Nonlinear dynamics of a heterogeneous quantum Commons’ tragedy

Abstract

This paper first builds a nonlinear dynamic system of the Commons’ tragedy with quantum strategies in discrete time. There are two heterogeneous expectations in the game, namely, one of the two players is boundedly rational and the other is of adaptive expectation. Secondly, we obtain a sufficient and necessary condition for quantum Nash equilibria to be stable equilibria. Finally, we investigate the impacts of quantum entanglement on the stable region of systems and analyze the behaviors of complex dynamical systems. The results show that quantum Nash equilibria are stable points of nonlinear dynamics for quantum Commons’ tragedy if both quantum entanglement and quantum strategy adjustment speed satisfy certain conditions. In addition, numerical simulations discuss the features of nonlinear dynamics, such as bifurcation, the largest Lyapunov exponent, fractal dimension, strange attractor, and sensitivity to initial conditions. The numerical results show that nonlinear dynamics appear chaotic if the adjustment speed of quantum strategy or the quantum entanglement degree is larger.