{}_{\u2322}}{x}=({\stackrel{{}_{\u2322}}{x}}_{1},{\stackrel{{}_{\u2322}}{x}}_{2},\dots ,{\stackrel{{}_{\u2322}}{x}}_{n})$ is an extended Nash equilibrium of this game. With the same argument as above, we can show that, for any $\stackrel{{}_{\u2322}}{z}\in [\stackrel{{}_{\u2322}}{x}]$, ${f}_{i}({\stackrel{{}_{\u2322}}{z}}_{i},{\stackrel{{}_{\u2322}}{z}}_{i}){\sim}^{U}{f}_{i}({\stackrel{{}_{\u2322}}{x}}_{i},{\stackrel{{}_{\u2322}}{x}}_{i})$ holds. Then, repeating the argument from (3.2) to (3.5), we obtain that $\stackrel{{}_{\u2322}}{z}$ is also an extended Nash equilibrium of this game. This completes the proof of this theorem. □
We can apply Theorem 2.8, which is a different version of the extensions of the AbianBrown fixed point theorem on preordered sets, to similarly show the following result. We list it as a theorem without giving the proof.
Theorem 3.5 Let $\mathrm{\Gamma}=(N,S,f,U)$ be an nperson nonmonetized noncooperative game. Suppose that for every player $i=1,2,3,\dots ,n$, ${f}_{i}:S\to U$ is a clusterpreserving singlevalued mapping. If, for any $x\in S$ in addition to that, ${f}_{i}$ satisfies assumptions G1, G3, and condition (3.1) given in Theorem 3.4, ${f}_{i}$ also satisfies the following condition:
G2′. The inverse image $\{{z}_{i}\in {S}_{i}:{P}_{i}({z}_{i},{x}_{i})\phantom{\rule{0.25em}{0ex}}\mathit{\text{is a maximal element of}}\phantom{\rule{0.25em}{0ex}}{P}_{i}({S}_{i},{x}_{i})\}$ is inductive with a finite number of maximal element clusters for each $x\in S$.
Then there is an ${\succcurlyeq}^{S}$cluster in S, in which every element is an extended Nash equilibrium of Γ.
In case the condition $\wedge {S}_{i}\ne \mathrm{\varnothing}$ holds for the strategy set $({S}_{i},{\succcurlyeq}_{i})$, for every i, condition (3.1) in Theorems 3.4 and 3.5 can be removed. As a consequence of Theorem 3.5, we have the following.
Corollary 3.6 Let $\mathrm{\Gamma}=(N,S,f,U)$ be an nperson nonmonetized noncooperative game. Suppose that for every player $i=1,2,3,\dots ,n$, $\wedge {S}_{i}\ne \mathrm{\varnothing}$ and for any $x\in S$, ${f}_{i}:S\to U$ is a clusterpreserving singlevalued mapping. If, for any $x\in S$ in addition to that, ${f}_{i}$ satisfies assumptions G1 and G3, ${f}_{i}$ also satisfies one of assumptions G2 or G2′ given in Theorem 3.4 and Theorem 3.5, then there is an ${\succcurlyeq}^{S}$cluster in S, in which every element is an extended Nash equilibrium of Γ.
Proof Take $p\in \wedge S$ and let q be a maximum element of $F(p)$. It is clear that p and q satisfy condition (3.1) given in Theorem 3.4. □
4 Applications to the generalized Nash equilibria of nonmonetized noncooperative games on chaincomplete preordered sets
It is clear that for $({t}_{i},{\stackrel{{}_{\u2322}}{x}}_{i}),({\stackrel{{}_{\u2322}}{x}}_{i},{\stackrel{{}_{\u2322}}{x}}_{i})\in S$, the order inequality ${f}_{i}({t}_{i},{\stackrel{{}_{\u2322}}{x}}_{i}){\preccurlyeq}^{U}{f}_{i}({\stackrel{{}_{\u2322}}{x}}_{i},{\stackrel{{}_{\u2322}}{x}}_{i})$ implies ${f}_{i}({t}_{i},{\stackrel{{}_{\u2322}}{x}}_{i}){\nsucc}^{U}{f}_{i}({\stackrel{{}_{\u2322}}{x}}_{i},{\stackrel{{}_{\u2322}}{x}}_{i})$. Hence, the generalized Nash equilibria can be considered as special cases of the extended Nash equilibria of nperson nonmonetized noncooperative games. The conditions for the existence of a generalized Nash equilibrium of an nperson nonmonetized noncooperative game should be stronger than the conditions for the existence of an extended Nash equilibrium applied to the theorems in the preceding section.
