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Volume 2018 |Article ID 3737253 | 5 pages | https://doi.org/10.1155/2018/3737253

Existence of Generalized Nash Equilibrium in Person Noncooperative Games under Incomplete Preference

Academic Editor: Xinguang Zhang
Received18 Apr 2018
Accepted25 Sep 2018
Published09 Oct 2018

Abstract

To prove the existence of Nash equilibrium by traditional ways, a common condition that the preference of players must be complete has to be considered. This paper presents a new method to improve it. Based on the incomplete preference corresponding to equivalence class set being a partial order set, we translate the incomplete preference problems into the partial order problems. Using the famous Zorn lemma, we get the existence theorems of fixed point for noncontinuous operators in incomplete preference sets. These new fixed point theorems provide a new way to break through the limitation. Finally, the existence of generalized Nash equilibrium is strictly proved in the person noncooperative games under incomplete preference.

1. Introduction and Preliminaries

As a kind of strategy combinations, Nash equilibrium is closely bound up with many important mathematical problems, and many problems in economy and engineering technology can also be described as a Nash equilibrium problem. Recently, the existence of the Nash equilibrium of noncooperative games has been studied [14]. In [1], the existence of uncertainty for generalized Nash equilibrium is proved by introducing the uncertainty to study generalized games. Using maximization theorem, the author presented the existence of Nash equilibrium in generalized games, and in these results, the strategy set is noncompact and has infinite players [2]. In [3] the existence theorem of generalized Nash equilibrium in games is given where strategy space has abstract convex structure. Assuming strategy set is a space, the equilibrium existence theorems have been given in [4]. In the above studies, either the partial order relation on policy sets is required to satisfy, or every total order subset in policy sets must have an upper bound or certain convexity condition.

For a long time, the preference of rational decision-makers on management and economics should satisfy the completeness. But in practice, they show the indecision on many major issues. Since the preference without completeness is a kind of more general order structure, it can make preference relation and the partial order relation unified completely. The research of preference without completeness is started from the von Neumann and Morgenstern in [5]. Aumman and Bewley have made the classic study in [6, 7]; Schmeidler has studied the existence of economic equilibrium with infinite number of institutions under incomplete preference in [8].

In noncooperative games, the policy set composed of player’s selection strategies is a set which cannot meet the needs of completeness. If preference does not meet the completeness, Pareto optimality is meaningless and the traditional method of partial order will lose effectiveness without the antisymmetry axiom inevitably. Therefore, it is consistent with the realistic decision-making environment to study the existence of generalized Nash equilibrium of noncooperative game, but the study of this part has seldom been seen in the past literature research.

In this article, based on the equivalence class set which corresponds to the elements of incomplete preference set being a partial order set, the problem under incomplete preference is translated into the problem with partial order. This method overcomes the difficulties which are brought about by the elements in the set without the completeness. Using the famous Zorn lemma, we get the existence theorems of fixed point for noncontinuous operators in incomplete preference sets. The fixed point theorems provide a new way for breaking through the limitations. The existence of generalized Nash equilibrium is strictly proven in the person noncooperative games under incomplete preference.

Here some concepts and theorems are given, which are related to incomplete preference.

Let be a nonempty set. An ordering relation on may satisfy the following axiom:

Reflexive: , for any ;

Symmetry: If and , then , for any ;

Transitive: If and , then ,for any ;

Complete: If , for any , there is at least one inequality to be established.

Definition 1 (see [9]). Let be a nonempty set. An order relation defined among certain elements of is said to be partial order if the order relation satisfies reflexive, transitivity, and antisymmetry axioms. Then is called a poset.

Definition 2 (see [10]). Let be a nonempty set. An order relation defined among certain elements of is said to be incomplete preference order if the order relation satisfies reflexive and transitivity axioms, which is denoted by . If completeness axiom is still satisfied for incomplete preference order, the order relation is said to be preference order, which is still denoted by . Then is called an incomplete preference set.

Definition 3 (see [11]). Let be an incomplete preference set. For any , we say that are indifference, which is denoted by , whenever both and hold.

Remark 4. , but .

Remark 5. The indifference relation the equivalence relation.

