Mixed equilibria in Tullock contests

Any symmetric mixed-strategy equilibrium in a Tullock contest with intermediate values of the decisiveness parameter ("2 < R < ∞") has countably infinitely many mass points. All probability weight is concentrated on those mass points, which have the zero bid as their sole point of accumulation. With contestants randomizing over a non-convex set, there is a cost of being "halfhearted," which is absent from both the lottery contest and the all-pay auction. Numerical bid distributions are generally negatively skewed, and exhibit, for some parameter values, a higher probability of ex-post overdissipation than the all-pay auction.