6. Stackelberg

A Century after Cournot von Stackelberg proposed a modification of the model. He suggested that any duopolist may learn the competitor’s Cournot reaction function, and so search for a better solution as a “leader”. This, of course, works provided the other, the “follower”, actually adheres to the proper reaction function. In this way two feasible leader/follower pairs can be formed. Both can also attempt leadership, in which case there would be economic warfare, from which presumably the firm with better financial backing might come out as winner. It was proved that Stackelberg equilibrium always is more profitable for the leader than Cournot equilibrium. Stackelberg never thought in dynamic terms—he could, for instance, have considered a variable supply policy over time to check if it might render a higher profit than a constant supply ∩ over time, but he did not. If we want to combine the static Stackelberg case with the dynamic Cournot process, things become really interesting. There may even seem to be a contradiction buried here. Stackelberg equilibrium is more profitable, no question about that, but, even the successful Stackelberg leader will ripe a higher temporary profit from jumping to Cournot action. This is so because the Cournot action was defined as the best move in any situation. However, this will not last for long—if the Cournot process is stable, then the orbit will approach Cournot equilibrium—and that is worse than Stackelberg leadership. So the stage is set for designing a rule for switching between Cournot and Stackelberg actions, which seems not to have been attempted before. The natural format is to let the competitors switch to Stackelberg leadership whenever the resulting profit is higher than, not the expected profit from Cournot action, because it never is, but, say, exceeds a fraction, say 50 or 75% of it. Incorporating such a rule of switching results in interesting scenarios, including both duopolists choosing Stackelberg leadership or both using Cournot best reply. The dynamics shows different periodic scenarios, multiplicity of attractors, and bifurcations. Some results are really unexpected, for instance that it may be more profitable than any other action to be a follower if the other becomes a leader, which, of course is not a choice under the follower’s control.