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- Open Access
Infinitely split Nash equilibrium problems in repeated games
- Jinlu Li^{1}Email author
https://doi.org/10.1186/s13663-018-0636-1
© The Author(s) 2018
- Received: 24 January 2018
- Accepted: 19 March 2018
- Published: 23 April 2018
Abstract
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 (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.
Keywords
- Repeated game
- Infinitely split Nash equilibrium
- Nash equilibrium
- Fixed point theorem on posets
MSC
- 49J40
- 49J52
- 91A10
- 91A25
- 91A80
1 Introduction and preliminaries
1.1 Definitions and notations in game theory
- 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 \(i \in N\).
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 n-person noncooperative strategic games below.
1.2 n-person dual games
An n-person game \(G = (N, S _{N}, f)\) 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 \(G = (N, S _{N}, f)\) 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 [6]). The dynamic model for n-person repeated games will be studied in Sect. 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 \(\mathrm{G}(N, S _{N},f)\) 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 \(S _{N}\). 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 n-person 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.
2 Split Nash equilibrium problems in dual games
2.1 Definitions and notations for split Nash equilibrium problems in dual games
Definition 1
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 chain-complete 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 [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])
- 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 \(y _{*}\succcurlyeq ^{P} v _{*}\).
- (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 \(x^{*}\succcurlyeq ^{P} y _{*}\).
2.2 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.
Definition 2
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
- (a)
For every \(i \in N\), \(f _{i}\) is order-positive 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 \(x^{\prime} \preccurlyeq u^{\prime}\);
- (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
2.3 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 [1], 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:
Corollary 1
- (a)
For every \(i \in N\), \(f _{i}\) is order-positive 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 \(x^{\prime} \preccurlyeq u^{\prime}\),
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.
3 Infinitely split Nash equilibrium problems in repeated games
3.1 Definitions and notations of n-person repeated games
Let \(\mathrm{G}(N, S _{N}, f)\) be an n-person 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 component-wise partial order of \(\succcurlyeq _{i}\)’s naturally equipped on \(S _{N}\).
- (M)
\(\vert f _{i}(x) \vert \leq M\), for every \(i\in N\) and for all \(x \in S _{N}\).
Definition 3
Definition 4
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 n-person repeated game is a Nash equilibrium of this repeated game.
Proof
Theorem 2
- (a)
For every \(i \in N\), \(f _{i}\) is order-positive 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 \(x^{\prime} \preccurlyeq u^{\prime}\).
- (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
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
- (a)
For every \(i \in N\), \(f _{i}\) is order-positive 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 n-person repeated games.
Corollary 3
Let \(\mathrm{G}(N,S_{N},f,A_{k})_{k = 0}^{\infty }\) 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.
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.
4 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 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.
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 2-person 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.
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 \(\mathrm{G}(N,S_{N},u,A_{k})_{k = 0}^{\infty } \).
Theorem 3
- (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
Then (35) is proved by induction. □
- (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
5 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.
- 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.
Notes
Declarations
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).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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.
Authors’ Affiliations
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