A nested contest: Tullock meets the all-pay auction

I present a two-player nested contest which is a convex combination of two widely studied contests: the Tullock (lottery) contest and the all-pay auction. A Nash equilibrium exists for all parameters of the nested contest. If and only if the contest is sufficiently asymmetric, then there is an equilibrium in pure strategies. In this equilibrium, individual and aggregate efforts are lower relative to the efforts in a Tullock contest. This leads to the surprising result that if aggregate efforts in the all-pay auction are higher than the aggregate efforts in the Tullock contest, then aggregate efforts in the nested contest may not lie between aggregate efforts in the all-pay auction and aggregate efforts in the Tullock contest. When the contest is symmetric or asymmetric, I find a mixed-strategy equilibrium and describe some properties of the equilibrium distribution function; I also find the equilibrium payoffs and expected bids. When the weight on the all-pay auction component of this nested contest lies in an intermediate range, then there exist multiple non-payoff-equivalent equilibria such that there is an all-pay auction equilibrium as defined in Alcade and Dahm (2010) and another equilibrium which is not an all-pay auction equilibrium; these equilibria cannot be ranked using the Pareto criterion. If the goal of a contest-designer is to reduce aggregate effort (i.e., wasteful rent-seeking efforts), then this nested contest may be better than both the Tullock contest and the all-pay auction.