 Research
 Open Access
 Published:
Infinitely split Nash equilibrium problems in repeated games
Fixed Point Theory and Applicationsvolume 2018, Article number: 12 (2018)
Abstract
In this paper, we introduce the concept of infinitely split Nash equilibrium in repeated games in which the profile sets are chaincomplete posets. Then by using a fixed point theorem on posets in (J. Math. Anal. Appl. 409:1084–1092, 2014), we prove an existence theorem. As an application, we study the repeated extended Bertrant duopoly model of price competition.
Introduction and preliminaries
Definitions and notations in game theory
In begin of this section, we review some concepts and notations in game theory that are used by many authors. The reader is referred to [1–11] for more details. Let n be a positive integer greater than 1. An nperson noncooperative strategic game, simply called an nperson game, consists of the following elements:

1.
the set of n players denoted by N with \(\vert N \vert = n\);

2.
the set of profiles \(S _{N} =\prod_{i\in N} S _{i}\), where \(S _{i}\) is the pure strategy set for player \(i \in N\);

3.
the utility vector mapping \(f = \prod_{i\in N} f _{i}:S _{N} \to R ^{n}\), where \(f _{i}\) is the utility (payoff) function of player i, for \(i \in N\).
This game is denoted by \(\mathrm{G}(N, S _{N}, f)\). Throughout this paper, we always assume that, in the products \(\prod_{i\in N} S _{i}\), \(\prod_{i\in N} f _{i}\) and \(\prod_{i\in N \backslash \{i\} } S _{k}\), the players appear in the same sequential orders. As usual, for every \(i\in N\), we often denote a profile of pure strategies for player i’s opponents by
The set of profiles of pure strategies for player i’s opponents is then denoted by
Hence we may write \(x \in S _{N}\) as
Moreover, for every \(x_{i} \in S_{i}\), we denote
From \(f = \prod_{i\in N} f _{i}\) in the game \(\mathrm{G}(N, S _{N}, f)\), for any \(x \in S\), we have
One of the most important topics in game theory is the study of Nash equilibrium problems. It has been widely studied by many authors and has been extensively applied to economic theory, business and related industries (see [1, 2, 6, 7, 9–12]). We recall the definition of Nash equilibrium in nperson noncooperative strategic games below.
Let \(\mathrm{G}(N, S _{N}, f)\) be an nperson game. A profile of pure strategies $\stackrel{\u2322}{x}=({\stackrel{\u2322}{x}}_{1},{\stackrel{\u2322}{x}}_{2},\dots ,{\stackrel{\u2322}{x}}_{n})\in {S}_{N}$ is a Nash equilibrium of this game if and only if, it satisfies
It can be rewritten as
In an nperson game \(\mathrm{G}(N, S _{N}, f )\), we define a mapping \(F: S _{N}\times S _{N}\to R ^{n}\) by
\(F(z, x)\) is called the utility vector at profile \(x \in S _{N}\) associated to \(z \in S _{N}\). It is clear that
Let \(\geq ^{n}\) be the componentwise partial order on \(R ^{n}\) satisfying that, for \(x, z \in S _{N}\),
From (2)–(5), the Nash equilibrium can be described thus: a profile $\stackrel{\u2322}{x}\in {S}_{N}$ is a Nash equilibrium of \(\mathrm{G}(N, S _{N}, f)\) if and only if,
nperson dual games
An nperson game \(G = (N, S _{N}, f)\) is static. Some games in the real world may not be static. That is, it may not be oneshot nature. It is more realistic for this game to be repeatedly played. The dynamic model of game based on an nperson game \(G = (N, S _{N}, f)\) is formulized by the process that this static game is repeated infinite periods (times). It is called an nperson repeated game, in which there is a discount factor involved for the utilities (see [6]). The dynamic model for nperson repeated games will be studied in Sect. 3.
In this paper, we first consider a special model: nperson dual game. An nperson dual game based on an nperson game \(\mathrm{G}(N, S _{N},f)\) is modeled as follows: At first, the players play the game as a static nperson noncooperative strategic game. After this game is played first time and before this game is played again, every player always considers the reaction of its competitors to its strategy applied in the first time. To seek the optimization of the player’s utilities, the players may make arrangements of strategies to use in the second play. Suppose that this performance is represented by a mapping A on \(S _{N}\). Hence, if \(x\in S _{N}\) is the profile used by the players in the first time, then \(Ax\in S _{N}\) will be the profile used by the players in the second time. This nperson dual game is denoted by \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\).
Then we ask: Is there a Nash equilibrium $\stackrel{\u2322}{x}\in {S}_{N}$ of the game \(\mathrm{G}(N, S _{N}, f )\) (first play) such that $A\stackrel{\u2322}{x}\in {S}_{N}$ is also a Nash equilibrium of this game in second play with respect to the translated profiles? It raises the so called split Nash equilibrium problems for dual games.
