#### Abstract

We provide several results on the existence of equilibria for discontinuous games in general topological spaces without any convexity structure. All of the theorems yielding existence of equilibria here are stated in terms of the player’s preference relations over joint strategies.

#### 1. Introduction

Nash equilibrium is a fundamental concept in the theory of games and the most widely used method of predicting the outcome of a strategic interaction in almost all areas of economics as well as in business and other social sciences.

Following Reny [1] and Tian [2], a game is simply a family of ordered tuples , where is a finite or infinite (countable or uncountable) set of players, and, for each , is the set of strategies of player , and is a binary relation on .

When can be represented by a payoff function , the game introduced by Nash in [3] is a special case of .

A strategy profile is a pure strategy Nash equilibrium of a game if for all and .

Nash [3] proved that an (Nash) equilibrium of the game exists if the set of pure strategies of player is a compact convex subset of an Euclidean space, and if payoff function of player is continuous and (quasi-)concave in , for each . However, it is known that many important games frequently exhibit discontinuities or non-quasi-concavity in payoffs, such as those in [4, 5]. Also, many economic models do not have convex strategy spaces, so payoff functions under consideration do not have any form of quasi-concavity.

Accordingly, many economists continually strive to seek to weaken the continuity and quasi-concavity of payoff functions. Dasgupta and Maskin [5], Reny [6], Nessah [7], Nessah and Tian [8], and others established the existence of pure strategy Nash equilibrium for discontinuous, compact, and quasi-concave games. Baye et al. [4], Yu [9], Tan et al. [10], Zhang [11], Lignola [12], Nessah and Tian [13, 14], Kim and Lee [15], Hou [16], Chang [17], and Tian [10] and others investigated the existence of pure strategy Nash equilibrium for discontinuous and/or non-quasi-concave games with finite or countable players by using the approach to consider a mapping of individual payoffs into an aggregator function (the aggregator function is defined by for each ), which is pioneered by Nikaido and Isoda [18]. To use these results, one must analyze the aggregator function. Such an analysis involves a high dimension and is hard to check in a particular game. Also, it was already indicated in [4, 19] that the quasi-concavity of individual payoffs is not sufficient to establish these concavities that appeared in [4, 9–12, 15–17] for the aggregator function. In addition, to use the method of [18], the countability of amount of players in the game considered is needed.

In this paper, we firstly establish a new existence result of Nash equilibria for discontinuous games in general topological spaces with binary relations. Then, we give some results on the existence of symmetric Nash equilibria and dominant strategy equilibria in general topological spaces without any convexity structure (geometrical or abstract). All of the theorems yielding existence of equilibria here are stated in terms of the players preference relations over joint strategies. It should be emphasized that the method we use is different in essence from those methods given in all results mentioned above.

The paper proceeds as follows. Section 2 provides some notations. Section 3 provides a new notion called generalized convex game and our main result, Theorem 8, as well as an example which holds our assumptions, but the old ones do not hold. Section 4 provides a theorem which is a generalization of Proposition 5.2 of Reny [1] to general topological spaces. Section 5 provides a new notion called generalized uniformly quasi-concavity which is a natural extension of the uniformly transfer quasi-concavity introduced by Bay et al. [4] to topological spaces, and a characterization of dominant strategy equilibrium for games in general topological spaces.

#### 2. Preliminaries

Throughout this work, all topological spaces are assumed to be Hausdorff. Let A be a subset of a topological space X. We denote by the closure of in . If A is a subset of a vector space, we denote by the convex hull of . We use to denote the set of all real numbers, to denote the dimensional Euclidean space, to denote the standard -dimensional simplex in , and for to denote the standard base in . Let be the set of players that is either finite or infinite (even uncountable). Each player ’s strategy space is a general topological space without any convexity (geometric or abstract). Denote by the Cartesian product of all ’s equipped with the product topology, which is the set of strategy profiles. For each player , denote by all other players rather than player . Also denote by the Cartesian product of the sets of strategies of players with , and we sample write for the set consisting of a single point . Let denote the asymmetric part of , i.e., if and only if but not .

#### 3. Existence of Nash Equilibrium for Generalized Convex Game with Single Player Deviation Property

In this section, we introduce the notion called generalized convex game which is a natural extension of the convex game of Reny [1] to topological spaces and is unrelated to the diagonal transfer quasi-concavity of Baye et al. [4], the concavity of Kim and Lee [15], the quasi-concavity of Hou [16], the 0-pair-concavity of Chang [17], and the 0-diagonal quasi-concavity that appeared in [5, 10–12] and establish an existence result of a pure strategy Nash equilibrium for noncooperative games in topological spaces.

*Definition 1 (see [1]). *Let be a convex subset of a topological vector space for each . If for each and each , is a convex set, then the game is said to be convex.

*Definition 2. *If for each and each finite subset of there exists a continuous mapping such that for any , where , then one says that the game * is generalized convex*.

