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Fair, Manipulation-Robust, and Transparent Sortition

Carmel Baharav, Bailey Flanigan

TL;DR

This work studies sortition under quotas and introduces Goldilocks, a convex equality objective that balances high and low selection probabilities to achieve simultaneous fairness and manipulation robustness while supporting transparency. The authors develop a unified model of fairness, manipulation, and transparency, derive formal bounds showing the fundamental trade-offs under coalitional misreporting, and prove that Goldilocks$_1$ achieves near-optimal robustness and fairness across challenging instances. They extend the framework to a rounding-based transparency mechanism, showing end-to-end preservation of guarantees when using a sufficiently large $m$, and validate the approach with empirical studies on real-world-like data where Goldilocks$_1$ reaches near instance-optimal minimum and maximum probabilities and outperforms existing objectives. The results suggest Goldilocks as a practically viable, manipulation-resilient, transparent-sortition method that harmonizes normative ideals for deliberative panels and quotas in real-world settings.

Abstract

Sortition, the random selection of political representatives, is increasingly being used around the world to choose participants of deliberative processes like Citizens' Assemblies. Motivated by sortition's practical importance, there has been a recent flurry of research on sortition algorithms, whose task it is to select a panel from among a pool of volunteers. This panel must satisfy quotas enforcing representation of key population subgroups. Past work has contributed an algorithmic approach for fulfilling this task while ensuring that volunteers' chances of selection are maximally equal, as measured by any convex equality objective. The question, then, is: which equality objective is the right one? Past work has mainly studied the objectives Minimax and Leximin, which respectively minimize the maximum and maximize the minimum chance of selection given to any volunteer. Recent work showed that both of these objectives have key weaknesses: Minimax is highly robust to manipulation but is arbitrarily unfair; oppositely, Leximin is highly fair but arbitrarily manipulable. In light of this gap, we propose a new equality objective, Goldilocks, that aims to achieve these ideals simultaneously by ensuring that no volunteer receives too little or too much chance of selection. We theoretically bound the extent to which Goldilocks achieves these ideals, finding that in an important sense, Goldilocks recovers among the best available solutions in a given instance. We then extend our bounds to the case where the output of Goldilocks is transformed to achieve a third goal, Transparency. Our empirical analysis of Goldilocks in real data is even more promising: we find that this objective achieves nearly instance-optimal minimum and maximum selection probabilities simultaneously in most real instances -- an outcome not even guaranteed to be possible for any algorithm.

Fair, Manipulation-Robust, and Transparent Sortition

TL;DR

This work studies sortition under quotas and introduces Goldilocks, a convex equality objective that balances high and low selection probabilities to achieve simultaneous fairness and manipulation robustness while supporting transparency. The authors develop a unified model of fairness, manipulation, and transparency, derive formal bounds showing the fundamental trade-offs under coalitional misreporting, and prove that Goldilocks achieves near-optimal robustness and fairness across challenging instances. They extend the framework to a rounding-based transparency mechanism, showing end-to-end preservation of guarantees when using a sufficiently large , and validate the approach with empirical studies on real-world-like data where Goldilocks reaches near instance-optimal minimum and maximum probabilities and outperforms existing objectives. The results suggest Goldilocks as a practically viable, manipulation-resilient, transparent-sortition method that harmonizes normative ideals for deliberative panels and quotas in real-world settings.

