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 from the unit square to itself, which is defined at each computable point of the square, such that 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 from a simplex to itself, with property that each open set contains a point such that , which means , and also at every point on the boundaries of the simplex , 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 -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 denote an -dimensional simplex. 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 in equal segments and draw the lines parallel to the sides of . is a finite natural number. Then, the 2-dimensional simplex is partitioned into triangles. We consider partition of inductively for cases of higher dimension. In a 3-dimensional case, each face of is a 2-dimensional simplex, and so it is partitioned into triangles in the above-mentioned way, and draw the planes parallel to the faces of . Then, the 3-dimensional simplex is partitioned into trigonal pyramids. And this is similar to cases of higher dimension.

Figure 1 

Partition and labeling of 2-dimensional simplex.

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

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

Lemma 1 (Sperner’s lemma). If one labels the vertices of 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 and partition of are finite, the number of small simplices constructed by partition is also finite. Thus, we can constructively find a fully labeled -dimensional simplex of through finite steps.

3. Nash Equilibrium in Strategic Game

Let be a point in an -dimensional simplex , and consider a function from to itself. Denote the th components of and by and or .

The definition of local nonconstancy of functions is as follows.

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

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. is compact in the sense that for each , there exists a finitely enumerable -approximation to (a set is finitely enumerable if there exist a natural number and a mapping of the set onto ). An -approximation to is a subset of such that for each , there exists in that -approximation with . Each face (boundary) of 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 , there exist totally bounded sets , each of diameter less than or equal to , such that .

The definition of sequential local nonconstancy is as follows.

Definition 4 (sequential local nonconstancy of functions). There exists with the following property. For each less than , there exist totally bounded sets , each of diameter less than or equal to , such that , and if for all sequences , in each , and , then .

We show the following lemma.

Lemma 5. Let be a uniformly continuous and sequentially locally nonconstant function from to itself. Assume that for defined above. If the following property holds: for each , there exists such that if , , and , then , then there exists a point such that .

Proof. Choose a sequence in such that . Compute such that for all . Then, for , we have . Since is arbitrary, is a Cauchy sequence in and converges to a limit . The continuity of yields , that is, .

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 -players strategic game with pure strategies for each player. and are finite natural numbers not smaller than 2. Let be the set of pure strategies of player , and denote each of his pure strategies by . His mixed strategy is defined as a probability distribution over and is denoted by . Let be a probability that player chooses , then we must have for all . A combination of mixed strategies of all players is called a profile. It is denoted by . Let be the expected payoff of player at profile , and let be his payoff when he chooses a strategy at that profile, where denotes a combination of mixed strategies of players other than at profile . is written as follows: 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, is uniformly continuous about .

For each and , let and define the following function: where for all . Let , . Since each is an -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 -dimensional simplex. is a combination of vectors ’s. It is a vector, such that its components are components of for all players. Thus, it is a vector with components, but since the number of independent components is , the range of is the -times product of -dimensional simplices. It is convex, and homeomorphic to an -dimensional simplex. is also a vector with components, and the number of its independent components is .

Let us consider a homeomorphism between an -dimensional simplex and the space of players’ mixed strategies which is denoted by . Figure 2 depicts an example of a case of two players with two pure strategies for each player. Vertices , , , and represent states where two players choose pure strategies, and points on edges , , , and represent states where one player chooses a pure strategy. Vertices of the simplex and points on faces (simplices whose dimension is lower than ) of the simplex correspond to the points on faces of . For example, in Figure 2, A, B and C correspond, respectively, to , , and . On the other hand, each vertex of , , , , and 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 with the following property. For each less than , there exist totally bounded sets , each of diameter less than or equal to , such that , and if for all sequences , in each , , for all for all , then .

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

For each less than , there exist totally bounded sets , each of diameter less than or equal to , such that , and if for all sequences , in each , and , then .

Thus, is sequentially locally nonconstant.

