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Meritocratic Lotteries under� Intersectional Diversity Quotas

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Yuval Heller �� Joint with: Ron Peretz & Amnon Schreiber

Bar Ilan University

Silvaplana Political Economy Workshop, July 2026

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MOTIVATION

  • Indivisible opportunities often allocated by weighted lotteries
    • Merit or grades (e.g., Dutch medical schools, Cohen-Schotanus et al., 2006)
    • Need or entitlement (e.g., Charter schools in New Jersey)
    • Ethical/clinical priority (e.g., COVID-19 scarce medical treatments, White et al., 2022)
  • Selected groups often face intersectional representation constraints
    • Citizens’ assemblies (e.g., Ireland - gender, age, region, social class; Farrell & Suiter, 2021)
    • Assignment markets (e.g., School choice; Hafalir et al., 2013; Ehlers et al., 2014)
    • Diversity quotas for juries (e.g., Chaco province, Argentina: 6 jurors from each gender, 6 jurors belong to defendant's indigenous community; Diamond et al., 2024)
  • Can we achieve both goals (meritocracy and diversity)?

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MODEL

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RESULT 1: EXACT DIVERSITY WITH 2 DIMENSIONS

  • Dimension = Collection of pairwise disjoint traits �(e.g., categories within gender)
  • A decomposition of traits to dimensions corresponds to a proper edge-coloring of the trait hypergraph (V,E)
  • Theorem 1: with at most 2 dimensions, �consistency 🡪 exact implementation
  • Proof idea (a la Balinski & Demange, 89):
    • Representation as a circulation graph
    • Start with fractional flow according to p(v)
    • Cycle rounding makes the lottery integral while preserving expectations

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EXAMPLE: IMPOSSIBILITY WITH 3 DIMENSIONS

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MAIN RESULT: APPROXIMATE DIVERSITY

  • Define D=deg(V,E) as the maximum number of traits possessed by any single agent (level of intersectionality)
  • Define r-approximately diverse: Each trait e has at least l(e)-r and at most u(e)+r selected members
  • Main Result: Exact meritocracy + quota violation ≤ D-1
    • Proof relies on Beck-Fiala (1981) counting argument

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PRACTICAL ASPECTS: COMPARISON WITH LEXIMIN

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Our algorithm

LEXIMIN (Flanigan et al., 2021)

Exact implementation

Meritocracy (always)

Diversity (if possible)

Approximate implementation

Diversity

Meritocracy

Bound on approximation

Each quota is violated by at most D-1 (=deg(V,E)-1)

No stated worst-case bound. Typically, selection probabilities of most agents are reduced by half

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COMPARISON WITH EXISTING ALGORITHMS

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Our algorithm

LEXIMIN (Flanigan et al., 2021)

Exact implementation

Meritocracy (always)

Diversity (if possible)

Approximate implementation

Diversity

Meritocracy

Bound on approximation

Each quota is violated by at most D-1 (=deg(V,E)-1)

No stated worst-case bound. Typically, selection probabilities of most agents are reduced by half

Example (CCA)�|V|=825, k=75

Selection probability

Exact uniform selection probability: 9.1%

Minimal selection probability is reduced to 2.4%

Traits’ quotas

Satisfies each quota up

to 3 seats

Exactly satisfies all

quotas

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COMPRESSED REPRESENTATION

  • All the algorithms have running times polynomial in |V|+|E|
  • What if there are millions of agents (|V|>>1)?
  • One could compress the hypergraph, by having each vertex represent an agent type (i.e., merging all agents with the same trait profile and same selection probability)
    • Number of vertices reduces to a few dozen �(number of distinct trait profiles in the population)
    • Algorithms remain essentially the same

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CLOSEST RELATED PAPER

  • Multidimensional apportionment - Cembrano, Correa & Verdugo (2022): 2-vs.-3 dimensional divide & approximate proportionality for �3+ dimensions. �Key differences:
      • They do not have target selection probabilities �(=no consideration of meritocracy)
      • Our approximation depends on deg(V,E), theirs on the number of dimensions, which is at least as large, and can be arbitrarily larger

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CONCLUSION

  • Exact meritocracy and exact diversity are�compatible in two dimensions
  • With 3+ dimensions, they may be incompatible
  • Main result: An efficient algorithm that preserves meritocracy� exactly and relaxes each diversity quota by strictly less than the level of intersectionality (maximal number of traits held by one agent)
  • Existing methods preserve diversity by distorting merit
  • We preserve merit and give explicit diversity guarantees

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Meritocratic Lotteries under Intersectional Diversity Quotas (Heller, Peretz, Schreiber)

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RELATED LITERATURE

  • Multidimensional apportionment (Grimmett, 2004; Correa et al., 2024;�Gölz, Peters & Procaccia, 2026)
  • Diverse committees (Aziz et al., 2017; Aziz, 2019; Bredereck et al., 2018; Mehrotra et al., 2022; Flanigan et al., 2021; Ebadian et al., 2022; Baharav & Flanigan, 2024; Assos et al., 2025; Halpern et al., 2025; Talmon & Shapiro, 2026).
  • Implementable lotteries (Hylland & Zeckhauser, 1979; Bogomolnaia & Moulin, 2001; Budish et al., 2013; Akbarpour and Nikzad, 2020; Balbuzanov, 2022; Han, 2024; Basteck and Ehlers, 2025)
  • Distributional constraints in matching (Abdulkadiroğlu & Sönmez, 2003; Hafalir et al., 13; Ehlers et al., 14; Kamada & Kojima, 15/18/24; Carvalho et al., 25)

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