Infinitely Split Nash Equilibrium Problems in Repeated Games

In this paper, we introduce the concept of infinitely split Nash equilibrium in repeated games in which the profile sets are chain-complete posets. Then by using a fixed point theorem on posets in [8], we prove an existence theorem. As an application, we study the repeated extended Bertrant duopoly model of price competition.


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 readers are referred to [1], [67], [1014] for more details. Let n be a positive integer greater than 1. An n-person noncooperative strategic game, simply called an nperson game, consists of the following elements: 1. the set of n players denoted by N with N = n; 2. the set of profiles SN = iN Si, where Si is the pure strategy set for player i  N; 3. the utility vector mapping f = iN fi: SN  R n , where fi is the utility (payoff) function of player i, for i  N.
This game is denoted by G (N, SN, f). Throughout this paper, we always assume that, in the products iN Si, iN fi and kN\{i} Sk, the players appear in the same sequential orders. As usual, for every i  N, we often denote a profile of pure strategies for player i's opponents by xi = (x1, x2, …, xi 1, xi +1, xn).
The set of profiles of pure strategies for player i's opponents is then denoted by Si = kN\{i} Sk.
Hence we may write x  SN as x = (xi, xi) with xi  Si, for i  N. (1) Moreover, for every xi  Si, we denote fi(Si, xi) = {fi(zi, xi): zi  Si}.
From f = iN fi in the game G(N, SN, f), for any x  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], [7], [1014]). We recall the definition of Nash equilibrium in n-person noncooperative strategic games below.
Let G(N, SN, f) be an n-person game. A profile of pure strategies It can be rewritten as , for every i  N and for every z  SN.
In an n-person game G(N, SN, f), we define a mapping F: SNSN  R n by F(z, x) = iN fi(zi, xi), for any x, z  SN.
F(z, x) is called the utility vector at profile x  SN associated to z  SN. It is clearly to see Let  n be the component-wise partial order on R n satisfying that, for x, z  SN, From (25), the Nash equilibrium can be rewritten by: a profile x   SN is a Nash equilibrium of G(N, SN, f) if and only if,

n-person dual games
An n-person game G = (N, SN, f) is static. Some games in the real world may not be static. That is, it may not be one-shot nature. It is more realistic for this game to be repeatedly played. The dynamic model of game based on an n-person game G = (N, SN, f) is formulized by the process that this static game is repeated infinite periods (times). It is called an n-person repeated game, in which there is a discount factor involved for the utilities (see [10]). The dynamic model for nperson repeated games will be studied in section 3 In this paper, we first consider a special model: n-person dual game. An n-person dual game based on an n-person game G(N, SN, f) is modeled as follows: At first, the players play the game as a static n-person 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 SN. Hence, if x  SN is the profile used by the players in the first time, then Ax  SN will be the profile used by the players in the second time. This n-person dual game is denoted by G(N, SN, f, A) 2 .
Then we ask: Is there a Nash equilibrium x   SN of the game G(N, SN, f) (first play) such that A x   SN 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 [25] and [9], 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 [6], 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 section 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 section 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.

Definitions and notations for split Nash equilibrium problems in dual games
Let G(N, SN, f) be an n-person game. Throughout this paper, unless otherwise stated, we assume that, for every player i  N, his strategy set Si is nonempty and is equipped with a partial order i. That is, for every i  N, player i's strategy set is assumed to be a poset (Si, i). As the product partially ordered set of (Si, i)'s, the profile set is also a poset (SN, ) in which the partial order  is the component-wise partial order of i's. That is, for x = (x1, x2, …, xn) and y = (y1, y2, …, yn)  SN, we have that y  x, if and only if, yi i xi, for all i  N.
For every i  N, (Si, i) is similarly defined to be the product poset of (Sj, j)'s, j  i, in which i is the corresponding component-wise partial order of j's, j  i.

Definition 1. Let G(N, SN, f, A) 2 be an n-person dual game. The split Nash equilibrium problem associated with this dual game, denoted by
such that the profile A x   SN solves the following From (6), a profile x   SN satisfying (78) can be rewritten as: and Such a profile x  in SN is called a split Nash equilibrium of this split Nash equilibrium problem 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 SN, NSNE(G(N, SN, f, I) 2 ) reduces to the classical Nash equilibrium problem for the game G(N, SN, 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 [8]. In this theorem, the underlying space is a chaincomplete poset and the considered mapping is just required to satisfy order-increasing 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 [8]. 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 [8])
. Let (P,  P ) be a chain-complete poset and let : P  2 P \{} be a set-valued mapping satisfying the following three conditions: There is an element y* in P and v*  (y*) with y*  P v*.
Let () denote the set of fixed points of . Then is a nonempty inductive poset; and  has an  P -maximal fixed point x* with x* P y*.

