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ISRN Computational Mathematics
Volume 2012 (2012), Article ID 459459, 8 pages
http://dx.doi.org/10.5402/2012/459459
Research Article

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

Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

Received 5 August 2011; Accepted 19 September 2011

Academic Editor: T. Karakasidis

Copyright © 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.

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 Brouwer’s fixed-point theorem cannot be constructively proved.

Refernce [1] provided a constructive proof of Brouwer’s fixed-point theorem. But it is not constructive from the viewpoint of constructive mathematics á la Bishop. It is sufficient to say that one-dimensional case of Brouwer’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 𝑓 from the unit square to itself, which is defined at each computable point of the square, such that 𝑓 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, Orevkov’s map does not have a total extension.

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 [3, 6]. Thus, Brouwer’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 𝑓 from a simplex to itself, with property that each open set contains a point 𝑥 such that 𝑥𝑓(𝑥), which means |𝑥𝑓(𝑥)|>0, and also at every point 𝑥 on the boundaries of the simplex 𝑥𝑓(𝑥), 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 Dalen’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 Sperner’s lemma for an 𝑛-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 Sperner’s lemma. We follow the Bishop-style constructive mathematics according to [2, 8, 9].

2. Sperner’s Lemma

Let Δ denote an 𝑛-dimensional simplex. 𝑛 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 Δ in 𝑚 equal segments and draw the lines parallel to the sides of Δ. 𝑚 is a finite natural number. Then, the 2-dimensional simplex is partitioned into 𝑚2 triangles. We consider partition of Δ inductively for cases of higher dimension. In a 3-dimensional case, each face of Δ is a 2-dimensional simplex, and so it is partitioned into 𝑚2 triangles in the above-mentioned way, and draw the planes parallel to the faces of Δ. Then, the 3-dimensional simplex is partitioned into 𝑚3 trigonal pyramids. And this is similar to cases of higher dimension.

459459.fig.001
Figure 1: Partition and labeling of 2-dimensional simplex.

Let 𝐾 denote the set of small 𝑛-dimensional simplices of Δ constructed by partition. Vertices of these small simplices of 𝐾 are labeled with the numbers 0,1,2,,𝑛 subject to the following rules. (1)The vertices of Δ are, respectively, labeled with 0 to 𝑛. We label a point (1,0,,0) with 0, a point (0,1,0,,0) with 1, a point (0,0,1,0) with 2,, and a point (0,,0,1) with 𝑛. That is, a vertex whose 𝑘th coordinate (𝑘=0,1,,𝑛) is 1 and all other coordinates are 0 is labeled with 𝑘.(2)If a vertex of 𝐾 is contained in an 𝑛1-dimensional face of Δ, 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 𝐾 is contained in an 𝑛2-dimensional face of Δ, 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 Δ is labeled with an arbitrary number among 0,1,,𝑛.

A small simplex of 𝐾 which is labeled with the numbers 0,1,,𝑛 is called a fully labeled simplex. Sperner’s lemma is stated as follows.

Lemma 1 (Sperner’s lemma). If one labels the vertices of 𝐾 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 Sperner’s lemma, see [10] or [11].

Since 𝑛 and partition of Δ are finite, the number of small simplices constructed by partition is also finite. Thus, we can constructively find a fully labeled 𝑛-dimensional simplex of 𝐾 through finite steps.

3. Nash Equilibrium in Strategic Game

Let 𝐩=(𝐩0,𝐩1,,𝐩𝑛) be a point in an 𝑛-dimensional simplex Δ, and consider a function 𝜑 from Δ to itself. Denote the 𝑖th components of 𝐩 and 𝜑(𝐩) by 𝐩𝑖 and 𝜑𝑖(𝐩) or 𝜑𝑖.

The definition of local nonconstancy of functions is as follows.

Definition 2 (local nonconstancy of functions). (1) At a point 𝐩 on the faces (boundaries) of a simplex 𝜑(𝐩)𝐩, this means that 𝜑𝑖(𝐩)>𝐩𝑖 or 𝜑𝑖(𝐩)<𝐩𝑖 for at least one 𝑖.
(2) In any open set in Δ, there exists a point 𝐩 such that 𝜑(𝐩)𝐩.

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. Δ is compact in the sense that for each 𝜀>0, there exists a finitely enumerable 𝜀-approximation to Δ (a set 𝑆 is finitely enumerable if there exist a natural number 𝑁 and a mapping of the set {1,2,,𝑁} onto 𝑆). An 𝜀-approximation to Δ is a subset of Δ such that for each 𝐩Δ, there exists 𝐪 in that 𝜀-approximation with |𝐩𝐪|<𝜀. Each face (boundary) of Δ is also a simplex, and so it is compact. According to Corollary  2.2.12 of [9], we have the following result.

