R E S E A R C H
Open Access
Order-clustered fixed point theorems on
chain-complete preordered sets and their
applications to extended and generalized
Nash equilibria
Linsen Xie
1, Jinlu Li
2*and Wenshan Yang
3*Correspondence: [email protected] 2Department of Mathematics,
Shawnee State University, Portsmouth, Ohio 45662, USA Full list of author information is available at the end of the article
Abstract
In this paper, we introduce the concept of order-clustered fixed point of set-valued mappings on preordered sets and give several generalizations of the extension of the Abian-Brown fixed point theorem provided in (Mas-Colellet al.in Microeconomic Theory, 1995), which is from chain-complete posets to chain-complete preordered sets. By using these generalizations and by applying the order-increasing upward property of set-valued mappings, we prove several existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chain-complete preordered sets.
MSC: 46B42; 47H10; 58J20; 91A06; 91A10
Keywords: chain-complete preordered sets; order-clustered fixed point;
Abian-Brown fixed point theorem; order-increasing 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 [–]). Recently, the concept of nonmonetized noncooperative games has been introduced where the pay-off 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 non-monetized noncooperative games has been studied by applying fixed point theorems to ordered sets. These games are also named generalized games by some authors (see [– ]). Naturally, fixed point theory on ordered sets has revealed its crucial importance in this new subject in game theory.
For single-valued mappings, Tarski provided a fixed point theorem on complete lattices; and Abian and Brown extended it to a fixed point theorem on chain-complete posets (see []). In [], Fujimoto generalized Tarski’s fixed point theorem from single-valued map-pings to set-valued mapmap-pings on complete lattices and gave some applications to vector-complementarity problems. Very recently, in [] and [], Li provided several versions of extension of both the Abian-Brown fixed point theorem and the Fujimoto-Tarski fixed point theorem to set-valued mappings on chain-complete posets, which are applied to
prove the existence of generalized and extended Nash equilibria of nonmonetized nonco-operative games on posets. We list one of them below for easy reference.
Theorem . in [] Let(P,)be a chain-complete poset and let F:P→P\{∅}be a set-valued mapping.If F satisfies the following three conditions:
A. Fis order-increasing upward;
A. F(x)is inductive with a finite number of maximal elements for everyx∈P; A. There is ayinPwithyufor someu∈F(y).
Then F has a fixed point,that is,there exists x∗∈P such that x∗∈F(x∗).
In some nonmonetized noncooperative games, both the domains and the ranges of pay-off functions may be preordered sets instead of posets; that is, some different elements in the domain or in the range may be order-indifferent or order-equivalent. This is the mo-tivation 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 integerk, letRkdenote thek-dimensional Euclidean space. Letkbe the binary relation onRk, which is defined as:xkywheneverx ≥ yforx,y∈Rk. It is clear thatkis a preorder relation onRk. Hence (Rk,k) is a preordered set. Then any noncooperative game with all the strategies in (Rm,m) and the values of payoff in (Rn,n), for some positive integersmandn, is a nonmonetized noncooperative game, which will be defined in Section .
In Section of this paper, we generalize the extensions of the Abian-Brown fixed point theorem provided in [] from chain-complete posets to chain-complete preordered sets for set-valued mappings. Evidently, they are also generalizations of the Fujimoto-Tarski fixed point theorem from complete lattices to chain-complete preordered sets. In Sec-tions and , we apply these generalizaSec-tions to prove some existence theorems of the ex-tended and generalized Nash equilibria of nonmonetized noncooperative games on chain-complete preordered sets.
2 Order-clustered fixed point
In this section, we recall and provide some concepts and properties of preordered sets, and we introduce the concept of order-clustered fixed point of set-valued mapping on pre-ordered sets. Then we generalize the Abian-Brown fixed point theorem from single-valued mappings to set-valued mappings, which is also from complete lattices to preordered sets. Here we closely follow the notations from [, , ], and [].
LetPbe a nonempty set. An ordering relationonPis said to be a preorder whenever it satisfies the following two conditions:
. (reflexive)xxfor everyx∈P;
. (transitive)xyandyzimplyxzfor allx,y,z∈P.
