Near-Optimal Dropout-Robust Sortition

TL;DR

This work addresses dropout resilience in citizen-assembly sortition by formulating a minimax problem where a chooser selects a panel that remains representative after worst-case dropouts drawn from a distribution within a $\gamma$-ball around estimated marginals. The authors propose a two-stage algorithm (MinMax-Pipage) that first solves a continuous minimax relaxation to obtain a fractional panel with $\gamma$-robustness and $[\alpha,\beta]$-equality, then dependently rounds it to an integral panel using Pipage rounding, with theoretical guarantees and empirical validation on real-world datasets. They develop a polynomial-time method that uses a projected-subgradient minimizer against a best-response dropout distribution computed via Ellipsoid on the dual LP, along with a primal recovery step, yielding convergence at rate $O(\sqrt{n/T})$ and ensuring quotas post-dropout are close to targets. The approach provides a principled, controllable trade-off between robustness, representation accuracy, and equality of selection probabilities, with practical implications for dropouts in deliberative democratic processes and beyond.

Abstract

Citizens' assemblies - small panels of citizens that convene to deliberate on policy issues - often face the issue of panelists dropping out at the last-minute. Without intervention, these dropouts compromise the size and representativeness of the panel, prompting the question: Without seeing the dropouts ahead of time, can we choose panelists such that after dropouts, the panel will be representative and appropriately-sized? We model this problem as a minimax game: the minimizer aims to choose a panel that minimizes the loss, i.e., the deviation of the ultimate panel from predefined representation targets. Then, an adversary defines a distribution over dropouts from which the realized dropouts are drawn. Our main contribution is an efficient loss-minimizing algorithm, which remains optimal as we vary the maximizer's power from worst case to average case. Our algorithm - which iteratively plays a projected gradient descent subroutine against an efficient algorithm for computing the best-response dropout distribution - also addresses a key open question in the area: how to manage dropouts while ensuring that each potential panelist is chosen with relatively equal probabilities. Using real-world datasets, we compare our algorithms to existing benchmarks, and we offer the first characterizations of tradeoffs between robustness, loss, and equality in this problem.

Figures (6)

• Figure 1: Here we show the loss of $\hat{\mathbf{p}}$ at each combination of $\gamma$ and $\gamma'$ (the adversary's strength we train and test on, respectively) in Instance 1.
• Figure 2: Here we show the loss of $\hat{\mathbf{p}}$ over $\kappa \in \{0,0.25,0.5,0.75,1\}$, i.e., as the range $[\alpha,\beta]$ expands. Errors are chosen to be moderate: $\gamma = \gamma' = 0.15$.
• Figure 3: Here we compare $\mathcal{L}(\hat{\mathbf{p}},\mathbf{d};\mathcal{I})$ (solid lines), $\mathcal{L}(\tilde{K},\mathbf{d};\mathcal{I})$ (dotted lines) and $\mathcal{L}(K^{\text{ERM}},\mathbf{d};\mathcal{I})$ (black line), over true error $\gamma'$ (x axis) and robustness level $\gamma$. For $\tilde{K}$, we show averages over 100 runs of MinMax-Pipage; shaded regions are standard errors.
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Theorems & Definitions (14)

• Theorem 3.1: Optimality
• Theorem 3.2: Efficiency
• Theorem 4.1: Representation and Panel Size
• Theorem 4.2: Equality
• Proposition A.1: Convexity in $\mathbf{p}$, concavity in $\delta$
• Theorem A.1: Sion's Minimax Theorem
• Theorem A.2: Dual optimization via Ellipsoid
• Theorem A.3: Primal recovery
• Corollary A.1: Polynomial-time best response
• Lemma B.1: Negative association, marginal preservation of Pipage rounding