Lemma 3. For each 𝜀>0, there exist totally bounded sets 𝐻1,𝐻2,,𝐻𝑛, each of diameter less than or equal to 𝜀, such that Δ=𝑛𝑖=1𝐻𝑖.

The definition of sequential local nonconstancy is as follows.

Definition 4 (sequential local nonconstancy of functions). There exists 𝜀 with the following property. For each 𝜀>0 less than 𝜀, there exist totally bounded sets 𝐻1,𝐻2,,𝐻𝑚, each of diameter less than or equal to 𝜀, such that Δ=𝑚𝑖=1𝐻𝑖, and if for all sequences (𝐩𝑛)𝑛1, (𝐪𝑛)𝑛1 in each 𝐻𝑖, |𝜑(𝐩𝑛)𝐩𝑛|0 and |𝜑(𝐪𝑛)𝐪𝑛|0, then |𝐩𝑛𝐪𝑛|0.

We show the following lemma.

Lemma 5. Let 𝜑 be a uniformly continuous and sequentially locally nonconstant function from Δ to itself. Assume that inf𝐩𝐻𝑖𝜑(𝐩)=0 for 𝐻𝑖Δ defined above. If the following property holds: for each 𝜀>0, there exists 𝛿>0 such that if 𝐩,𝐪𝐻𝑖, |𝜑(𝐩)𝐩|<𝛿, and |𝜑(𝐪)𝐪|<𝛿, then |𝐩𝐪|𝜀, then there exists a point 𝐫𝐻𝑖 such that 𝜑(𝐫)=𝐫.

Proof. Choose a sequence (𝐩𝑛)𝑛1 in 𝐻𝑖 such that |𝜑(𝐩𝑛)𝐩𝑛|0. Compute 𝑁 such that |𝜑(𝐩𝑛)𝐩𝑛|<𝛿 for all 𝑛𝑁. Then, for 𝑚,𝑛𝑁, we have |𝐩𝑚𝐩𝑛|𝜀. Since 𝜀>0 is arbitrary, (𝐩𝑛)𝑛1 is a Cauchy sequence in 𝐻𝑖 and converges to a limit 𝐫𝐻𝑖. The continuity of 𝜑 yields |𝜑(𝐫)𝐫|=0, that is, 𝜑(𝐫)=𝐫.

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 𝑛-players strategic game with 𝑚 pure strategies for each player. 𝑛 and 𝑚 are finite natural numbers not smaller than 2. Let 𝑆𝑖 be the set of pure strategies of player 𝑖, and denote each of his pure strategies by 𝑠𝑖𝑗. His mixed strategy is defined as a probability distribution over 𝑆𝑖 and is denoted by 𝐩𝑖. Let 𝑝𝑖𝑗 be a probability that player 𝑖 chooses 𝑠𝑖𝑗, then we must have 𝑚𝑗=1𝑝𝑖𝑗=1 for all 𝑖. A combination of mixed strategies of all players is called a profile. It is denoted by 𝐩. Let 𝜋𝑖(𝐩) be the expected payoff of player 𝑖 at profile 𝐩, and let 𝜋𝑖(𝑠𝑖𝑗,𝐩𝑖) be his payoff when he chooses a strategy 𝑠𝑖𝑗 at that profile, where 𝐩𝑖 denotes a combination of mixed strategies of players other than 𝑖 at profile 𝐩. 𝜋𝑖(𝐩) is written as follows:𝜋𝑖(𝐩)=𝜋𝑖𝐩𝑖,𝐩𝑖={𝑗𝑝𝑖𝑗>0}𝑝𝑖𝑗𝜋𝑖𝑠𝑖𝑗,𝐩𝑖.(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, 𝜋𝑖(𝐩) is uniformly continuous about 𝐩.

For each 𝑖 and 𝑗, let𝑣𝑖𝑗=𝑝𝑖𝑗𝜋+max𝑖𝑠𝑖𝑗,𝐩𝑖𝜋𝑖,(𝐩),0(2) and define the following function:𝜓𝑖𝑗𝑣(𝐩)=𝑖𝑗𝑣𝑖1+𝑣𝑖2++𝑣𝑖𝑚,(3) where 𝑚𝑗=1𝜓𝑖𝑗=1 for all 𝑖. Let 𝜓𝑖(𝐩)=(𝜓𝑖1,𝜓𝑖2,,𝜓𝑖𝑚), 𝜓(𝐩)=(𝜓1,𝜓2,,𝜓𝑛). Since each 𝜓𝑖 is an 𝑚-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 𝑚1-dimensional simplex. 𝜓(𝐩) is a combination of vectors 𝜓𝑖’s. It is a vector, such that its components are components of 𝜓𝑖(𝐩) for all players. Thus, it is a vector with 𝑛×𝑚 components, but since the number of independent components is 𝑛(𝑚1), the range of 𝜓 is the 𝑛-times product of 𝑚1-dimensional simplices. It is convex, and homeomorphic to an 𝑛(𝑚1)-dimensional simplex. 𝐩=(𝐩1,𝐩2,,𝐩𝑛) is also a vector with 𝑛×𝑚 components, and the number of its independent components is 𝑛(𝑚1).

