Robust Tournaments

We characterize robust tournament design – the prize scheme that maximizes the lowest effort in a rank-order tournament where the distribution of noise is unknown, except for an upper bound, $\bar{H}$, on its Shannon entropy. The robust tournament scheme awards positive prizes to all ranks except the last, with a distinct top prize. Asymptotically, the prizes follow the harmonic number sequence and induce an exponential distribution of noise with rate parameter $e^{-\bar{H}}$. The robust prize scheme is highly unequal, especially in small tournaments, but becomes more equitable as the number of participants grows, with the Gini coefficient approaching $1\mathrm{/}2$.