Definition 6 (see [11]). Let be an incomplete preference set. If for any complete preference subset of , there is denumerable set such that if , there is which satisfies , then is said to be pseudo separable in incomplete preference.
Let be an incomplete preference set, and is a subset in . The order relation in quotient set is elicited by the incomplete preference relation in . Let , and is an equivalence class set in .

Definition 7 (see [11]). For any , if there are such that , we write

Lemma 8 (see [11]). Let be an incomplete preference set, and is a subset in . The order relation in quotient set which is elicited by the incomplete preference relation in is a partial order. Then the quotient set is a poset.

Lemma 9 (see [11]). If is incomplete preference pseudo separable, then is incomplete preference pseudo separable.

Lemma 10 (see [12] (Zorn Lemma)). Let be a nonempty partial ordered set. If every total ordered subset in has an upper bound in , then there is a maximal element in .

2. Existence Theorems for Fixed Point on Incomplete Preference Sets

Partial order method is discussed and applied greatly in mathematics, and the conclusion on the partial order is becoming a very complete system [1323]. But few scholars study the fixed point and extreme value theorems on incomplete preference set.

Definition 11. Let be incomplete preference sets, and let be an order-increasing set-valued mapping. is said to be order-increasing upward, if in , for any ; there is such that ; is said to be order-increasing downward, if in , for any; there is such that . If is both order-increasing upward and order-increasing downward, is said to be order-increasing.

Definition 12. Let be incomplete preference set, and let be an order-increasing set-valued mapping. An element is called a generalized fixed point of , if there are such that .
Let be incomplete preference set, and let be an order-increasing set-valued mapping. The following notation will be used in Theorem 13:

Theorem 13. Let be an incomplete preference pseudo separable set, and let be an order-increasing set-valued mapping. If satisfies the following conditions:
Every increasing sequence in has an upper bound in
There is a with , for some
then has a generalized fixed point; that is, there are such that .

Proof. Let . From the condition , it implies that is a nonempty set in . Take an arbitrary total ordered subset . Since is also an arbitrary total ordered subset of incomplete preference pseudo separable set , there is denumerable set such that if , there is which satisfy .
Let Since is an arbitrary total ordered subset, is well defined. So For any , by the condition , there is a point such that , and since is an incomplete preference pseudo separable set, there is a such that . By the definition of , we get That is, is an upper bound of total ordered subset .
Let ; then is equivalence class set in . Assuming that is a quotient set corresponding to the equivalence relation , then applying Lemmas 8 and 9, we get that the order relation in quotient set which is elicited by the incomplete preference relation in is a partial order.
Take an arbitrary total ordered subset , next, to show that the set has an upper bound in .
Let It is easy to know that is total ordered subset in . In fact, for any , there are . Since is total ordered subset in , we can get that either or is valid. According to the relation between partial order and incomplete preference, we have that either or is valid. So is total ordered subset in .
For any , by the definition of , we get . Since every total ordered subset in has an upper bound, there is such that . So ; that is, every total ordered subset in has an upper bound. Then applying Zorn lemma, we have that there is a maximal element in . By the definition of , we have that is the maximal element in .
Since is the maximal element in , there is such that . Supposing that , the monotonicity of together with implies that there is such that . It means , and it is contradictory with which is the maximal element in . So is proved. Hence there are such that ; that is, has a generalized fixed point.

Corollary 14. Let be an incomplete preference pseudo separable set, and let be an order-increasing set-valued mapping. If satisfies the following conditions:
Every increasing sequence in has an upper bound in
There is with , for some
If , then
then has a fixed point; that is, there are such that .

3. Existence of Nash Equilibrium Points in Generalized Games under Incomplete Preferences

The incomplete preference we present in the paper is more general order relation than the preference in the field of economic management and coincident with the reality of economic phenomenon. It can be applied to the existence of generalized Nash equilibrium of noncooperative game theory.

Definition 15. Let be a positive integer greater than . An person noncooperative game consists of the following elements:
(1) The set of players is denoted by ;
(2) For any , let be the strategy set of player and be an incomplete preference pseudo separable set; denote ;
(3) Let be the payoff function for player ; denote .
The game is denoted by .
Every player in the person noncooperative game independently chooses his own strategy , , to maximize his payoff function . For any , denote Then , and can be written as .