Let us consider a homeomorphism between an 𝑛(𝑚1)-dimensional simplex and the space of players’ mixed strategies which is denoted by 𝐏. Figure 2 depicts an example of a case of two players with two pure strategies for each player. Vertices 𝐷, 𝐸, 𝐹, and G represent states where two players choose pure strategies, and points on edges 𝐷𝐸, 𝐸𝐹, 𝐹𝐺, and 𝐺𝐷 represent states where one player chooses a pure strategy. Vertices of the simplex and points on faces (simplices whose dimension is lower than 𝑛(𝑚1)) of the simplex correspond to the points on faces of 𝐏. For example, in Figure 2, A, B and C correspond, respectively, to 𝐼, 𝐽, and 𝐻. On the other hand, each vertex of 𝐏, 𝐷, 𝐸, 𝐹, and 𝐺 corresponds, respectively, to itself on a face of the simplex which contains it.

459459.fig.002
Figure 2: Homeomorphism between simplex and combination of strategies.

Next, we assume the following condition.

Definition 6 (sequential local nonconstancy of payoff functions). There exists 𝜀 with the following property. For each 𝜀>0 less than 𝜀, there exist totally bounded sets 𝐻1,𝐻2,,𝐻𝑚, each of diameter less than or equal to 𝜀, such that 𝐏=𝑚𝑖=1𝐻𝑖, and if for all sequences (𝐩𝑛)𝑛1, (𝐪𝑛)𝑛1 in each 𝐻𝑖, max(𝜋𝑖(𝑠𝑖𝑗,(𝐩𝑛)𝑖)𝜋𝑖(𝐩𝑛),0)0, max(𝜋𝑖(𝑠𝑖𝑗,(𝐪𝑛)𝑖)𝜋𝑖(𝐪𝑛),0)0 for all 𝑠𝑖𝑗𝑆𝑖 for all 𝑖, then |𝐩𝑛𝐪𝑛|0.

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

For each 𝜀>0 less than 𝜀, there exist totally bounded sets 𝐻1,𝐻2,,𝐻𝑚, each of diameter less than or equal to 𝜀, such that 𝐏=𝑚𝑖=1𝐻𝑖, and if for all sequences (𝐩𝑛)𝑛1, (𝐪𝑛)𝑛1 in each 𝐻𝑖, |𝜓(𝐩𝑛)𝐩𝑛|0 and |𝜓(𝐪𝑛)𝐪𝑛|0, then |𝐩𝑛𝐪𝑛|0.

Thus, 𝜓 is sequentially locally nonconstant.

