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

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 𝑥≠𝑓(𝑥), which means |𝑥−𝑓(𝑥)|>0, and also at every point 𝑥 on the boundaries of the simplex 𝑥≠𝑓(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.

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 𝑛−1-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 𝑛−2-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,…,𝐩_𝑛) 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 𝐩_𝑖 and 𝜑_𝑖(𝐩) or 𝜑_𝑖.

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 𝜑_𝑖(𝐩)>𝐩_𝑖 or 𝜑_𝑖(𝐩)<𝐩_𝑖 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 𝐩∈Δ, there exists 𝐪 in that 𝜀-approximation with |𝐩−𝐪|<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,…,𝐻_𝑛, each of diameter less than or equal to 𝜀, such that Δ=⋃^𝑛𝑖=1𝐻_𝑖.

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,…,𝐻_𝑚, each of diameter less than or equal to 𝜀, such that Δ=⋃^𝑚𝑖=1𝐻_𝑖, and if for all sequences (𝐩_𝑛)_𝑛≥1, (𝐪_𝑛)_𝑛≥1 in each 𝐻_𝑖, |𝜑(𝐩_𝑛)−𝐩_𝑛|→0 and |𝜑(𝐪_𝑛)−𝐪_𝑛|→0, then |𝐩_𝑛−𝐪_𝑛|→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 inf_{𝐩∈𝐻𝑖}𝜑(𝐩)=0 for 𝐻_𝑖⊂Δ defined above. If the following property holds: for each 𝜀>0, there exists 𝛿>0 such that if 𝐩,𝐪∈𝐻_𝑖, |𝜑(𝐩)−𝐩|<𝛿, and |𝜑(𝐪)−𝐪|<𝛿, then |𝐩−𝐪|≤𝜀, then there exists a point 𝐫∈𝐻_𝑖 such that 𝜑(𝐫)=𝐫.

Proof. Choose a sequence (𝐩_𝑛)_𝑛≥1 in 𝐻_𝑖 such that |𝜑(𝐩_𝑛)−𝐩_𝑛|→0. Compute 𝑁 such that |𝜑(𝐩_𝑛)−𝐩_𝑛|<𝛿 for all 𝑛≥𝑁. Then, for 𝑚,𝑛≥𝑁, we have |𝐩_𝑚−𝐩_𝑛|≤𝜀. Since 𝜀>0 is arbitrary, (𝐩_𝑛)_𝑛≥1 is a Cauchy sequence in 𝐻_𝑖 and converges to a limit 𝐫∈𝐻_𝑖. 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 𝑆_𝑖 be the set of pure strategies of player 𝑖, and denote each of his pure strategies by 𝑠_𝑖𝑗. His mixed strategy is defined as a probability distribution over 𝑆_𝑖 and is denoted by 𝐩_𝑖. Let 𝑝_𝑖𝑗 be a probability that player 𝑖 chooses 𝑠_𝑖𝑗, then we must have ∑^𝑚𝑗=1𝑝_𝑖𝑗=1 for all 𝑖. A combination of mixed strategies of all players is called a profile. It is denoted by 𝐩. Let 𝜋_𝑖(𝐩) be the expected payoff of player 𝑖 at profile 𝐩, and let 𝜋_𝑖(𝑠_𝑖𝑗,𝐩_−𝑖) be his payoff when he chooses a strategy 𝑠_𝑖𝑗 at that profile, where 𝐩_−𝑖 denotes a combination of mixed strategies of players other than 𝑖 at profile 𝐩. 𝜋_𝑖(𝐩) is written as follows:𝜋_𝑖(𝐩)=𝜋_𝑖𝐩_𝑖,𝐩_−𝑖=_{{𝑗∶𝑝𝑖𝑗>0}}𝑝_𝑖𝑗𝜋_𝑖𝑠_𝑖𝑗,𝐩_−𝑖.(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, 𝜋_𝑖(𝐩) is uniformly continuous about 𝐩.