ThenP, together with the preorder, is called a preordered set, which is denoted by (P,). Furthermore, a preorderon a preordered setPis said to be a partial order if, in addition to the above two properties, it also satisfies
. (antisymmetric)xyandyximplyx=yfor everyx,y∈P.
Remark . It is worth mentioning for clarification that a preordered set (P,) equipped with the preorderonPis defined as a partially ordered system (p.o.s.) in [].
Let (P,) be a preordered set. IfAis a subset ofP, then an elementuofPis called an upper bound ofAifxufor eachx∈A; ifu∈A, thenuis called a greatest element (or a maximum element) ofA. The collection of all greatest elements (maximum elements) ofA is denoted bymaxA. A lower bound ofAand a smallest element (or a minimum element) ofAcan be defined similarly. The collection of all smallest elements (minimum elements) ofAis denoted byminA. If the set of all upper bounds ofAhas a smallest element, we call it a supremum ofA; and the collection of all supremum elements ofAis denoted by supAor∨A. An infimum ofAis similarly defined as a greatest element of the set of all lower bounds ofA, provided that it exists; and the collection of all infimum elements ofA is denoted byinfAor∧A.
It is important to note that, from the above definitions, ifAis a subset of a preordered set (P,), thenmaxA,minA,supAandinfAare the subsets ofP. In particular, ifAis a subset of a poset (P,), thenmaxA,minA,supAandinfAare singletons, provided that they are nonempty. In this case,maxA,minA,supAandinfAare simply written as the contained elements, respectively.
An elementy∈Ais said to be a maximal element of the subsetAof the preordered set (P,) if for anyz∈Awithyzimplieszy. Similar to the definition of maximal element, a minimal element ofAcan be defined. Then every greatest element (smallest element) of the subsetAof the preordered set (P,) is a maximal element (minimum element) ofA; the converse does not hold.
A subsetCof the preordered set (P,) is said to be totally ordered (or a chain inP) whenever, for every pairx,y∈C, eitherxyoryx. Following [] and [], we have the following definition.
Definition . A preordered set (P,) is said to be
(i) inductive if every totally ordered subset of (chain in)Phas an upper bound inP; (ii) chain-complete whenever, for every totally ordered subsetCof (a chain in)P, the
set of all supremum elements ofCis a nonempty subset ofP; that is,∨C=∅.
In game theory and decision theory, the decision makers may consider having indif-ference (same) utilities at some order-equivalent elements in a preordered set. It is the motivation to introduce the following concepts of equivalent elements and order-equivalent classes in a preordered set.
Let (P,P) be a preordered set. For anyx,y∈P, we say thatx,yareP-order equiva-lent (P-order indifferent), which is denoted byx∼Py, whenever bothxPyandyPx hold. It is clear that∼Pis an equivalence relation onP. For anyx∈P, let [x] denote the order-equivalent class (order-indifferent class) containingx, which is called aP-cluster (or simply anorder cluster, or acluster, if there is no confusion caused). LetP/∼PorP˜ denote the collection of all order clusters in the preordered set (P,P). So,x∈[x]∈ ˜Pfor everyx∈P.
X→U\{∅}is order-increasing upward ifxXyinXandu∈F(x) imply that there is w∈F(y) such thatuUw.Fis said to be order-increasing downward ifxXyinXand w∈F(y) imply that there isu∈F(x) such thatuUw. A set-valued mappingFis said to be order-increasing wheneverFis order-increasing both upward and downward.
Order-increasing mappings from a preordered set to a preordered set have the following equivalent classes preserving property.
Lemma . Let(X,X)and(U,U)be two preordered sets and let F:X→U be an order-increasing single-valued mapping.Then,for every x in X,y∈[x]implies F(y)∈[F(x)]in U.