Let us replace 𝑛 by 𝑛(𝑚1). 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 𝑛(𝑚1)-dimensional simplex Δ, so that the conditions for Sperner’s lemma are satisfied. We partition Δ according to the method in the proof of Sperner’s lemma and label the vertices of simplices constructed by partition of Δ. It is important how to label the vertices contained in the faces of Δ. Let 𝐾 be the set of small simplices constructed by partition of Δ, let 𝐩=(𝐩0,𝐩1,,𝐩𝑛(𝑚1)) be a vertex of a simplex of 𝐾, and denote the 𝑖th coordinate of 𝜓(𝐩) by 𝜓𝑖 or 𝜓𝑖(𝐩). We label a vertex 𝐩 according to the following rule: if𝐩𝑘+𝜏>𝜓𝑘,welabel𝐩with𝑘.(4)𝜏 is an arbitrary positive number. If there are multiple 𝑘’s which satisfy this condition, we label 𝐩 conveniently for the conditions for Sperner’s lemma to be satisfied.
For example, let 𝐩 be a point contained in an 𝑛(𝑚1)1-dimensional face of Δ such that 𝑝𝑖=0 for one 𝑖 among 0,1,2,,𝑛(𝑚1). With 𝜏>0, we have 𝑓𝑖>0 or 𝑓𝑖<𝜏 (in constructive mathematics for any real number 𝑥 we can not prove that 𝑥0 or 𝑥<0, that, 𝑥>0, 𝑥=0, or 𝑥<0. But for any distinct real numbers 𝑥, 𝑦, and 𝑧 such that 𝑥>𝑧, we can prove that 𝑥>𝑦 or 𝑦>𝑧). When 𝜓𝑖>0, from 𝑛(𝑚1)𝑗=0𝐩𝑗=1, 𝑛(𝑚1)𝑗=0𝜓𝑗=1, and 𝐩𝑖=0, 𝑛(𝑚1)𝑗=0,𝑗𝑖𝐩𝑗>𝑛(𝑚1)𝑗=0,𝑗𝑖𝜓𝑗.(5) Then, for at least one 𝑗 (denote it by 𝑘), we have 𝐩𝑘>𝜓𝑘, and we label 𝐩 with 𝑘, where 𝑘 is one of the numbers which satisfy 𝐩𝑘>𝜓𝑘. Since 𝜓𝑖>𝐩𝑖=0, 𝑖 does not satisfy this condition. Assume that 𝜓𝑖<𝜏. 𝐩𝑖=0 implies 𝑛(𝑚1)𝑗=0,𝑗𝑖𝐩𝑗=1. Since 𝑛(𝑚1)𝑗=0,𝑗𝑖𝜓𝑗1, we obtain 𝑛(𝑚1)𝑗=0,𝑗𝑖𝐩𝑗𝑛(𝑚1)𝑗=0,𝑗𝑖𝜓𝑗.(6) Then, for a positive number 𝜏, we have 𝑛(𝑚1)𝑗=0,𝑗𝑖𝐩𝑗>+𝜏𝑛(𝑚1)𝑗=0,𝑗𝑖𝜓𝑗.(7) There is at least one 𝑗(𝑖) which satisfies 𝐩𝑗+𝜏>𝜓𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘. 𝑘 is one of the numbers other than 𝑖 such that 𝐩𝑘+𝜏>𝜓𝑘 is satisfied. 𝑖 itself satisfies this condition (𝐩𝑖+𝜏>𝜓𝑖). But, since there is a number other than 𝑖 which satisfies this condition, we can select a number other than 𝑖. We have proved that we can label the vertices contained in an 𝑛(𝑚1)1-dimensional face of Δ such that 𝐩𝑖=0 for one 𝑖 among 0,1,2,,𝑛(𝑚1) with the numbers other than 𝑖. By similar procedures, we can show that we can label the vertices contained in an 𝑛(𝑚1)2-dimensional face of Δ such that 𝐩𝑖=0 for two 𝑖’s among 0,1,2,,𝑛(𝑚1) with the numbers other than those 𝑖’s, and so on.
Consider the case where 𝐩𝑖=𝐩𝑖+1=0. We see that when 𝜓𝑖>0 or 𝜓𝑖+1>0, 𝑛(𝑚1)𝑗=0,𝑗𝑖,𝑖+1𝐩𝑗>𝑛(𝑚1)𝑗=0,𝑗𝑖,𝑖+1𝜓𝑗,(8) and so for at least one 𝑗(denote it by 𝑘), we have 𝐩𝑘>𝜓𝑘, and we label 𝐩 with 𝑘. On the other hand, when 𝜓𝑖<𝜏 and 𝜓𝑖+1<𝜏, we have 𝑛(𝑚1)𝑗=0,𝑗𝑖,𝑖+1𝐩𝑗𝑛(𝑚1)𝑗=0,𝑗𝑖,𝑖+1𝜓𝑗.(9) Then, for a positive number 𝜏, we have 𝑛(𝑚1)𝑗=0,𝑗𝑖,𝑖+1𝐩𝑗>+𝜏𝑛(𝑚1)𝑗=0,𝑗𝑖,𝑖+1𝜓𝑗.(10) Thus, there is at least one 𝑗(𝑖,𝑖+1) which satisfies 𝐩𝑗+𝜏>𝜓𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘.