For each 𝑖 and 𝑗, let𝑣_𝑖𝑗=𝑝_𝑖𝑗+max𝜋_𝑖𝑠_𝑖𝑗,𝐩_−𝑖−𝜋_𝑖(𝐩),0,(2) and define the following function:𝜓_𝑖𝑗(𝐩)=𝑣^𝑖𝑗𝑣_𝑖1+𝑣_𝑖2+⋯+𝑣_𝑖𝑚,(3) where ∑^𝑚𝑗=1𝜓_𝑖𝑗=1 for all 𝑖. Let 𝜓_𝑖(𝐩)=(𝜓_𝑖1,𝜓_𝑖2,…,𝜓_𝑖𝑚), 𝜓(𝐩)=(𝜓₁,𝜓₂,…,𝜓_𝑛). Since each 𝜓_𝑖 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 𝑚−1-dimensional simplex. 𝜓(𝐩) is a combination of vectors 𝜓_𝑖’s. It is a vector, such that its components are components of 𝜓_𝑖(𝐩) for all players. Thus, it is a vector with 𝑛×𝑚 components, but since the number of independent components is 𝑛(𝑚−1), the range of 𝜓 is the 𝑛-times product of 𝑚−1-dimensional simplices. It is convex, and homeomorphic to an 𝑛(𝑚−1)-dimensional simplex. 𝐩=(𝐩₁,𝐩₂,…,𝐩_𝑛) is also a vector with 𝑛×𝑚 components, and the number of its independent components is 𝑛(𝑚−1).

Let us consider a homeomorphism between an 𝑛(𝑚−1)-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(𝑚−1)) 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.

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,…,𝐻_𝑚, each of diameter less than or equal to 𝜀, such that 𝐏=⋃^𝑚𝑖=1𝐻_𝑖, and if for all sequences (𝐩_𝑛)_𝑛≥1, (𝐪_𝑛)_𝑛≥1 in each 𝐻_𝑖, max(𝜋_𝑖(𝑠_𝑖𝑗,(𝐩_𝑛)_−𝑖)−𝜋_𝑖(𝐩_𝑛),0)→0, max(𝜋_𝑖(𝑠_𝑖𝑗,(𝐪_𝑛)_−𝑖)−𝜋_𝑖(𝐪_𝑛),0)→0 for all 𝑠_𝑖𝑗∈𝑆_𝑖 for all 𝑖, then |𝐩_𝑛−𝐪_𝑛|→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,…,𝐻_𝑚, each of diameter less than or equal to 𝜀, such that 𝐏=⋃^𝑚𝑖=1𝐻_𝑖, and if for all sequences (𝐩_𝑛)_𝑛≥1, (𝐪_𝑛)_𝑛≥1 in each 𝐻_𝑖, |𝜓(𝐩_𝑛)−𝐩_𝑛|→0 and |𝜓(𝐪_𝑛)−𝐪_𝑛|→0, then |𝐩_𝑛−𝐪_𝑛|→0.

Thus, 𝜓 is sequentially locally nonconstant.

