Complexity

Volume 2018, Article ID 9649863, 13 pages

https://doi.org/10.1155/2018/9649863

## The Evolution of Price Competition Game on Complex Networks

^{1}School of Modern Posts, Xi’an University of Posts and Telecommunications, Xi’an, ShaanXi, China^{2}Industrial Economics Research Institute, Xi’an University of Posts and Telecommunications, Xi’an, ShaanXi, China^{3}Department of Mechanical and Materials Engineering, College of Engineering & Applied Science, University of Cincinnati, Cincinnati, OH, USA

Correspondence should be addressed to Jing Shi; ude.cu@ihs.gnij

Received 29 November 2017; Revised 22 April 2018; Accepted 12 May 2018; Published 9 July 2018

Academic Editor: Shahadat Uddin

Copyright © 2018 Feng Jie Xie and Jing Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The well-known “Bertrand paradox” describes a price competition game in which two competing firms reach an outcome where both charge a price equal to the marginal cost. The fact that the Bertrand paradox often goes against empirical evidences has intrigued many researchers. In this work, we study the game from a new theoretical perspective—an evolutionary game on complex networks. Three classic network models, square lattice, WS small-world network, and BA scale-free network, are used to describe the competitive relations among the firms which are bounded rational. The analysis result shows that full price keeping is one of the evolutionary equilibriums in a well-mixed interaction situation. Detailed experiment results indicate that the price-keeping phenomenon emerges in a square lattice, small-world network and scale-free network much more frequently than in a complete network which represents the well-mixed interaction situation. While the square lattice has little advantage in achieving full price keeping, the small-world network and the scale-free network exhibit a stronger capability in full price keeping than the complete network. This means that a complex competitive relation is a crucial factor for maintaining the price in the real world. Moreover, competition scale, original price, degree of cutting price, and demand sensitivity to price show a significant influence on price evolution on a complex network. The payoff scheme, which describes how each firm’s payoff is calculated in each round game, only influences the price evolution on the scale-free network. These results provide new and important insights for understanding price competition in the real world.

#### 1. Introduction

The well-known “Bertrand paradox” describes a game situation in which two firms engage in price competition in a static setting [1]. They sell a homogeneous good and have identical unit costs. The outcome is that both charge a price equal to the marginal cost. The game is called a paradox because the two firms could easily earn positive profits by charging a higher price. It has been studied by many researchers, indicating that some reasonable modifications to the Bertrand setting indeed resolve the paradox. Firstly, within the homogeneous-good framework, the Bertrand paradox can be resolved by introducing capacity constraints [2], sluggish consumers [3], the endogenous choice of production technologies [4], or the endogenous timing of price decisions [5]. Furthermore, differentiation [6, 7], uncertainty with respect to demand [8, 9] or product quality [10, 11], and nonconstant unit cost [12–15] are found to be the important factors for eliminating the Bertrand paradox. Thirdly, many works in laboratory settings focus on the oligopoly competition but do not observe the Bertrand equilibrium. This can be mainly attributed to the relaxation of the assumption regarding the rationality of economic agents in the laboratory environment [16–18].

Recently, the studies about the evolutionary game on complex networks reveal that topological structures can directly influence the evolution equilibrium of the game. In a pioneer work, Nowak and May introduced a two-dimensional spatial lattice, that is, a square lattice, to analyze the evolution of a prisoner’s dilemma game (PDG) [19]. They observed a remarkable cooperation level in this noncooperative game. Following this work, a wealth of studies provides additional evidence for the positive effect of various spatial lattices on cooperation in PDG [20–24]. Starting from the discovery of the small-world network [25, 26] and scale-free network [27], many studies about the evolution of PDG on these two network styles have emerged [28–32]. Furthermore, other game models, such as the snowdrift game, public good game, and stag hunt game, have been introduced to characterize different game situations among players. The studies about the evolution of these games on the small-world network and scale-free network yield abundant outcomes [33–38].

Inspired by the above works, we attempt to explore the Bertrand game from a new theoretical perspective—an evolutionary game on complex networks. The justification of such a perspective lies in two aspects. First, firms in the real world are bounded rational, but not complete rational. Huck et al. provided experimental tests for various learning theories in Bertrand games and concluded that firms imitate the most successful behavior [39, 40]. This is evidence of the bounded rationality of firms facing a price decision in the real world. Second, the previous works find that competitive relations of firms in the real world have typical structural properties of complex networks [41–43]. Integrating the two aspects, we confirm that the evolutionary game theory on complex networks is suitable to study the price competition problem of bounded rational firms which have complex competitive relations.

