Complex dynamics of a symmetric quantum Stackelberg duopoly game model with heterogeneous expectations

This paper proposes a symmetric quantum Stackelberg duopoly model based on the Frankiewicz quantum method with linear demand and cost functions. The Frankiewicz quantum technique has seldom been applied in the Stackelberg duopoly, and, to the best of the authors' knowledge, this is the first study to use a linear cost function in this context. The game focuses on two players with different expectations. The leader is bounded rational, whereas the follower is adaptable. The equilibrium points of the game model are computed using the backward induction approach, and their stability is numerically examined in terms of various factors. It has two equilibrium points: the subgame perfect equilibrium point and the quantum boundary equilibrium point, which has no economic significance. This model shows chaotic behavior, as evidenced by time series analysis (sensitivity to initial conditions) and phase plots. Furthermore, the complex dynamics of the model, as impacted by various model parameters, are thoroughly examined using one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents, and Kaplan-Yorke dimension charts. This research can aid in analyzing and comprehending unexpected actions and strategic circumstances in this game. Understanding the factors that cause chaotic behaviors allows participants (financial firms) to optimize their strategy when confronted with such scenarios.