Period adding bifurcations in dynamic pricing processes

Price information enables consumers to anticipate a price and to make purchasing decisions based on their price expectations, which are critical for agents with pricing decisions or price regulations. A company with pricing decisions can aim to optimise the short-term or the long-term revenue, each of which leads to different pricing strategies thereby different price expectations. Two key ingredients play important roles in the choosing of the short-term or the long-term optimisation objectives: the maximal revenue and the robustness of the chosen pricing strategy against market volatility. However the robustness is rarely identified in a volatile market. Here, we investigate the robustness of optimal pricing strategies with the short-term or long-term optimisation objectives through the analysis of nonlinear dynamics of price expectations. Bifurcation diagrams and period diagrams are introduced to compare the change in dynamics of the optomal pricing strategies. Our results highlight that period adding bifurcations occur during the dynamic pricing processes studied. These bifurcations would challenge the robustness of an optimal pricing strategy. The consideration of the long-term revenue allows a company to charge a higher price, which in turn increases the revenue. However, the consideration of the short-term revenue can reduce the occurrence of period adding bifurcations, contributing to a robust pricing strategy. For a company, this strategy is a robust guarantee of optimal revenue in a volatile market; for consumers, this strategy avoids rapid changes in price and reduce their dissatisfaction of price variations.