Let us replace 𝑛 by 𝑛(𝑚−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 𝑛(𝑚−1)-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,…,𝐩_{𝑛(𝑚−1)}) be a vertex of a simplex of 𝐾, and denote the 𝑖th coordinate of 𝜓(𝐩) by 𝜓_𝑖 or 𝜓_𝑖(𝐩). We label a vertex 𝐩 according to the following rule: if𝐩_𝑘+𝜏>𝜓_𝑘,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(𝑚−1)−1-dimensional face of Δ such that 𝑝_𝑖=0 for one 𝑖 among 0,1,2,…,𝑛(𝑚−1). With 𝜏>0, we have 𝑓_𝑖>0 or 𝑓_𝑖<𝜏 (in constructive mathematics for any real number 𝑥 we can not prove that 𝑥≥0 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 𝜓_𝑖>0, from ∑^{𝑛(𝑚−1)𝑗=0}𝐩_𝑗=1, ∑^{𝑛(𝑚−1)𝑗=0}𝜓_𝑗=1, and 𝐩_𝑖=0, ^𝑛(𝑚−¹⁾_{𝑗=0,𝑗≠𝑖}𝐩_𝑗>^{𝑛(𝑚−1)}_𝑗=_{0,𝑗≠𝑖}𝜓_𝑗.(5) Then, for at least one 𝑗 (denote it by 𝑘), we have 𝐩_𝑘>𝜓_𝑘, and we label 𝐩 with 𝑘, where 𝑘 is one of the numbers which satisfy 𝐩_𝑘>𝜓_𝑘. Since 𝜓_𝑖>𝐩_𝑖=0, 𝑖 does not satisfy this condition. Assume that 𝜓_𝑖<𝜏. 𝐩_𝑖=0 implies ∑^{𝑛(𝑚−1)𝑗=0,𝑗≠𝑖}𝐩_𝑗=1. Since ∑^{𝑛(𝑚−1)𝑗=0,𝑗≠𝑖}𝜓_𝑗≤1, we obtain ^𝑛(𝑚−¹⁾_{𝑗=0,𝑗≠𝑖}𝐩_𝑗≥^{𝑛(𝑚−1)}_𝑗=_{0,𝑗≠𝑖}𝜓_𝑗.(6) Then, for a positive number 𝜏, we have ^𝑛(𝑚−¹⁾_{𝑗=0,𝑗≠𝑖}𝐩_𝑗+𝜏>^{𝑛(𝑚−1)}_𝑗=_{0,𝑗≠𝑖}𝜓_𝑗.(7) There is at least one 𝑗(≠𝑖) which satisfies 𝐩_𝑗+𝜏>𝜓_𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘. 𝑘 is one of the numbers other than 𝑖 such that 𝐩_𝑘+𝜏>𝜓_𝑘 is satisfied. 𝑖 itself satisfies this condition (𝐩_𝑖+𝜏>𝜓_𝑖). 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(𝑚−1)−1-dimensional face of Δ such that 𝐩_𝑖=0 for one 𝑖 among 0,1,2,…,𝑛(𝑚−1) with the numbers other than 𝑖. By similar procedures, we can show that we can label the vertices contained in an 𝑛(𝑚−1)−2-dimensional face of Δ such that 𝐩_𝑖=0 for two 𝑖’s among 0,1,2,…,𝑛(𝑚−1) with the numbers other than those 𝑖’s, and so on.