Proof Without any confusion, [x] is understood to be an order-equivalence class in (X,X) with respect to the preorderXand [F(x)] is understood to be an order-equivalence class in (U,U) with respect to the preorderU. For everyxinX,y∈[x] if and only if both of xXyandyXxhold inX. Then the order-increasing property ofFimplies that both of F(x)UF(y) andF(y)UF(x) hold inU. That is,F(y)∈[F(x)].
In game theory and decision theory, there are useful mappingsF:X→Ubetween two preordered sets which map order-indifferent elements inXto order-indifferent elements inU. That is, if the inputs of the mappingFareX-order equivalent elements inX, then the outputs ofF areU-order equivalent elements inU. This leads us to the following definition.
Definition . Let (X,X) and (U,U) be two preordered sets and letF:X→Ube a single-valued mapping.Fis said to be order cluster-preserving, whenever
x∼Xy implies F(x)∼UF(y) forx,y∈X. (.)
Lemma . Every order-preserving single-valued mapping from a preordered set to a pre-ordered set is cluster-preserving.
Proof The order cluster-preserving property (.) immediately follows from Lemma ..
Definition . Let (X,X) be a preordered set and letF:X→X\{∅}be a set-valued mapping. An elementx∈Xis called
. a fixed point ofF, wheneverx∈F(x);
. anX-clustered fixed point (an order-clustered fixed point, or simply a clustered
fixed point) ofF, whenever there is aw∈[x]such thatw∈F(x).
Definition . can be explained as follows: an elementx∈Xis an order-clustered fixed point ofF, whenever there is aw∈Xwithw∼Xxsuch thatw∈F(x). Sincex∼Xx, then we have the following property:
xis a fixed point ofF ⇒ xis an order-clustered fixed point ofF.
Let (P,) be a chain-complete preordered set and letF:P→P\{∅}be a set-valued mapping. Following Fujimoto [], we have the following notation:
SF(x) =z∈P:zufor someu∈F(x) for everyx∈P. (.)
Theorem . Let(P,)be a chain-complete preordered set and let F:P→P\{∅}be a set-valued mapping.If F satisfies the following three conditions:
A. Fis order-increasing upward;
A. SF(x)is an inductive subset ofPfor eachx∈P; A. There is ayinPwithyufor someu∈F(y).
Then F has an order-clustered fixed point,that is,there exists x∗∈P with w∈[x∗]such that w∈F(x∗).
Proof Let
A=x∈P:xufor someu∈F(x). (.)
It is clear that the elementygiven in assumption A is inA; and thereforeA=∅. Very similarly to the proof of the extension of Tarski’s fixed point theorem by Fujimoto in [], by applying isotonic property A and assumption A in this theorem, we can show that the setAis inductive. To this end, take an arbitrary totally ordered subset (a chain)C⊆A. SinceC is also a chain of the chain-complete preordered set (P,), then ∨C=∅ and
∨C⊆P. Take an arbitrary elementb∈ ∨C. So, for anyx∈C⊆A, from (.), there is a u∈F(x) withxu. On the other hand, fromxbandu∈F(x), applying assumption A, there isv∈F(b) such thatxuv; and hencex∈SF(b) for everyx∈C. It implies
C⊆SF(b).
From assumption A,SF(b) is inductive; and therefore the totally ordered subset (chain) CofSF(b) has an upper bound inSF(b), sayc, withc∈SF(b). From (.), it yields that there is au∈F(b) such thatcu.
Sinceb∈ ∨Candcis an upper bound ofC, we havebc. Then we obtain
bcu for someu∈F(b),
which implies thatb∈A. Hencebis an upper bound of the given chainCinA; and there-foreAis inductive.
Then applying Zorn’s lemma (Theorem I.. in []), the inductive setAhas a maximal element, sayx∗∈A. Equation (.) implies that there isw∈F(x∗) such thatx∗w. For this elementw∈F(x∗) withx∗w, from assumption A in this theorem, there isz∈F(w) withwz, which implies thatw∈A. Sincex∗is a maximal element ofA, fromx∗w, we must havex∗∼w∈F(x∗). Hence,x∗is an order-clustered fixed point ofF. This theorem
is proved.