Next, consider the case where 𝐩𝑖=0 for all 𝑖 other than 𝑛(𝑚1). If for some 𝑖  𝜓𝑖>0, then we have 𝐩𝑛(𝑚1)>𝜓𝑛(𝑚1) and label 𝐩 with 𝑛(𝑚1). On the other hand, if 𝜓𝑗<𝜏 for all 𝑗𝑛(𝑚1), then we obtain 𝐩𝑛(𝑚1)𝜓𝑛(𝑚1). It implies 𝐩𝑛(𝑚1)+𝜏>𝜓𝑛(𝑚1). Thus, we can label 𝐩 with 𝑛(𝑚1).
Therefore, the conditions for Sperner’s lemma are satisfied, and there exists an odd number of fully labeled simplices in 𝐾.
(2) Suppose that we partition Δ sufficiently fine so that the distance between any pair of the vertices of simplices of 𝐾 is sufficiently small. Let 𝛿𝑛(𝑚1) be a fully labeled 𝑛(𝑚1)-dimensional simplex of 𝐾, and let 𝐩0,𝐩1, and 𝐩𝑛(𝑚1) be the vertices of 𝛿𝑛(𝑚1). We name these vertices so that 𝐩0,𝐩1,,𝐩𝑛(𝑚1) are labeled, respectively, with 0, 1,, 𝑛(𝑚1). The values of 𝜓 at these vertices are 𝜓(𝐩0),𝜓(𝐩1), and 𝜓(𝐩𝑛(𝑚1)). The 𝑗th coordinates of 𝐩𝑖 and 𝜓(𝐩𝑖),𝑖=0,1,,𝑛(𝑚1), are, respectively, denoted by 𝐩𝑖𝑗 and 𝜓𝑗(𝐩𝑖). About 𝐩0, from the labeling rules, we have 𝐩00+𝜏>𝜓0(𝐩0). About 𝐩1, also from the labeling rules, we have 𝐩11+𝜏>𝜓1(𝐩1). Since 𝑛 and 𝑚 are finite, by the uniform continuity of 𝜓 there exists 𝛿>0 such that if |𝐩𝑖𝐩𝑗|<𝛿, then |𝜓(𝐩𝑖)𝜓(𝐩𝑗)|<𝜀/2𝑛(𝑚1)[𝑛(𝑚1)+1] for 𝜀>0 and 𝑖𝑗. |𝜓(𝐩0)𝜓(𝐩1)|<𝜀/2𝑛(𝑚1)[𝑛(𝑚1)+1] means 𝜓1(𝐩1)>𝜓1(𝐩0)𝜀/2𝑛(𝑚1)[𝑛(𝑚1)+1]. On the other hand, |𝐩0𝐩1|<𝛿 means that 𝐩01>𝐩11𝛿. We can make 𝛿 satisfy 𝛿<𝜀/2𝑛(𝑚1)[𝑛(𝑚1)+1]. Thus, from 𝐩01>𝐩11𝛿,𝐩11+𝜏>𝜓1𝐩1,𝜓1𝐩1>𝜓1𝐩0𝜀[𝑛],2𝑛(𝑚1)(𝑚1)+1(11) we obtain 𝐩01>𝜓1𝐩0𝜀𝛿𝜏[]2𝑛(𝑚1)𝑛(𝑚1)+1>𝜓1𝐩0𝜀[]𝑛(𝑚1)𝑛(𝑚1)+1𝜏.(12) By similar arguments, for each 𝑖 other than 0, 𝐩0𝑖>𝜓𝑖𝐩0𝜀[]𝑛(𝑚1)𝑛(𝑚1)+1𝜏.(13) For 𝑖=0, we have 𝐩00>𝜓0𝐩0𝜏.(14) Adding (13) and (14) side by side except for some 𝑖 (denote it by 𝑘) other than 0, 𝑛(𝑚1)𝑗=0,𝑗𝑘𝐩0𝑗>𝑛(𝑚1)𝑗=0,𝑗𝑘𝜓𝑗𝐩0[𝑛]𝜀(𝑚1)1[]𝑛(𝑚1)𝑛(𝑚1)+1𝑛(𝑚1)𝜏.(15) From 𝑛(𝑚1)𝑗=0𝐩0𝑗=1, 𝑛(𝑚1)𝑗=0𝜓𝑗(𝐩0)=1, we have 1𝐩0𝑘>1𝜓𝑘(𝐩0)([𝑛(𝑚1)1]𝜀/(𝑛(𝑚1)[𝑛(𝑚1)+1]))𝑛(𝑚1)𝜏, which is rewritten as 𝐩0𝑘<𝜓𝑘𝐩0+[]𝜀𝑛(𝑚1)1[]𝑛(𝑚1)𝑛(𝑚1)+1+𝑛(𝑚1)𝜏.(16) Since (13) implies 𝐩0𝑘>𝜓𝑘(𝐩0)𝜀/(𝑛(𝑚1)[𝑛(𝑚1)+1])𝜏, we have 𝜓𝑘𝐩0𝜀[]𝑛(𝑚1)𝑛(𝑚1)+1𝜏<𝐩0𝑘<𝜓𝑘𝐩0+[]𝜀𝑛(𝑚1)1[]𝑛(𝑚1)𝑛(𝑚1)+1+𝑛(𝑚1)𝜏.(17) Thus, ||𝐩0𝑘𝜓𝑘𝐩0||<[]𝜀𝑛(𝑚1)1[]𝑛(𝑚1)𝑛(𝑚1)+1+𝑛(𝑚1)𝜏(18) is derived. On the other hand, adding (13) from 1 to 𝑛(𝑚1) yields 𝑛(𝑚1)𝑗=1𝐩0𝑗>𝑛(𝑚1)𝑗=1𝜓𝑗𝐩0𝜀𝑛(𝑚1)+1𝑛(𝑚1)𝜏.(19) From 𝑛(𝑚1)𝑗=0𝐩0𝑗=1, 𝑛(𝑚1)𝑗=0𝜓𝑗(𝐩0)=1, we have 1𝐩00>1𝜓0𝐩0𝜀𝑛(𝑚1)+1𝑛(𝑚1)𝜏.(20) Then, from (14) and (20), we get ||𝐩00𝜓0𝐩0||<𝜀𝑛(𝑚1)+1+𝑛(𝑚1)𝜏.(21) From (18) and (21), we obtain the following result: ||𝐩0𝑖𝜓𝑖𝐩0||<𝜀𝑛(𝑚1)+1+𝑛(𝑚1)𝜏𝑖.(22) Thus, ||𝐩0𝐩𝜓0||[]<𝜀+𝑛(𝑚1)𝑛(𝑚1)+1𝜏.(23) Since 𝜀 and 𝜏 are arbitrary, we have inf𝐩Δ|𝜓(𝐩)𝐩|=0.
