@misc{ERS14,
author = { Steven N. Evans and Ronald L. Rivest and Philip B. Stark },
title = { Leading the Field: Fortune Favors the Bold in Thurstonian Choice Models },
date = { 2014-09-20 },
OPTyear = { 2014 },
OPTmonth = { Sep. 20 },
urla = { arxiv },
abstract = { Consider a random real $n$-vector $Z$ of preference
scores. Item $i$ is preferred to item $j$ if $Z_i <
Z_j$. Suppose $\mathbb{P}\{Z_i = Z_j\} = 0$ for $i
\ne j$. Item $k$ is the favorite if $Z_k <
\min_{i\ne k} Z_i$. This model generalizes
Thurstone's Law of Comparative Judgment. Let $p_i$
denote the chance that item $i$ is the favorite. We
characterize a large class of distributions for $Z$
for which $p_1 > p_2 > \ldots > p_n$. Our results
are most surprising when $\mathbb{P}\{Z_i < Z_j\} =
\mathbb{P}\{Z_i > Z_j\} = \frac{1}{2}$ for $i \ne
j$, so neither of any two items is likely to be
preferred over the other in a pairwise comparison,
and we find that, under suitable assumptions, this
ordering of $\{p_i\}$ holds when the variability of
$Z_i$ decreases with $i$. Our conclusions echo the
proverb "Fortune favors the bold." It follows that
if $n$ treatments are assigned randomly to $n$
subjects; if no treatment systematically helps or
hurts; and if the treatment effects are
stochastically ordered in the same way for all
subjects; then the treatment with most variable
effect is most likely to appear to be best---even
though a randomized trial of any two treatments on
any two subjects has the same chance of finding the
first or the second treatment better. Moreover, our
results explain "sophomore slump" without the
regression effect and show why heteroscedasticity
can result in misleadingly small $p$-values in
nonparametric tests of association. },
}