Stability of Equilibrium Prices in a Dynamic Duopoly Bertrand Game with Asymmetric Information and Cluster Spillovers

Bounded rationality, asymmetric information and spillover effects are widespread in the economic market, and had been studied extensively in oligopoly games, but few references discussed incomplete information in a duopoly market with rationality expectations. Considering the positive externalities brought by the spillover effect between enterprises in a cluster, a duopoly Bertrand game with bounded rationality and asymmetric information is proposed in this paper. In our model, a firm with private information, high or low marginal cost, is introduced. Interestingly, our theoretical analysis reveals that: (1) In a dynamic duopoly Bertrand game with perfect rationality and asymmetric information, the equilibrium price is positively correlated with product substitution rate and the probability of a high marginal cost, while it is negatively correlated with the cluster spillover. (2) In a dynamic duopoly Bertrand game with asymmetric information and adaptive expectation adopted by both firms, the Nash equilibrium prices are always asymptotically stable. (3) In a dynamic duopoly Bertrand game with heterogenous expectation and asymmetric information, where two firms use adaptive expectation and boundedly rational expectation respectively, the Nash equilibrium prices are locally stable under certain conditions. Furthermore, results indicate that, high product substitution rate or large probability of high marginal cost for firm 2 with private information may make the market price unstable, bifurcating or even falling into chaos, while high technology spillover is conducive to stabilize the market by contrast. It is also shown that the chaos can be controlled by a hybrid control strategy with the state variables feedback and parameter variation. Our research has an important theoretical and practical significance to the price competition in oligopoly markets.

Complex Dynamical Behaviors in a Bertrand Game with Service Factor and Differentiated Products

In this paper, taking the factor of service level provided by the manufacturers into consideration, a static duopolistic Bertrand game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic game model of duopoly Bertrand with boundedly rational is established with using the gradient mechanism. By using numerical simulation tools, there are two paths for the system to drop into chaos, that is, flip bifurcation and Neimark-Sacker bifurcation. The symmetric structures can be found from two-parameter bifurcation diagrams. Saddle-homoclinic bifurcation also can be observed from the evolution process of phase portraits. In addition, the emergence of intermittent chaos implies that the established system has the capability of self-regulating, where PM-I intermittency, PM-III intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated. At last, it can be found that the values of product differentiation degree and service spillover effect are not the bigger the better. Keeping these two parameters in a relatively small range will be conducive to the long-term stable operation of the two manufacturers.

Managing strategies of product quality and price based on trust under different power structures

This paper investigates the quality and pricing decisions in a supply chain consists of one retailer and two manufacturers. The retailer updates the trust value to manufacturers based on the historical information of product quality to determine the actual order quantity. We propose four game models by considering two situations, i.e., manufacturer-dominant scenario and retailer-dominant scenario. Based on theoretical and numerical analysis， we obtain a series of valuable management insights. For example, when a manufacturer with unreliable product quality is the leader and competes with another reliable manufacturer, the retailer’s profit would eventually surpass that of two stronger manufacturers. When the retailer is the leader, the retailer’s profit is often greater when two manufacturers engage in the Stackelberg game rather than that in Bertrand game, and the manufacturer with unreliable product quality can make profit no less than that of another manufacturer with reliable quality.

Tripartite game models in a dual-channel supply chain: competition and cooperation

This paper considers tripartite games in a dual-channel supply chain which involves a manufacturer, an offline retailer and an online retailer. Both competition and cooperation issues are analyzed. In the competition model, a Stackelberg game between the manufacturer and two retailers and a Bertrand game between two retailers occur simultaneously. It is shown that the channel which attracts more consumers’ purchase preference is charged a higher wholesale price and it meanwhile declares a higher sales price. In the presence of revenue sharing, cooperation issues between the three participants are studied and the change of the revenue of each participant is analyzed when partial cooperation exists. Further, the definition of the optimum two-player coalition is proposed. We demonstrate that the channel which attracts more preference of consumers is definitely in the optimum coalition. The structure of the two-player coalition is analyzed. Finally, under revenue sharing and cost apportionments, the change of each participant’s profit is examined.

Production and Pricing Strategies of Energy-Saving Products in the Presence of Duopolistic Manufacturers

This paper considers optimal production and pricing strategies of energy-saving products in the presence of duopolistic manufacturers. First, we analyze the free competition case by a Bertrand game. A sufficient condition for guaranteeing the existence and the uniqueness of the equilibrium solution is proposed. The change rate of the benefit function of environment with regard to purchasing preference proportions is examined. Second, we investigate the case in the presence of energy-saving incentive. A two-layer decision model is constructed by considering the decision order of each participant. Optimal strategies between the two cases are compared. We provide theoretical foundations for the government to formulate policies of energy-saving incentive under a financial budget constraint. Finally, a numerical example is presented to verify the obtained conclusions and make some supplements.