(3) Since, by Lemma 3,  Δ=𝑛𝑖=1𝐻𝑖, where each 𝐻𝑖 is a totally bounded set whose diameter is less than or equal to 𝜀, we have inf𝐩𝐻𝑖|𝜓(𝐩)𝐩|=0 for at least one 𝐻𝑖. Choose a sequence (𝐫𝑛)𝑛1 such that |𝜓(𝐫𝑛)𝐫𝑛|0 in such 𝐻𝑖. In view of Lemma 5, it is enough to prove that the following condition holds.
For each 𝜀>0, there exists 𝛿>0 such that if 𝐩,𝐪𝐻𝑖, |𝜓(𝐩)𝐩|<𝛿, and |𝜓(𝐪)𝐪|<𝛿, then |𝐩𝐪|𝜀.
Assume that the set 𝑇=(𝐩,𝐪)𝐻𝑖×𝐻𝑖||||𝐩𝐪𝜀(24) is nonempty and compact (see Theorem  2.2.13 of [9]). Since the mapping (𝐩,𝐪)max(|𝜓(𝐩)𝐩|,|𝜓(𝐪)𝐪|) is uniformly continuous, we can construct an increasing binary sequence (𝜆𝑛)𝑛1 such that 𝜆𝑛=0inf(𝐩,𝐪)𝑇||𝜓||,||𝜓||max(𝐩)𝐩(𝐪)𝐪<2𝑛,𝜆𝑛=1inf(𝐩,𝐪)𝑇||||,||||max𝜓(𝐩)𝐩𝜓(𝐪)𝐪>2𝑛1.(25) It suffices to find 𝑛 such that 𝜆𝑛=1. In that case, if |𝜓(𝐩)𝐩|<2𝑛1, |𝜓(𝐪)𝐪|<2𝑛1, we have (𝐩,𝐪)𝑇 and |𝐩𝐪|𝜀. Assume that 𝜆1=0. If 𝜆𝑛=0, choose (𝐩𝑛,𝐪𝑛)𝑇 such that max(|𝜓(𝐩𝑛)𝐩𝑛|,|𝜓(𝐪𝑛)𝐪𝑛|)<2𝑛, and if 𝜆𝑛=1, set 𝐩𝑛=𝐪𝑛=𝐫𝑛. Then, |𝜓(𝐩𝑛)𝐩𝑛|0 and |𝜓(𝐪𝑛)𝐪𝑛|0, so |𝐩𝑛𝐪𝑛|0. Computing 𝑁 such that |𝐩𝑁𝐪𝑁|<𝜀, we must have 𝜆𝑁=1. We have completed the proof of the existence of a point which satisfies 𝜓(𝐩)=𝐩.
(4) Denote one of the points which satisfy 𝜓(𝐩)=𝐩 by ̃𝐩=(̃𝑝1,̃𝑝2,,̃𝑝𝑛) and the components of ̃𝑝𝑖 by ̃𝑝𝑖𝑗. Then, we have 𝜓𝑖𝑗=̃𝑝𝑖𝑗,𝑖,𝑗.(26) By the definition of 𝜓𝑖𝑗, ̃𝑝𝑖𝑗𝜋+max𝑖𝑠i𝑗,̃𝐩𝑖𝜋𝑖(̃𝐩),01+𝑚𝑘=1𝜋max𝑖𝑠𝑖𝑘,̃𝐩𝑖𝜋𝑖(̃𝐩),0=̃𝑝𝑖𝑗.(27) Let 𝜆=𝑚𝑘=1max(𝜋𝑖(𝑠𝑖𝑘,̃𝐩𝑖)𝜋𝑖(̃𝐩),0), then 𝜋max𝑖𝑠𝑖𝑗,̃𝐩𝑖𝜋𝑖(̃𝐩),0=𝜆̃𝑝𝑖𝑗,(28) where ̃𝐩𝑖 denotes a combination of mixed strategies of players other than 𝑖 at profile ̃𝐩.
Since 𝜋𝑖(̃𝐩)={𝑗̃𝑝𝑖𝑗>0}̃𝑝𝑖𝑗𝜋𝑖(𝑠𝑖𝑗,̃𝐩𝑖), it is impossible that max(𝜋𝑖(𝑠𝑖𝑗,̃𝐩𝑖)𝜋𝑖(̃𝐩),0)=𝜋𝑖(𝑠𝑖𝑗,̃𝐩𝑖)𝜋𝑖(̃𝐩)>0 for all 𝑗 satisfying ̃𝑝𝑖𝑗>0. Thus, 𝜆=0, and max(𝜋𝑖(𝑠𝑖𝑗,̃𝐩𝑖)𝜋𝑖(̃𝐩),0)=0 holds for all 𝑠𝑖𝑗’s whether ̃𝑝𝑖𝑗>0 or not, and it holds for all players. Then, strategies of all players in ̃𝐩 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 Prisoners’ Dilemma. Pure strategies of players 1 and 2 are 𝑋 and 𝑌. 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 𝑝𝑋 and 1𝑝𝑋 denote the probabilities that player 1 chooses, respectively, 𝑋 and 𝑌, and 𝑞𝑋 and 1𝑞𝑋 denote the probabilities for player 2. Denote the expected payoffs of players 1 and 2 by 𝜋1(𝑝𝑋,𝑞𝑋) and 𝜋2(𝑝𝑋,𝑞𝑋), then 𝜋1𝑝𝑋,𝑞𝑋=2𝑝𝑋𝑞𝑋+31𝑝𝑋𝑞𝑋+1𝑝𝑋1𝑞𝑋=1𝑝𝑋+2𝑞𝑋,𝜋2𝑝𝑋,𝑞𝑋=2𝑝𝑋𝑞𝑋+3𝑝𝑋1𝑞𝑋+1𝑝𝑋1𝑞𝑋=1𝑞𝑋+2𝑝𝑋.(29) Denote the payoff of player 1 when he chooses 𝑋 by 𝜋1(𝑋,𝑞𝑋) and that when he chooses 𝑌 by 𝜋1(𝑌,𝑞𝑋). Do similarly for Player B, then𝜋1𝑌,𝑞𝑋=1+2𝑞𝑋>𝜋1𝑝𝑋,𝑞𝑋forany𝑞𝑋,𝑝𝑋𝜋>0,2𝑝𝑋,𝑌=1+2𝑝𝑋>𝜋2𝑝𝑋,𝑞𝑋forany𝑝𝑋,𝑞𝑋>0.(30)