An existence theorem for split Nash equilibrium in dual games
We need the following concept, order-positive, 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.
and (U,  U ) be posets. Let C, D be nonempty subsets of X and Y, respectively. A mapping g: XY U is said to be order-positive from XY to U whenever, for Let G(N, SN, f, A) 2 be an n-person dual game. To prove an existence theorem for split Nash equilibrium problem NSNE(G (N, SN, f, A) 2 ), we need to define a mapping :  can be equivalently written, for x  SN, as for every i  N and for all z  SN}. Observation 1. In Theorem 1 given below, it is assumed that, for every x  SN, (x)  . It means that, for any given profile x  SN and for every player i  N, when player i's opponents take xi to play, there exists a strategy ti  Si such that player i will optimize his utility at the profile (ti, xi). Hence, the condition that (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.  The elements x   SN and u   (x  ) given in condition d) in this theorem satisfy that u   (x  ) such that x   u  . So  satisfies all conditions in the Fixed Point Theorem A. It follows that ()   and it satisfies the properties (i) and (ii) in Theorem A. From (9) and (10), the definition of (G (N, SN, f, A) 2 ), and (14), the definition of , we obtain (G (N, SN, f, A) 2 ) = ().
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 n-person 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 [8], it was proved that every partially ordered compact Hausdorff topological space is both chain-complete and universally inductive, as a consequence of Theorem 1, we have  N, SN, f, A) 2 ) has the properties (i) and (ii) listed in Theorem 1.

Remarks 2.
In Corollary 1, even though the profile set in the dual game G(N, SN, f, A) 2 is a subset of a Banach space, the operator A: SN  SN is not required to be linear. It may be a nonlinear operator.

Definitions and notations of n-person repeated games
Let G (N, SN, f)  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 n-person simultaneous-move 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 Ak: SN  SN, for k = 1, 2, 3, … (Since (Si, i) is just a poset, it may not be equipped with any algebraic structure. So the linearity of Ak is may not be defined). To summarizing this process, if x  SN is the profile used by the players in the first time, then A1x  SN will be the profile used by the players in the second time; A2A1x  SN will be the profile used by the players in the third time. Hence, for k = 1, 2, 3, …, Ak …A2A1x  SN will be the profile used by the players in the (k +1) th play. For simplicity, we write k = Ak Ak1 …A1A0, for k = 0, 1, 2, …. There is a discount factor 0 <  < 1. For every i  N, player i's discounted value of utility at a profile x  SN is Player i's discounted value of utility at the profile x associated with a profile z  SN is (20)

It implies
Hi(x, x) = hi(x), for every i  N and for all x  SN.
The utility vector with discounted values for this repeated game at the profile x associated with a profile z  SN is ).

Proposition 1. Every infinitely split Nash equilibrium of an n-person repeated game is a Nash equilibrium of this repeated game.
Proof. Suppose that, for every i  N, for k = 0, 1, 2, … , the following inequality holds Since 0 <  < 1, it implies It completes the proof of this proposition.  Similar to (13) for the definition of the mapping , regarding to , we need to define a mapping : Then the repeated game has an infinitely split Nash equilibrium. Moreover 2 \{} is a well-defined set-valued mapping with universally inductive values in SN. Next we show that  is -increasing upward. From condition a), for every i  N, fi is order-positive from (Si, i)(Si, i) to (R, ). From condition c), it implies that, for every k = 0, 1, 2, … k: SN  SN is -increasing. Then, for arbitrary x, y  SN with x  y, similarly to (15) and (16), we can show that F(kz, kx)  n F(kt, kx)  F(kz, ky)  n F(kt, ky), for any z, t  SN, k = 0, 1, …. (25) (25) implies that if x  y, then (x)  (y). Hence  is -increasing upward. The elements x   SN and u   (x  ) given in condition d) in this theorem implies that u   (x  ) with x   u  . So  satisfies all conditions of Fixed Point Theorem A. Rest of the proof is the same to the proof of Theorem 1.  ) 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 n-person repeated games.
be an n-person 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.
Remarks 2. Theorems 1, 2 and Corollary 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 assured that there is one, if these conditions are not satisfied.

Applications to repeated extended Bertrant duopoly model of price competition
In [6], the present author generalized the Bertrant duopoly model of price competition with two firms from the same price model (see [10]) 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 [6]. We review this duopoly model below.
The extended Bertrant duopoly model of price competition is a model of oligopolistic competition that deals with two profit-maximizing 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 c1 > 0 and c2 > 0, per unit produced, respectively, where the costs c1 and c2 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 pj be the price of the products by firm j, for j = 1, 2. Let (p1, p2) be the demand function in this duopoly market. Let j(p1, p2) be the sale function for firm j, for j = 1, 2. f and 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 j p , for j = 1, 2, such that, for all pk j(pj, pk)  0, for all pj  [0, j p ) and j(pj, pk) = 0, for all pj  j p .
Suppose that the socially optimal (competitive) output level in this market is strictly positive and finite for every firm 0 < (c1, c2) < .
Let  = , / 2 1 c c that defines the ratio of the qualities of the products by firm 1 to firm 2. From the assumption (17), we have   (0, 1]. Considered as a noncooperative strategic game, the competition takes place as follows: The two firms simultaneously name their prices p1, p2, respectively. The sales 1(p1, p2) and 2(p1, p2) are then satisfied