A. F(x)is inductive with a finite number of maximal element clusters for everyx∈P,
then F has an order-clustered fixed point.
Proof Take the same setA={x∈P:xufor someu∈F(x)}as defined in the proof of Theorem .. Assumption A in this theorem also impliesA=∅. Then we show thatA is inductive. To this end, take an arbitrary chainCofA. Since (P,) is a chain-complete preordered set, then∨C=∅and∨C⊆P. Pick an arbitrary elementb∈ ∨C. For every x∈C⊆A, there is a pointux∈F(x) satisfyingxux. On the other hand, fromxb, assumption A implies that there is ane(x)∈F(b) such thatuxe(x). Then we obtain a mappinge:C→F(b) satisfying the following order inequality:
xuxe(x) withux∈F(x) ande(x)∈F(b) for everyx∈C. (.)
From assumption Ain this theorem, we can write the set of maximal element clusters ofF(b) by{[u], [u], . . . , [um]} ⊆F(b), for some positive integerm, with{u,u, . . . ,um} ⊆ F(b). We claim that there are elementsx∈Canduj, for somejwith ≤j≤m, such that
xuj for allx∈Cwithxx. (.)
In fact, from (.), we have, for everyx∈C,xe(x)∈F(∨C). SinceF(b) is inductive and{[u], [u], . . . , [um]}is the set of all the maximal element clusters ofF(b), then, for any
givenx∈C, we must have
xe(x)ui for someiwith ≤i≤m. (.)
By applying (.), the proof of (.) is similar to the proof of Theorem . in []. Then, from the claim (.), we havebuj∈F(b) for somejwith ≤j≤m; and henceb∈A. This implies thatChas an upper boundbinA; and therefore,Ais inductive. The rest of
the proof is very similar to the proof of Theorem ..
As a consequence of Theorem ., we obtain the following special cases of the extensions of the Abian-Brown fixed point theorem in preordered sets.
Corollary . Let(P,)be a chain-complete preordered set and let F:P→P\{∅}be a set-valued mapping.If F satisfies conditionsAandAgiven in Theorem.and
A. F(x)has a maximum element for everyx∈P,
then F has an order-clustered fixed point.
Corollary . Let(P,)be a chain-complete preordered set with∧P=∅in P.Let F : P→P\{∅}be a set-valued mapping.If F satisfies conditionAgiven in Theorem.and one of conditionsA, A,orAgiven in Theorems., .and Corollary.,respectively, then F has an order-clustered fixed point.
Proof Takey∈ ∧PinP, then for anyu∈F(y), we obviously haveyu. So,y∈ ∧Psatisfies
assumption A in Theorem ..
As consequences, when posets are considered as special preordered sets, the extensions of the Abian-Brown fixed point theorem in posets provided in [] immediately follow from Corollary .. As a matter of fact, we have the following result on a chain-complete poset, which is more general than the extension of Tarski’s fixed point theorem on com-plete lattice by Fujimoto [].
Corollary . Let(P,)be a chain-complete poset with ∧P exists in P. Let F :P→ P\{∅}be a set-valued map.If F satisfies conditionAgiven in Theorem.and one of conditionsA, A,orAgiven in Theorems., .,and Corollary.,respectively,then F has a fixed point.
3 Applications to the extended Nash equilibria of nonmonetized noncooperative games on chain-complete 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 [–]. Then we apply the extensions of the Abian-Brown fixed point the-orem provided in the last section to prove some existence thethe-orems for the extended Nash equilibrium of nonmonetized noncooperative games on chain-complete preordered sets, which can be considered as extensions of the results proved by Li in [], which are on chain-complete posets.
Definition . Letnbe a positive integer greater than . Ann-person nonmonetized non-cooperative game consists of the following elements:
. the set ofnplayers, which is denoted byN={, , , . . . ,n};
. the collection of n strategy sets{S,S, . . . ,Sn}, for thenplayers respectively, such that (Si,i)is a chain-complete preordered set for playeri= , , , . . . ,n, with notation
S=S×S× · · · ×Sn;
. the outcome space(U;U)that is a preordered set;
. thenpayoff functionsf,f, . . . ,fn, wherefiis the payoff function for the playerithat
is a mapping fromS×S× · · · ×Snto the preordered set(U;U), for i= , , , . . . ,n. We definef ={f,f, . . . ,fn}.