Consider the case where 𝐩_𝑖=𝐩_𝑖+1=0. We see that when 𝜓_𝑖>0 or 𝜓_𝑖+1>0, ^{𝑛(𝑚−1)}_{𝑗=0,𝑗≠𝑖,𝑖+1}𝐩_𝑗>^{𝑛(𝑚−1)}_𝑗=0,_{𝑗≠𝑖,𝑖+1}𝜓_𝑗,(8) and so for at least one 𝑗(denote it by 𝑘), we have 𝐩_𝑘>𝜓_𝑘, and we label 𝐩 with 𝑘. On the other hand, when 𝜓_𝑖<𝜏 and 𝜓_𝑖+1<𝜏, we have ^{𝑛(𝑚−1)}_{𝑗=0,𝑗≠𝑖,𝑖+1}𝐩_𝑗≥^{𝑛(𝑚−1)}_𝑗=0,_{𝑗≠𝑖,𝑖+1}𝜓_𝑗.(9) Then, for a positive number 𝜏, we have ^{𝑛(𝑚−1)}_{𝑗=0,𝑗≠𝑖,𝑖+1}𝐩_𝑗+𝜏>^{𝑛(𝑚−1)}_𝑗=0,_{𝑗≠𝑖,𝑖+1}𝜓_𝑗.(10) Thus, there is at least one 𝑗(≠𝑖,𝑖+1) which satisfies 𝐩_𝑗+𝜏>𝜓_𝑗. Denote it by 𝑘, and we label 𝐩 with 𝑘.
Next, consider the case where 𝐩_𝑖=0 for all 𝑖 other than 𝑛(𝑚−1). If for some 𝑖  𝜓_𝑖>0, then we have 𝐩_{𝑛(𝑚−1)}>𝜓_{𝑛(𝑚−1)} and label 𝐩 with 𝑛(𝑚−1). On the other hand, if 𝜓_𝑗<𝜏 for all 𝑗≠𝑛(𝑚−1), then we obtain 𝐩_{𝑛(𝑚−1)}≥𝜓_{𝑛(𝑚−1)}. It implies 𝐩_{𝑛(𝑚−1)}+𝜏>𝜓_{𝑛(𝑚−1)}. Thus, we can label 𝐩 with 𝑛(𝑚−1).
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 𝛿^{𝑛(𝑚−1)} be a fully labeled 𝑛(𝑚−1)-dimensional simplex of 𝐾, and let 𝐩⁰,𝐩¹,… and 𝐩^{𝑛(𝑚−1)} be the vertices of 𝛿^{𝑛(𝑚−1)}. We name these vertices so that 𝐩⁰,𝐩¹,…,𝐩^{𝑛(𝑚−1)} are labeled, respectively, with 0, 1,…, 𝑛(𝑚−1). The values of 𝜓 at these vertices are 𝜓(𝐩⁰),𝜓(𝐩¹),… and 𝜓(𝐩^{𝑛(𝑚−1)}). The 𝑗th coordinates of 𝐩^𝑖 and 𝜓(𝐩^𝑖),𝑖=0,1,…,𝑛(𝑚−1), are, respectively, denoted by 𝐩^𝑖𝑗 and 𝜓_𝑗(𝐩^𝑖). 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 |𝐩^𝑖−𝐩^𝑗|<𝛿, then |𝜓(𝐩^𝑖)−𝜓(𝐩^𝑗)|<𝜀/2𝑛(𝑚−1)[𝑛(𝑚−1)+1] for 𝜀>0 and 𝑖≠𝑗. |𝜓(𝐩⁰)−𝜓(𝐩¹)|<𝜀/2𝑛(𝑚−1)[𝑛(𝑚−1)+1] means 𝜓₁(𝐩¹)>𝜓₁(𝐩⁰)−𝜀/2𝑛(𝑚−1)[𝑛(𝑚−1)+1]. On the other hand, |𝐩⁰−𝐩¹|<𝛿 means that 𝐩⁰¹>𝐩¹¹−𝛿. We can make 𝛿 satisfy 𝛿<𝜀/2𝑛(𝑚−1)[𝑛(𝑚−1)+1]. Thus, from 𝜓₁𝐩⁰¹>𝐩¹¹−𝛿,𝐩¹¹+𝜏>𝜓₁𝐩¹,𝐩¹>𝜓₁𝐩⁰−2𝑛(𝑚−1)[𝑛𝜀(𝑚−1)+1],(11) we obtain 𝐩⁰¹>𝜓₁𝐩⁰−𝛿−𝜏−𝜀>𝜓₁2𝑛(𝑚−1)[𝑛(𝑚−1)+1]𝐩⁰−𝑛(𝑚−1)[𝑛𝜀(𝑚−1)+1]−𝜏.