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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.

The well-known “Bertrand paradox” describes a game situation in which two firms engage in price competition in a static setting [

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) [

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 [

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 [

The remainder of this paper is organized as follows. Section

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 [

Based on the Bertrand model, the payoff

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

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 [

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 [

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

According to payoff function (

According to the evolutionary game theory, the relative order of four elements of payoff matrix

Since the parameters satisfy

Given

While

While

While

Given

With the above analysis results, we obtain the order of

Order of

Range of |
Range of |
Order of |
Evolutionary equilibrium |
---|---|---|---|

Full price cutting | |||

Full price cutting | |||

Full price cutting | |||

Full price cutting | |||

Full price cutting or full price keeping |

It can be seen in Table

In the study of the evolutionary game on complex networks, a complete network represents the random interactions in a well-mixed situation [

All the parameters in our model and their associated ranges for simulation experiments are summarized in Table

Summary of model parameters for simulation experiment.

Parameter | Description | Values used in experiments |
---|---|---|

Competition scale | ||

Original price | ||

Cutting price | ||

Slope of demand function | ||

The maximum of demand |

In all simulations, the initial prices (

In Figure

Probability distribution of the density of price-keeping

An outstanding feature of the distribution in Figure

Based on the above results, we know that although the price-keeping phenomena emerge in four different network styles, the small-world network and scale-free network are more beneficial for price keeping than the complete network and square lattice. The complex competitive relation networks in the real world generally have a small-world property and scale-free degree distribution at the same time [

The above results are derived from the combination of all parameters (

Under the two different payoff schemes, we first explore the level of density

Density of price keeping,

It can be observed in Figures

The payoffs of firms with a different degree

Moreover, in Figures

Based on the above results, we attempt to specify the effects of the other four parameters (

Figure

Density of price keeping,

It can be seen in Figure

Firstly, according to the theoretical analysis in Section

The relation between the values of

It can be seen in Figure

Secondly, in Table

Values of

10 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 | |
---|---|---|---|---|---|---|---|---|---|---|---|

3 | |||||||||||

4 |

It can be seen in Table

According to the results in Section

Figure

Density of price-keeping,

It can be seen in Figure

Firstly, with

Secondly, we transfer the equation

The evolutionary game is the theory of dynamic adaption and learning in repeated games played by bounded rational players. It is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. When the interacting players in a game are linked in a specific complex network style, the evolutionary game on complex networks, which integrates the evolutionary game theory and complex network theory, provides an effective method to obtain the solution of the game.

In this work, based on the Bertrand model, we study the price competition problem from the perspective of the evolutionary game on complex networks. To the best of our knowledge, this is the first work that attempts to understand the price competition problem under the consideration of the complex competition relations among the firms who charge the price. We find that once the firms are considered as a bounded rationality and just make a simple decision on keeping the price or cutting the price, full price keeping is one of the evolutionary equilibriums. More importantly, the price-keeping phenomenon emerges in the small-world network, scale-free network, and square lattice much more frequently than in the complete network. In the fierce competition environment where more than 90% of the firms cut their price, the small-world network, scale-free network, and square lattice are a prominently beneficial example for the other less than 10% of the firms to keep the price. While the square lattice has little advantage to achieve full price keeping, the small-world network and scale-free network exhibit a stronger capacity in full price keeping than the complete network. These results indicate that the complex competition relation among firms is a crucial factor to maintain the price in the real world. Besides, competition scale, original price, degree of price cutting, and demand sensitivity to price also influence the price evolution. Specifically, the larger the competition scale, the easier for the whole system to keep the original price; the more severe the price cutting of some firms at the beginning, the easier for all the firms to keep the original price at the evolutionary equilibrium state; the more sensitive the demand to the price cutting, the more difficult for all the firms to keep the original price. The effect of original price on price evolution is relatively complex. Both extremely low and high original prices are not beneficial for price keeping. There exists a medium range of original price under which an optimal price-keeping equilibrium is achieved. Lastly, the payoff scheme, which describes how each firm’s payoff is calculated in each round game, influences the price evolution on the scale-free network. Under the accumulated payoff and average payoff, there are significant differences in the density of firms keeping the price in the evolution system. These results provide new and important insights for understanding price competition in the real world.

Based on the current results, we can also envision some important extension work in the future. Firstly, the WS small-world network model and BA scale-free network model are two of the most classic network models in the complex network theory. Several network models are based on the two models for characterizing various structural properties of a complex network, such as degree correlation and community structure and mixing pattern [

The authors declare that there is no conflict of interest regarding the publication of this paper.

The authors thank the financial support of the National Social Science Foundation of China (no. 15BGL014) and Social Science Youth Foundation of the Ministry of Education of China (no. 12YJCZH226) and for the comments and suggestions from anonymous reviewers.