To this end, some modifications to the Bertrand model are inevitable. Firstly, the evolutionary game theory studies the strategy evolution of large populations who are bounded rational [44, 45]. Then, “two complete rational players” in a static setting should be modified as “a lot of bounded rational players” in an evolutionary dynamic setting. Furthermore, the interactions between players are supposed to happen at random in the evolutionary game theory, that is, the players interact in a well-mixed situation [44, 45]. Also, while the complex network is introduced, the interaction of firms should be based on the complex competitive relations. We study price evolution under these new modifications. This is the first work that understands the price competition problem with the consideration of competition relations among the firms who charge the price. Our study explores the emergence of price keeping and identifies the positive effect of a complex competition relation on price keeping. Some important economic factors which influence the price evolution are also found and analyzed.

The remainder of this paper is organized as follows. Section 2 describes the model. Section 3 presents the theoretical analysis. Section 4 provides the simulation results and explanations. Section 5 summarizes our findings and concludes the paper.

#### 2. The Evolution of the Price Competition Game on Complex Networks

As the first attempt of exploring the price competition problem from the theoretical perspective of the evolutionary game on complex networks, three widely applied network models are used to characterize the competitive interactions between firms, namely, square lattice [19], WS small-world network [25, 26], and BA scale-free network [27]. The average connectivity among WS networks and BA networks is set as four which is identical to that of a square lattice. Each node in networks is occupied by a firm. Each edge defines a competitive relation between two firms.

Based on the Bertrand model, the payoff of firm who has a competitive relation with firm depends on their prices and . That is where is the constant unit cost and is the demand function which generally has an expression . In the expression, is the maximum of demand quantity while and are the slope of demand function which indicates the sensitivity of demand to price. Without losing the generality, we set the constant unit cost as such that the model is simplified. Accordingly, the payoff becomes

Under the framework of the evolutionary game theory, firms are bounded rational and thus have no capability to make the perfect decision of setting the price at marginal cost. They just make a simple decision: keep the original price or cut the price to , where is the degree of price cutting.

All firms simultaneously decide what prices they should offer. Each firm uses the same price for all of its competitive relations, that is, for all of its neighbor firms. The payoff of a firm can be measured by two payoff schemes: accumulated payoff or average payoff. It is worth mentioning that under different payoff schemes, the effects of scale-free networks on cooperation are different accordingly [31, 46–48]. Therefore, the payoff scheme is one of the important factors to be examined in the current work.

Price evolution is carried out implementing the rule of imitate best. In the previous studies of the evolutionary game on complex networks, various imitation rules are provided, such as imitate best [46, 49–51], imitate better with probability [46, 52–54], and imitate better with probability and error [46, 55, 56]. Here, we use the rule of imitate best, because the work of Huck et al. provides experimental evidence that firms imitate the most successful behavior, that is, the best one [39, 40].

The above price competition game and evolution mechanism can be described more specifically as follows. In each round of the game, that is, at each game time , each firm offers a price . The payoff of a firm at game time is calculated by accumulated payoff or average payoff. Under the accumulated payoff scheme, the payoff of firm is the sum over all interactions of its neighbor firms, and can be written as , where is the set of neighbor firms of . Under the average payoff scheme, the payoff of firm is obtained by dividing the accumulated payoff by the number of its neighbor firms, that is, . After all firms obtain payoffs in game time , they update prices simultaneously. When firm updates its price, it compares the payoffs between itself and all of its neighbor firms and adopts the price that yields the highest payoff in game time . After all firms have updated their prices, the next game time begins.

#### 3. Analysis

According to payoff function (2), for each pair of firms who have a competitive relation, they both receive upon mutual price , or upon mutual price . One offering price receives an amount while the other offering price receives 0 payoff. Such a price-competitive situation can be described by payoff matrix . That is

According to the evolutionary game theory, the relative order of four elements of payoff matrix can lead to different evolutionary equilibriums [57, 58]. Denoting four payoffs in the matrix , respectively, as , , , and , the relative order of matrix elements *R*, *S*, *T*, and is analyzed in the following:
(1)Since the parameters satisfy , , , , and , the order “” is supported. Besides, indicates . It follows that “*The order**is supported*.”(2)Given , , denoted as inequality (1), is supported. Since , we can set . Then, inequality (1) transforms to , denoted as inequality (2). Furthermore, since , inequality (2) transforms to , denoted as inequality (3). It follows that , denoted as inequality (4), is supported. While , inequalities and are supported. Besides, since , inequality is supported. Accordingly, , denoted as inequality (5), is supported. However, inequality (5) contradicts with inequality (4), indicating that “*While*,* is supported*.” While , is supported. This contradicts with inequality (3), indicating that “*While *,* is supported*.” While , is supported. Inequality (4) transforms to . This means that “*While*,*is supported if**; otherwise,**is supported*.”(3)Given , , denoted as inequality (6), is supported. With and , inequality (6) transforms to . Since , is supported. Thus, “*While*,*is supported; otherwise,**is supported*.”

With the above analysis results, we obtain the order of *R*, *S*, *T*, and in Table 1 and present the corresponding evolutionary equilibriums. Full price cutting and full price keeping mean that all firms offer price and price , respectively.