Competition on Many Fronts: A Stackelberg Signaling Equilibrium

An economic agent, the incumbent, is operating in many environments at the same time. These may be locations, markets, or specific activities. He is informed of the particular conditions relevant to each situation. His action in each case is observable by another agent, the entrant, who does not have the private information. Because the incumbent is in operation in many environments simultaneously, the entrant has the ability to discern the exact statistical relationship between the incumbent's action and information, and we assume that he cannot commit not to draw the inference. At each location the entrant must choose one of two actions, which we call "attack" or "no attack." This paper concerns the interplay between these sets of decisions. We characterize the optimal actions of the incumbent under the assumption that he can commit to his decisions, and thus that he will behave as a Stackelberg leader by manipulating the inferences drawn by the entrant. The solution obtained is compared with and contrasted to the Bayesian perfect equilibria of a game where both players move simultaneously. That game is a more appropriate model of the "no commitment" case. Thus, the value of the possibility of commitment and manupulation of the entrant's beliefs is assessed. Some applications are discussed.