(12) By similar arguments, for each 𝑖 other than 0, 𝐩^0𝑖>𝜓_𝑖𝐩⁰−𝜀𝑛(𝑚−1)[𝑛(𝑚−1)+1]−𝜏.(13) For 𝑖=0, we have 𝐩⁰⁰>𝜓₀𝐩⁰−𝜏.(14) Adding (13) and (14) side by side except for some 𝑖 (denote it by 𝑘) other than 0, ^𝑛(𝑚−¹⁾_{𝑗=0,𝑗≠𝑘}𝐩^0𝑗>^{𝑛(𝑚−1)}_𝑗=_{0,𝑗≠𝑘}−[𝜓𝑛(_𝑗𝑚𝐩−⁰1)−1]𝜀𝑛(𝑚−1)[𝑛(𝑚−1)+1]−𝑛(𝑚−1)𝜏.(15) From ∑^{𝑛(𝑚−1)𝑗=0}𝐩^0𝑗=1, ∑^{𝑛(𝑚−1)𝑗=0}𝜓_𝑗(𝐩⁰)=1, we have 1−𝐩^0𝑘>1−𝜓_𝑘(𝐩⁰)−([𝑛(𝑚−1)−1]𝜀/(𝑛(𝑚−1)[𝑛(𝑚−1)+1]))−𝑛(𝑚−1)𝜏, which is rewritten as 𝐩^0𝑘<𝜓_𝑘𝐩⁰+[𝑛(𝑚−1)−1]𝜀𝑛(𝑚−1)[𝑛(𝑚−1)+1]+𝑛(𝑚−1)𝜏.(16) Since (13) implies 𝐩^0𝑘>𝜓_𝑘(𝐩⁰)−𝜀/(𝑛(𝑚−1)[𝑛(𝑚−1)+1])−𝜏, we have 𝜓_𝑘𝐩⁰−𝜀<𝐩_0𝑘𝑛(𝑚−1)[𝑛(𝑚−1)+1]−𝜏<𝜓_𝑘𝐩⁰+[𝑛(𝑚−1)−1]𝜀𝑛(𝑚−1)[𝑛(𝑚−1)+1]+𝑛(𝑚−1)𝜏.(17) Thus, ||𝐩^0𝑘−𝜓_𝑘𝐩⁰||<[𝑛(𝑚−1)−1]𝜀𝑛(𝑚−1)[𝑛(𝑚−1)+1]+𝑛(𝑚−1)𝜏(18) is derived. On the other hand, adding (13) from 1 to 𝑛(𝑚−1) yields ^𝑛(^𝑚−1)_𝑗=1𝐩^0𝑗>^{𝑛(𝑚−1)}_𝑗=1𝜓_𝑗𝐩⁰−𝜀𝑛(𝑚−1)+1−𝑛(𝑚−1)𝜏.(19) From ∑^{𝑛(𝑚−1)𝑗=0}𝐩^0𝑗=1, ∑^{𝑛(𝑚−1)𝑗=0}𝜓_𝑗(𝐩⁰)=1, we have 1−𝐩⁰⁰>1−𝜓₀𝐩⁰−𝜀𝑛(𝑚−1)+1−𝑛(𝑚−1)𝜏.(20) Then, from (14) and (20), we get ||𝐩⁰⁰−𝜓₀𝐩⁰||<𝜀𝑛(𝑚−1)+1+𝑛(𝑚−1)𝜏.(21) From (18) and (21), we obtain the following result: ||𝐩^0𝑖−𝜓_𝑖𝐩⁰||<𝜀𝑛(𝑚−1)+1+𝑛(𝑚−1)𝜏∀𝑖.(22) Thus, ||𝐩⁰−𝜓𝐩⁰||<𝜀+𝑛(𝑚−1)[𝑛(𝑚−1)+1]𝜏.(23) Since 𝜀 and 𝜏 are arbitrary, we have inf_𝐩∈Δ|𝜓(𝐩)−𝐩|=0.
(3) Since, by Lemma 3,  Δ=∑^𝑛𝑖=1𝐻_𝑖, where each 𝐻_𝑖 is a totally bounded set whose diameter is less than or equal to 𝜀, we have inf_{𝐩∈𝐻𝑖}|𝜓(𝐩)−𝐩|=0 for at least one 𝐻_𝑖. Choose a sequence (𝐫_𝑛)_𝑛≥1 such that |𝜓(𝐫_𝑛)−𝐫_𝑛|→0 in such 𝐻_𝑖. 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,𝐪∈𝐻_𝑖, |𝜓(𝐩)−𝐩|<𝛿, and |𝜓(𝐪)−𝐪|<𝛿, then |𝐩−𝐪|≤𝜀.
