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.
Highlights
The well-known “Bertrand paradox” describes a game situation in which two firms engage in price competition in a static setting [1]
The evolutionary game is the theory of dynamic adaption and learning in repeated games played by bounded rational players
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
Summary
The well-known “Bertrand paradox” describes a game situation in which two firms engage in price competition in a static setting [1]. 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. 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.
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