Let us replace by . 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 -dimensional simplex , so that the conditions for Sperner’s lemma are satisfied. We partition according to the method in the proof of Sperner’s lemma and label the vertices of simplices constructed by partition of . It is important how to label the vertices contained in the faces of . Let be the set of small simplices constructed by partition of , let be a vertex of a simplex of , and denote the th coordinate of by or . We label a vertex according to the following rule: is an arbitrary positive number. If there are multiple ’s which satisfy this condition, we label conveniently for the conditions for Sperner’s lemma to be satisfied.
For example, let be a point contained in an -dimensional face of such that for one among . With , we have or (in constructive mathematics for any real number we can not prove that or , that, , , or . But for any distinct real numbers , , and such that , we can prove that or ). When , from , , and , Then, for at least one (denote it by ), we have , and we label with , where is one of the numbers which satisfy . Since , does not satisfy this condition. Assume that . implies . Since , we obtain Then, for a positive number , we have There is at least one which satisfies . Denote it by , and we label with . is one of the numbers other than such that is satisfied. itself satisfies this condition (). But, since there is a number other than which satisfies this condition, we can select a number other than . We have proved that we can label the vertices contained in an -dimensional face of such that for one among with the numbers other than . By similar procedures, we can show that we can label the vertices contained in an -dimensional face of such that for two ’s among with the numbers other than those ’s, and so on.
Consider the case where . We see that when or , and so for at least one (denote it by ), we have , and we label with . On the other hand, when and , we have Then, for a positive number , we have Thus, there is at least one which satisfies . Denote it by , and we label with .
Next, consider the case where for all other than . If for some   , then we have and label with . On the other hand, if for all , then we obtain . It implies . Thus, we can label with .
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 sufficiently fine so that the distance between any pair of the vertices of simplices of is sufficiently small. Let be a fully labeled -dimensional simplex of , and let and be the vertices of . We name these vertices so that are labeled, respectively, with 0, 1,, . The values of at these vertices are and . The th coordinates of and , are, respectively, denoted by and . About , from the labeling rules, we have . About , also from the labeling rules, we have . Since and are finite, by the uniform continuity of there exists such that if , then for and . means . On the other hand, means that . We can make satisfy . Thus, from we obtain By similar arguments, for each other than 0, For , we have Adding (13) and (14) side by side except for some (denote it by ) other than 0, From , , we have , which is rewritten as Since (13) implies , we have Thus, is derived. On the other hand, adding (13) from 1 to yields From , , we have Then, from (14) and (20), we get From (18) and (21), we obtain the following result: Thus, Since and are arbitrary, we have .
(3) Since, by Lemma 3,  , where each is a totally bounded set whose diameter is less than or equal to , we have for at least one . Choose a sequence such that in such . In view of Lemma 5, it is enough to prove that the following condition holds.
For each , there exists such that if , , and , then .
Assume that the set is nonempty and compact (see Theorem  2.2.13 of [9]). Since the mapping is uniformly continuous, we can construct an increasing binary sequence such that It suffices to find such that . In that case, if , , we have and . Assume that . If , choose such that , and if , set . Then, and , so . Computing such that , we must have . We have completed the proof of the existence of a point which satisfies .
(4) Denote one of the points which satisfy by and the components of by . Then, we have By the definition of , Let , then where denotes a combination of mixed strategies of players other than at profile .
Since , it is impossible that for all satisfying . Thus, , and holds for all ’s whether or not, and it holds for all players. Then, strategies of all players in 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 and . 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 and denote the probabilities that player 1 chooses, respectively, and , and and denote the probabilities for player 2. Denote the expected payoffs of players 1 and 2 by and , then Denote the payoff of player 1 when he chooses by and that when he chooses by . Do similarly for Player B, then

Consider two sequences of , and , such that and . If and , then , , and .

Consider two sequences of , and , such that and . If and , then , , and .

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: Then, (i)when , for , (ii)when , for , (iii) when , for , (iv) when , for .