In [3, 13–16] and [17], multitudinous of iterative algorithms have provided for the approximations of split Nash equilibria for two games. In all results about estimating Nash equilibria in the listed papers, there is a common essential prerequisite: The existence of a Nash equilibrium in the considered problem is assumed. It is indubitable that the existence of solutions for split Nash equilibrium problems is always the crux of the matter for solving these problems.
In [4], the present author proved an existence theorem of split Nash equilibrium problems for related games by using the Fan–KKM Theorem. Since the present author has studied the fixed point theory on posets for several years and has made some applications to Nash equilibrium problems, so, in this paper, we will apply some fixed point theorems on posets to study the solvability of split Nash equilibrium problems for dual games. To this end, the profile sets of games must be equipped with partial orders that may be neither linear spaces, nor topological spaces. The positive aspect of this research is that the utility functions in the considered games are unnecessary to be continuous and the mapping A that defines the split Nash equilibrium problems is unnecessary to be linear.
In Sect. 3, we extend the concept of split Nash equilibrium problems for dual games to infinitely split Nash equilibrium problems for repeated games and prove an existence theorem. As applications, in Sect. 4, we study the existence of infinitely split Nash equilibrium and Nash equilibrium for the repeated extended Bertrant duopoly model of price competition that is a special repeated game.
Split Nash equilibrium problems in dual games
Definitions and notations for split Nash equilibrium problems in dual games
Let \(\mathrm{G}(N, S _{N}, f )\) be an nperson game. Throughout this paper, unless otherwise stated, we assume that, for every player \(i\in N\), his strategy set \(S _{i}\) is nonempty and is equipped with a partial order \(\succcurlyeq _{i}\). That is, for every \(i\in N\), player i’s strategy set is assumed to be a poset \((S _{i}, \succcurlyeq _{i})\). As the product partially ordered set of \((S _{i}, \succcurlyeq _{i})\)’s, the profile set is also a poset \((S _{N}, \succcurlyeq)\) in which the partial order ≽ is the componentwise partial order of \(\succcurlyeq _{i}\)’s. That is, for \(x = ( x _{1}, x _{2}, \ldots , x _{n} )\) and \(y = ( y _{1}, y _{2}, \ldots , y _{n} ) \in S _{N}\), we have
For every \(i\in N\), \((S_{i}, \succcurlyeq _{i})\) is similarly defined to be the product poset of \((S _{j}, \succcurlyeq _{j})\)’s, \(j \neq i\), in which \(\succcurlyeq _{i}\) is the corresponding componentwise partial order of \(\succcurlyeq _{j}\)’s, \(j \neq i\).
Definition 1
Let \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) be an nperson dual game. The split Nash equilibrium problem associated with this dual game, denoted by \(\operatorname{NSNE}(\mathrm{G}(N, S _{N}, f, A ) ^{2} )\), is formalized as: to find a profile $\stackrel{\u2322}{x}\in {S}_{N}$ satisfying
such that the profile $A\stackrel{\u2322}{x}\in {S}_{N}$ solves the following:
From (6), a profile $\stackrel{\u2322}{x}\in {S}_{N}$ satisfying (7)–(8) can be rewritten as:
and
Such a profile $\stackrel{\u2322}{x}$ in \(S _{N}\) is called a split Nash equilibrium of this split Nash equilibrium problem \(\operatorname{NSNE}(\mathrm{G}(N, S _{N}, f, A ) ^{2} )\). The set of all split Nash equilibriums is denoted by \(S ( \mathrm{G}(N, S _{N}, f, A ) ^{2} )\).
When looking at the equilibrium problems (5) and (6) separately, the problem (5) is the classical Nash equilibriums problem of strategic games. When considering a special case, A = I (A is unnecessary to be linear), that is, the identity mapping on \(S _{N}\), \(\operatorname{NSNE}( \mathrm{G}(N, S _{N}, f, I ) ^{2} )\) reduces to the classical Nash equilibrium problem for the game \(\mathrm{G}(N, S _{N}, f)\). In this view, split Nash equilibrium problems for dual games can be considered as the natural extensions of the classical Nash equilibrium problems.
A fixed point theorem on posets is proved in [1]. In this theorem, the underlying space is a chaincomplete poset and the considered mapping is just required to satisfy orderincreasing upward condition without any continuity condition (as a matter of fact, the underlying space is just equipped with a partial order and it may not have any topological structure). The values of the considered mapping are universally inductive that is a relatively broad concept. Some properties and examples of universally inductive posets have been provided in [1]. We recall this theorem below that will be used in the proof of the main theorems in this paper.
Fixed Point Theorem A
(Theorem 3.2 in [5])
Let (P, \(\succcurlyeq ^{P}\)) be a chaincomplete poset and let \(\Gamma : P\to 2^{P}\backslash \{\emptyset \}\) be a setvalued mapping satisfying the following three conditions:

A1.
Γ is \(\succcurlyeq ^{P}\)increasing upward;

A2.
\((\Gamma (x), \succcurlyeq ^{P})\) is universally inductive, for every \(x \in P\);

A3.
There is an element \(y_{*}\) in P and v\(_{*}\in \Gamma (y _{*})\) with \(y _{*}\succcurlyeq ^{P} v _{*}\).
Let \(\mathcal{F}(\Gamma )\) denote the set of fixed points of Γ. Then

(i)
(\(\mathcal{F}(\Gamma )\), \(\succcurlyeq ^{P}\)) is a nonempty inductive poset;

(ii)
(\(\mathcal{F}(\Gamma )\cap [y _{*})\), \(\succcurlyeq^{P}\)) is a nonempty inductive poset; and Γ has an \(\succcurlyeq ^{P}\)maximal fixed point \(x^{*}\) with \(x^{*}\succcurlyeq ^{P} y _{*}\).
An existence theorem for split Nash equilibrium in dual games
We need the following concept, orderpositive, for mappings from posets to posets. It is an important condition for the mapping A for the existence of split Nash equilibrium in split Nash equilibrium problems.
Definition 2
Let (X, \(\succcurlyeq ^{X}\)), (Y, \(\succcurlyeq ^{Y}\)) and (U, \(\succcurlyeq ^{U}\)) be posets. Let C, D be nonempty subsets of X and Y, respectively. A mapping \(g: X\times Y \rightarrow U\) is said to be orderpositive from \(X\times Y\) to U whenever, for \(x, y\in D\), if \(x \preccurlyeq ^{Y} y\), then
In particular, if \((U,\succcurlyeq ^{U}) = (R ^{m}, \geq ^{m})\), where m is a natural number, a mapping g is orderpositive from \(X\times Y\) to \((R ^{m}, \geq ^{m})\) whenever, for \(x, y\in C\), if \(x \preccurlyeq ^{X} y\), then
Let \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) be an nperson dual game. To prove an existence theorem for split Nash equilibrium problem \(\operatorname{NSNE}( \mathrm{G}(N, S_{N}, f, A ) ^{2} )\), we need to define a mapping \(\pi : S_{N} \rightarrow 2^{S_{N}}\), for \(x \in S _{N}\), by
π can be equivalently written, for \(x \in S_{N}\), as
Observation 1
In Theorem 1 given below, it is assumed that, for every \(x \in S _{N}\), \(\pi (x) \neq \emptyset \). It means that, for any given profile \(x \in S _{N}\) and for every player \(i\in N\), when player i’s opponents take \(x_{i}\) to play, there exists a strategy \(t _{i}\in S _{i}\) such that player i will optimize his utility at the profile (\(t _{i}\), \(x_{i}\)). Hence, the condition that \(\pi (x)\) is nonempty is a reasonable condition and it should not be too strong.
Now we prove one of the main theorems of this paper.
Theorem 1
Let \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) be an nperson dual game. Suppose that, for every \(i\in N\), \((S _{i}, \succcurlyeq _{i})\) is a nonempty chaincomplete poset. Let \((S _{N}, \succcurlyeq)\) be the product poset of \((S _{i}, \succcurlyeq _{i})\)’s equipped with the componentwise partial order ≽. If f and A satisfy the following conditions:

(a)
For every \(i \in N\), \(f _{i}\) is orderpositive from \((S _{i}, \succcurlyeq _{i}) \times (S_{i},\succcurlyeq _{i})\) to \((R, \geq )\).

(b)
For every \(x \in S _{N}\), \(\pi(x)\) is a universally inductive subset of \(S_{N}\).

(c)
The operator \(A: S _{N}\to S _{N}\) is ≽increasing.

(d)
There are elements \(x^{\prime} \in S_{N}\) and \(u^{\prime} \in\pi (x^{\prime})\) satisfying \(x^{\prime} \preccurlyeq u^{\prime}\);
then the dual game \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) has a split Nash equilibrium. Moreover

(i)
\((S ( \mathrm{G}(N, S _{N}, f, A ) ^{2}), \succcurlyeq )\) is a nonempty inductive poset;

(ii)
\((S(\mathrm{G}(N, S _{N}, f, A ) ^{2} ) \cap [ x ^{\prime}), \succcurlyeq)\) is a nonempty inductive poset.
Proof
Since, for every \(i\in N\), \((S _{i}, \succcurlyeq _{i})\) is a nonempty chaincomplete poset, then the profile set, as a product space of chaincomplete spaces \((S _{i}, \succcurlyeq _{i})\)’s, \((S _{N}, \succcurlyeq)\) is a nonempty chaincomplete poset, where ≽ is the componentwise partial order of \(\succcurlyeq_{i}\)’s. Define a setvalued mapping \(\Gamma: S _{N} \rightarrow 2^{S_{N}}\) by
From condition (b) in this theorem, it implies that, for every \(x \in S_{N}\), \(\Gamma (x) \neq \emptyset \). Hence the mapping \(\Gamma : S _{N} \rightarrow 2^{S_{N}}\backslash \{\emptyset \}\) is a welldefined setvalued mapping with universally inductive values in \(S_{N}\).
Next we show that Γ is ≽increasing upward. Notice that the partial order ≽ on \(S _{N}\) is the componentwise partial order of \(\succcurlyeq _{i}\)’s on \(S _{i}\)’s, respectively. It implies that, for any x, \(y\in S _{N}\), \(x \preccurlyeq y\) is equivalent to \(x _{i} \preccurlyeq _{i} y _{i}\) and \(x_{i} \preccurlyeq _{i} y_{i}\), for every \(i \in N\). From condition (a), for every \(i \in N\), \(f _{i}\) is orderpositive from \((S _{i}, \succcurlyeq_{i})\times (S_{i}, \succcurlyeq _{i})\) to \((R, \geq )\). From (3), it implies that F is orderpositive from \((S _{N}, \succcurlyeq)\times (S _{N}, \succcurlyeq)\) to \((R^{n}, \geq ^{n})\). Then, for arbitrary x, \(y\in S _{N}\) with \(x \preccurlyeq y\), it implies \(x_{i}\preccurlyeq _{i} y_{i}\), for every \(i \in N\). From condition (a), we then have
It follows that
From condition (c), the operator A: \(S _{N}\to S _{N}\) is ≽increasing. It implies that if \(x \preccurlyeq y\), then \(Ax \preccurlyeq Ay\). From condition (a) again, similar to (15), we have
(15) and (16) together imply that if \(x \preccurlyeq y\), then \(\Gamma (x) \subseteq \Gamma (y)\). Hence Γ is ≽increasing upward.
The elements \(x^{\prime} \in S _{N}\) and \(u^{\prime} \in \pi (x^{\prime})\) given in condition (d) in this theorem satisfy \(u^{\prime} \in \Gamma (x^{\prime})\) such that \(x^{\prime} \preccurlyeq u^{\prime}\). So Γ satisfies all conditions in the Fixed Point Theorem A. It follows that \(\mathcal{F}(\Gamma ) \neq \emptyset \) and it satisfies the properties (i) and (ii) in Theorem A. From (9) and (10), the definition of \(S(\mathrm{G}(N, S _{N}, f, A)^{2})\), and (14), the definition of Γ, we obtain
By Applying Theorem A and (17), the proof of this theorem is completed immediately. □
Applications to partially ordered Banach spaces
In this subsection, we consider a special case of nperson dual games in which the strategy set for every player is a nonempty and compact subset of a partially ordered Banach space. This case should be very useful in the applications. In [1], it was proved that every partially ordered compact Hausdorff topological space is both chaincomplete and universally inductive, as a consequence of Theorem 1, we have:
Corollary 1
Let \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) be an nperson dual game. Suppose that, for every \(i\in N\), \(S _{i}\) is a nonempty compact subset of a partially ordered Banach space \((B _{i}, \succcurlyeq _{i})\). Let \(B _{N} =\prod_{i\in N} B _{i}\) equipped with the componentwise partial order \(\succcurlyeq of \succcurlyeq _{i}\)’s. If f and A satisfy the following conditions:

(a)
For every \(i \in N\), \(f _{i}\) is orderpositive from \((S _{i}, \succcurlyeq _{i})\times (S_{i},\succcurlyeq _{i})\) to \((R, \geq )\).

(b)
For every \(x \in S _{N}\), \(\pi(x)\) is a nonempty closed subset of \(S_{N}\).

(c)
The operator \(A: S _{N}\to S _{N}\) is ≽increasing.

(d)
There are elements \(x^{\prime} \in S_{N}\) and \(u^{\prime} \in\pi (x^{\prime})\) satisfying \(x^{\prime} \preccurlyeq u^{\prime}\),
then the dual game \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) has a split Nash equilibrium. Moreover, \(S ( \mathrm{G}(N, S_{N}, f, A ) ^{2} )\) has the properties (i) and (ii) listed in Theorem 1.
Remark 1
In Corollary 1, even though the profile set in the dual game \(\mathrm{G}(N, S _{N}, f, A ) ^{2}\) is a subset of a Banach space, the operator \(A: S _{N} \rightarrow S _{N}\) is not required to be linear. It may be a nonlinear operator.
Infinitely split Nash equilibrium problems in repeated games
Definitions and notations of nperson repeated games
Let \(\mathrm{G}(N, S _{N}, f)\) be an nperson game. Recall that, for every \(i\in N\), player i’s strategy set is assumed to be a poset \((S _{i}, \succcurlyeq _{i})\). \((S _{N}, \succcurlyeq)\) is the product poset, where ≽ is the componentwise partial order of \(\succcurlyeq _{i}\)’s naturally equipped on \(S _{N}\).
For every natural number k, after the players repeated play the game k times and, for each time, the game is played as a static nperson simultaneousmove game, before they play this static game again, every player always considers the reaction of its competitors to its strategy applied in the previous time. To optimize their utilities, the players may make arrangements of strategies to use in the next play (the (k+1)th play). Suppose that the profile of the arranged strategies is represented by the value of a mapping \(A _{k}\): \(S _{N}\to S _{N}\), for \(k = 1, 2, 3, \ldots \) . (Since (\(S _{k}\), \(\succcurlyeq _{k}\)) is just a poset, it may not be equipped with any algebraic structure. So the linearity of \(A _{k}\) is may not be defined.) To summarizing this process, if \(x \in S _{N}\) is the profile used by the players in the first time, then \(A _{1} x \in S _{N}\) will be the profile used by the players in the second time; \(A _{2} A _{1} x \in S _{N}\) will be the profile used by the players in the third time. Hence, for \(k = 1, 2, 3, \ldots, A _{k} \cdots A _{2} A _{1} x \in S _{N}\) will be the profile used by the players in the \((k +1)\)th play. For simplicity, we write
where \(A _{0} = I\), that is, the identity operator on \(S _{N}\). Then \(\Pi_{k}: S _{N}\to S _{N}\) is a singlevalued mapping. In particular, if \(A _{k} = A_{k1} =\cdots = A _{1} = A\), then we denote
Suppose that the utilities of this game are bounded. That is, there is a number \(M > 0\) such that

