Contests with multiple rounds

This paper studies contests where players have the flexibility to add to their previous efforts after observing their rivals' most recent effort in an intermediate stage. It is found that (1) contrary to previous findings, the Stackelberg outcome where the underdog leads and the favorite follows cannot be an equilibrium. (2) There are multiple subgame perfect equilibria all occurring on the underdog's usual one-shot reaction function in-between and including the one-shot Cournot-Nash and Stackelberg outcome with the favorite leading. (3) The total equilibrium effort is typically greater than or equal to what a one-shot Cournot-Nash play would predict; and (4) in settings where players can choose whether or not to disclose their early actions to the rival, both the favorite and the underdog disclose in equilibrium. Applications in sports, lobbying, and R&D races are discussed.