International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 459459 | 8 pages | https://doi.org/10.5402/2012/459459

Constructive Proof of the Existence of Nash Equilibrium in a Finite Strategic Game with Sequentially Locally Nonconstant Payoff Functions

Academic Editor: T. Karakasidis
Received05 Aug 2011
Accepted19 Sep 2011
Published24 Nov 2011

Abstract

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.

1. Introduction

It is often said that Brouwerโ€™s fixed-point theorem cannot be constructively proved.

Refernce [1] provided a constructive proof of Brouwerโ€™s fixed-point theorem. But it is not constructive from the viewpoint of constructive mathematics รก la Bishop. It is sufficient to say that one-dimensional case of Brouwerโ€™s fixed-point theorem, that is, the intermediate value theorem, is nonconstructive. See [2] or [3]. On the other hand, in [4] Orevkov constructed a computably coded continuous function ๐‘“ from the unit square to itself, which is defined at each computable point of the square, such that ๐‘“ has no computable fixed point. His map consists of a retract of the computable elements of the square to its boundary followed by a rotation of the boundary of the square. As pointed out by Hirst in [5], since there is no retract of the square to its boundary, Orevkovโ€™s map does not have a total extension.

The existence of a Nash equilibrium in a finite strategic game also cannot be constructively proved. Spernerโ€™s lemma which is used to prove Brouwerโ€™s theorem, however, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwerโ€™s theorem using Spernerโ€™s lemma. See [3, 6]. Thus, Brouwerโ€™s fixed-point theorem can be constructively proved in its constructive version. Also van Dalen in [3] states a conjecture that a uniformly continuous function ๐‘“ from a simplex to itself, with property that each open set contains a point ๐‘ฅ such that ๐‘ฅโ‰ ๐‘“(๐‘ฅ), which means |๐‘ฅโˆ’๐‘“(๐‘ฅ)|>0, and also at every point ๐‘ฅ on the boundaries of the simplex ๐‘ฅโ‰ ๐‘“(๐‘ฅ), has an exact fixed point. We call such a property of functions local nonconstancy. Further, we define a stronger property sequential local nonconstancy. In another paper [7], we have constructively proved Dalenโ€™s conjecture with sequential local nonconstancy.

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 Spernerโ€™s lemma for an ๐‘›-dimensional simplex whose constructive proof is omitted indicating references. In Section 3, we present a proof of 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 Spernerโ€™s lemma. We follow the Bishop-style constructive mathematics according to [2, 8, 9].

2. Spernerโ€™s Lemma

Let ฮ” denote an ๐‘›-dimensional simplex. ๐‘› is a finite natural number. For example, a 2-dimensional simplex is a triangle. Let us partition or triangulate the simplex. Figure 1 is an example of partitioning (triangulation) a 2-dimensional simplex. In a 2-dimensional case, we divide each side of ฮ” in ๐‘š equal segments and draw the lines parallel to the sides of ฮ”. ๐‘š is a finite natural number. Then, the 2-dimensional simplex is partitioned into ๐‘š2 triangles. We consider partition of ฮ” inductively for cases of higher dimension. In a 3-dimensional case, each face of ฮ” is a 2-dimensional simplex, and so it is partitioned into ๐‘š2 triangles in the above-mentioned way, and draw the planes parallel to the faces of ฮ”. Then, the 3-dimensional simplex is partitioned into ๐‘š3 trigonal pyramids. And this is similar to cases of higher dimension.

Let ๐พ denote the set of small ๐‘›-dimensional simplices of ฮ” constructed by partition. Vertices of these small simplices of ๐พ are labeled with the numbers 0,1,2,โ€ฆ,๐‘› subject to the following rules. (1)The vertices of ฮ” are, respectively, labeled with 0 to ๐‘›. We label a point (1,0,โ€ฆ,0) with 0, a point (0,1,0,โ€ฆ,0) with 1, a point (0,0,1โ€ฆ,0) with 2,โ€ฆ, and a point (0,โ€ฆ,0,1) with ๐‘›. That is, a vertex whose ๐‘˜th coordinate (๐‘˜=0,1,โ€ฆ,๐‘›) is 1 and all other coordinates are 0 is labeled with ๐‘˜.(2)If a vertex of ๐พ is contained in an ๐‘›โˆ’1-dimensional face of ฮ”, then this vertex is labeled with some number which is the same as the number of one of the vertices of that face.(3)If a vertex of ๐พ is contained in an ๐‘›โˆ’2-dimensional face of ฮ”, then this vertex is labeled with some number which is the same as the number of one of the vertices of that face, and so on for cases of lower dimension.(4)A vertex contained inside of ฮ” is labeled with an arbitrary number among 0,1,โ€ฆ,๐‘›.

A small simplex of ๐พ which is labeled with the numbers 0,1,โ€ฆ,๐‘› is called a fully labeled simplex. Spernerโ€™s lemma is stated as follows.

Lemma 1 (Spernerโ€™s lemma). If one labels the vertices of ๐พ following the rules (1)~(4), then there are an odd number of fully labeled simplices, and so there exists at least one fully labeled simplex.

Proof. About constructive proofs of Spernerโ€™s lemma, see [10] or [11].

Since ๐‘› and partition of ฮ” are finite, the number of small simplices constructed by partition is also finite. Thus, we can constructively find a fully labeled ๐‘›-dimensional simplex of ๐พ through finite steps.

3. Nash Equilibrium in Strategic Game

Let ๐ฉ=(๐ฉ0,๐ฉ1,โ€ฆ,๐ฉ๐‘›) be a point in an ๐‘›-dimensional simplex ฮ”, and consider a function ๐œ‘ from ฮ” to itself. Denote the ๐‘–th components of ๐ฉ and ๐œ‘(๐ฉ) by ๐ฉ๐‘– and ๐œ‘๐‘–(๐ฉ) or ๐œ‘๐‘–.

The definition of local nonconstancy of functions is as follows.

Definition 2 (local nonconstancy of functions). (1) At a point ๐ฉ on the faces (boundaries) of a simplex ๐œ‘(๐ฉ)โ‰ ๐ฉ, this means that ๐œ‘๐‘–(๐ฉ)>๐ฉ๐‘– or ๐œ‘๐‘–(๐ฉ)<๐ฉ๐‘– for at least one ๐‘–.
(2) In any open set in ฮ”, there exists a point ๐ฉ such that ๐œ‘(๐ฉ)โ‰ ๐ฉ.

Next, by reference to the notion of sequentially at most one maximum in [12], we define the property of sequential local nonconstancy.