Assume that the set 𝑇=(𝐩,𝐪)∈𝐻_𝑖×𝐻_𝑖∶||𝐩−𝐪||≥𝜀(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 (𝜆_𝑛)_𝑛≥1 such that 𝜆𝜆_𝑛𝑛=0⟹inf_{(𝐩,𝐪)∈𝑇}max||𝜓(𝐩)−𝐩||,||𝜓(𝐪)−𝐪||<2^−𝑛,=1⟹inf_{(𝐩,𝐪)∈𝑇}max||𝜓(𝐩)−𝐩||,||𝜓(𝐪)−𝐪||>2^−𝑛−1.(25) It suffices to find 𝑛 such that 𝜆_𝑛=1. In that case, if |𝜓(𝐩)−𝐩|<2^−𝑛−1, |𝜓(𝐪)−𝐪|<2^−𝑛−1, we have (𝐩,𝐪)∉𝑇 and |𝐩−𝐪|≤𝜀. Assume that 𝜆₁=0. If 𝜆_𝑛=0, choose (𝐩_𝑛,𝐪_𝑛)∈𝑇 such that max(|𝜓(𝐩_𝑛)−𝐩_𝑛|,|𝜓(𝐪_𝑛)−𝐪_𝑛|)<2^−𝑛, and if 𝜆_𝑛=1, set 𝐩_𝑛=𝐪_𝑛=𝐫_𝑛. Then, |𝜓(𝐩_𝑛)−𝐩_𝑛|→0 and |𝜓(𝐪_𝑛)−𝐪_𝑛|→0, so |𝐩_𝑛−𝐪_𝑛|→0. Computing 𝑁 such that |𝐩_𝑁−𝐪_𝑁|<𝜀, we must have 𝜆_𝑁=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 𝐩̃=(𝑝₁̃,𝑝₂̃,…,𝑝_𝑛̃) and the components of 𝑝_𝑖̃ by 𝑝̃_𝑖𝑗. Then, we have 𝜓_𝑖𝑗=𝑝̃_𝑖𝑗,∀𝑖,𝑗.(26) By the definition of 𝜓_𝑖𝑗, 𝑝̃_𝑖𝑗+max𝜋_𝑖𝑠_i𝑗,𝐩̃_−𝑖−𝜋_𝑖(𝐩̃),01+∑^𝑚𝑘=1max𝜋_𝑖𝑠_𝑖𝑘,𝐩̃_−𝑖−𝜋_𝑖(𝐩̃),0=𝑝̃^𝑖𝑗.(27) Let 𝜆=∑^𝑚𝑘=1max(𝜋_𝑖(𝑠_𝑖𝑘,𝐩̃_−𝑖)−𝜋_𝑖(𝐩̃),0), then max𝜋_𝑖𝑠_𝑖𝑗,𝐩̃_−𝑖−𝜋_𝑖(𝐩̃),0=𝜆𝑝̃_𝑖𝑗,(28) where 𝐩̃_−𝑖 denotes a combination of mixed strategies of players other than 𝑖 at profile 𝐩̃.
Since 𝜋_𝑖(𝐩̃)=∑_{𝑗∶𝑝̃_𝑖𝑗>0}𝑝̃_𝑖𝑗𝜋_𝑖(𝑠_𝑖𝑗,𝐩̃_𝑖), it is impossible that max(𝜋_𝑖(𝑠_𝑖𝑗,𝐩̃_−𝑖)−𝜋_𝑖(𝐩̃),0)=𝜋_𝑖(𝑠_𝑖𝑗,𝐩̃_−𝑖)−𝜋_𝑖(𝐩̃)>0 for all 𝑗 satisfying 𝑝̃_𝑖𝑗>0. Thus, 𝜆=0, and max(𝜋_𝑖(𝑠_𝑖𝑗,𝐩̃_−𝑖)−𝜋_𝑖(𝐩̃),0)=0 holds for all 𝑠_𝑖𝑗’s whether 𝑝̃_𝑖𝑗>0 or not, and it holds for all players. Then, strategies of all players in 𝐩̃ 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 𝑝_𝑋 and 1−𝑝_𝑋 denote the probabilities that player 1 chooses, respectively, 𝑋 and 𝑌, and 𝑞_𝑋 and 1−𝑞_𝑋 denote the probabilities for player 2. Denote the expected payoffs of players 1 and 2 by 𝜋₁(𝑝_𝑋,𝑞_𝑋) and 𝜋₂(𝑝_𝑋,𝑞_𝑋), then 𝜋₁𝑝_𝑋,𝑞_𝑋==12𝑝−_𝑋𝑞𝑝_𝑋𝑋+31−𝑝_𝑋𝑞_𝑋+1−𝑝_𝑋1−𝑞_𝑋𝜋₂+2𝑞_𝑋,𝑝_𝑋,𝑞_𝑋==12𝑝−_𝑋𝑞𝑞_𝑋𝑋+3𝑝_𝑋1−𝑞_𝑋+1−𝑝_𝑋1−𝑞_𝑋+2𝑝_𝑋.(29) Denote the payoff of player 1 when he chooses 𝑋 by 𝜋₁(𝑋,𝑞_𝑋) and that when he chooses 𝑌 by 𝜋₁(𝑌,𝑞_𝑋). Do similarly for Player B, then𝜋𝜋₁𝑌,𝑞_𝑋=1+2𝑞_𝑋>𝜋₁𝑝_𝑋,𝑞_𝑋forany𝑞_𝑋,𝑝_𝑋>0,₂𝑝_𝑋,𝑌=1+2𝑝_𝑋>𝜋₂𝑝_𝑋,𝑞_𝑋forany𝑝_𝑋,𝑞_𝑋>0.(30)

