Diversimax: Maximizing Intersectional Diversity in Sortition

Oren Matar
Democracy 3.0
Code

Abstract

Sortition algorithms select citizens assembly panels from a pool of eligible candidates while satisfying demographic quotas that mirror the broader population. However, multiple panel compositions can satisfy the same quota constraints, requiring additional criteria to select between them.

We introduce Diversimax, an algorithm that maximizes intersectional diversity, thereby promoting representation across combinations of demographic categories (e.g., age × gender × education), not just within individual categories. Using mixed integer programming, Diversimax balances representation across all demographic intersections, minimizing disparities between well-represented and underrepresented groups. When applied to data from municipal citizens assemblies conducted in Israel, Diversimax achieves lower Gini coefficients across intersections and reduces the number of unrepresented identities compared to commonly used alternative algorithms, while maintaining identical quota compliance.

What is a Citizens' Assembly?

A citizens' assembly is a democratic institution in which randomly selected citizens come together to study a policy question in depth, hear from experts, deliberate among themselves, and produce recommendations for decision-makers. This model has grown significantly in popularity over the past fifteen years—what the OECD has termed a "deliberative wave" (OECD, 2020).

The power of this approach lies in how participants are selected. Rather than relying on elections or self-nomination, assemblies use sortition: random selection from the population. In practice, a large number of citizens are first contacted and invited to join a lottery; from those who respond, a selection algorithm samples the assembly members, ensuring the final panel meets demographic quotas for criteria such as age, gender, geography, and education.

This process creates something parliaments often lack: a body that genuinely mirrors the broader population. When an assembly reflects society's diversity, the resulting deliberation draws on a wider range of life experiences and perspectives, and the recommendations carry democratic legitimacy precisely because they come from "people like us."

Comparing Diversimax to Leximin

Different algorithms can produce substantially different panels while satisfying identical quota constraints. To demonstrate this, we compare Diversimax to Leximin (Flanigan et al. 2021), a commonly used sortition algorithm that prioritizes "fairness." Both algorithms were applied to data from Israeli municipal citizens assemblies in Ra'anana and Kfar Saba, using identical quota constraints.

The visualizations below show how participants are distributed across demographic intersections in panels selected by each algorithm.

Diversimax
Leximin

Diversimax consistently achieves lower Gini coefficients across all intersections, indicating more balanced representation. This results in fewer empty cells - fewer unique identities left unrepresented in the deliberation.

The Case for Diversity

Research demonstrates that diverse groups exchange a wider range of information during deliberation, make fewer factual errors, and conduct more thorough discussions than homogeneous groups (Sommers, 2006). This benefit extends beyond simple demographic representation: the concept of intersectionality, introduced by Crenshaw (1989), highlights that experiences at the intersection of multiple identities—such as being both Black and a woman—are "greater than the sum" of those individual categories. Each unique intersection brings distinct perspectives that cannot be captured by representing categories separately.

We therefore adopt the assumption that the first representative from an underrepresented demographic group provides greater value to deliberation than the tenth representative from an already well-represented group. Because of this assumption, Diversimax maximizes the number of unique intersectional identities represented in the panel while maintaining required demographic quotas. This ensures that marginalized groups, especially those underrepresented in the candidate pool, gain a voice in the assembly.

Methodology

Diversimax uses Mixed Integer Programming (MIP) to optimize for maximum diversity. The algorithm first generates intersection tables for all demographic dimensions: single dimensions (e.g., age, gender, education) as well as all combinations between them (e.g., age × gender, age × education, gender × education × income). For each intersection table \(t\), it calculates the optimal value per cell if representation were perfectly balanced:

\[V_{opt}^{(t)} = \frac{N}{k_t}\]

where:

  • \(N\) = Total panel size (number of participants)
  • \(k_t\) = Number of cells in intersection table t

The algorithm then creates a binary decision variable for each participant in the pool (selected = 1, not selected = 0). It optimizes this selection to: (1) satisfy all required demographic representation ranges (minimum and maximum values for each category), and (2) minimize the total deviation from the optimal balanced distribution across all intersection cells:

\[\min \sum_{t \in T} \sum_{c \in C_t} \left| n_c^{(t)} - V_{opt}^{(t)} \right|\]

where:

  • \(T\) = Set of all intersection tables (all 1D, 2D, and 3D combinations)
  • \(C_t\) = Set of cells in intersection table t
  • \(n_c^{(t)}\) = Actual number of selected participants in cell c of table t

This approach minimizes empty or sparse cells across all intersections, ensuring broader representation of unique identities.

Visualization of intersection table generation and diversity objective function

Diversimax selects participants to minimize differences from optimal values across all intersection tables

Summary

Diversimax is a sortition algorithm that optimizes intersectional diversity while maintaining demographic quotas. By balancing the number of participants across all demographic intersections, the algorithm ensures that underrepresented identities—particularly those at the margins of multiple categories—gain voice in deliberation.

The results from Israeli municipal assemblies demonstrate that different sortition algorithms, even when satisfying identical quota constraints, can produce substantially different outcomes. Diversimax has been successfully used in six assemblies in 2025, and a survey conducted by Democracy 3.0 found that the panels' diversity was one of the most appealing aspects to participants and positively correlated with their support for the assembly’s methods and outcomes. Diversimax’s value may be particularly relevant for assemblies deliberating topics of community and belonging, such as in the Kfar Saba assembly, where diverse intersectional identities significantly enhance the quality of discussion.

Future work should study the effect that choice of algorithm has on deliberation in practice. The most appropriate algorithm may depend on the specific goals and context of each assembly.

References

Crenshaw, K. (1989). Demarginalizing the Intersection of Race and Sex: A Black Feminist Critique of Antidiscrimination Doctrine, Feminist Theory and Antiracist Politics. University of Chicago Legal Forum, 1989(1), 139-167.

Flanigan, B., Gölz, P., Gupta, A., Hennig, B., & Procaccia, A. D. (2021). Fair algorithms for selecting citizens' assemblies. Nature, 596(7873), 548-552.

OECD (2020). Innovative Citizen Participation and New Democratic Institutions: Catching the Deliberative Wave. OECD Publishing, Paris. https://doi.org/10.1787/339306da-en

Sommers, S. R. (2006). On racial diversity and group decision making: Identifying multiple effects of racial composition on jury deliberations. Journal of Personality and Social Psychology, 90(4), 597-612.