Dynamic analysis of a two-dimensional Cournot duopoly model involving social welfare and bounded rationality

In this manuscript, a Cournot duopoly game whose competitors adopt a nonlinear cost and a linear inverse demand function is studied. Competitors in this game aim to optimize their respective objective functions, which consist of profits and weighted social welfare. The progression of the game is modeled using a two-dimensional nonlinear discrete dynamic map characterized by eight parameters. We simplify the parameters of this map to better study its dynamics, given its complicated analytical form. The map of the game includes four fixed points, one of which corresponds to the intersection of the marginal objective functions that is the Nash point. We illustrate that the manifold of the map can be examined using a 1D map, which exhibits similarities to the logistic map. The Nash point's stability or instability is analyzed, showing that there are three zones in the game's map phase portrait, Z₄, Z₂, and Z₀. and hence it is non-invertible. Simulations indicate that the attraction basin for certain attracting sets contributes to the emergence of contact bifurcations when different values of speed parameters are selected. This contact bifurcation is absent in the symmetric scenario where identical speed parameters are applied. The absorbing area, which is raised on synchronization phenomena, is discussed.