First, we recapitulate the compactness (total boundedness with completeness) of a set in constructive mathematics. ฮ” is compact in the sense that for each ๐œ€>0, there exists a finitely enumerable ๐œ€-approximation to ฮ” (a set ๐‘† is finitely enumerable if there exist a natural number ๐‘ and a mapping of the set {1,2,โ€ฆ,๐‘} onto ๐‘†). An ๐œ€-approximation to ฮ” is a subset of ฮ” such that for each ๐ฉโˆˆฮ”, there exists ๐ช in that ๐œ€-approximation with |๐ฉโˆ’๐ช|<๐œ€. Each face (boundary) of ฮ” is also a simplex, and so it is compact. According to Corollaryโ€‰โ€‰2.2.12 of [9], we have the following result.

Lemma 3. For each ๐œ€>0, there exist totally bounded sets ๐ป1,๐ป2,โ€ฆ,๐ป๐‘›, each of diameter less than or equal to ๐œ€, such that โ‹ƒฮ”=๐‘›๐‘–=1๐ป๐‘–.

The definition of sequential local nonconstancy is as follows.

Definition 4 (sequential local nonconstancy of functions). There exists ๐œ€ with the following property. For each ๐œ€>0 less than ๐œ€, there exist totally bounded sets ๐ป1,๐ป2,โ€ฆ,๐ป๐‘š, each of diameter less than or equal to ๐œ€, such that โ‹ƒฮ”=๐‘š๐‘–=1๐ป๐‘–, and if for all sequences (๐ฉ๐‘›)๐‘›โ‰ฅ1, (๐ช๐‘›)๐‘›โ‰ฅ1 in each ๐ป๐‘–, |๐œ‘(๐ฉ๐‘›)โˆ’๐ฉ๐‘›|โ†’0 and |๐œ‘(๐ช๐‘›)โˆ’๐ช๐‘›|โ†’0, then |๐ฉ๐‘›โˆ’๐ช๐‘›|โ†’0.

We show the following lemma.

Lemma 5. Let ๐œ‘ be a uniformly continuous and sequentially locally nonconstant function from ฮ” to itself. Assume that inf๐ฉโˆˆ๐ป๐‘–๐œ‘(๐ฉ)=0 for ๐ป๐‘–โŠ‚ฮ” defined above. If the following property holds: for each ๐œ€>0, there exists ๐›ฟ>0 such that if ๐ฉ,๐ชโˆˆ๐ป๐‘–, |๐œ‘(๐ฉ)โˆ’๐ฉ|<๐›ฟ, and |๐œ‘(๐ช)โˆ’๐ช|<๐›ฟ, then |๐ฉโˆ’๐ช|โ‰ค๐œ€, then there exists a point ๐ซโˆˆ๐ป๐‘– such that ๐œ‘(๐ซ)=๐ซ.

Proof. Choose a sequence (๐ฉ๐‘›)๐‘›โ‰ฅ1 in ๐ป๐‘– such that |๐œ‘(๐ฉ๐‘›)โˆ’๐ฉ๐‘›|โ†’0. Compute ๐‘ such that |๐œ‘(๐ฉ๐‘›)โˆ’๐ฉ๐‘›|<๐›ฟ for all ๐‘›โ‰ฅ๐‘. Then, for ๐‘š,๐‘›โ‰ฅ๐‘, we have |๐ฉ๐‘šโˆ’๐ฉ๐‘›|โ‰ค๐œ€. Since ๐œ€>0 is arbitrary, (๐ฉ๐‘›)๐‘›โ‰ฅ1 is a Cauchy sequence in ๐ป๐‘– and converges to a limit ๐ซโˆˆ๐ป๐‘–. The continuity of ๐œ‘ yields |๐œ‘(๐ซ)โˆ’๐ซ|=0, that is, ๐œ‘(๐ซ)=๐ซ.

Now, we look at the problem of the existence of a Nash equilibrium in a finite strategic game according to [13]. A Nash equilibrium of a finite strategic game is a state where all players choose their best responses to strategies of other players.

Consider an ๐‘›-players strategic game with ๐‘š pure strategies for each player. ๐‘› and ๐‘š are finite natural numbers not smaller than 2. Let ๐‘†๐‘– be the set of pure strategies of player ๐‘–, and denote each of his pure strategies by ๐‘ ๐‘–๐‘—. His mixed strategy is defined as a probability distribution over ๐‘†๐‘– and is denoted by ๐ฉ๐‘–. Let ๐‘๐‘–๐‘— be a probability that player ๐‘– chooses ๐‘ ๐‘–๐‘—, then we must have โˆ‘๐‘š๐‘—=1๐‘๐‘–๐‘—=1 for all ๐‘–. A combination of mixed strategies of all players is called a profile. It is denoted by ๐ฉ. Let ๐œ‹๐‘–(๐ฉ) be the expected payoff of player ๐‘– at profile ๐ฉ, and let ๐œ‹๐‘–(๐‘ ๐‘–๐‘—,๐ฉโˆ’๐‘–) be his payoff when he chooses a strategy ๐‘ ๐‘–๐‘— at that profile, where ๐ฉโˆ’๐‘– denotes a combination of mixed strategies of players other than ๐‘– at profile ๐ฉ. ๐œ‹๐‘–(๐ฉ) is written as follows:๐œ‹๐‘–(๐ฉ)=๐œ‹๐‘–๎€ท๐ฉ๐‘–,๐ฉโˆ’๐‘–๎€ธ=๎“{๐‘—โˆถ๐‘๐‘–๐‘—>0}๐‘๐‘–๐‘—๐œ‹๐‘–๎€ท๐‘ ๐‘–๐‘—,๐ฉโˆ’๐‘–๎€ธ.(1) Assume that the values of payoffs of all players are finite, then since pure strategies are finite, and expected payoffs are linear functions about probability distributions over the sets of pure strategies of all players, ๐œ‹๐‘–(๐ฉ) is uniformly continuous about ๐ฉ.

