Robust Tournaments

TL;DR

This paper characterizes a robust tournament design that maximizes effort under uncertain noise distribution, proposing a prize scheme that adapts to entropy bounds and becomes more equitable with more participants.

Contribution

It introduces a prize scheme optimized for unknown noise distributions, linking it to entropy bounds and asymptotic harmonic prize sequences.

Findings

Prizes follow harmonic numbers asymptotically.

The scheme induces an exponential noise distribution.

Gini coefficient approaches 1/2 as participants increase.

Abstract

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 $1/2$.