tab1
Table 1: Example of game 1.

Consider two sequences of 𝑝𝑋, (𝑝𝑋(𝑚))𝑚1 and (𝑝𝑋(𝑚))𝑚1, such that 𝑝𝑋(𝑚)>0 and 𝑝𝑋(𝑚)>0. If max(max(𝜋1(𝑋,𝑞𝑋),𝜋1(𝑌,𝑞𝑋))𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)=max(𝜋1(𝑌,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0 and max(max(𝜋1(𝑋,𝑞𝑋),𝜋1(𝑌,𝑞𝑋))𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)=max(𝜋1(𝑌,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0, then 𝑝𝑋(𝑚)0, 𝑝𝑋(𝑚)0, and |𝑝𝑋(𝑚)𝑝𝑋(𝑚)|0.

Consider two sequences of 𝑞𝑋, (𝑞𝑋(𝑚))𝑚1 and (𝑞𝑋(𝑚))𝑚1, such that 𝑞𝑋(𝑚)>0 and 𝑞𝑋(𝑚)>0. If max(max(𝜋2(𝑝𝑋,𝑋),𝜋2(𝑝𝑋,𝑌))𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)=max(𝜋2(𝑝𝑋,𝑌)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0 and max(max(𝜋2(𝑝𝑋,𝑋),𝜋2(𝑝𝑋,𝑌))𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)=max(𝜋2(𝑝𝑋,𝑌)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0, then 𝑞𝑋(𝑚)0, 𝑞𝑋(𝑚)0, and |𝑞𝑋(𝑚)𝑞𝑋(𝑚)|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: 𝜋1𝑝𝑋,𝑞𝑋=2𝑝𝑋𝑞𝑋+1𝑝𝑋1𝑞𝑋=1+𝑝𝑋3𝑞𝑋1𝑞𝑋,𝜋1𝑋,𝑞𝑋=2𝑞𝑋,𝜋1𝑌,𝑞𝑋=1𝑞𝑋,𝜋2𝑝𝑋,𝑞𝑋=𝑝𝑋𝑞𝑋+21𝑝𝑋1𝑞𝑋=2+𝑞𝑋3𝑝𝑋22𝑝𝑋,𝜋2𝑝𝑋,𝑋=𝑝𝑋,𝜋2𝑝𝑋,𝑌=22𝑝𝑋.(31) Then, (i)when 𝑞𝑋>1/3, 𝜋1(𝑋,𝑞𝑋)>𝜋1(𝑝𝑋,𝑞𝑋) for 𝑝𝑋<1, (ii)when 𝑞𝑋<1/3, 𝜋1(𝑌,𝑞𝑋)>𝜋1(𝑝𝑋,𝑞𝑋) for 𝑝𝑋>0, (iii) when 𝑝𝑋>2/3, 𝜋2(𝑝𝑋,𝑋)>𝜋2(𝑝𝑋,𝑞𝑋) for 𝑞𝑋<1, (iv) when 𝑝𝑋<2/3, 𝜋2(𝑝𝑋,𝑌)>𝜋2(𝑝𝑋,𝑞𝑋) for 𝑞𝑋>0.

