When will the Range of Prizes in Tournaments Increase in the Noise or in the Number of Players?

The symmetric equilibrium resulting from the celebrated Tournament model of Lazear and Rosen has a range of compensation between winner and loser which is inversely proportional to E[f(X)], the expectation of the additive noise’s density. There seems to be a belief that this functional is always increasing in the noise’s variability, which would agree with economic intuition when output is noisier it should be less worthwhile to work hard. We show such is not the case for all distributions, and characterize classes where such is or is not the case. When the number of players n grows, winning is more difficult so we would expect the required range of compensation to be larger. That would require that E[f(Y )], where Y = max(X1, · · · , Xn−1), will decrease in n. We examine the generality of this property. Finally we explore the same issues within a multiplicative model.