Applications of fixed point theory to extended Nash equilibriums of nonmonetized noncooperative games on posets
© Xie et al.; licensee Springer. 2013
Received: 4 June 2013
Accepted: 15 August 2013
Published: 13 September 2013
We say that a noncooperative game is nonmonetized if the ranges of the utilities of the players are posets. In this paper, we examine some nonmonetized noncooperative games of which both the collection of strategies and the ranges of the utilities for the players are posets. Then we carry the concept of generalized Nash equilibriums of noncooperative games defined in (Li in J. Nonlinear Anal. Forum 18:1-11, 2013; Li and Park in Br. J. Econ. Manag. Trade 4(1), 2014) to extended Nash equilibriums of nonmonetized noncooperative games. By applying some fixed point theorems in posets and by using the order-preserving property of mappings, we prove an existence theorem of extended Nash equilibriums for nonmonetized noncooperative games.
MSC:46B42, 47H10, 58J20, 91A06, 91A10.
In economic theory, social science, military science, or other fields, the outcomes of some games may not be in a complete (totally) ordered set; that is, the utilities of the players may not be represented by real-valued functions, which is different from most noncooperative games in game theory. We give an example below to more precisely demonstrate our arguments.
This preference relation ⪰ on U is not a complete order, which is a partial order on U. In case the budget of this military industry is limited (this is always true in the real world), the decision makers of this industry would seek the war production plan to optimize the fighting capability under some given warfare environment.
It is clear that such an optimization problem is not a normal optimization problem (with real-valued functions). As a matter of fact, these optimization ideas, introduced and studied by Giannessi  in 1980, are called vector optimization problems and vector variational inequalities, where the outcome space is in a finite-dimensional vector space. Since then, the theory as well as algorithms of vector variational inequalities and vector optimization have been extensively studied. See, e.g., [2–9] and the references therein.
Understanding the existence of optimization problems with respect to a partial order on a set of objects U is of considerable interest for Li , Li and Park  to introduce the concepts of the generalized Nash equilibrium of nonmonetized noncooperative games, in which the utilities of the players may not be real-valued; that is, the payoffs of the players may not be represented by real-valued functions. Now we recall the concept of generalized Nash equilibrium of nonmonetized noncooperative games from [10, 11].
Existence theorems of generalized Nash equilibriums of some nonmonetized noncooperative games are provided in [10, 11]. In  the space of outcomes is a Banach lattice, and in  it is just a lattice, which is not required to be equipped with neither topological structure nor algebraic structure.
It is clear that in both pure mathematical theory and real applications the order inequality (ii) is more general than the order inequality (i). In Section 3, we prove an existence theorem of extended Nash equilibriums of some nonmonetized noncooperative games.
Let be a poset. An element u of P is called an upper bound of a subset A of P if for each . If , then u is called the greatest element of A and is denoted by . If the set of all upper bounds of A has the smallest element, we call it the supremum of A and denote it by supA or ∨A. An element y is called a maximal element of A if and if and imply that . A lower bound of A, the smallest element minA of A, the infimum of A, and a minimal element of A can be defined similarly.
A poset is called a lattice if and exist for all . Define and . A subset C of a poset is called a chain if or for all .
inductive if every chain in P has an upper bound in P;
inversely inductive if every chain in P has a lower bound in P;
bi-inductive whenever it is both inductive and inversely inductive.
strongly inductive whenever for every chain C in P, the supremum of C, supC, exists in P;
strongly inversely inductive whenever for every chain C in P, the infimum of C, infC, exists in P;
strongly bi-inductive whenever it is both strongly inductive and strongly inversely inductive.
Extension of Zorn’s lemma Every bi-inductive poset has both maximal and minimal elements.
Given posets and , we say that a mapping is order increasing upward if in X and imply that is nonempty, that is, if in X and , then there is such that . The mapping F is order increasing downward if in X and imply that is nonempty. F is said to be order increasing whenever F is both order increasing upward and downward.
As a special case, a single-valued mapping F from a poset to the poset is said to be order increasing whenever implies . An increasing mapping is said to be strictly order increasing whenever implies .
A nonempty subset A of a subset Y of a poset is said to be order compact upward in Y if for every chain C of Y that has a supremum in P, the intersection is nonempty whenever is nonempty for every . The set A is order compact downward in Y if for every chain C of Y that has the infimum in P, the intersection is nonempty whenever is nonempty for every . If A is both order compact upward and order compact downward in Y, then A is said to be order compact in Y.
Let A be a subset of a poset . An element is called a sup-center of A in P if exists in P for each . If exists in P for each , then c is called an inf-center of A in P. If c is both a sup-center and an inf-center of A in P, then c is called an order center of A in P. In particular, if , then c is simply called a sup-center or an inf-center of P, respectively.
Let A be a nonempty subset of a poset . The set is the collection of all possible supremums and infimums of chains of A, which is called the order closure of A. If , then A is said to be order closed.
Remark 2.3 Every nonempty strongly bi-inductive subset of a poset is order closed.
Now we recall a fixed point theorem on posets from . It will be used in the proof of the existence of extended Nash equilibrium of nonmonetized noncooperative games in Section 3.
Theorem 2.4 [, Theorem 2.12]
Let be a poset. Assume that a set-valued mapping is order increasing, and that its values are order compact in . If chains of have supremums and infimums (in P), and if has a sup-center or an inf-center in P, then F has minimal and maximal fixed points.