Theorem 4.1 Let $\mathrm{\Gamma}=(N,S,f,U)$ be an nperson nonmonetized noncooperative game. Suppose that for every player $i=1,2,3,\dots ,n$, ${f}_{i}:S\to U$ is a clusterpreserving singlevalued mapping, and for any $x\in S$, in addition to that, ${f}_{i}$ satisfies the following two conditions:
g1. ${f}_{i}({S}_{i},{x}_{i})$ is a subset of $(U,{\succcurlyeq}^{U})$ with a maximum element;
g3. For $x{\preccurlyeq}^{S}y$ in S, if ${z}_{i}\in {S}_{i}$ with ${f}_{i}({z}_{i},{x}_{i})$ is a maximum element of ${f}_{i}({S}_{i},{x}_{i})$, then there is ${w}_{i}\in {S}_{i}$ with ${z}_{i}{\preccurlyeq}_{i}{w}_{i}$ such that ${f}_{i}({w}_{i},{y}_{i})$ is a maximum element of ${f}_{i}({S}_{i},{y}_{i})$,
${f}_{i}$ also satisfies one of the following conditions:
g2. The ${\succcurlyeq}_{i}$downward set of the inverse image {${z}_{i}\in {S}_{i}$: ${P}_{i}({z}_{i},{x}_{i})$ is a maximum element of ${f}_{i}({S}_{i},{x}_{i})$} is an inductive subset of ${S}_{i}$;
g2′. The inverse image $\{{z}_{i}\in {S}_{i}:{P}_{i}({z}_{i},{x}_{i})\phantom{\rule{0.25em}{0ex}}\mathit{\text{is a maximum element of}}\phantom{\rule{0.25em}{0ex}}{f}_{i}({S}_{i},{x}_{i})\}$ is inductive with a finite number of maximal element clusters for each $x\in S$.
then there is an ${\succcurlyeq}^{S}$cluster in S, in which every element is a generalized Nash equilibrium of Γ.
This shows that $\stackrel{{}_{\u2322}}{x}=({\stackrel{{}_{\u2322}}{x}}_{1},{\stackrel{{}_{\u2322}}{x}}_{2},\dots ,{\stackrel{{}_{\u2322}}{x}}_{n})$ is a generalized Nash equilibrium of this game. Similarly to the proof of Theorem 3.4, we can show that for any $\stackrel{{}_{\u2322}}{z}\in [\stackrel{{}_{\u2322}}{x}]$, $\stackrel{{}_{\u2322}}{z}$ is also a generalized Nash equilibrium of this game. This completes the proof of this theorem. □
Similar to Corollary 3.6, if we consider some special collections of strategies with lower bound, then we obtain the following consequence of Theorem 4.1, where condition (3.1)′ can be removed.
Corollary 4.2 Let $\mathrm{\Gamma}=(N,S,f,U)$ be an nperson nonmonetized noncooperative game. Suppose that for every player $i=1,2,3,\dots ,n$, $\wedge {S}_{i}\ne \mathrm{\varnothing}$, and for any $x\in S$, ${f}_{i}$ satisfies assumptions g1 and g3. If ${f}_{i}$ satisfies one of assumptions g2 or g2′ given in Theorem 4.1, then there is an ${\succcurlyeq}^{S}$cluster in S, in which every element is a generalized Nash equilibrium of Γ.
Declarations
Acknowledgements
The authors are grateful to the anonymous reviewers for their valuable suggestions, which really improved the presentation of this paper. First author was partially supported by the National Natural Science Foundation of China (11171137). Third author was partially supported by the National Natural Science Foundation of China (Grant 11101379).
Authors’ Affiliations
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