tab2
Table 2: Example of game 2.

Consider sequences (𝑝𝑋(𝑚))𝑚1, (𝑝𝑋(𝑚))𝑚1, (𝑞𝑋(𝑚))𝑚1, and (𝑞𝑋(𝑚))𝑚1. (1)When 𝑝𝑋>2/3, 𝑞𝑋>1/3, if max(𝜋1(𝑋,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0 and max(𝜋1(𝑋,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0, then 𝑝𝑋(𝑚)1, 𝑝𝑋(𝑚)1, and |𝑝𝑋(𝑚)𝑝𝑋(𝑚)|0. If max(𝜋2(𝑝𝑋,𝑋)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0 and max(𝜋2(𝑝𝑋,𝑋)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0, then 𝑞𝑋(𝑚)1, 𝑞𝑋(𝑚)1, and |𝑞𝑋(𝑚)𝑞𝑋(𝑚)|0.(2)When 𝑝𝑋<2/3, 𝑞𝑋<1/3, if max(𝜋1(𝑌,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0 and max(𝜋1(𝑌,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0, then 𝑝𝑋(𝑚)0, 𝑝𝑋(𝑚)0, and |𝑝𝑋(𝑚)𝑝𝑋(𝑚)|0. If max(𝜋2(𝑝𝑋,𝑌)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0 and max(𝜋2(𝑝𝑋,𝑌)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0, then 𝑞𝑋(𝑚)0, 𝑞𝑋(𝑚)0, and |𝑞𝑋(𝑚)𝑞𝑋(𝑚)|0.(3) When 𝑝𝑋<2/3, 𝑞𝑋>1/3, there exist no pair of sequences (𝑝𝑋(𝑚))𝑚1 and (𝑞𝑋(𝑚))𝑚1 such that max(𝜋1(𝑋,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0 and max(𝜋2(𝑝𝑋,𝑌)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0.(4) When 𝑝𝑋>2/3, 𝑞𝑋<1/3, there exist no pair of sequences (𝑝𝑋(𝑚))𝑚1 and (𝑞𝑋(𝑚))𝑚1 such that max(𝜋1(𝑌,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0 and max(𝜋2(𝑝𝑋,𝑋)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0.(5) When (2/3)𝜀<𝑝𝑋<(2/3)+𝜀, (1/3)𝜀<𝑞𝑋<(1/3)+𝜀 with 0<𝜀<1/3, if max(𝜋1(𝑋,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0, max(𝜋1(𝑌,𝑞𝑋)𝜋1(𝑝𝑋(𝑚),𝑞𝑋),0)0, max(𝜋2(𝑝𝑋,𝑋)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0, and max(𝜋2(𝑝𝑋,𝑌)𝜋2(𝑝𝑋,𝑞𝑋(𝑚)),0)0, then (𝑝𝑋(𝑚),𝑞𝑋(𝑚))(2/3,1/3) for all sequences (𝑝𝑋(𝑚))𝑚1 and (𝑞𝑋(𝑚))𝑚1. 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 á 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 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 [1418].

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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