(M)
\(\vert f _{i}(x) \vert \leq M\), for every \(i\in N\) and for all \(x \in S _{N}\).
There is a discount factor \(0 < \rho < 1\). For every \(i\in N\), player i’s discounted value of utility at a profile \(x \in S _{N}\) is
Player i’s discounted value of utility at the profile x associated with a profile \(z \in S _{N}\) is
It implies
The utility vector with discounted values for this repeated game at the profile x associated with a profile \(z \in S _{N}\) is
Under the boundedness condition (M), for every \(i\in N\), both of \(h _{i}\) and \(H _{i}\) are welldefined real valued functions on \(S_{N}\) and \(S _{N} \times S _{N}\), respectively. Then it forms an nperson dynamic model based on an nperson game. It is called an nperson repeated game based on the nperson static game \(\mathrm{G}(N, S_{N}, f )\) and is denoted by
Definition 3
Let \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) be an nperson repeated game. A profile \(\hat{x}\in S _{N}\) is called a Nash equilibrium of this repeated game, if the following inequalities are satisfied:
Definition 4
The infinitely split Nash equilibrium problem associated with the repeated game \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } \), denoted by \(\operatorname{NSNE}(\mathrm{G}(N,S_{N},f,A_{k})_{k =0}^{\infty })\), is formalized as: to find a profile $\stackrel{\u2322}{x}\in {S}_{N}$ satisfying
Such a profile $\stackrel{\u2322}{x}\in {S}_{N}$ is called an infinitely split Nash equilibrium. The set of all infinitely split Nash equilibriums of \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) is denoted by \(S(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } )\).
Proposition 1
Every infinitely split Nash equilibrium of an nperson repeated game is a Nash equilibrium of this repeated game.
Proof
Suppose that, for every \(i\in N\), for \(k =0, 1, 2,\ldots \) , the following inequality holds:
Since \(0 < \lambda < 1\), it implies
It completes the proof of this proposition. □
Similar to (13) for the definition of the mapping π, regarding \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } \), we need to define a mapping \(\psi:S _{N} \rightarrow 2^{S_{N}}\), for \(x\in S _{N}\), by
ψ can be rewritten, for \(x \in S _{N}\), as
Theorem 2
Let \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) be an nperson repeated game. Suppose that, for every \(i\in N\), \((S _{i}, \succcurlyeq _{i})\) is a nonempty chaincomplete poset. Let \((S _{N}, \succcurlyeq)\) be the product poset of \((S _{i}, \succcurlyeq _{i})\)’s equipped with the componentwise partial order ≽. Suppose that the following conditions are satisfied:

(a)
For every \(i \in N\), \(f _{i}\) is orderpositive from \((S _{i}, \succcurlyeq _{i})\times (S_{i},\succcurlyeq _{i}) \) to \((R, \geq )\).

(b)
For every \(x \in S _{N}\), \(\psi (x)\) is a nonempty universally inductive subset of \(S_{N}\).

(c)
For every \(k = 1, 2,\ldots\,, A _{k}: S _{N}\to S _{N}\) is an ≽increasing operator.

(d)
There are elements \(x^{\prime} \in S_{N}\) and \(u^{\prime} \in \psi (x^{\prime})\) satisfying \(x^{\prime} \preccurlyeq u^{\prime}\).
Then the repeated game \(\mathrm{G}(N,S_{N},f,A_{k})_{k =0}^{\infty } \) has an infinitely split Nash equilibrium. Moreover

(i)
\((S ( \mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } ),\succcurlyeq )\) is a nonempty inductive poset;

(ii)
\(( S ( \mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } ) \cap [x^{\prime}),\succcurlyeq ) \) is a nonempty inductive poset.
Proof
The proof of this theorem is similar to the proof of Theorem 1. As a product space of chaincomplete posets \((S _{i}, \succcurlyeq _{i})\)’s, the profile set \((S _{N}, \succcurlyeq)\) is also a nonempty chaincomplete poset, where ≽ is the componentwise partial orders \(\succcurlyeq _{i}\)’s. By using (23), we define a setvalued mapping \(\Gamma:S _{N} \rightarrow 2^{S_{N}}\), for \(x \in S _{N}\), by
From (3) and (24), \(\Gamma(x)\) can be rewritten as
From condition (b) in this theorem, the mapping \(\Gamma:S _{N} \rightarrow 2^{S_{N}}\backslash \{\emptyset \}\) is a welldefined setvalued mapping with universally inductive values in \(S _{N}\). Next we show that Γ is ≽increasing upward. From condition (a), for every \(i \in N\), \(f _{i}\) is orderpositive from \((S _{i}, \succcurlyeq _{i}) \times ( S_{i}, \succcurlyeq _{i})\) to \(( R, \geq )\). From condition (c), it implies that, for every \(k = 0, 1, 2,\ldots \Pi _{k}: S _{N}\to S _{N}\) is ≽increasing. Then, for arbitrary \(x, y\in S _{N}\) with \(x\preccurlyeq y\), similarly to (15) and (16), we can show that
Then (25) implies that if \(x \preccurlyeq y\), then \(\Gamma(x)\subseteq \Gamma(y)\). Hence Γ is ≽increasing upward. The elements \(x^{\prime} \in S _{N}\) and \(u^{\prime} \in \psi (x^{\prime})\) given in condition (d) in this theorem implies that \(u^{\prime} \in\Gamma (x^{\prime})\) with \(x^{\prime}\preccurlyeq u^{\prime}\). So Γ satisfies all conditions of Fixed Point Theorem A. The rest of the proof is the same as the proof of Theorem 1. □
Similar to Corollary 1, as an application of Theorem 2 to partially ordered Banach spaces, we have
Corollary 2
Let \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) be an nperson repeated game. Suppose that, for every \(i\in N\), \(S _{i}\) is a nonempty compact subset of a partially ordered Banach space (\(B _{i}\), \(\succcurlyeq _{i}\)). Let \((B _{N}, \succcurlyeq )\) be the product partially ordered Banach space of \((B _{i}, \succcurlyeq _{i})\)’s, where ≽ is the componentwise partial order of \(\succcurlyeq _{i}\)’s. Suppose that the following conditions are satisfied:

