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We will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally nonconstant payoff functions. The proof is based on the existence of approximate Nash equilibria which is proved by Sperner's lemma. We follow the Bishop-style constructive mathematics.

It is often said that Brouwer’s fixed-point theorem cannot be constructively proved.

Refernce [

The existence of a Nash equilibrium in a finite strategic game also cannot be constructively proved. Sperner’s lemma which is used to prove Brouwer’s theorem, however, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer’s theorem using Sperner’s lemma. See [

In this paper, we present a proof of the existence of a Nash equilibrium in a finite strategic game with sequentially locally nonconstant payoff functions. In the next section, we present Sperner’s lemma for an

Let

Partition and labeling of 2-dimensional simplex.

Let

The vertices of

If a vertex of

If a vertex of

A vertex contained inside of

A small simplex of

If one labels the vertices of

About constructive proofs of Sperner’s lemma, see [

Since

Let

The definition of local nonconstancy of functions is as follows.

(1) At a point

(2) In any open set in

Next, by reference to the notion of

First, we recapitulate the compactness (total boundedness with completeness) of a set in constructive mathematics.

For each

The definition of sequential local nonconstancy is as follows.

There exists

We show the following lemma.

Let

Choose a sequence

Now, we look at the problem of the existence of a Nash equilibrium in a finite strategic game according to [

Consider an

For each

Let us consider a homeomorphism between an

Homeomorphism between simplex and combination of strategies.

Next, we assume the following condition.

There exists

By the sequential local nonconstancy of payoff functions we obtain the following result.

For each

Thus,

Let us replace

In any finite strategic game with sequentially locally nonconstant payoff functions, there exists a Nash equilibrium.

Let us prove this theorem through some steps.

(1) First we show that we can partition an

For example, let

Consider the case where

Next, consider the case where

Therefore, the conditions for Sperner’s lemma are satisfied, and there exists an odd number of fully labeled simplices in

(2) Suppose that we partition

(3) Since, by Lemma

For each

Assume that the set

(4) Denote one of the points which satisfy

Since

Consider two examples. See a game in Table

Example of game 1.

Player 2 | |||
---|---|---|---|

X | Y | ||

Player 1 | X | 2, 2 | 0, 3 |

Y | 3, 0 | 1, 1 |

Consider two sequences of

Consider two sequences of

Therefore, the payoff functions are sequentially locally nonconstant.

Let us consider another example. See a game in Table

when

when

when

when

Example of game 2.

Player 2 | |||
---|---|---|---|

X | Y | ||

Player 1 | X | 2, 1 | 0, 0 |

Y | 0, 0 | 1, 2 |

Consider sequences

When

If

When

If

When

When

When

The payoff functions are sequentially locally nonconstant.

In this paper, we have presented a constructive procedure to prove the existence of Nash equilibrium in finite strategic games from the viewpoint of constructive mathematics á la Bishop, that is, mathematics based on intuitionistic logic. As a future research program, we are studying the following themes:

an application of the method of this paper to economic theory, in particular, the problem of the existence of an equilibrium in competitive economy with excess demand functions which have the property that is similar to sequential local nonconstancy;

a generalization of the result of this paper to Kakutani’s fixed-point theorem for multivalued functions with property of sequential local nonconstancy and its application to economic theory.

For other researches about computability of Nash equilibrium, see [

This work was supported in part by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (C) no. 20530165.