General competing mechanism games with strategy-proof punishment

This paper studies a general competing mechanism game of incomplete information where each seller can offer a contract that determines a menu of any complex mechanisms conditional on buyers’ messages, and then chooses a mechanism that he wants from the menu. The focus is the class of robust equilibria in which sellers punish a deviating seller with dominant strategy incentive compatible direct mechanisms. We show how to characterize the class of such equilibria. In applications, the number of sellers is endogenized given a number of buyers and fixed entry costs. In a large market, a seller’s equilibrium ex-ante expected net profit is always equal to zero but any price in an interval can be supported as an equilibrium price. The equilibrium ratio of the number of buyers to the number of sellers is lowest at the monopoly price, and increases as the price moves farther away from it in either direction.