(a)
For every \(i \in N\), \(f _{i}\) is orderpositive from \((S _{i}, \succcurlyeq _{i})\times (S_{i},\succcurlyeq _{i}) \) to \((R, \geq )\);

(b)
For every \(x \in S _{N}\), \(\psi (x)\) is a nonempty closed subset of S_{ N };

(c)
For every \(k = 1, 2,\ldots\,, A _{k}: S _{N}\to S _{N}\) is an ≽increasing operator;

(d)
There are elements \(x^{\prime} \in S_{N}\) and \(u^{\prime} \in \psi (x^{\prime})\) satisfying \(x^{\prime} \preccurlyeq u^{\prime}\).
Then the repeated game \(\mathrm{G}(N,S_{N},f,A_{k})_{k =0}^{\infty } \) has an infinitely split Nash equilibrium. Moreover, \(S(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty })\) has the properties (i) and (ii) given in Theorem 2.
By using Proposition 1, as applications of Theorem 2, or in Corollary 2, we obtain the following existence results about Nash equilibrium of nperson repeated games.
Corollary 3
Let \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) be an nperson repeated game as given in Theorem 2 (or in Corollary 2). If conditions (a)–(d) listed in Theorem 2 (or in Corollary 2) are satisfied, then this repeated game has a Nash equilibrium.
Remark 2
Theorems 1, 2 and Corollaries 1, 2, 3 provide some conditions for the existence of infinitely split Nash equilibrium or Nash equilibrium in repeated games. Notice that these conditions are just necessary conditions and are not sufficient conditions. Hence, if the conditions of these existence results do not hold for some repeated games, there still may exist an infinitely split Nash equilibrium. It only means that it cannot be ensured that there is one, if these conditions are not satisfied.
Applications to repeated extended Bertrant duopoly model of price competition
In [4], the present author generalized the Bertrant duopoly model of price competition with two firms from the same price model (see [6]) to the model with possibly different prices. Then the dual extended Bertrant model is introduced and an existence theorem of split Nash equilibrium for the Markov dual extended Bertrant duopoly model of price competition is proved in [4]. We review this duopoly model below.
The extended Bertrant duopoly model of price competition is a model of oligopolistic competition that deals with two profitmaximizing firms, named by 1 and 2, in a market. In this model, it is assumed that the two firms have constant returns to scale technologies with costs \(c _{1}\) > 0 and \(c _{2}\) > 0, per unit produced, respectively, where the costs \(c _{1}\) and \(c _{2}\) are possibly different. Without loss of the generality, we assume
The inequality (26) means that the qualities of the products by these two firms may be different. More precisely, the quality of the products in firm 1 may not be as good as the quality of the products in firm 2.
Let \(p _{j}\) be the price of the products by firm j, for \(j = 1, 2\). Let \(\delta ( p _{1}, p _{2} )\) be the demand function in this duopoly market. Let \(\delta _{j} ( p _{1}, p _{2})\) be the sale function for firm j, for \(j = 1, 2\). f and \(\delta_{j}\) are assumed to be continuous functions of two variables and strictly decreasing with respect to every given variable. Suppose that there are positive numbers \(\bar{p}_{j}\), for \(j = 1, 2\), such that, for all \(p _{k}\)
Suppose that the socially optimal (competitive) output level in this market is strictly positive and finite for every firm
For given prices \(p _{1}\), \(p _{2}\), set by firms 1 and 2, respectively, the market is assumed to be clear. That is,
Let \(\lambda =c_{1}/c_{2}\), that defines the ratio of the qualities of the products by firm 1 to firm 2. From the assumption (17), we have \(\lambda \in (0, 1]\). Considered as a noncooperative strategic game, the competition takes place as follows: The two firms simultaneously name their prices \(p _{1}\), \(p _{2}\), respectively. The sales \(\delta_{1}(p _{1}, p _{2})\) and \(\delta_{2}(p _{1}, p _{2})\) are then satisfied
and
We assume that the firms produce to order and so they incur production costs only for an output level equal to their actual sales. Therefore, for given prices \(p _{1}\), \(p _{2}\), the firm j has profits
In [4], an existence theorem for Nash equilibrium of the extended Bertrant duopoly model is proved. We recalled it below for easy reference.
Theorem 6.1 in [4]
In the extended Bertrant duopoly model, there is a unique Nash equilibrium (\(\hat{p}_{1},\hat{p}_{2}\)). In this equilibrium, both firms set their prices equal to their costs, respectively: \(\hat{p}_{1} = c _{1} \), \(\hat{p}_{2} = c _{2} \).
This extended Bertrant duopoly model of price competition with two firms is a 2person static game. It is denoted by \(\mathrm{G}(N, S _{N}, u )\), where \(N= \{1, 2\}\), \(S _{j} \in [0,\bar{p}_{j}]\), \(u = (u _{1}, u _{2})\), and \(u _{j}\) is defined by (30), for \(j = 1, 2\), respectively.
For every natural number k, after the two firms repeated play the game k times and, for each time, the game is played as a static 2person simultaneousmove game and before they name their prices to play again, every firm always considers the reaction of its competitor to its strategy (price) applied in the previous time. Suppose that when this game is played in the kth time, the two firms names their prices as \(p_{j}^{k  1}\), for \(j= 1, 2\), respectively. To optimize their utilities, for example, firm 1 could try to increase its profits by increasing the price from \(p_{1}^{k  1}\) to \(p_{1}^{k}\) (never excess \(p_{2}^{k  1}\)), even though decreasing its sales. Meanwhile, firm 2 could try to increase its profits by decreasing the price from \(p_{2}^{k  1}\) to \(p_{2}^{k}\) (never lower than \(p_{1}^{k  1}\)) for increasing its sales. Suppose that such performance is defined by a linear transformation \(A _{k}\) from \(( p_{1}^{k  1},p_{2}^{k  1} ) \) to \(( p_{1}^{k},p_{2}^{k} ) \). Here we assume \(p_{1}^{0}=p _{1}\), and \(p_{2}^{0}= p _{2}\), which are the prices set by the two firms in the very first time. So, for \(k= 1, 2,\ldots \) , there is a \(2\times 2 \) matrix \(M _{k}\):
where \(0 \leq \alpha _{k}\), \(\beta _{k}\leq 1\), such that
It implies
Hence the process of repeatedly playing the static game \(\mathrm{G}(N, S _{N},u)\) with the sequence of linear transformations \(\{ A_{k}\}_{k =0}^{\infty } \) defined by (32) is a dynamic game, that is, the repeated extended Bertrant duopoly model of price competition. It is a special repeated game denoted by \(\mathrm{G}(N,S_{N},u,A_{k})_{k = 0}^{\infty } \).
From Definition 4, $\stackrel{\u2322}{p}=({\stackrel{\u2322}{p}}_{1},{\stackrel{\u2322}{p}}_{1})\in {S}_{N}$ is an infinitely split Nash equilibriums of the repeated game \(\mathrm{G}(N,S_{N},u,A_{k})_{k = 0}^{\infty } \), if it satisfies
Theorem 3
For the infinitely split Nash equilibrium problem of the repeated extended Bertrant duopoly model \(\mathrm{G}(N,S_{N},u,A_{k})_{k = 0}^{\infty } \), we have