For each ๐‘– and ๐‘—, let๐‘ฃ๐‘–๐‘—=๐‘๐‘–๐‘—๎€ท๐œ‹+max๐‘–๎€ท๐‘ ๐‘–๐‘—,๐ฉโˆ’๐‘–๎€ธโˆ’๐œ‹๐‘–๎€ธ,(๐ฉ),0(2) and define the following function:๐œ“๐‘–๐‘—๐‘ฃ(๐ฉ)=๐‘–๐‘—๐‘ฃ๐‘–1+๐‘ฃ๐‘–2+โ‹ฏ+๐‘ฃ๐‘–๐‘š,(3) where โˆ‘๐‘š๐‘—=1๐œ“๐‘–๐‘—=1 for all ๐‘–. Let ๐œ“๐‘–(๐ฉ)=(๐œ“๐‘–1,๐œ“๐‘–2,โ€ฆ,๐œ“๐‘–๐‘š), ๐œ“(๐ฉ)=(๐œ“1,๐œ“2,โ€ฆ,๐œ“๐‘›). Since each ๐œ“๐‘– is an ๐‘š-dimensional vector such that the values of its components are between 0 and 1, and the sum of its components is 1, it represents a point on an ๐‘šโˆ’1-dimensional simplex. ๐œ“(๐ฉ) is a combination of vectors ๐œ“๐‘–โ€™s. It is a vector, such that its components are components of ๐œ“๐‘–(๐ฉ) for all players. Thus, it is a vector with ๐‘›ร—๐‘š components, but since the number of independent components is ๐‘›(๐‘šโˆ’1), the range of ๐œ“ is the ๐‘›-times product of ๐‘šโˆ’1-dimensional simplices. It is convex, and homeomorphic to an ๐‘›(๐‘šโˆ’1)-dimensional simplex. ๐ฉ=(๐ฉ1,๐ฉ2,โ€ฆ,๐ฉ๐‘›) is also a vector with ๐‘›ร—๐‘š components, and the number of its independent components is ๐‘›(๐‘šโˆ’1).

Let us consider a homeomorphism between an ๐‘›(๐‘šโˆ’1)-dimensional simplex and the space of playersโ€™ mixed strategies which is denoted by ๐. Figure 2 depicts an example of a case of two players with two pure strategies for each player. Vertices ๐ท, ๐ธ, ๐น, and G represent states where two players choose pure strategies, and points on edges ๐ท๐ธ, ๐ธ๐น, ๐น๐บ, and ๐บ๐ท represent states where one player chooses a pure strategy. Vertices of the simplex and points on faces (simplices whose dimension is lower than ๐‘›(๐‘šโˆ’1)) of the simplex correspond to the points on faces of ๐. For example, in Figure 2, A, B and C correspond, respectively, to ๐ผ, ๐ฝ, and ๐ป. On the other hand, each vertex of ๐, ๐ท, ๐ธ, ๐น, and ๐บ corresponds, respectively, to itself on a face of the simplex which contains it.

Next, we assume the following condition.

Definition 6 (sequential local nonconstancy of payoff functions). There exists ๐œ€ with the following property. For each ๐œ€>0 less than ๐œ€, there exist totally bounded sets ๐ป1,๐ป2,โ€ฆ,๐ป๐‘š, each of diameter less than or equal to ๐œ€, such that โ‹ƒ๐=๐‘š๐‘–=1๐ป๐‘–, and if for all sequences (๐ฉ๐‘›)๐‘›โ‰ฅ1, (๐ช๐‘›)๐‘›โ‰ฅ1 in each ๐ป๐‘–, max(๐œ‹๐‘–(๐‘ ๐‘–๐‘—,(๐ฉ๐‘›)โˆ’๐‘–)โˆ’๐œ‹๐‘–(๐ฉ๐‘›),0)โ†’0, max(๐œ‹๐‘–(๐‘ ๐‘–๐‘—,(๐ช๐‘›)โˆ’๐‘–)โˆ’๐œ‹๐‘–(๐ช๐‘›),0)โ†’0 for all ๐‘ ๐‘–๐‘—โˆˆ๐‘†๐‘– for all ๐‘–, then |๐ฉ๐‘›โˆ’๐ช๐‘›|โ†’0.

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

For each ๐œ€>0 less than ๐œ€, there exist totally bounded sets ๐ป1,๐ป2,โ€ฆ,๐ป๐‘š, each of diameter less than or equal to ๐œ€, such that โ‹ƒ๐=๐‘š๐‘–=1๐ป๐‘–, and if for all sequences (๐ฉ๐‘›)๐‘›โ‰ฅ1, (๐ช๐‘›)๐‘›โ‰ฅ1 in each ๐ป๐‘–, |๐œ“(๐ฉ๐‘›)โˆ’๐ฉ๐‘›|โ†’0 and |๐œ“(๐ช๐‘›)โˆ’๐ช๐‘›|โ†’0, then |๐ฉ๐‘›โˆ’๐ช๐‘›|โ†’0.

Thus, ๐œ“ is sequentially locally nonconstant.

Let us replace ๐‘› by ๐‘›(๐‘šโˆ’1). We show the following theorem.

Theorem 7. In any finite strategic game with sequentially locally nonconstant payoff functions, there exists a Nash equilibrium.