This game is denoted by= (N,S,f,U).
The rule to play in an n-person nonmonetized noncooperative game= (N,S,f,U) is that when allnplayers , , , . . . ,nsimultaneously and independently choose their own strategiesx,x, . . . ,xn, wherexi∈Si, fori= , , , . . . ,n, then the playeriwill receive his or her utility (payoff )fi(x,x, . . . ,xn)∈U. For anyx= (x,x, . . . ,xn)∈S, and for every given i= , , , . . . ,n, as usual, we define
Thenx–i∈S–iandxcan simply be written asx= (xi,x–i). Moreover, we define
fi(Si,x–i) =
fi(ti,x–i) :ti∈Si
.
Ann-person nonmonetized noncooperative game defined in this section is also called a generalized game (see []). Now we recall the extensions of the concept of Nash equi-librium of noncooperative games to the generalized Nash equiequi-librium and the extended Nash equilibrium of nonmonetized noncooperative games.
Definition . In ann-person nonmonetized noncooperative game= (N,S,f,U), a se-lection of strategies (x,
x, . . . ,
xn)∈S×S× · · · ×Snis called
. a generalized Nash equilibrium of this game if, for everyi= , , , . . . ,n, the following order inequality holds:
fi(xi,
x–i)Ufi( xi,
x–i) for allxi∈Si;
. an extended Nash equilibrium of this game if, for everyi= , , , . . . ,n, the following order inequality holds:
fi(xi,
x–i)Ufi( xi,
x–i) for allxi∈Si.
It is clear that any generalized Nash equilibrium of ann-person nonmonetized nonco-operative game is an extended Nash equilibrium of this game; and the converse may not be true.
Lemma . Let(Si,i)be a preordered set for every i= , , . . . ,n.LetSbe the coordinate ordering on S induced by the preordersi,that is,for any x,y∈S with x= (x,x, . . . ,xn)
and y= (y,y, . . . ,xn),
xSy if and only if xiiyi for all i= , , . . . ,n.
Then S is a preorder on S;and hence(S,S)is a preordered set.Furthermore,if every (Si,i)is chain-complete(inductive),then(S,S)is also chain-complete(inductive). More-over,we have
˜
S=S˜× ˜S× · · · × ˜Sn.
Furthermore, for any x= (x,x, . . . ,xn)∈S, we have[x] = ([x], [x], . . . , [xn]),where[x]
stands for anS-cluster inS and˜ [x
], [x], . . . , [xn]are order clusters inS˜,S˜, . . . ,S˜n,
re-spectively.
Proof The proof is straightforward and is omitted here.
Let Abe a subset of a preordered set (P,). The following subset ofP is called the
-downward set ofAdenoted byD(A),
The following theorem provides a result for the existence of an extended Nash equilibrium of nonmonetized noncooperative games on preordered sets.
Theorem . Let = (N,S,f,U)be an n-person nonmonetized noncooperative game. Suppose that for every player i= , , , . . . ,n,and for any x∈S,fi:S→U is a cluster-preserving single-valued mapping that satisfies the following conditions:
G. fi(Si,x–i)is an inductive subset of the preordered set(U,U);
G. Thei-downward set of the inverse image
{zi∈Si:fi(zi,x–i)is a maximal element of fi(Si,x–i)}is an inductive subset ofSi;
G. ForxSyinS,ifz
i∈Siwithfi(zi,x–i)is a maximal element offi(Si,x–i),then there
iswi∈Siwithziiwisuch thatfi(wi,y–i)is a maximal element offi(Si,y–i).
If there are p= (pi,p–i),q= (qi,q–i)∈S satisfying that
pSq and fi(qi,p–i)is a maximal element of fi(Si,p–i), (.)
then this game has an extended Nash equilibriumx= (x,
x, . . . ,
xn).Moreover,every element in[x]is an extended Nash equilibrium of.