(i)
If \(c_{1} = c _{2} = c\), then, for any sequence of linear transformations \(\{ A_{k}\}_{k = 0}^{\infty }\) defined in (32), $\stackrel{\u2322}{p}=({\stackrel{\u2322}{p}}_{1},{\stackrel{\u2322}{p}}_{1})=(c,c)$ is the unique infinitely split Nash equilibrium;

(ii)
If \(c_{1} < c _{2}\), then there exists a unique infinitely split Nash equilibrium $\stackrel{\u2322}{p}=({\stackrel{\u2322}{p}}_{1},{\stackrel{\u2322}{p}}_{1})=({c}_{1},{c}_{2})$, only if all linear transformations \(A_{k}\)’s equal to the identity, that is,
Proof
Part (i) is an immediate consequence of Theorem 6.1 in [4]. To prove part (ii), notice that every infinitely split Nash equilibrium of the repeated game \(\mathrm{G}(N,S_{N},u,A_{k})_{k = 0}^{\infty } \) is a split Nash equilibrium of the dual game \(\mathrm{G}(N,S_{N},u,A_{1})^{2}\) studied in [4]. From Theorem 6.2 in [4], $({\stackrel{\u2322}{p}}_{1},{\stackrel{\u2322}{p}}_{1})=({c}_{1},{c}_{2})$ is the unique split Nash equilibrium of the dual game \(\mathrm{G}(N,S_{N},u,A_{1})^{2}\), only if
It implies \(\Pi _{2}\) = \(A _{2}\). Applying Theorem 6.1 in [4] again, it follows that $({\stackrel{\u2322}{p}}_{1},{\stackrel{\u2322}{p}}_{1})=({c}_{1},{c}_{2})$ is the unique split Nash equilibrium of the triple game \(\mathrm{G}(N,S_{N},u,I,A_{2})^{3}\), only if
Then (35) is proved by induction. □
Since \(\delta_{1}\) and \(\delta_{2}\) are continuous functions, from the condition (27) and the definition (30) of the utility functions, it implies that there exists \(M > 0\), such that

(M)
\(\vert u _{i}(p) \vert \leq M\), for \(i= 1, 2\) and for all \(p \in S _{N}\).
Let ρ be the discount factor of this dynamic game. By Proposition 1, as a consequence of Theorem 3, we have:
Corollary 4
In the repeated extended Bertrant duopoly model \(\mathrm{G}(N,S_{N},u,A_{k})_{k = 0}^{\infty } \), there is a Nash equilibrium $({\stackrel{\u2322}{p}}_{1},{\stackrel{\u2322}{p}}_{1})=({c}_{1},{c}_{2})$ at which, every firm has zero discounted value of utility, that is,
Conclusions and open problems
Nash equilibrium problems in static games has been extensively studied by many authors and the equilibrium theory has become an important branch in both of mathematics and economic theory. The concept of split Nash equilibrium problems was introduced for studying two related static games or a static game repeated twice.
In the real world, a strategic game may be infinitely repeated to play that arises a repeated game of dynamic model. For every natural number k, after the players repeated the static game k times and before they play this static game again, every player always considers the reaction of its competitors to its strategy applied in the previous time. The players may make arrangements of strategies to use in the next play to optimize their utilities. If the profile of the arranged strategies is represented by a mapping on the profile set, it arises the infinitely split Nash equilibrium problems for repeated games.
In this paper, we prove the solvability of some infinitely split Nash equilibrium problems for repeated games by applying a fixed point theorem on posets, in which the considered utility functions are not required to have any continuity conditions.
To conclude this paper, we list the following problems below for consideration:

1.
Prove the existence of infinitely split Nash equilibrium for repeated games by applying fixed point theorems on topological vector spaces, in which the considered utility functions may be required to satisfy some continuity conditions.

