Dynamical Cournot game with bounded rationality and time delay for marginal profit

• Delay for marginal profit is considered in a duopoly game with bounded rationality.
• Each producer adjusts its output according to the weighted marginal profit.
• Equilibrium solutions of the model and their local stability are investigated.
• Proper time delay has positive effects on system stability.
• Intermediate values of model parameters can speed up the convergence to equilibrium.

In this work, a kind of delayed structure on marginal profit is introduced in a dynamical Cournot game with bounded rationality. Time delay is considered for producers’ marginal profits so that each producer follows a local adjustment process to adjust its output via a smoothed marginal profit, which averages previous marginal profits with different weights. Delayed dynamics is built for such a process and analysis of local stability is mathematically done for it. Its boundary equilibria are proved to be unstable and the conditions for local stability of its unique interior equilibrium are obtained by Schur–Cohn Criterion. To show how the delayed system evolves and what influence the model parameters including the delay weight (a memory parameter) have on the system stability, numerical simulations are done for different kinds of dynamical behaviors such as bifurcation diagram, phase portrait, chaotic attractor, convergence speed and stability region. It is demonstrated that a proper delay weight to the memory plays an important role in expanding the stability region and delaying the occurrence of complex behaviors such as bifurcation and chaos. It is also demonstrated that properly medium delay weights and properly medium adjustment rates may speed up the convergence to equilibrium.