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

2026-07-11

AI Summary

This study investigates how information content in order statistics evolves with increasing sample size, with a focus on their capacity to aggregate information in auction and voting settings. By leveraging Blackwell informativeness, hazard rate functions, and log-supermodularity, the paper establishes the first precise link between the informational precision of order statistics and the tail properties of the underlying distribution. The main contributions include proving that, except for the exponential distribution, enlarging the sample size almost always enhances the information conveyed by a fixed-rank order statistic; demonstrating that central order statistics become fully informative in large samples; and showing that intermediate ranks are advantageous under incomplete information. These results unify and extend existing theories of information aggregation in auctions and voting, offering a novel analytical framework grounded in order statistics.

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.

Problem

• order statistics
• information comparison
• hazard rate
• log-supermodularity
• conditional independence

Innovation

• order statistics
• log-supermodularity
• information aggregation
• hazard rate
• Blackwell informativeness