This paper proposes an extension of the valuation equilibrium concept (Jehiel and Samet, 2007), which partly endogenises the underlying grouping of actions. The effect on equilibrium predictions is illustrated in a burning money game.

We consider a two-player all-pay auction with symmetric independent private values that are uniformly distributed. The designer chooses the size of a head start that is given to one of the players. The designer’s objective is to maximize a convex combination of the expected highest effort and...