Proof. Let us prove this theorem through some steps.
(1) First we show that we can partition an ๐‘›(๐‘šโˆ’1)-dimensional simplex ฮ”, so that the conditions for Spernerโ€™s lemma are satisfied. We partition ฮ” according to the method in the proof of Spernerโ€™s lemma and label the vertices of simplices constructed by partition of ฮ”. It is important how to label the vertices contained in the faces of ฮ”. Let ๐พ be the set of small simplices constructed by partition of ฮ”, let ๐ฉ=(๐ฉ0,๐ฉ1,โ€ฆ,๐ฉ๐‘›(๐‘šโˆ’1)) be a vertex of a simplex of ๐พ, and denote the ๐‘–th coordinate of ๐œ“(๐ฉ) by ๐œ“๐‘– or ๐œ“๐‘–(๐ฉ). We label a vertex ๐ฉ according to the following rule: if๐ฉ๐‘˜+๐œ>๐œ“๐‘˜,welabel๐ฉwith๐‘˜.(4)๐œ is an arbitrary positive number. If there are multiple ๐‘˜โ€™s which satisfy this condition, we label ๐ฉ conveniently for the conditions for Spernerโ€™s lemma to be satisfied.
For example, let ๐ฉ be a point contained in an ๐‘›(๐‘šโˆ’1)โˆ’1-dimensional face of ฮ” such that ๐‘๐‘–=0 for one ๐‘– among 0,1,2,โ€ฆ,๐‘›(๐‘šโˆ’1). With ๐œ>0, we have ๐‘“๐‘–>0 or ๐‘“๐‘–<๐œ (in constructive mathematics for any real number ๐‘ฅ we can not prove that ๐‘ฅโ‰ฅ0 or ๐‘ฅ<0, that, ๐‘ฅ>0, ๐‘ฅ=0, or ๐‘ฅ<0. But for any distinct real numbers ๐‘ฅ, ๐‘ฆ, and ๐‘ง such that ๐‘ฅ>๐‘ง, we can prove that ๐‘ฅ>๐‘ฆ or ๐‘ฆ>๐‘ง). When ๐œ“๐‘–>0, from โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0๐ฉ๐‘—=1, โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0๐œ“๐‘—=1, and ๐ฉ๐‘–=0, ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–๐ฉ๐‘—>๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–๐œ“๐‘—.(5) Then, for at least one ๐‘— (denote it by ๐‘˜), we have ๐ฉ๐‘˜>๐œ“๐‘˜, and we label ๐ฉ with ๐‘˜, where ๐‘˜ is one of the numbers which satisfy ๐ฉ๐‘˜>๐œ“๐‘˜. Since ๐œ“๐‘–>๐ฉ๐‘–=0, ๐‘– does not satisfy this condition. Assume that ๐œ“๐‘–<๐œ. ๐ฉ๐‘–=0 implies โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0,๐‘—โ‰ ๐‘–๐ฉ๐‘—=1. Since โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0,๐‘—โ‰ ๐‘–๐œ“๐‘—โ‰ค1, we obtain ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–๐ฉ๐‘—โ‰ฅ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–๐œ“๐‘—.(6) Then, for a positive number ๐œ, we have ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–๎€ท๐ฉ๐‘—๎€ธ>+๐œ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–๐œ“๐‘—.(7) There is at least one ๐‘—(โ‰ ๐‘–) which satisfies ๐ฉ๐‘—+๐œ>๐œ“๐‘—. Denote it by ๐‘˜, and we label ๐ฉ with ๐‘˜. ๐‘˜ is one of the numbers other than ๐‘– such that ๐ฉ๐‘˜+๐œ>๐œ“๐‘˜ is satisfied. ๐‘– itself satisfies this condition (๐ฉ๐‘–+๐œ>๐œ“๐‘–). But, since there is a number other than ๐‘– which satisfies this condition, we can select a number other than ๐‘–. We have proved that we can label the vertices contained in an ๐‘›(๐‘šโˆ’1)โˆ’1-dimensional face of ฮ” such that ๐ฉ๐‘–=0 for one ๐‘– among 0,1,2,โ€ฆ,๐‘›(๐‘šโˆ’1) with the numbers other than ๐‘–. By similar procedures, we can show that we can label the vertices contained in an ๐‘›(๐‘šโˆ’1)โˆ’2-dimensional face of ฮ” such that ๐ฉ๐‘–=0 for two ๐‘–โ€™s among 0,1,2,โ€ฆ,๐‘›(๐‘šโˆ’1) with the numbers other than those ๐‘–โ€™s, and so on.
Consider the case where ๐ฉ๐‘–=๐ฉ๐‘–+1=0. We see that when ๐œ“๐‘–>0 or ๐œ“๐‘–+1>0, ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐ฉ๐‘—>๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐œ“๐‘—,(8) and so for at least one ๐‘—(denote it by ๐‘˜), we have ๐ฉ๐‘˜>๐œ“๐‘˜, and we label ๐ฉ with ๐‘˜. On the other hand, when ๐œ“๐‘–<๐œ and ๐œ“๐‘–+1<๐œ, we have ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐ฉ๐‘—โ‰ฅ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐œ“๐‘—.(9) Then, for a positive number ๐œ, we have ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๎€ท๐ฉ๐‘—๎€ธ>+๐œ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘–,๐‘–+1๐œ“๐‘—.(10) Thus, there is at least one ๐‘—(โ‰ ๐‘–,๐‘–+1) which satisfies ๐ฉ๐‘—+๐œ>๐œ“๐‘—. Denote it by ๐‘˜, and we label ๐ฉ with ๐‘˜.
Next, consider the case where ๐ฉ๐‘–=0 for all ๐‘– other than ๐‘›(๐‘šโˆ’1). If for some ๐‘–โ€‰โ€‰๐œ“๐‘–>0, then we have ๐ฉ๐‘›(๐‘šโˆ’1)>๐œ“๐‘›(๐‘šโˆ’1) and label ๐ฉ with ๐‘›(๐‘šโˆ’1). On the other hand, if ๐œ“๐‘—<๐œ for all ๐‘—โ‰ ๐‘›(๐‘šโˆ’1), then we obtain ๐ฉ๐‘›(๐‘šโˆ’1)โ‰ฅ๐œ“๐‘›(๐‘šโˆ’1). It implies ๐ฉ๐‘›(๐‘šโˆ’1)+๐œ>๐œ“๐‘›(๐‘šโˆ’1). Thus, we can label ๐ฉ with ๐‘›(๐‘šโˆ’1).
Therefore, the conditions for Spernerโ€™s lemma are satisfied, and there exists an odd number of fully labeled simplices in ๐พ.
(2) Suppose that we partition ฮ” sufficiently fine so that the distance between any pair of the vertices of simplices of ๐พ is sufficiently small. Let ๐›ฟ๐‘›(๐‘šโˆ’1) be a fully labeled ๐‘›(๐‘šโˆ’1)-dimensional simplex of ๐พ, and let ๐ฉ0,๐ฉ1,โ€ฆ and ๐ฉ๐‘›(๐‘šโˆ’1) be the vertices of ๐›ฟ๐‘›(๐‘šโˆ’1). We name these vertices so that ๐ฉ0,๐ฉ1,โ€ฆ,๐ฉ๐‘›(๐‘šโˆ’1) are labeled, respectively, with 0, 1,โ€ฆ, ๐‘›(๐‘šโˆ’1). The values of ๐œ“ at these vertices are ๐œ“(๐ฉ0),๐œ“(๐ฉ1),โ€ฆ and ๐œ“(๐ฉ๐‘›(๐‘šโˆ’1)). The ๐‘—th coordinates of ๐ฉ๐‘– and ๐œ“(๐ฉ๐‘–),๐‘–=0,1,โ€ฆ,๐‘›(๐‘šโˆ’1), are, respectively, denoted by ๐ฉ๐‘–๐‘— and ๐œ“๐‘—(๐ฉ๐‘–). About ๐ฉ0, from the labeling rules, we have ๐ฉ00+๐œ>๐œ“0(๐ฉ0). About ๐ฉ1, also from the labeling rules, we have ๐ฉ11+๐œ>๐œ“1(๐ฉ1). Since ๐‘› and ๐‘š are finite, by the uniform continuity of ๐œ“ there exists ๐›ฟ>0 such that if |๐ฉ๐‘–โˆ’๐ฉ๐‘—|<๐›ฟ, then |๐œ“(๐ฉ๐‘–)โˆ’๐œ“(๐ฉ๐‘—)|<๐œ€/2๐‘›(๐‘šโˆ’1)[๐‘›(๐‘šโˆ’1)+1] for ๐œ€>0 and ๐‘–โ‰ ๐‘—. |๐œ“(๐ฉ0)โˆ’๐œ“(๐ฉ1)|<๐œ€/2๐‘›(๐‘šโˆ’1)[๐‘›(๐‘šโˆ’1)+1] means ๐œ“1(๐ฉ1)>๐œ“1(๐ฉ0)โˆ’๐œ€/2๐‘›(๐‘šโˆ’1)[๐‘›(๐‘šโˆ’1)+1]. On the other hand, |๐ฉ0โˆ’๐ฉ1|<๐›ฟ means that ๐ฉ01>๐ฉ11โˆ’๐›ฟ. We can make ๐›ฟ satisfy ๐›ฟ<๐œ€/2๐‘›(๐‘šโˆ’1)[๐‘›(๐‘šโˆ’1)+1]. Thus, from ๐ฉ01>๐ฉ11โˆ’๐›ฟ,๐ฉ11+๐œ>๐œ“1๎€ท๐ฉ1๎€ธ,๐œ“1๎€ท๐ฉ1๎€ธ>๐œ“1๎€ท๐ฉ0๎€ธโˆ’๐œ€[๐‘›],2๐‘›(๐‘šโˆ’1)(๐‘šโˆ’1)+1(11) we obtain ๐ฉ01>๐œ“1๎€ท๐ฉ0๎€ธ๐œ€โˆ’๐›ฟโˆ’๐œโˆ’[]2๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1>๐œ“1๎€ท๐ฉ0๎€ธโˆ’๐œ€[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1โˆ’๐œ.