Table 1 

Example of game 1.

Consider two sequences of 𝑝_𝑋, (𝑝_𝑋(𝑚))_𝑚≥1 and (𝑝^𝑋(𝑚))_𝑚≥1, such that 𝑝_𝑋(𝑚)>0 and 𝑝^𝑋(𝑚)>0. If max(max(𝜋₁(𝑋,𝑞_𝑋),𝜋₁(𝑌,𝑞_𝑋))−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)=max(𝜋₁(𝑌,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0 and max(max(𝜋₁(𝑋,𝑞_𝑋),𝜋₁(𝑌,𝑞_𝑋))−𝜋₁(𝑝^𝑋(𝑚),𝑞_𝑋),0)=max(𝜋₁(𝑌,𝑞_𝑋)−𝜋₁(𝑝^𝑋(𝑚),𝑞_𝑋),0)→0, then 𝑝_𝑋(𝑚)→0, 𝑝^𝑋(𝑚)→0, and |𝑝_𝑋(𝑚)−𝑝^𝑋(𝑚)|→0.

Consider two sequences of 𝑞_𝑋, (𝑞_𝑋(𝑚))_𝑚≥1 and (𝑞^𝑋(𝑚))_𝑚≥1, such that 𝑞_𝑋(𝑚)>0 and 𝑞^𝑋(𝑚)>0. If max(max(𝜋₂(𝑝_𝑋,𝑋),𝜋₂(𝑝_𝑋,𝑌))−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)=max(𝜋₂(𝑝_𝑋,𝑌)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0 and max(max(𝜋₂(𝑝_𝑋,𝑋),𝜋₂(𝑝_𝑋,𝑌))−𝜋₂(𝑝_𝑋,𝑞^𝑋(𝑚)),0)=max(𝜋₂(𝑝_𝑋,𝑌)−𝜋₂(𝑝_𝑋,𝑞^𝑋(𝑚)),0)→0, then 𝑞_𝑋(𝑚)→0, 𝑞^𝑋(𝑚)→0, and |𝑞_𝑋(𝑚)−𝑞^𝑋(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: 𝜋₁𝑝_𝑋,𝑞_𝑋==21𝑝+_𝑋𝑞𝑝_𝑋𝑋+1−𝑝_𝑋1−𝑞_𝑋𝜋₁3𝑞_𝑋−1−𝑞_𝑋,𝜋₁𝑋,𝑞_𝑋=2𝑞_𝑋,𝜋₂𝑌,𝑞_𝑋=1−𝑞_𝑋,𝑝_𝑋,𝑞_𝑋==2𝑝_𝑋+𝑞_𝑋𝑞_𝑋+21−𝑝_𝑋1−𝑞_𝑋𝜋₂3𝑝_𝑋−2−2𝑝_𝑋,𝜋₂𝑝_𝑋,𝑋=𝑝_𝑋,𝑝_𝑋,𝑌=2−2𝑝_𝑋.(31) Then, (i)when 𝑞_𝑋>1/3, 𝜋₁(𝑋,𝑞_𝑋)>𝜋₁(𝑝_𝑋,𝑞_𝑋) for 𝑝_𝑋<1, (ii)when 𝑞_𝑋<1/3, 𝜋₁(𝑌,𝑞_𝑋)>𝜋₁(𝑝_𝑋,𝑞_𝑋) for 𝑝_𝑋>0, (iii) when 𝑝_𝑋>2/3, 𝜋₂(𝑝_𝑋,𝑋)>𝜋₂(𝑝_𝑋,𝑞_𝑋) for 𝑞_𝑋<1, (iv) when 𝑝_𝑋<2/3, 𝜋₂(𝑝_𝑋,𝑌)>𝜋₂(𝑝_𝑋,𝑞_𝑋) for 𝑞_𝑋>0.

