The Complexion of Multi-period Stackelberg Triopoly Game with Bounded Rationality

摘要

Stackelberg model is a dynamic model, in which two players with different scales and power players act sequentially. However, there are few literatures that apply complex oligopoly dynamics theory in this model. In this paper, based on a traditional Stackelberg model, we improve the model in Peng and Lu (Appl Math Comput 271:259-268, 2015) and construct a multi-period Stackelberg triopoly game model. One leader firm and two followers with bounded rationality behavior are considered. The leader's decision-making variable, which is simplified as a constant in Peng and Lu's paper, is observed by the followers in stage 1 in every period in this model. We arrive at the conclusion that the leader would have the first-move advantage even when the players adopt a gradient output adjustment process in a multi-period Stackelberg triopoly game model. The speeds of output adjustment form a three-dimensional stability region. In the equilibrium state, the outputs of the followers are one-third of the leader's. With adjustment speed of the leader increasing, Stackelberg equilibrium would be broken at a certain point. The effect of adjustment speed on speed of convergence to equilibrium is also analyzed. Theoretical result and numerical simulation both demonstrate that the speed converging to equilibrium is slowing when the Lyapunov exponent increases. Strange attractor and the sensitivity on initial values are presented by numerical simulation, while feedback control method is used to eliminate chaos. Moreover, in the stage of periodic bifurcation outside the stability region, the increase of the adjustment speed of the leader could be incentive for choosing chaos. While in the chaos stage, the average profits of three firms are uncertain, which shows that the relative benefit is closely related to adjustment speed of bounded rationality.