For the generalized convexity, we have the following proposition which shows that the generalized convexity is a natural extension of Reny’s convexity to topological spaces without any convexity structure.

Proposition 3. *For each , let be a convex subset of a topological vector space and be complete and transitive. If the game is convex, then it is generalized convex.*

*Proof. *Let and be a finite subset of . Define a mapping by for each . Obviously, is continuous. For any , let Let and . Since is complete and transitive, there exists such that for all . Since is convex, one has that . Therefore, for any , one has for all .

Motivated by the proof of Corollary 2.2 of Guillerme [20], we have the following Lemma which, albeit simple, provides seemingly a new approach on the investigation for the existence of Nash equilibria.

Lemma 4. *Let be a game. Then has a Nash equilibrium if and only if , where .*

*Proof. **Sufficiency*. Since , we pick up an element . Then, for each and , we have , and thus .*Necessity.* Suppose that is a Nash equilibrium point of . Then for each and , we have , and thus .

*Definition 5. *A game is said to have -single player deviation property if, whenever is not a Nash equilibrium, there exist player , , and a neighborhood of such that for each , where is a subset of .

*Remark 6. *If , then the -single player deviation property is the single player deviation property due to Prokopovych [21, pp. 387] (see also Remark 4 of Reny [1] and Definition 3.2 of Nessah and Tian [13] where it was called weak transfer continuity). The single player deviation property holds in a large class of discontinuous games and is often quite simple to check in a particular game.

Lemma 7 (see [22]). *Let and be a natural number for each . For each , let be a closed set-valued map, where denotes the standard basis of . If, for any and any finite subset of , one has , then *

Theorem 8. *Let have -single player deviation property such that is compact, where is a nonempty finite subset of and is a nonempty finite subset of for each . Suppose that, for each , is complete and transitive. If is generalized convex, then has a Nash equilibrium.*

*Proof. *Assume, by way of contradiction, that has no Nash equilibrium in pure strategies. For each and each , we use to denote the set We show that which means that has Nash equilibrium by Lemma 4, a contradiction.

Since is compact, it is immediate to verify that is compact.

We first show that Obviously, Now we show that If not, then there exist a an , and an such that . By Lemma 4, it follows that is not a Nash equilibrium. Clearly, Since has -single player deviation property, there exist a , an , and a neighborhood of such that for each . It follows that This contradicts (9), and so we have that In order to complete the proof, we only need to show that

Since is compact, we only need to show that the family has the finite intersection property. Toward this end, let be an arbitrary finite subset of , , and be a finite subset of for each .

By the generalized convexity condition, for each , there exists a continuous mapping such that, for any , one has that for each , where

Take an arbitrary point . Define a mapping as follows: for each .

Obviously, is a continuous mapping from into .

For each , we take an arbitrary finite subset of .

We show that Indeed, if , then there exists an such that for each . Thus for each . Particularly, for each .

Let . Then for each , and , by the definition of .

Since for each , one has that for each . We show that . If not, then , where for each and . By (14), where

Obviously, . By (17), This is impossible. Therefore, for any and any subset of , we have that By using Lemma 7 with for each and , we have that Pick up an element Then This completes the proof of the theorem.

From Theorem 8, we obtain immediately the following corollary which improves and generalizes Theorem 3.2 of [13], Corollary 2.2 of [20], and Corollary 3.1 of [7].

Corollary 9. *Let be compact and have the single player deviation property such that is complete and transitive. If is generalized convex, then has a Nash equilibrium.*

Now, we give an example of problem of existence of pure strategy Nash equilibrium for discontinuous games, which holds our assumptions, but the old ones do not hold.

*Example 10. *Consider the game that consists of two players where and are the closed intervals and , respectively, and the player ’s preference relations are defined as follows: for each . We firstly show the single player deviation property of the game. If is not a Nash equilibrium, then there is an and an such that . If , then it is obvious that there is a neighborhood of such that for each . If , then . Let . Then is a neighborhood of , and thus for each ; it implies that for all .

Now we check the generalized convexity of the game. Let be a finite subset of . Define a mapping by for each . Clearly, is continuous, and = + = + + = = , and thus , for some and all , where .

Let be a finite subset of . Define a mapping by for each . Clearly, is continuous. Let and . Without loss the generality, we assume that for each . If for all , then , and thus If there exists a such that , then . Therefore, the game is generalized convex. By Theorem 8, the game has a Nash equilibrium. In fact, the strategy profile is a Nash equilibrium of the game.

On the other hand, we show that the game is not convex. Indeed, if we pick up a point , and take and , then . Obviously, and . Therefore, the set is not convex. So the game is not convex.

#### 4. Existence of Symmetric Pure Strategy Nash Equilibria

Throughout this section, we assume that the strategy spaces for all players are the same. As such, let for each , and let denote the strategy profile in which player chooses and every other player chooses . If, in addition, for every pair of players and , if and only if , then we say that is a quasi-symmetric game. A Nash equilibrium of a game is said to be symmetric iff for each . For each , we use to denote the strategy profiles in which for all .

