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Adversarial Selection

Alma Cohen, Alon Klement, Zvika Neeman, Eilon Solan

Abstract

In many institutional settings, $k$ items are selected with the goal of representing the underlying distribution of claims, opinions, or characteristics in a large population. We study environments with two adversarial parties whose preferences over the selected items are commonly known and opposed. We propose the Quantile Mechanism: one party partitions the population into $k$ disjoint subsets, and the other selects one item from each subset. We show that this procedure is optimally representative among all feasible mechanisms, and illustrate its use in jury selection, multi-district litigation, and committee formation.

Adversarial Selection

Abstract

In many institutional settings, items are selected with the goal of representing the underlying distribution of claims, opinions, or characteristics in a large population. We study environments with two adversarial parties whose preferences over the selected items are commonly known and opposed. We propose the Quantile Mechanism: one party partitions the population into disjoint subsets, and the other selects one item from each subset. We show that this procedure is optimally representative among all feasible mechanisms, and illustrate its use in jury selection, multi-district litigation, and committee formation.

Paper Structure

This paper contains 10 sections, 4 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Application to Jury Selection, MDL, and Redistricting Committees
  3. Related Literature
  4. Model
  5. Maximizing Representativeness
  6. Extensions
  7. Any Selection of Quantiles is Possible
  8. Non-Antagonistic Players
  9. One Player
  10. Conclusion

Key Result

Theorem 1

Suppose that $n=(2m+1)k$. Then, the Quantile mechanism is KS-optimal, ${L_1}$-optimal, and CvM-optimal. Moreover, for every population $x$ and ranking $\succsim$, every sample $y'$ that is not equivalent to the sample $y$ that is produced by the Quantile mechanism $y$

Figures (2)

  • Figure 1: $F_x$ and $F_y$ under the Quantile mechanism for $n=972, k=12, m=40$.
  • Figure 2: Comparison of the Quantile, Median-Sample, Strike and Replace, and Random mechanisms with $k=12$ and $n^*=259$ chosen items in terms of KS statistic for sampling $100$ times from $n=972, k=12, m=40$.

Theorems & Definitions (26)

  • Definition 1: Mechanism
  • Remark 1
  • Definition 2: Utility function
  • Remark 2
  • Remark 3
  • Definition 3: Mechanism game
  • Definition 4: Equilibrium
  • Example 1: Random Sample
  • Example 2: Strike and Replace Mechanism
  • Example 3: Median-Sample Mechanism
  • ...and 16 more