A discrete fractional-order Cournot duopoly game

This paper focuses on associating the Cournot duopoly problem with long-memory effects. First of all, we generate a discrete fractional-order Cournot duopoly game by introducing the Caputo fractional-order difference calculus to the classical duopoly theory. In the fractional-order game, participants can make their decisions by taking full advantage of their historical information. Then we discuss both Nash equilibria and local stability of the game by employing the linear approximation theory. At last, we numerically validate the main results by using bifurcation diagrams, phase portraits, the largest Lyapunov exponent, and the 0–1 test algorithms.