Table 2 

Example of game 2.

Consider sequences (𝑝_𝑋(𝑚))_𝑚≥1, (𝑝^𝑋(𝑚))_𝑚≥1, (𝑞_𝑋(𝑚))_𝑚≥1, and (𝑞^𝑋(𝑚))_𝑚≥1. (1)When 𝑝_𝑋>2/3, 𝑞_𝑋>1/3, if max(𝜋₁(𝑋,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0 and max(𝜋₁(𝑋,𝑞_𝑋)−𝜋₁(𝑝^𝑋(𝑚),𝑞_𝑋),0)→0, then 𝑝_𝑋(𝑚)→1, 𝑝^𝑋(𝑚)→1, and |𝑝_𝑋(𝑚)−𝑝^𝑋(𝑚)|→0. If max(𝜋₂(𝑝_𝑋,𝑋)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0 and max(𝜋₂(𝑝_𝑋,𝑋)−𝜋₂(𝑝_𝑋,𝑞^𝑋(𝑚)),0)→0, then 𝑞_𝑋(𝑚)→1, 𝑞^𝑋(𝑚)→1, and |𝑞_𝑋(𝑚)−𝑞^𝑋(𝑚)|→0.(2)When 𝑝_𝑋<2/3, 𝑞_𝑋<1/3, if max(𝜋₁(𝑌,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0 and max(𝜋₁(𝑌,𝑞_𝑋)−𝜋₁(𝑝^𝑋(𝑚),𝑞_𝑋),0)→0, then 𝑝_𝑋(𝑚)→0, 𝑝^𝑋(𝑚)→0, and |𝑝_𝑋(𝑚)−𝑝^𝑋(𝑚)|→0. If max(𝜋₂(𝑝_𝑋,𝑌)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0 and max(𝜋₂(𝑝_𝑋,𝑌)−𝜋₂(𝑝_𝑋,𝑞^𝑋(𝑚)),0)→0, then 𝑞_𝑋(𝑚)→0, 𝑞^𝑋(𝑚)→0, and |𝑞_𝑋(𝑚)−𝑞^𝑋(𝑚)|→0.(3) When 𝑝_𝑋<2/3, 𝑞_𝑋>1/3, there exist no pair of sequences (𝑝_𝑋(𝑚))_𝑚≥1 and (𝑞_𝑋(𝑚))_𝑚≥1 such that max(𝜋₁(𝑋,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0 and max(𝜋₂(𝑝_𝑋,𝑌)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0.(4) When 𝑝_𝑋>2/3, 𝑞_𝑋<1/3, there exist no pair of sequences (𝑝_𝑋(𝑚))_𝑚≥1 and (𝑞_𝑋(𝑚))_𝑚≥1 such that max(𝜋₁(𝑌,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0 and max(𝜋₂(𝑝_𝑋,𝑋)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0.(5) When (2/3)−𝜀<𝑝_𝑋<(2/3)+𝜀, (1/3)−𝜀<𝑞_𝑋<(1/3)+𝜀 with 0<𝜀<1/3, if max(𝜋₁(𝑋,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0, max(𝜋₁(𝑌,𝑞_𝑋)−𝜋₁(𝑝_𝑋(𝑚),𝑞_𝑋),0)→0, max(𝜋₂(𝑝_𝑋,𝑋)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0, and max(𝜋₂(𝑝_𝑋,𝑌)−𝜋₂(𝑝_𝑋,𝑞_𝑋(𝑚)),0)→0, then (𝑝_𝑋(𝑚),𝑞_𝑋(𝑚))→(2/3,1/3) for all sequences (𝑝_𝑋(𝑚))_𝑚≥1 and (𝑞_𝑋(𝑚))_𝑚≥1. 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 [14–18].

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.