@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 =         { <a href="http://arxiv.org/abs/1409.5924">arxiv</a> },
  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.  },
}