3 Nonmonetized noncooperative games on posets
In the section of Introduction, we described the motivations to extend the noncooperative games and the concept of generalized Nash equilibrium to nonmonetized noncooperative games. In this section, we give some definitions for these extensions, which are the generalized notions of Nash equilibriums defined and studied in games theory (see, e.g., [16–21]).
the set of n players, which is denoted by ;
the collection of n strategy sets , for the n players respectively, which is also written as ;
the outcome space that is a poset;
the n utilities functions (payoff mappings) , where is the utility function for player i that is a mapping from to the poset for . We define .
This game is denoted by .
Now we extend the concept of Nash equilibrium of noncooperative games and generalized Nash equilibrium of nonmonetized noncooperative games to the extended Nash equilibrium of nonmonetized noncooperative games.
Then is a poset. Furthermore, if every is (strongly) inductive, then is also (strongly) inductive. If every is (strongly) bi-inductive, then is also (strongly) bi-inductive.
Proof The proof is straightforward and is omitted here. □
The following theorem is the main result of this paper. It provides some conditions for the existence of the extended Nash equilibrium of nonmonetized noncooperative games.
is a strongly inversely inductive poset;
has a sup-center (or an inf-center simultaneously for all i);
is (single-valued) order increasing with respect to the product order ;
for any fixed , is an inductive subset of ;
for any fixed , and for any , the inverse image is a strongly bi-inductive subset of ;
- (4)for any satisfying , the maximal elements have the following monotone properties:
if is a maximal element of , then there is with such that is a maximal element of ;
if is a maximal element of , then there is with such that is a maximal element of .
Then this nonmonetized noncooperative game Γ has an extended Nash equilibrium. Furthermore, this nonmonetized noncooperative game Γ has minimal and maximal (with respect to the product lattice order ) extended Nash equilibriums.
From condition (3), is a strongly bi-inductive subset of . It implies that the chain has both supremum and infimum in the inverse image , which is contained in . They clearly are the supremum and infimum of this chain in , respectively. It proves claim (2).
From (2) and applying Lemma 3.3, we obtain that is a nonempty strongly bi-inductive subset of for every .
In order to apply Theorem 2.12 in  to show the existence of a fixed point for this mapping , we have to show that the set-valued mapping T is increasing, that its values are order compact in , that chains of have supremums and infimums, and that has a sup-center or an inf-center in S.
Let . By combining (3) and (4), we obtain and . It proves that T is an order increasing upward mapping on S.
Secondly, we prove the order increasing downward property for T. For any given , with , and for any , we need to show that there is such that . Write . The hypothesis implies for every . That is, is a maximal element of . Applying part (b) of condition (4) of this theorem, and from that is implied by the condition , we obtain that is a maximal element of ; that is, . Notice for every . It implies ; and hence T is order increasing downward. Thus T is an order increasing set-valued mapping.
Hence, is order compact upward in .
By the strongly bi-inductive property (2), very similarly to the above proof, we can show that for every , is order compact downward in . Hence is order compact in .
Hence ν is the supremum of this chain C in S. (Furthermore, the second part of (13) implies that , that is, the supremum of this chain C in is in .)
exists and is in . Let . From the above definition, b is the infimum of this chain C in S. Hence, for any arbitrary chain C in , both supC and infC exist in S.
which shows that is an extended Nash equilibrium of this game. Furthermore, from Theorem 2.4, the mapping T has minimal and maximal fixed points. Based on the above argument, they are the minimal and maximal (with respect to the product lattice order ) extended Nash equilibriums for this game. This completes the proof of this theorem. □
4 Nonmonetized noncooperative games on lattices
In this section, we consider a special case of nonmonetized noncooperative games, defined in Definition 3.1 in the last section, for the outcome space and strategy sets to be lattices. Then we examine that the generalized Nash equilibriums of nonmonetized noncooperative games defined in [10, 11] are the special cases of the extended Nash equilibriums defined in Definition 3.2.
Definition 4.1 [, Definition 3.2]
for all and for every .
It is clear that any generalized Nash equilibrium of a nonmonetized noncooperative game is an extended Nash equilibrium of this game. The converse may not be true. We show below that Theorem 3.4 in  about the existence of generalized Nash equilibrium can be obtained as a corollary of Theorem 3.4 in this paper.
Corollary 4.2 [, Theorem 3.4]
is (single-valued) order increasing with respect to the product order ;
for any fixed , is an order bounded and Dedekind complete subset of ;
for any fixed , and for any , the inverse image is an order bounded and Dedekind complete subset of ;
if , then for any , there is such that and .
Then this nonmonetized noncooperative game Γ has a generalized Nash equilibrium. Furthermore, Γ has minimal and maximal (with respect to the product lattice order ) generalized Nash equilibriums.
So, is also the supremum of in , which is the unique maximal element of . Taking , then part (b) of condition (4) in Theorem 3.4 holds.
Since , from the above order inequality, we get . Hence is the supremum of in , which is the unique maximal element of . It shows that part (a) of condition (4) in Theorem 3.4 holds. Hence condition (4) in Theorem 3.4 is satisfied.
Condition (ii) for the set of strategies in Theorem 3.4 is satisfied. Every element of is both a sup-center and an inf-center of . It is because is a strongly inversely inductive lattice. So, condition (i) for the set of strategies in Theorem 3.4 is also satisfied.
It is a contradiction to (16) and this corollary is proved. □
The first author was partially supported by National Natural Science Foundation of China (11171137). The third author was partially supported by a grant from NSC 102-2115-M-037-001.
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