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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.

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 [

When

A strategy profile

Nash [

Accordingly, many economists continually strive to seek to weaken the continuity and quasi-concavity of payoff functions. Dasgupta and Maskin [

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

Throughout this work, all topological spaces are assumed to be Hausdorff. Let A be a subset of a topological space X. We denote by

In this section, we introduce the notion called generalized convex game which is a natural extension of the convex game of Reny [

Let

If for each

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.

For each

Let

Motivated by the proof of Corollary 2.2 of Guillerme [

Let

A game

If

Let

Let

Assume, by way of contradiction, that

Since

We first show that

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

Since

By the generalized convexity condition, for each

Take an arbitrary point

Obviously,

For each

We show that

Indeed, if

Let

Since

Obviously,

Pick up an element

From Theorem

Let

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.

Consider the game

Now we check the generalized convexity of the game. Let

Let

On the other hand, we show that the game is not convex. Indeed, if we pick up a point

Throughout this section, we assume that the strategy spaces for all players are the same. As such, let

The following notion of a diagonally point secure game was introduced in Reny [

A quasi-symmetric game

Let

For each

Since

We first show that

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

Since

By the generalized convexity condition, there exists a continuous mapping

Define a mapping

Obviously,

We take an arbitrary finite subset

We show that

Indeed, if

Obviously,

Pick up an element

From Theorem

Let

Bay et al. [

Let

A game

A game

When

Let

For any

We show firstly that

Now we show that

Let

Let

Indeed, if

In Theorem

No data were used to support this study.

The author declares that they have no conflicts of interest.

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).