Multistability in a dynamic Cournot game with three oligopolists

The time evolution of a dynamic oligopoly game with three competing firms is modeled by a discrete dynamical system obtained by the iteration of a three-dimensional non-invertible map. For the symmetric case of identical players a complete analytical study of the stability conditions for the fixed points, which are Nash equilibria of the game, is given. For the situation of several coexisting stable Nash equilibria a numerical study of their basins of attraction is provided. This gives, evidence of the occurrence of global bifurcations at which the basins are transformed from simply connected sets into non-connected sets, a basin structure which is peculiar of non-invertible maps. The presence of several coexisting attractors (or multistability) is observed even when complex attractors exist. Two different routes to complexity are presented: one related to the creation of more and more complex attractors; the other related to the creation of more and more complex structures of the basins. Starting from the benchmark case of identical players, the effects of heterogeneous behavior of the players, causing the loss of the symmetry properties of the dynamical system, are investigated through numerical explorations.