(12) By similar arguments, for each ๐‘– other than 0, ๐ฉ0๐‘–>๐œ“๐‘–๎€ท๐ฉ0๎€ธโˆ’๐œ€[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1โˆ’๐œ.(13) For ๐‘–=0, we have ๐ฉ00>๐œ“0๎€ท๐ฉ0๎€ธโˆ’๐œ.(14) Adding (13) and (14) side by side except for some ๐‘– (denote it by ๐‘˜) other than 0, ๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘˜๐ฉ0๐‘—>๐‘›(๐‘šโˆ’1)๎“๐‘—=0,๐‘—โ‰ ๐‘˜๐œ“๐‘—๎€ท๐ฉ0๎€ธโˆ’[๐‘›]๐œ€(๐‘šโˆ’1)โˆ’1[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1โˆ’๐‘›(๐‘šโˆ’1)๐œ.(15) From โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0๐ฉ0๐‘—=1, โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0๐œ“๐‘—(๐ฉ0)=1, we have 1โˆ’๐ฉ0๐‘˜>1โˆ’๐œ“๐‘˜(๐ฉ0)โˆ’([๐‘›(๐‘šโˆ’1)โˆ’1]๐œ€/(๐‘›(๐‘šโˆ’1)[๐‘›(๐‘šโˆ’1)+1]))โˆ’๐‘›(๐‘šโˆ’1)๐œ, which is rewritten as ๐ฉ0๐‘˜<๐œ“๐‘˜๎€ท๐ฉ0๎€ธ+[]๐œ€๐‘›(๐‘šโˆ’1)โˆ’1[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1+๐‘›(๐‘šโˆ’1)๐œ.(16) Since (13) implies ๐ฉ0๐‘˜>๐œ“๐‘˜(๐ฉ0)โˆ’๐œ€/(๐‘›(๐‘šโˆ’1)[๐‘›(๐‘šโˆ’1)+1])โˆ’๐œ, we have ๐œ“๐‘˜๎€ท๐ฉ0๎€ธโˆ’๐œ€[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1โˆ’๐œ<๐ฉ0๐‘˜<๐œ“๐‘˜๎€ท๐ฉ0๎€ธ+[]๐œ€๐‘›(๐‘šโˆ’1)โˆ’1[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1+๐‘›(๐‘šโˆ’1)๐œ.(17) Thus, ||๐ฉ0๐‘˜โˆ’๐œ“๐‘˜๎€ท๐ฉ0๎€ธ||<[]๐œ€๐‘›(๐‘šโˆ’1)โˆ’1[]๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1+๐‘›(๐‘šโˆ’1)๐œ(18) is derived. On the other hand, adding (13) from 1 to ๐‘›(๐‘šโˆ’1) yields ๐‘›(๐‘šโˆ’1)๎“๐‘—=1๐ฉ0๐‘—>๐‘›(๐‘šโˆ’1)๎“๐‘—=1๐œ“๐‘—๎€ท๐ฉ0๎€ธโˆ’๐œ€๐‘›(๐‘šโˆ’1)+1โˆ’๐‘›(๐‘šโˆ’1)๐œ.(19) From โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0๐ฉ0๐‘—=1, โˆ‘๐‘›(๐‘šโˆ’1)๐‘—=0๐œ“๐‘—(๐ฉ0)=1, we have 1โˆ’๐ฉ00>1โˆ’๐œ“0๎€ท๐ฉ0๎€ธโˆ’๐œ€๐‘›(๐‘šโˆ’1)+1โˆ’๐‘›(๐‘šโˆ’1)๐œ.(20) Then, from (14) and (20), we get ||๐ฉ00โˆ’๐œ“0๎€ท๐ฉ0๎€ธ||<๐œ€๐‘›(๐‘šโˆ’1)+1+๐‘›(๐‘šโˆ’1)๐œ.(21) From (18) and (21), we obtain the following result: ||๐ฉ0๐‘–โˆ’๐œ“๐‘–๎€ท๐ฉ0๎€ธ||<๐œ€๐‘›(๐‘šโˆ’1)+1+๐‘›(๐‘šโˆ’1)๐œโˆ€๐‘–.(22) Thus, ||๐ฉ0๎€ท๐ฉโˆ’๐œ“0๎€ธ||[]<๐œ€+๐‘›(๐‘šโˆ’1)๐‘›(๐‘šโˆ’1)+1๐œ.(23) Since ๐œ€ and ๐œ are arbitrary, we have inf๐ฉโˆˆฮ”|๐œ“(๐ฉ)โˆ’๐ฉ|=0.
(3) Since, by Lemma 3,โ€‰โ€‰โˆ‘ฮ”=๐‘›๐‘–=1๐ป๐‘–, where each ๐ป๐‘– is a totally bounded set whose diameter is less than or equal to ๐œ€, we have inf๐ฉโˆˆ๐ป๐‘–|๐œ“(๐ฉ)โˆ’๐ฉ|=0 for at least one ๐ป๐‘–. Choose a sequence (๐ซ๐‘›)๐‘›โ‰ฅ1 such that |๐œ“(๐ซ๐‘›)โˆ’๐ซ๐‘›|โ†’0 in such ๐ป๐‘–. In view of Lemma 5, it is enough to prove that the following condition holds.
For each ๐œ€>0, there exists ๐›ฟ>0 such that if ๐ฉ,๐ชโˆˆ๐ป๐‘–, |๐œ“(๐ฉ)โˆ’๐ฉ|<๐›ฟ, and |๐œ“(๐ช)โˆ’๐ช|<๐›ฟ, then |๐ฉโˆ’๐ช|โ‰ค๐œ€.
Assume that the set ๎€ฝ๐‘‡=(๐ฉ,๐ช)โˆˆ๐ป๐‘–ร—๐ป๐‘–โˆถ||||๎€พ๐ฉโˆ’๐ชโ‰ฅ๐œ€(24) is nonempty and compact (see Theoremโ€‰โ€‰2.2.13 of [9]). Since the mapping (๐ฉ,๐ช)โ†’max(|๐œ“(๐ฉ)โˆ’๐ฉ|,|๐œ“(๐ช)โˆ’๐ช|) is uniformly continuous, we can construct an increasing binary sequence (๐œ†๐‘›)๐‘›โ‰ฅ1 such that ๐œ†๐‘›=0โŸนinf(๐ฉ,๐ช)โˆˆ๐‘‡๎€ท||๐œ“||,||๐œ“||๎€ธmax(๐ฉ)โˆ’๐ฉ(๐ช)โˆ’๐ช<2โˆ’๐‘›,๐œ†๐‘›=1โŸนinf(๐ฉ,๐ช)โˆˆ๐‘‡๎€ท||||,||||๎€ธmax๐œ“(๐ฉ)โˆ’๐ฉ๐œ“(๐ช)โˆ’๐ช>2โˆ’๐‘›โˆ’1.(25) It suffices to find ๐‘› such that ๐œ†๐‘›=1. In that case, if |๐œ“(๐ฉ)โˆ’๐ฉ|<2โˆ’๐‘›โˆ’1, |๐œ“(๐ช)โˆ’๐ช|<2โˆ’๐‘›โˆ’1, we have (๐ฉ,๐ช)โˆ‰๐‘‡ and |๐ฉโˆ’๐ช|โ‰ค๐œ€. Assume that ๐œ†1=0. If ๐œ†๐‘›=0, choose (๐ฉ๐‘›,๐ช๐‘›)โˆˆ๐‘‡ such that max(|๐œ“(๐ฉ๐‘›)โˆ’๐ฉ๐‘›|,|๐œ“(๐ช๐‘›)โˆ’๐ช๐‘›|)<2โˆ’๐‘›, and if ๐œ†๐‘›=1, set ๐ฉ๐‘›=๐ช๐‘›=๐ซ๐‘›. Then, |๐œ“(๐ฉ๐‘›)โˆ’๐ฉ๐‘›|โ†’0 and |๐œ“(๐ช๐‘›)โˆ’๐ช๐‘›|โ†’0, so |๐ฉ๐‘›โˆ’๐ช๐‘›|โ†’0. Computing ๐‘ such that |๐ฉ๐‘โˆ’๐ช๐‘|<๐œ€, we must have ๐œ†๐‘=1. We have completed the proof of the existence of a point which satisfies ๐œ“(๐ฉ)=๐ฉ.
(4) Denote one of the points which satisfy ๐œ“(๐ฉ)=๐ฉ by ฬƒ๐ฉ=(ฬƒ๐‘1,ฬƒ๐‘2,โ€ฆ,ฬƒ๐‘๐‘›) and the components of ฬƒ๐‘๐‘– by ฬƒ๐‘๐‘–๐‘—. Then, we have ๐œ“๐‘–๐‘—=ฬƒ๐‘๐‘–๐‘—,โˆ€๐‘–,๐‘—.(26) By the definition of ๐œ“๐‘–๐‘—, ฬƒ๐‘๐‘–๐‘—๎€ท๐œ‹+max๐‘–๎€ท๐‘ i๐‘—,ฬƒ๐ฉโˆ’๐‘–๎€ธโˆ’๐œ‹๐‘–(ฬƒ๐ฉ๎€ธ),0โˆ‘1+๐‘š๐‘˜=1๎€ท๐œ‹max๐‘–๎€ท๐‘ ๐‘–๐‘˜,ฬƒ๐ฉโˆ’๐‘–๎€ธโˆ’๐œ‹๐‘–(ฬƒ๎€ธ๐ฉ),0=ฬƒ๐‘๐‘–๐‘—.(27) Let โˆ‘๐œ†=๐‘š๐‘˜=1max(๐œ‹๐‘–(๐‘ ๐‘–๐‘˜,ฬƒ๐ฉโˆ’๐‘–)โˆ’๐œ‹๐‘–(ฬƒ๐ฉ),0), then ๎€ท๐œ‹max๐‘–๎€ท๐‘ ๐‘–๐‘—,ฬƒ๐ฉโˆ’๐‘–๎€ธโˆ’๐œ‹๐‘–(ฬƒ๐ฉ๎€ธ),0=๐œ†ฬƒ๐‘๐‘–๐‘—,(28) where ฬƒ๐ฉโˆ’๐‘– denotes a combination of mixed strategies of players other than ๐‘– at profile ฬƒ๐ฉ.
Since ๐œ‹๐‘–(ฬƒโˆ‘๐ฉ)={๐‘—โˆถฬƒ๐‘๐‘–๐‘—>0}ฬƒ๐‘๐‘–๐‘—๐œ‹๐‘–(๐‘ ๐‘–๐‘—,ฬƒ๐ฉ๐‘–), it is impossible that max(๐œ‹๐‘–(๐‘ ๐‘–๐‘—,ฬƒ๐ฉโˆ’๐‘–)โˆ’๐œ‹๐‘–(ฬƒ๐ฉ),0)=๐œ‹๐‘–(๐‘ ๐‘–๐‘—,ฬƒ๐ฉโˆ’๐‘–)โˆ’๐œ‹๐‘–(ฬƒ๐ฉ)>0 for all ๐‘— satisfying ฬƒ๐‘๐‘–๐‘—>0. Thus, ๐œ†=0, and max(๐œ‹๐‘–(๐‘ ๐‘–๐‘—,ฬƒ๐ฉโˆ’๐‘–)โˆ’๐œ‹๐‘–(ฬƒ๐ฉ),0)=0 holds for all ๐‘ ๐‘–๐‘—โ€™s whether ฬƒ๐‘๐‘–๐‘—>0 or not, and it holds for all players. Then, strategies of all players in ฬƒ๐ฉ are the best responses to each other, and a state where all players choose these strategies is a Nash equilibrium.

