Information Comparison of Order Statistics, with Applications to Auctions and Voting

Abstract

We compare the informativeness of order statistics in a sample of conditionally independent draws from a distribution \(F(x\midθ)\) as the sample size n increases. The k-th highest of n+1 draws is more accurate than the k-th highest of n if and only if the cumulative reverse hazard -\log F(x\midθ) is log-supermodular. Symmetrically, the k-th lowest is more accurate if and only if the cumulative hazard -\log(1-F(x\midθ)) is log-supermodular. Reversals are exceptional, occurring only for experiments that are, up to increasing transformations, exponential location experiments. In large samples, middle order statistics are asymptotically fully informative, while bounded lower and upper ranks require unbounded informativeness tail conditions. When full learning fails, bounded ranks converge to location experiments, and more central ranks are Blackwell more informative. Extending the analysis from scalar order statistics to blocks of selected data, we obtain multidimensional comparisons under log-supermodularity of hazard rates. The results unify and extend information-aggregation results in auctions and provide a new order-statistic approach to strategic voting.