2.
Construct some iterated algorithms to approximate infinitely split Nash equilibriums for repeated games.

3.
Introduce the concept of infinitely split variational inequality problems and prove the solvability of some infinitely split variational inequality problems.

4.
Construct some iterated algorithms to approximate the solutions to some infinitely split variational inequality problems.

5.
Similarly, to Theorem 3 in this paper about the repeated extended Bertrant duopoly model, prove the existence of infinitely split Nash equilibrium for Cournot model of repeated pricing games.

6.
Extend the competitive equilibrium growth models (see [7]) to dynamic model of repeated games and prove the existence of infinitely split equilibrium. That may be considered as some new methods in recursive macroeconomic theory.
Abbreviations
 \(\mathcal{F}(\Gamma)\) :

the set of fixed points of a mapping Γ
 N :

the set of players in a game
 \(S _{N}\) :

the set of profiles in a game
 f :

the utility vector mapping in a game
 \(\mathrm{G}(N, S _{N}, f)\) :

an nperson game in which N is the set of players, \(S _{N}\) is the set of profiles and f is the utility vector mapping
 \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) :

a repeated game
 \(\operatorname{NSNE}(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty })\) :

an infinitely split Nash equilibrium problem associate with a repeated game \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } \)
 \(S(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty })\) :

the set of infinitely split Nash equilibriums of \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty } \)
References
 1.
Li, J.L.: Several extensions of the Abian–Brown fixed point theorem and their applications to extended and generalized Nash equilibria on chaincomplete posets. J. Math. Anal. Appl. 409, 1084–1092 (2014). https://doi.org/10.1016/j.jmaa.2013.07.070
 2.
Bade, S.: Nash equilibrium in games with incomplete preferences. Econ. Theory 26, 309–332 (2005). https://doi.org/10.1007/s0019900405411
 3.
Chang, S.S., Wang, L., Kun, Y., Wang, G.: Moudafis open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems. Fixed Point Theory Appl. 2014, 215 (2014). https://doi.org/10.1186/168718122014215
 4.
Li, J.L.: Split equilibrium problems for related games and applications to economic theory. https://arxiv.org/ftp/arxiv/papers/1610/1610.05409.pdf
 5.
Li, J.L.: Inductive properties of fixed point sets of mappings on posets and on partially ordered topological spaces. Fixed Point Theory Appl. 2015, 211 (2015). https://doi.org/10.1186/s1366301504618
 6.
MasColell, A., Whinston, M.D.: Microeconomics Theory. Oxford University Press, Oxford (1995)
 7.
Osborne, M.J.: An Introduction to Game Theory. Oxford University Press, Oxford (2004)
 8.
Stokey, N.L., Lucas, R. Jr.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge (1989)
 9.
Xie, L.S., Li, J.L., Yang, W.S.: Orderclustered fixed point theorems on chaincomplete preordered sets and their applications to extended and generalized Nash equilibria. Fixed Point Theory Appl. 2013, Article ID 192 (2013). https://doi.org/10.1186/168718122013192
 10.
Xu, H.K.: A variable Krasnoselskii–Mann algorithm and the multiplesets split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)
 11.
Xie, L., Li, J., Petruşel, A., Yao, J.C.: Orderclustered fixed point theorems and their applications to Pareto equilibrium problems. Fixed Point Theory 18(2), 755–772. (2017)
 12.
Zhang, C.J.: SetValued Analysis with Applications in Economics. Sciences Press, Beijing (2004) (in Chinese)
 13.
Bnouhachem, A.: Strong convergence algorithms for split equilibrium problems and hierarchical fixed point theorems. Sci. World J. 2014, Article ID 390956 (2014). https://doi.org/10.1155/2014/390956
 14.
Bui, D., Son, D.X., Jiao, L., Kim, D.S.: Line search algorithms for split equilibrium problems and nonexpansive mappings. Fixed Point Theory Appl. 2016, 27 (2016)
 15.
Carl, S., Heikkila, S.: Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory. Springer, New York (2010)
 16.
Censor, Y., Gibali, A., Reich, S.: Algorithms for split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
 17.
Ma, Z., Wang, L., Chang, S.S., Duan, W.: Convergence theorems for split equality mixed equilibrium problems with applications. Fixed Point Theory Appl. 2015, 31 (2015). https://doi.org/10.1186/s136630150281x
Acknowledgements
The author is very grateful to Professor Hongkun Xu for communications and his valuable suggestions, which improved the presentation of this paper. The author also thanks the National Natural Science Foundation of China (11771194) for partially support about this research.
Availability of data and materials
This manuscript is available for the reader after January 20, 2018.
Funding
This research was partially supported by the National Natural Science Foundation of China (11771194).
Author information
Affiliations
Contributions
All authors read and approved the final manuscript.
Corresponding author
Correspondence to Jinlu Li.
Ethics declarations
Competing interests
The author declares that he has no competing interests.
Additional information
Dedicated to Professor Hongkun Xu
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
MSC
 49J40
 49J52
 91A10
 91A25
 91A80
Keywords
 Repeated game
 Infinitely split Nash equilibrium
 Nash equilibrium
 Fixed point theorem on posets