The following notion of a diagonally point secure game was introduced in Reny [1, Definition 5.1]. Let be a finite set of players. A quasi-symmetric convex game is diagonally point secure if whenever is not a Nash equilibrium, there is a point and a neighborhood of such that for every , for every . To show that for every is excessive, we introduce the following definition.

*Definition 11. *A quasi-symmetric game is said to have the -diagonal deviation property if, whenever is not a Nash equilibrium, there is a point and a neighborhood of such that, for every , , where is a subset of .

Theorem 12. *Let be a quasi-symmetric game and have -diagonal deviation property such that is compact for some nonempty finite subset of and is complete and transitive. If is generalized convex, then has a symmetric Nash equilibrium.*

*Proof. *For each , we use to denote the set We show that which means that has a symmetric Nash equilibrium.

Since is compact, it is obvious that is compact.

We first show that Obviously, Now we show that If not, then there exists a such that . We show that . If not, then there exist such that . Since is Hausdorff, there exist a neighborhood of and a neighborhood of such that . Obviously, is a neighborhood of with , which contradicts (27). It is easy to see that is not a Nash equilibrium and belongs to . By the -diagonal deviation property, there exist an and a neighborhood of such that for each . It follows that , i.e., This contradicts (27), and so

In order to complete the proof, we only need to show that

Since is compact, we only need to show that the family has the finite intersection property. Toward this end, let be a finite subset of .

By the generalized convexity condition, there exists a continuous mapping such that, for any , one has the fact that for each , where

Define a mapping as follows: , for each .

Obviously, is a continuous mapping from into .

We take an arbitrary finite subset of .

We show that

Indeed, if , then for each . Particularly, for each , and thus For each . We show that . If not, then , where for each and . By (28), , where

Obviously, . Since is complete and transitive, we have This is impossible. Therefore, for any subset of , we have that By the classic KKM theorem, we have that

Pick up an element Then This completes the proof of the theorem.

From Theorem 12, we obtain immediately the following corollary which improves and generalizes Proposition 5.2 of Reny [1].

Corollary 13. *Let be a quasi-symmetric compact game and have the -diagonal deviation property with . Suppose that is complete and transitive. If is generalized convex, then has a symmetric Nash equilibrium.*

#### 5. Existence of Dominant Strategy Equilibria

Bay et al. [4] gave a complete characterization for the existence of dominant strategy equilibrium in games with the set of pure strategies of player being a compact convex subset of a topological vector space, by introducing a concavity notion called uniformly transfer quasi-concavity.

Let be a game. A point is said to be a* dominant strategy equilibrium* if, for all , for all .

*Definition 14. *A game is said to be* generalized uniformly quasi-concave* if, for any and any finite subset of , there exists a continuous mapping such that, for any , for all , where

*Definition 15. *A game is said to be* transfer upper semicontinuous * if, for every , , and , implies that there exists a point and a neighborhood of such that for all .

*Remark 16. *When is a convex subset of a topological vector space and can be represented by a payoff function , Definition 15 is due to Bay et al. [4].

Theorem 17. *Let be a transfer upper semicontinuous game with being complete and transitive. Suppose for each there exists a finite subset of such that is compact where . Then has a dominant strategy equilibrium if and only if is generalized uniformly quasi-concave.*

*Proof. **Necessity*. Suppose that the game has a dominant strategy equilibrium . We want to show that is generalized uniformly quasi-concave. Let and be an arbitrary finite subset of . Define a mapping as follows:for each . Obviously, is continuous.

For any , For all , where .*Sufficiency.* Let . Define a mapping as follows: for each .

We show firstly that . To this end, we only need to show . Indeed, if , then there exists an such that . By the definition of , one has that . By the transfer upper semicontinuity, there exists an and a neighborhood of such that for all . So, . Therefore, . Thus Now we show that Toward this end, we firstly show that the family has the finite intersection property.

Let be an arbitrary finite subset of . By the generalized uniform quasi-concavity, there exists a continuous mapping such that, for any , one has that for all , where

Let be an arbitrary subset of . We show that Indeed, if , then . So, , i.e., for each , one has that . By the definition of , we have that for all . Now we demonstrate that . If not, then . By (37), one has that for all . This contradicts (35). Therefore, . By the classic KKM theorem, Pick up an element . Then . Therefore, . Therefore, the family has the finite intersection property. Particularly, the family has the finite intersection property. By the supposition, the set is compact. So, By (34), we have that . Take an element . It is easy to verify that is a dominant strategy equilibrium of .

*Remark 18. *In Theorem 17, the compactness of implies that is compact. So, Theorem 17 extends Theorems 4 and 5 of Bay et al. [4] to general topological spaces without any convexity (geometrical or abstract) structure.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that they have no conflicts of interest.

#### Acknowledgments

This work was supported by Qin Xin Talents Cultivation Program (no. QXTCP A201702) of Beijing Information Science and Technology University and the National Natural Science Foundation of China (NSFC-11271178).