Definition 16. Let be an person noncooperative game. The strategy is said to be a generalized Nash equilibrium in the noncooperative game under the incomplete preference, if there is strategy , for every ; the following order inequality holds

Lemma 17. Let be an incomplete preference pseudo separable set. is a coordinate ordering set composed of , for any , the order relation in induced by the partial order , denoted as the following: Then is an incomplete preference pseudo separable set.

Proof. First we show that is an incomplete preference set. Since is incomplete preference set, for any , we can get , . So . Hence the order relation satisfies reflexive axiom.
For any , , and , which satisfy , By Definition 15, we have , Since is incomplete preference set, we have . So ; that is, is incomplete preference set.
Next we prove that the incomplete preference set is pseudo separable, for any , since is pseudo separable,
Let be an arbitrary total ordered subset of set ; then there is denumerable set such that if , there is which satisfies .
Define the following Let , ; then by the definition of , we have This together with implies that . Hence the incomplete preference set is pseudo separable.

Theorem 18. Let be an person noncooperative game. Suppose that, for any , the payoff function , satisfies the following conditions:
Every total ordered subset in has an upper bound in ;
Every increasing sequence in the inverse image has an upper bound;
For any , if there is with to be a maximal element of , then there is with such that is a maximal element of ;
If there are such that and is a maximal element of ;
If , such that .
Then there is a generalized Nash equilibrium in the person noncooperative game .

Proof. Since is an incomplete preference pseudo separable set, for every , then, from Lemma 17, is also an incomplete preference pseudo separable set equipped with the product order .
For every fixed , define a set-valued mapping as the following: for all .
From assumption of this theorem, for every fixed element , every total ordered subset in has an upper bound in . Then applying Zorn Lemma, the set has a maximal element. Therefore, is a nonempty subset of . Then we define For any arbitrary , with respect to the set , we write For every , we have Now we prove that the operator satisfies the conditions in Theorem 18. Firstly, we will show that the operator is order-increasing. For any given in , and for any , for every , we have ; that is, is a maximal element of . Then from hypothesis of this theorem, there is with such that is a maximal element of ; that is, . Let . We obtain that and . Hence the operator is order-increasing.
From assumption of this theorem, every increasing sequence in has an upper bound. Then we can similarly show that every increasing sequence in has an upper bound. In fact, take an arbitrary total ordered subset . Since is also a total ordered subset of incomplete preference pseudo separable set , there is denumerable set such that for any , there are , which satisfy , .
From the above discussion, we have that there is such that . Thus we obtain a mapping satisfying the following order inequality: , with . For any an increasing sequence in , we have that there is such that . By the definition of , we get So we have that is an increasing sequence; then is an increasing sequence in . since every increasing sequence in has an upper bound in , we have that there is such that . Since , we have . Hence every increasing sequence in has an upper bound in .
Then applying Theorem 13, it implies that the operator has a generalized point; that is, there are such that . Since the operator is order-increasing, there are and such that .
From assumption of this theorem, if , we can get Since , we have that is a maximal element of . This together with implies that is a maximal element of . It implies that for any , , the following inequality is established This shows that is a generalized Nash equilibrium in the person noncooperative game .

4. Conclusion

Incomplete preference is more general order relation than complete preference in the field of economic management, because restriction on order relation is eased. So it is more consistent with the reality of economic management phenomenons. The generalized game model under the incomplete preference can play an important application in economic management problems. Generalized game plays an important role to prove existence of general equilibrium. But many economic problems ultimately come down to nonlinear problems which are denoted by the utility function without preference in an order infinite dimensional space. The traditional general game model cannot deal with the problems such as the utility function without preference, incomplete preference, order infinite dimension space, or nonlinear problem. Now since there are no ready-made methods to deal with the problems, new research methods must be sought. The generalized game model under the incomplete preference, which is proposed in this paper, can deal with the problems. But in this paper, the research is limited to the existence of the equilibrium. The stability of equilibrium and new game model which are more close to the reality are our next step research direction.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The author was supported by the National Social Science Fund of China (13CJL006), National Science Foundation for Post-doctoral Scientists of China (2014M551264), and Research Fund for the Doctoral Program of Changshu Institute of Technology (XE1509).

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Copyright © 2018 Xingchang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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