Consider two examples. See a game in Table 1. It is an example of the so-called Prisonersโ€™ Dilemma. Pure strategies of players 1 and 2 are ๐‘‹ and ๐‘Œ. The left-side number in each cell represents the payoff of player 1, and the right-side number represents the payoff of player 2. Let ๐‘๐‘‹ and 1โˆ’๐‘๐‘‹ denote the probabilities that player 1 chooses, respectively, ๐‘‹ and ๐‘Œ, and ๐‘ž๐‘‹ and 1โˆ’๐‘ž๐‘‹ denote the probabilities for player 2. Denote the expected payoffs of players 1 and 2 by ๐œ‹1(๐‘๐‘‹,๐‘ž๐‘‹) and ๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹), then ๐œ‹1๎€ท๐‘๐‘‹,๐‘ž๐‘‹๎€ธ=2๐‘๐‘‹๐‘ž๐‘‹๎€ท+31โˆ’๐‘๐‘‹๎€ธ๐‘ž๐‘‹+๎€ท1โˆ’๐‘๐‘‹๎€ธ๎€ท1โˆ’๐‘ž๐‘‹๎€ธ=1โˆ’๐‘๐‘‹+2๐‘ž๐‘‹,๐œ‹2๎€ท๐‘๐‘‹,๐‘ž๐‘‹๎€ธ=2๐‘๐‘‹๐‘ž๐‘‹+3๐‘๐‘‹๎€ท1โˆ’๐‘ž๐‘‹๎€ธ+๎€ท1โˆ’๐‘๐‘‹๎€ธ๎€ท1โˆ’๐‘ž๐‘‹๎€ธ=1โˆ’๐‘ž๐‘‹+2๐‘๐‘‹.(29) Denote the payoff of player 1 when he chooses ๐‘‹ by ๐œ‹1(๐‘‹,๐‘ž๐‘‹) and that when he chooses ๐‘Œ by ๐œ‹1(๐‘Œ,๐‘ž๐‘‹). Do similarly for Player B, then๐œ‹1๎€ท๐‘Œ,๐‘ž๐‘‹๎€ธ=1+2๐‘ž๐‘‹>๐œ‹1๎€ท๐‘๐‘‹,๐‘ž๐‘‹๎€ธforany๐‘ž๐‘‹,๐‘๐‘‹๐œ‹>0,2๎€ท๐‘๐‘‹๎€ธ,๐‘Œ=1+2๐‘๐‘‹>๐œ‹2๎€ท๐‘๐‘‹,๐‘ž๐‘‹๎€ธforany๐‘๐‘‹,๐‘ž๐‘‹>0.(30)


