Fair algorithms for selecting citizens’ assemblies

The study addresses how to select members of citizens’ assemblies by sortition in a way that simultaneously ensures descriptive representation (panels reflecting population composition) and fairness in individual selection probabilities. In practice, two-stage selection (invitation to form a volunteer pool followed by a quota-constrained random selection) leads to pools that overrepresent certain groups, necessitating quotas on features (for example, gender, age, education) to ensure representativeness. Existing algorithms focus on satisfying quotas but neglect the democratic ideal that individuals should have equal selection probabilities. Because quotas usually make perfectly equal probabilities impossible, the authors propose striving for maximal fairness: making selection probabilities as equal as possible subject to the quotas. They show that a widely used benchmark algorithm can yield markedly unequal probabilities, even near-zero for some volunteers, despite feasible alternatives with much higher minimum probabilities. The paper introduces theory and practice to compute and implement selection algorithms that maximize fairness while satisfying quotas.

Prior practical selection methods emphasize descriptive representation via quotas and are more general than stratified sampling approaches. Political theorists highlight equal selection probabilities as central to sortition for equality of opportunity, democratic equality, and allocative justice, and practitioner groups have advocated equal probabilities. However, due to biased volunteer pools, quotas often necessitate unequal probabilities. Two prior mathematical models suggested reconciling equal probabilities with representativeness, but their assumptions are incompatible with current practice. The authors connect their problem to fair division literature and established fairness metrics, showing these can be applied by viewing selection probability allocation as fair resource allocation.

The authors formalize the panel selection problem as follows: an instance consists of a volunteer pool of size n, a desired panel size k, and quota constraints over feature categories. A selection algorithm induces an output distribution over feasible panels and thus a probability allocation over individuals. A fairness measure maps a probability allocation to a scalar score, and an algorithm is optimal if its induced allocation maximizes the measure. Unlike prior heuristic, sequential algorithms, their framework explicitly optimizes over output distributions. The algorithm proceeds in two steps: (1) compute a maximally fair distribution over quota-compliant panels; (2) sample a panel from this distribution. To make step (1) tractable despite the exponential number of feasible panels, they use that fairness depends only on individual probabilities and that there exists an optimal portfolio (a small support set of panels) by Carathéodory’s theorem. They employ column generation: iteratively optimize over a current portfolio and, via an integer linear program (ILP), check optimality or generate a new panel (column) to add, until no improving column exists, yielding a maximally fair distribution. For deployment, they implement a specific algorithm, LEXIMIN, which optimizes the leximin fairness criterion: maximize the minimum individual selection probability; break ties by maximizing the second-lowest probability, and so on. LEXIMIN is particularly sensitive to protecting the least-advantaged individuals and is motivated by observed near-zero probabilities under the benchmark algorithm (LEGACY). The implementation is made available as open-source and via a web tool, and has been deployed by multiple organizations.

• Across ten real-world instances, LEXIMIN consistently yields substantially fairer probability allocations than the benchmark (LEGACY).
• Minimum selection probabilities: LEGACY assigns near-zero probabilities to some volunteers in all but one instance (e.g., ≤0.32%, ≤0.17%, ≤0.15%, ≤0.11%, ≤0.03%), while LEXIMIN increases the minimum to between 2.4% and 32.5% depending on the instance (e.g., 6.7% for sf(a), 4.0% for sf(b), 8.6% for sf(c), 4.7% for sf(d), 2.6% for sf(e), 2.4% for cca, 5.1% for hd, 20.0% for mass, 32.5% for nexus, 4.7% for obf). LEXIMIN’s minimum probabilities range from 26% to 65% (median 49%) of the ideal k/n.
• Breadth of impact: Between 13% and 56% of pool members (median 46%) receive a LEGACY probability lower than LEXIMIN’s minimum probability, indicating widespread improvement, not only for a few worst-off individuals.
• Inequality metrics: LEXIMIN reduces the Gini coefficient by 5–16 percentage points across instances (median 12; negligible change for the mild-constraint “mass” instance). For example, on “obf” the Gini drops from 59% to 43%. LEXIMIN also improves the geometric mean of probabilities on all instances.
• Practicality: Running times on consumer hardware range from about 1 second to 67 minutes across instances of varying sizes and numbers of quota categories, fitting within practical time constraints.
• Transparency: The LEXIMIN output distribution can be transformed into a uniform lottery over a finite set of panels for live, observable draws with minimal loss of fairness, enabling participants to verify probabilities.

The results show that maximizing fairness subject to quotas can eliminate the practical exclusion of some volunteers observed under prior algorithms, aligning selection with democratic ideals of equal opportunity. The algorithmic framework provides optimality guarantees for a broad class of fairness measures by explicitly optimizing output distributions rather than relying on myopic, quota-satisfying heuristics. Deployment of LEXIMIN demonstrates both feasibility and significant fairness gains in real settings. The collaboration with practitioners highlights opportunities for enhanced transparency, such as live uniform lotteries derived from optimized distributions, which allow participants to observe and verify probabilities. Further work is needed to formalize conditions under which such transformations preserve strong fairness guarantees generally.

This paper introduces an optimal algorithmic framework for fair selection of citizens’ assemblies and deploys LEXIMIN, a practical algorithm that substantially improves fairness over the prior state of the art while maintaining representativeness. By grounding sortition in fair division principles and delivering deployable tools, the work strengthens the foundations of practical sortition and advances democratic equality in access to representation. Future research directions include formal guarantees for transforming optimized distributions into simple, transparent lotteries, exploring additional fairness measures within the framework, and further scaling and optimizing implementations.

Perfect equality of selection probabilities is generally impossible under quota constraints and biased volunteer pools; the objective is maximal fairness subject to these constraints. The transformation of optimized distributions into uniform live lotteries, while effective in at least one deployment, requires further mathematical guarantees to ensure fairness preservation in general. Computationally, solving the optimization can take up to tens of minutes for larger instances, though still practical; performance may vary with instance complexity. The evaluation focuses on ten real-world datasets; broader testing across more contexts would further assess generalizability.