Learning To Be Leader

Consider a dynamic game where, at each stage: (a) the player P1 physically acts first, without knowing all the elements of the problem faced by his opponent; (b) the player P2 physically acts second, with full knowledge of P1's problem. Thus, P1 cannot compute exactly the reaction function of P2, while P2 knows the reaction function of P1. Suppose that at each stage of the game P2 announces that it will implement a given action u^a after P1 has played (The announcement u^a is assumed to be an additional, rationally chosen action available to P2 besides its physical action u. In such a situation, P1 has two possibilities: (i) implement its best reaction to the announcement u^a, that is, act as if it was certain that P2 would really implement the announced action when it comes to play; or (ii) implement some other action based on all the information available to him.In this setting, the choice of decision by any one of the players will typically have a dual impact. It will affect not only the state of the underlying dynamic system, but also the information state of P2. These two aspects have to be taken into account by the players whenever they choose their current decisions, taking into account all future consequences of these decisions.In this paper, we investigate the dual problem facing the players in a simple game of the type presented above. We show that, typically, it will be rational for to P1 learn over time the private information of P2, thus discarding fully the announcements made by P2 and becoming a standard Stackelberg leader. In some cases, however, it may be rational for P1 not to make the effort of active learning. Furthermore, even in the case where it learns all the information available to P1, it may in its interest not to make use of this information, but to keep accepting prima facies the announcements made by P2.