Player 2

XY
Player 1X2, 20, 3
Y 3, 0 1, 1

Consider two sequences of ๐‘๐‘‹, (๐‘๐‘‹(๐‘š))๐‘šโ‰ฅ1 and (๐‘๎…ž๐‘‹(๐‘š))๐‘šโ‰ฅ1, such that ๐‘๐‘‹(๐‘š)>0 and ๐‘๎…ž๐‘‹(๐‘š)>0. If max(max(๐œ‹1(๐‘‹,๐‘ž๐‘‹),๐œ‹1(๐‘Œ,๐‘ž๐‘‹))โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)=max(๐œ‹1(๐‘Œ,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0 and max(max(๐œ‹1(๐‘‹,๐‘ž๐‘‹),๐œ‹1(๐‘Œ,๐‘ž๐‘‹))โˆ’๐œ‹1(๐‘๎…ž๐‘‹(๐‘š),๐‘ž๐‘‹),0)=max(๐œ‹1(๐‘Œ,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๎…ž๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0, then ๐‘๐‘‹(๐‘š)โ†’0, ๐‘๎…ž๐‘‹(๐‘š)โ†’0, and |๐‘๐‘‹(๐‘š)โˆ’๐‘๎…ž๐‘‹(๐‘š)|โ†’0.

Consider two sequences of ๐‘ž๐‘‹, (๐‘ž๐‘‹(๐‘š))๐‘šโ‰ฅ1 and (๐‘ž๎…ž๐‘‹(๐‘š))๐‘šโ‰ฅ1, such that ๐‘ž๐‘‹(๐‘š)>0 and ๐‘ž๎…ž๐‘‹(๐‘š)>0. If max(max(๐œ‹2(๐‘๐‘‹,๐‘‹),๐œ‹2(๐‘๐‘‹,๐‘Œ))โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)=max(๐œ‹2(๐‘๐‘‹,๐‘Œ)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0 and max(max(๐œ‹2(๐‘๐‘‹,๐‘‹),๐œ‹2(๐‘๐‘‹,๐‘Œ))โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๎…ž๐‘‹(๐‘š)),0)=max(๐œ‹2(๐‘๐‘‹,๐‘Œ)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๎…ž๐‘‹(๐‘š)),0)โ†’0, then ๐‘ž๐‘‹(๐‘š)โ†’0, ๐‘ž๎…ž๐‘‹(๐‘š)โ†’0, and |๐‘ž๐‘‹(๐‘š)โˆ’๐‘ž๎…ž๐‘‹(๐‘š)|โ†’0.

Therefore, the payoff functions are sequentially locally nonconstant.

