Mixed Equilibria in Tullock Contests

Abstract

Abstract. Any symmetric mixed-strategy equilibrium in a Tullock contest with intermediate values of the decisiveness parameter ("2 < R < 1") has countably in…nitely 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 expost overdissipation than the all-pay auction.

Contests with a Non-Convex Strategy Space

“…The tradeoff for adopting greater generality is that we must exclude private 3 See Alcade and Dahm (2010) and Ewerhart (2015) for the complete characterization of these equilibria. 4 Without loss of generality, we also assume that marginal cost equals 1. information as a possibility.…”

Linking individual and collective contests through noise level and sharing rules

“…The two-player case is the most common in the literature since it allows one to provide with closed form solutions, clear comparative statics and a graphical representation of the results. Baik (1994) and Nti (1999) solved the two-player r-contest when an equilibrium in pure strategies exists (Baye et al, 1994;Alcalde and Dahm, 2010;Ewerhart, 2014) for the equilibrium in mixed strategies). As shown by Nti (1999), as long as V r 1 + V r 2 > rV r 2 (letting V 1 ≤ V 2 then r < 1 is a sufficient condition, while r < 2 is a necessary condition), the two-player r-contest has a unique equilibrium in pure strategies where both players are active (i.e., they exert strictly positive effort) with:…”

An envelope approach to tournament design

“…However, allowing for randomization would not re-establish efficiency. Also, the characterization of mixed-strategy equilibria in tournaments with little noise seems to require different methods (Ewerhart, 2015), and therefore lies beyond the scope of the present analysis. 23 Intuitively, the value function increases by at least as much as the value at any given global maximum.…”