On the stability of quadratic dynamics in discrete time nn-player Cournot games

Motivated by analyzing the dynamic adjustment process of players in an nn-player Cournot game, in this paper a discrete-time quadratic dynamical system is proposed and the stability of its equilibrium is analyzed. Several output adjustment mechanisms (e.g. best reply, adaptive adjustment and myopic) and expectations (e.g. naïve and adaptive) in a Cournot game with linear price function and quadratic costs, which form a quadratic dynamical system, are the special cases of the proposed model. The stability of the proposed quadratic dynamical system is analyzed with (1) time invariant parameters, (2) time invariant parameters in the presence of disturbance and (3) bounded time varying parameters in the presence of disturbance. In each case, the sufficient condition to find the region that belongs to the basin of attraction is derived using some discrete-time converse Lyapunov theorems. In addition, the proposed model and theorems are utilized to analyze the stability of the boundary and Nash equilibrium points of a Cournot game with three heterogeneous players.