Let us consider another example. See a game in Table 2. It is an example of the so-called Battle of the Sexes Game. Notations are the same as those in the previous example. The expected payoffs of players are as follows: ๐œ‹1๎€ท๐‘๐‘‹,๐‘ž๐‘‹๎€ธ=2๐‘๐‘‹๐‘ž๐‘‹+๎€ท1โˆ’๐‘๐‘‹๎€ธ๎€ท1โˆ’๐‘ž๐‘‹๎€ธ=1+๐‘๐‘‹๎€ท3๐‘ž๐‘‹๎€ธโˆ’1โˆ’๐‘ž๐‘‹,๐œ‹1๎€ท๐‘‹,๐‘ž๐‘‹๎€ธ=2๐‘ž๐‘‹,๐œ‹1๎€ท๐‘Œ,๐‘ž๐‘‹๎€ธ=1โˆ’๐‘ž๐‘‹,๐œ‹2๎€ท๐‘๐‘‹,๐‘ž๐‘‹๎€ธ=๐‘๐‘‹๐‘ž๐‘‹๎€ท+21โˆ’๐‘๐‘‹๎€ธ๎€ท1โˆ’๐‘ž๐‘‹๎€ธ=2+๐‘ž๐‘‹๎€ท3๐‘๐‘‹๎€ธโˆ’2โˆ’2๐‘๐‘‹,๐œ‹2๎€ท๐‘๐‘‹๎€ธ,๐‘‹=๐‘๐‘‹,๐œ‹2๎€ท๐‘๐‘‹๎€ธ,๐‘Œ=2โˆ’2๐‘๐‘‹.(31) Then, (i)when ๐‘ž๐‘‹>1/3, ๐œ‹1(๐‘‹,๐‘ž๐‘‹)>๐œ‹1(๐‘๐‘‹,๐‘ž๐‘‹) for ๐‘๐‘‹<1, (ii)when ๐‘ž๐‘‹<1/3, ๐œ‹1(๐‘Œ,๐‘ž๐‘‹)>๐œ‹1(๐‘๐‘‹,๐‘ž๐‘‹) for ๐‘๐‘‹>0, (iii) when ๐‘๐‘‹>2/3, ๐œ‹2(๐‘๐‘‹,๐‘‹)>๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹) for ๐‘ž๐‘‹<1, (iv) when ๐‘๐‘‹<2/3, ๐œ‹2(๐‘๐‘‹,๐‘Œ)>๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹) for ๐‘ž๐‘‹>0.


Player 2

XY
Player 1X2, 10, 0
Y 0, 0 1, 2

Consider sequences (๐‘๐‘‹(๐‘š))๐‘šโ‰ฅ1, (๐‘๎…ž๐‘‹(๐‘š))๐‘šโ‰ฅ1, (๐‘ž๐‘‹(๐‘š))๐‘šโ‰ฅ1, and (๐‘ž๎…ž๐‘‹(๐‘š))๐‘šโ‰ฅ1. (1)When ๐‘๐‘‹>2/3, ๐‘ž๐‘‹>1/3, if max(๐œ‹1(๐‘‹,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0 and max(๐œ‹1(๐‘‹,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๎…ž๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0, then ๐‘๐‘‹(๐‘š)โ†’1, ๐‘๎…ž๐‘‹(๐‘š)โ†’1, and |๐‘๐‘‹(๐‘š)โˆ’๐‘๎…ž๐‘‹(๐‘š)|โ†’0. If max(๐œ‹2(๐‘๐‘‹,๐‘‹)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0 and max(๐œ‹2(๐‘๐‘‹,๐‘‹)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๎…ž๐‘‹(๐‘š)),0)โ†’0, then ๐‘ž๐‘‹(๐‘š)โ†’1, ๐‘ž๎…ž๐‘‹(๐‘š)โ†’1, and |๐‘ž๐‘‹(๐‘š)โˆ’๐‘ž๎…ž๐‘‹(๐‘š)|โ†’0.(2)When ๐‘๐‘‹<2/3, ๐‘ž๐‘‹<1/3, if max(๐œ‹1(๐‘Œ,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0 and max(๐œ‹1(๐‘Œ,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๎…ž๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0, then ๐‘๐‘‹(๐‘š)โ†’0, ๐‘๎…ž๐‘‹(๐‘š)โ†’0, and |๐‘๐‘‹(๐‘š)โˆ’๐‘๎…ž๐‘‹(๐‘š)|โ†’0. If max(๐œ‹2(๐‘๐‘‹,๐‘Œ)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0 and max(๐œ‹2(๐‘๐‘‹,๐‘Œ)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๎…ž๐‘‹(๐‘š)),0)โ†’0, then ๐‘ž๐‘‹(๐‘š)โ†’0, ๐‘ž๎…ž๐‘‹(๐‘š)โ†’0, and |๐‘ž๐‘‹(๐‘š)โˆ’๐‘ž๎…ž๐‘‹(๐‘š)|โ†’0.(3) When ๐‘๐‘‹<2/3, ๐‘ž๐‘‹>1/3, there exist no pair of sequences (๐‘๐‘‹(๐‘š))๐‘šโ‰ฅ1 and (๐‘ž๐‘‹(๐‘š))๐‘šโ‰ฅ1 such that max(๐œ‹1(๐‘‹,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0 and max(๐œ‹2(๐‘๐‘‹,๐‘Œ)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0.(4) When ๐‘๐‘‹>2/3, ๐‘ž๐‘‹<1/3, there exist no pair of sequences (๐‘๐‘‹(๐‘š))๐‘šโ‰ฅ1 and (๐‘ž๐‘‹(๐‘š))๐‘šโ‰ฅ1 such that max(๐œ‹1(๐‘Œ,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0 and max(๐œ‹2(๐‘๐‘‹,๐‘‹)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0.(5) When (2/3)โˆ’๐œ€<๐‘๐‘‹<(2/3)+๐œ€, (1/3)โˆ’๐œ€<๐‘ž๐‘‹<(1/3)+๐œ€ with 0<๐œ€<1/3, if max(๐œ‹1(๐‘‹,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0, max(๐œ‹1(๐‘Œ,๐‘ž๐‘‹)โˆ’๐œ‹1(๐‘๐‘‹(๐‘š),๐‘ž๐‘‹),0)โ†’0, max(๐œ‹2(๐‘๐‘‹,๐‘‹)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0, and max(๐œ‹2(๐‘๐‘‹,๐‘Œ)โˆ’๐œ‹2(๐‘๐‘‹,๐‘ž๐‘‹(๐‘š)),0)โ†’0, then (๐‘๐‘‹(๐‘š),๐‘ž๐‘‹(๐‘š))โ†’(2/3,1/3) for all sequences (๐‘๐‘‹(๐‘š))๐‘šโ‰ฅ1 and (๐‘ž๐‘‹(๐‘š))๐‘šโ‰ฅ1. The payoff functions are sequentially locally nonconstant.

4. Concluding Remarks

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 รก la Bishop, that is, mathematics based on intuitionistic logic. As a future research program, we are studying the following themes: (1)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; (2)a generalization of the result of this paper to Kakutaniโ€™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 [14โ€“18].

Acknowledgment

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.

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Copyright ยฉ 2012 Yasuhito Tanaka. 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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