Abstract

Sortition, the random selection of political representatives, is increasingly being used around the world to choose participants of deliberative processes like Citizens' Assemblies. Motivated by sortition's practical importance, there has been a recent flurry of research on sortition algorithms, whose task it is to select a panel from among a pool of volunteers. This panel must satisfy quotas enforcing representation of key population subgroups. Past work has contributed an algorithmic approach for fulfilling this task while ensuring that volunteers' chances of selection are maximally equal, as measured by any convex equality objective. The question, then, is: which equality objective is the right one? Past work has mainly studied the objectives Minimax and Leximin, which respectively minimize the maximum and maximize the minimum chance of selection given to any volunteer. Recent work showed that both of these objectives have key weaknesses: Minimax is highly robust to manipulation but is arbitrarily unfair; oppositely, Leximin is highly fair but arbitrarily manipulable. In light of this gap, we propose a new equality objective, Goldilocks, that aims to achieve these ideals simultaneously by ensuring that no volunteer receives too little or too much chance of selection. We theoretically bound the extent to which Goldilocks achieves these ideals, finding that in an important sense, Goldilocks recovers among the best available solutions in a given instance. We then extend our bounds to the case where the output of Goldilocks is transformed to achieve a third goal, Transparency. Our empirical analysis of Goldilocks in real data is even more promising: we find that this objective achieves nearly instance-optimal minimum and maximum selection probabilities simultaneously in most real instances -- an outcome not even guaranteed to be possible for any algorithm.
Paper Structure (44 sections, 19 theorems, 41 equations, 5 figures, 4 tables)

This paper contains 44 sections, 19 theorems, 41 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Approach and Contributions
  3. Related Work
  4. Model
  5. Ideals: Manipulation Robustness, Fairness, and Transparency
  6. Handling Structural Exclusion
  7. Existence of "Good" Solutions
  8. Intuition: Problems 1 and 2 Prevent Good Solutions
  9. Formal Bounds on the Existence of Good Solutions
  10. Goldilocks$_1$: Manipulation Robustness and Fairness
  11. On the Optimality of Goldilocks
  12. Comparison to Existing Algorithms
  13. Goldilocks$_1$: Manipulation Robustness, Fairness, and Transparency
  14. Empirical Evaluation
  15. Maximum and Minimum Probabilities
  16. ...and 29 more sections

Key Result

Theorem 1

Fix any even $k \in \{6,\dots,n_{min}-3\}$, any $n_{min} \in \{k+3, \dots, \lfloor n/2\rfloor\}$, and any $c \in \{3,\dots,n_{min}-k\}$. Then there exists $\mathcal{I}$ with $k$, $n_{min}(\mathcal{I}) = n_{min}$ satisfying Assumption ass:inclusion, $C \subseteq [n]$ with $|C|=c$, and $\boldsymbol{\

Figures (5)

  • Figure 2: The solid, dashed lines respectively represent maximum, minimum probabilities per algorithm. The shaded region lies between the optimal maximum probability and optimal minimum probability, establishing the region where no algorithm's extremal probabilities can exist.
  • Figure 3: Gini coefficient across algorithms and instances. Lower Gini Coefficient means greater fairness.
  • Figure 4: The maximum amount of probability any single MU manipulator can gain, for 1 and 2 pool copies.
  • Figure 5: Deviations from $\textsc{Goldilocks} _1$-optimal selection probability assignments by Pipage. The values for Pipage correspond to averages of minimum, maximum probability per run over 1000 runs. Error bars are plotted to indicate standard deviation, but they are so small that they are not visible. Gray boxes extend vertically from the minimum (resp. maximum) probability given by $\textsc{Goldilocks} _1$ to the theoretical bound given by \ref{['thm:transparency-oldpaper']}. Optimal minimum, maximum probabilities per instance are shown for reference.
  • Figure 6: In instances 7-9, we use $\textsc{Maximin}$ instead of $\textsc{Leximin}$ to indicate the optimal minimum marginal probability because of computational costs due to the size of these instances. We additionally drop only 3 features instead of 4 because instance 9 only has 4 features.

Theorems & Definitions (22)

  • Theorem 1: Lower Bound
  • Theorem 2: Upper Bound
  • Theorem 3: Upper Bound
  • Theorem 4: Lower Bound
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 3: Thms 3.2 and 3.3, flanigan2021transparent
  • Theorem 5: Upper Bound
  • Proposition 4
  • ...and 12 more