Proof The first part of the proof is similar to the proof of Theorem . in []. Since (Si,i) is a chain-complete preordered set for everyi= , , . . . ,n, then from Lemma ., (S,S) is also a chain-complete preordered set equipped with the product orderS.
For every fixedi= , , , . . . ,n, we define a set-valued mappingTi:S→Si\{∅}by
Ti(x) =zi∈Si:fi(zi,x–i) is a maximal element offi(Si,x–i)
for allx= (x,x, . . . ,xn)∈S.
From assumption G of this theorem, for every fixed elementx∈S,fi(Si,x–i) is an inductive
subset ofU. By applying Zorn’s lemma, the rangefi(Si,x–i) has a maximal element; and
thereforeTi(x) is a nonempty subset ofSi. Hence, the mapTi:S→Si\{∅}is well defined. Then we define a set-valued mappingT:S→S\{∅}by
T(x) =T(x)×T(x)× · · · ×Tn(x) for allx∈S.
Now we prove that the mappingTsatisfies all conditions A, A and A in the first exten-sion of the Abian-Brown fixed point theorem in a chain-complete preordered set provided in the preceding section.
At first, we show thatT satisfies assumption A:T isS-increasing upward. For any givenxSyinS and for anyz= (z
,z, . . . ,zn)∈T(x), for everyi= , , . . . ,n, we have zi∈Ti(x), that is,fi(zi,x–i) is a maximal element offi(Si,x–i). Then from hypothesis G
of this theorem, there iswi∈Siwithziiwisuch thatfi(wi,y–i) is a maximal element of
fi(Si,y–i), that is,wi∈Ti(y). Takew= (w,w, . . . ,wn). We obtain thatzSwandw∈T(y).
Hence,T is isotone.
Secondly, we prove that the mappingTsatisfies assumption A in Theorem .. For an arbitraryxinS, with respect to the mappingT and from (.), we write
For everyi= , , . . . ,n, with respect to the mappingTi, we have
STi(x) =ti∈Si:tiizifor somezi∈Ti(x)
=ti∈Si:tiizifor somezi∈Sisuch thatfi(zi,x–i)
is a maximal element offi(Si,x–i)
=Dzi∈Si:fi(zi,x–i) is a maximal element offi(Si,x–i)
.
From assumption G of this theorem, thei-downward set of{zi∈Si:Pi(zi,x–i) is a
max-imal element ofPi(Si,x–i)}is inductive, which implies thatSTi(x) is an inductive subset of
S, so isST(x).
Finally, for the given elements p,q∈S in this theorem, from condition (.), we have pSqandfi(qi,p–i) is a maximal element of fi(Si,p–i). This implies thatq∈T(p) with
pSq; and hence the mappingTsatisfies assumption A.
Thus the mapping T :S→S\{∅} satisfies all conditions A, A and A in Theo-rem .. SinceSis a chain-complete preordered set, applying Theorem .,Thas an order-clustered fixed pointx= (x,
x, . . . ,
xn)∈S. Hence, there ist∈[x] such thatt ∈T(x). This impliesti∈Ti(
x); that is,
fi(ti,x–i) is a maximal element offi(Si,
x–i) for every fixedi= , , . . . ,n.
For everyi= , , , . . . ,n, it is equivalent to
fi(ti,
x–i)Ufi( ti,
x–i) for allti∈Si. (.)
The following is from the definition of the product orderSon (S,S):
t∼Sx ⇒ ti∼i
xi ⇒ (
ti,x–i)∼S(
xi,x–i) =
x fori= , , . . . ,n. (.)
Then from (.), t ∈[x] implies (ti,x–i)∼S (
xi,x–i) fori= , , . . . ,n. Since the payoff
functionfi:S→U is a cluster-preserving single-valued mapping, from (
ti,x–i)∼S(
xi,
x–i) and Definition ., it yields that
fi(ti,x–i)∈
fi(xi,x–i)
; that is, fi(ti,x–i)∼Ufi(
xi,x–i). (.)
