Winner-Take-All Tournaments ∗ Dmitry Ryvkin † Mikhail Drugov ‡ This version: March 26, 2018 Abstract We provide new general results for winner-take-all rank-order tournaments with additive and multiplicative noise. We show that the individual equilibrium effort is unimodal in the number of players when the density of noise is unimodal. A similar property holds for aggregate equilibrium effort when the distribution of noise has a unimodal failure (hazard) rate. These results extend to tournaments with a stochastic number of players. Under similar assumptions and an additional log-supermodularity restriction, individual and aggregate equilibrium effort levels are unimodal in the usual stochastic order of the distribution of the number of players. Equilibrium effort decreases as noise becomes more dispersed, in the sense of appropriately defined uncertainty orders and entropy. Keywords: tournament, unimodality, stochastic number of players, log-supermodularity, failure rate, uncertainty orders, entropy JEL codes: C72, D72, D82 * We are grateful to Johannes H¨ orner, Bruno Jullien, Michel Le Breton, Margaret Meyer, Marc M¨oller, Armin Schmutzler, Marco Serena and seminar participants at various universities. This paper was written when the second author was visiting the Toulouse School of Economics and he is very thankful for its hospitality. † Department of Economics, Florida State University, Tallahassee, FL 32306-2180, USA, dryvkin@ fsu.edu. ‡ New Economic School and CEPR, mdrugov@nes.ru. 1