Nonlinear dynamics of discrete time multi-level leader–follower games

We study dynamic multiple-player multiple-level discrete time leader–follower games in the vein of Cournot or Stackelberg games; these games generalize two-player dynamic Stackelberg or Cournot duopoly games which have been considered recently. A given player acts as a leader toward players in lower levels, and as a follower toward players in higher levels. We consider the case of either perfect or incomplete information, which in this context means that players either have complete information about other players within their level (perfect information) or lack information at the present timestep about other players within their level (incomplete information). Players always have perfect information about all players which are (relative) followers, and incomplete information about players which are (relative) leaders. The Cournot-type adjustment process under these information structures at each timestep results in the temporal dynamics of the game. As we consider dynamic games, we observe a variety of behaviors in time, including convergence to steady state or equilibrium quantities, cycles or periodic oscillations, and chaotic dynamics. We find that the costs facing each player strongly influence the form of the long-time dynamics, as will the information structure (perfect or incomplete) selected. One interesting finding is that under perfect information players tend to quickly converge upon their respective equilibrium values, while incomplete information can result in loss of regularity and the emergence of periodic or chaotic dynamics. However, in cases where players may be pushed out of the game in the presence of high relative costs and perfect information, we find that non-equilibrium dynamics under incomplete information allow such players to retain positive production, hence they are able to remain in the game.