Combining (.) and (.), we have
fi(ti,
x–i)Ufi( xi,
x–i) for allti∈Si. (.)
This shows thatx= (x,
x, . . . ,
xn) is an extended Nash equilibrium of this game. With the same argument as above, we can show that, for anyz∈[x],fi(
zi,
z–i)∼Ufi( xi,
x–i) holds. Then, repeating the argument from (.) to (.), we obtain thatz is also an extended
Nash equilibrium of this game. This completes the proof of this theorem.
Theorem . Let = (N,S,f,U) be an n-person nonmonetized noncooperative game. Suppose that for every player i= , , , . . . ,n,fi:S→U is a cluster-preserving single-valued mapping.If,for any x∈S in addition to that,fisatisfies assumptionsG, G,and condition (.)given in Theorem.,fialso satisfies the following condition:
G. The inverse image{zi∈Si:Pi(zi,x–i)is a maximal element ofPi(Si,x–i)}is inductive
with a finite number of maximal element clusters for eachx∈S.
Then there is anS-cluster in S,in which every element is an extended Nash equilibrium of.
In case the condition∧Si=∅holds for the strategy set (Si,i), for everyi, condition (.) in Theorems . and . can be removed. As a consequence of Theorem ., we have the following.
Corollary . Let = (N,S,f,U)be an n-person nonmonetized noncooperative game. Suppose that for every player i= , , , . . . ,n,∧Si=∅and for any x∈S,fi:S→U is a cluster-preserving single-valued mapping.If,for any x∈S in addition to that,fisatisfies assumptionsGandG,fialso satisfies one of assumptionsGorGgiven in Theorem. and Theorem.,then there is anS-cluster in S,in which every element is an extended Nash equilibrium of.
Proof Takep∈ ∧Sand letqbe a maximum element ofF(p). It is clear thatpandqsatisfy
condition (.) given in Theorem ..
4 Applications to the generalized Nash equilibria of nonmonetized noncooperative games on chain-complete preordered sets
It is clear that for (ti,x–i), (
xi,x–i)∈S, the order inequalityfi(ti,
x–i)Ufi(
xi,x–i) implies
fi(ti,x–i)Ufi(
xi,x–i). Hence, the generalized Nash equilibria can be considered as special
cases of the extended Nash equilibria ofn-person nonmonetized noncooperative games. The conditions for the existence of a generalized Nash equilibrium of ann-person non-monetized 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 . Let= (N,S,f,U)be an n-person nonmonetized noncooperative game. Sup-pose that for every player i= , , , . . . ,n,fi:S→U is a cluster-preserving single-valued mapping,and for any x∈S,in addition to that,fisatisfies the following two conditions:
g. fi(Si,x–i)is a subset of(U,U)with a maximum element;
g. ForxSyinS,ifz
i∈Siwithfi(zi,x–i)is a maximum element offi(Si,x–i),then there
iswi∈Siwithziiwisuch thatfi(wi,y–i)is a maximum element offi(Si,y–i), fialso satisfies one of the following conditions:
g. Thei-downward set of the inverse image{zi∈Si:Pi(zi,x–i)is a maximum element of fi(Si,x–i)}is an inductive subset ofSi;
g. The inverse image{zi∈Si:Pi(zi,x–i)is a maximum element of fi(Si,x–i)}is inductive
with a finite number of maximal element clusters for eachx∈S.
If there are p= (pi,p–i),q= (qi,q–i)∈S satisfying that
then there is anS-cluster in S,in which every element is a generalized Nash equilibrium of.
Proof The proof of this theorem is similar to the proof of Theorem .. For every fixed i= , , , . . . ,n, we define a mappingTi:S→Si\{∅}by
Ti(x) =zi∈Si:fi(zi,x–i) is a maximum element offi(Si,x–i)
for allx= (x,x, . . . ,xn)∈S.
We also define the product mappingT:S→S\{∅}by
T(x) =T(x)×T(x)× · · · ×Tn(x) for allx∈S.
Then, similarly to the proof of Theorem ., we can show that the mappingThas an order-clustered fixed pointx= (x,
x, . . . ,
xn)∈S. Hence, there ist∈[x] such thatt ∈T(x). This impliesti∈Ti(
x); that is,
fi(ti,x–i) is a maximum element offi(Si,
x–i) for every fixedi= , , . . . ,n.
It is equivalent to the following: for everyi= , , , . . . ,n, we have
fi(ti,
x–i)Ufi( ti,
x–i) for allti∈Si, (.)
From (.),t∼Sximplies (ti,x
–i)∼S(
xi,x–i) for every fixedi= , , . . . ,n. Sincefi:S→ Uis a cluster-preserving single-valued mapping from (.) in Definition ., it implies
fi(ti,x–i)∼Ufi(
xi,x–i). (.)
Combining (.) and (.), we have
fi(ti,x–i)Ufi(
xi,x–i) for allti∈Si.
This shows thatx= (x,
x, . . . ,
xn) is a generalized Nash equilibrium of this game. Simi-larly to the proof of Theorem ., we can show that for anyz∈[x],zis also a generalized
Nash equilibrium of this game. This completes the proof of this theorem.
Similar to Corollary ., if we consider some special collections of strategies with lower bound, then we obtain the following consequence of Theorem ., where condition (.) can be removed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Authors LX and JL initialed the basic ideas and definitions of this manuscript, and then all the three authors LX, JL and WY worked together in proving all theorems.
Author details
1Department of Mathematics, Lishui University, Lishui, Zhejiang 323000, China.2Department of Mathematics, Shawnee
State University, Portsmouth, Ohio 45662, USA.3Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321000, China.
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).
Received: 27 April 2013 Accepted: 8 July 2013 Published: 22 July 2013
References
1. Aubin, JP: Mathematical Methods of Games and Economic Theory. North-Holland, Amsterdam (1979) 2. Bohnenblust, HF, Karlin, S: Contributions to the Theory of Games. Princeton University Press, Princeton (1950) 3. Mas-Colell, A, Whinston, MD, Green, JR: Microeconomic Theory. Oxford University Press, Oxford (1995) 4. Samuelson, P: Foundations of Economic Analysis. Harvard University Press, Cambridge (1947) 5. Zhang, CJ: Set-Valued Analysis with Applications in Economics. Sciences Press, Beijing (2004) (Chinese)
6. Von Neumann, J, Morgenstern, O: The Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)
7. Carl, S, Heikkilä, S: Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory. Springer, New York (2010)
8. Li, JL: Extended and generalized Nash equilibria of nonmonetized noncooperative games on complete lattices (submitted)
9. Li, JL: Applications of fixed point theory to generalized Nash-equilibriums of nonmonetized noncooperative games on Banach lattices. Nonlinear Anal. Forum18, 1-11 (2013)
10. Li, JL: Extended Nash equilibria of nonmonetized noncooperative games on preordered sets. Int. Game Theory Rev. (to appear)
11. Li, JL, Park, S: Generalized Nash-equilibriums of nonmonetized noncooperative games on lattices. Br. J. Econ. Manag. Trade (to appear)
12. Li, JL: Several extensions of the Abian-Brown fixed point theorem and their applications to extended and generalized Nash equilibria on chain-complete posets. J. Math. Anal. Appl. (to appear)
13. Xie, LS, Li, JL, Wen, CF: Applications of fixed point theory to extended Nash-equilibriums of nonmonetized noncooperative games on posets (to appear)
14. Ok, EA: Order theory (forthcoming)
15. Fujimoto, T: An extension of Tarski’s fixed point theorem and its applications to isotone complementarity problems. Math. Program.28, 116-118 (1984)
16. Aliprantis, CD, Burkinshaw, O: Positive Operators. Springer, Berlin (2006) 17. Dunford, N, Schwartz, JT: Linear Operators. Part I. Wiley, New York (1988)
doi:10.1186/1687-1812-2013-192
