Ranked Method of Equal Shares

• Ranked Method of Equal Shares is a framework that extends proportional representation to ranked outcomes by allocating virtual budgets for fair candidate selection.
• It applies sequential allocation and budget splitting techniques to guarantee justified representation and balance between minority and majority preferences.
• Empirical studies demonstrate that RMES enhances fairness in diverse settings such as committee selection, resource allocation, and participatory budgeting.

The Ranked Method of Equal Shares (RMES) is a class of social choice algorithms extending the principle of proportional representation from committee selection and participatory budgeting to settings involving ranked outcomes. These include ranked candidate lists, aggregated rankings from multiple inputs, probabilistic assignments, and sequential resource allocation. RMES aims to ensure that every voter or input ranking receives proportional influence over the ordering or allocation, typically by allocating and deducting a virtual budget (or “equal share”) as items are placed or resources assigned. Unlike traditional majoritarian or purely utilitarian rules, RMES incorporates explicit mathematical guarantees of proportional fairness, often grounded in fairness concepts such as justified representation or its analogs.

1. Proportionality Beyond Committee Selection

The original notion of proportional representation—ensuring cohesive minority groups receive fair representation—was developed for the selection of subsets (committees). In RMES and related methods, this concept is generalized to ranked outcomes:

• For ranked lists, proportionality requires that every prefix contains a number of representatives from cohesive groups proportional both to their size and to the set of alternatives they unanimously approve. If $N'$ is a voter group with $\alpha$ fraction of the electorate and cohesiveness $c(N')$, then any top-$k$ prefix should deliver at least $jd(N',k) = \min(\lfloor \alpha k \rfloor, c(N'))$ candidates they approve (Skowron et al., 2016).
• In rank aggregation, as in the Proportional Sequential Borda rule and RMES variant, proportionality is interpreted in terms of pairwise agreement: an input ranking with weight $\alpha$ should agree with approximately $\alpha$ of the pairwise comparisons in the output ranking (Lederer, 22 Aug 2025).

This generalized framework subsumes committee/approval-based rules, resource allocation, and assignment settings, providing a unified approach for fair influence across ranking and allocation domains.

2. Algorithmic Formulations

Technical instances of RMES are distinguished by how they allocate and spend virtual budgets, the utility or “price” function used, and whether the outcome is deterministic or fractional/probabilistic.

• Rank Aggregation (RMES in (Lederer, 22 Aug 2025)): Each input ranking $\succ$ of weight $R(\succ)$ receives an initial budget $b_1(\succ) = R(\succ) \binom{m}{2}$, corresponding to the number of pairwise comparisons. Candidates are selected sequentially: in step $i$, the method computes a minimal per-unit price $\rho_i$ such that the collectively covered utility for candidate $x_i$ equals its cost $(m-i)$. Each ranking’s budget is reduced by its contribution. This procedure continues until the ranking is complete, with majority voting used for the last two candidates. The selection procedure is utilitarian for early positions: for roughly $\lfloor m/4 \rfloor$ steps, RMES picks candidates with maximal (initial) Borda score (Lederer, 22 Aug 2025).
• Approval-based Rankings (Skowron et al., 2016): Adaptations such as Phragmén’s Rule, Sequential Proportional Approval Voting (SeqPAV), Reverse SeqPAV, and p-Geometric RAV proceed by sequentially building the ranking, allocating “load” or share to each voter as candidates are added, and ensuring proportionality in every initial segment. The quality guarantee is formalized as $q_P(r)$, the worst-case ratio between group representation and justified demand in prefixes.
• Probabilistic Assignment (Chen et al., 2021): The probabilistic rank rule lets each agent “consume” their most-preferred available object at equal speed, exhausting their probability budget only when higher-priority options are depleted, characterized axiomatically by “sd-rank-fairness” and “equal-rank envy-freeness.” This can be interpreted as a ranked variant of equal shares—each agent receives an equal “share” at each rank.
• Resource Sharing (Lexicographically Maximin Fairness, (Li et al., 2021)): Here, the “rank” is interpreted as the ordering of rounds or objects, with a maximin-over-lexicographically-ordered utilities objective and fair-share guarantees precisely defined. Implementation exploits reductions to network flows.

3. Formal Fairness Guarantees

RMES and its variants offer formal proportionality and fairness guarantees:

• Group Representation: For every $(\alpha,\lambda)$-significant group of voters, there exists a position $\kappa(\alpha, \lambda)$ such that the group’s average representation in the top $\kappa$ positions is at least $\lambda$ (Skowron et al., 2016). For Phragmén’s rule, for example, $$\kappa(\alpha,\lambda) = \lceil(5\lambda/\alpha^2) + 1/\alpha\rceil$$.
• Pairwise Justified Representation: In rank aggregation (Lederer, 22 Aug 2025), RMES satisfies unanimous proportional justified representation (uPJR): each input ranking of weight $\alpha$ agrees with at least $\lfloor \alpha \binom{m}{2} \rfloor$ pairwise comparisons in the aggregate ranking.
• Priceability and Budget Splitting: Fairness is often justified via an explicit “payment” scheme, i.e., for each candidate added, each agent “pays” an amount in proportion to their utility for the candidate (as in participatory budgeting MES), or their budget is decremented accordingly.

4. Representative Algorithms: Comparison and Properties

A range of ranking methods inspired by RMES principles have been extensively analyzed, both theoretically and empirically:

Method	Proportionality Guarantee	Utilitarian Alignment
p-Geometric RAV (p ≈ 2)	Strong	Good in early rounds
SeqPAV, Reverse SeqPAV	Strong	Good trade-off
Phragmén’s Rule	Strong	Shares “load”
Greedy Chamberlin–Courant	Weak	Highly utilitarian
Approval Voting (AV)	Weak/minority ignored	Majoritarian
RMES (Rank Aggregation)	uPJR, priceable	Maximizes Borda initially
Probabilistic Rank (PR Rule)	Sd-rank-fairness, EREF	Equal shares at each rank

In experimental analysis, methods such as 2-Geometric RAV, SeqPAV, Reverse SeqPAV, and Phragmén’s rule consistently produce rankings that better satisfy justified demand for all group sizes, outperforming Approval Voting and greedy methods for minority and cohesive group protection (Skowron et al., 2016).

5. Connections to Related Domains

RMES analogues pervade several subfields:

• Participatory Budgeting (MES/BOS/AMES, e.g., (Kraiczy et al., 2023, Papasotiropoulos et al., 23 Sep 2024)): Each voter’s virtual budget is spent sequentially to “buy” project funding, ensuring proportional influence on aggregate decisions.
• Assignment Problems (Chen et al., 2021): PR Rule’s equal rate consumption by agents across ranked options generalizes equal shares to probabilistic, ordinal environments.
• Flow-based Rank Aggregation (Lederer, 22 Aug 2025): Flow-adjusting Borda rule achieves “stronger” proportionality (sPJR) through network flow-based payments—an extension of RMES/cost-splitting techniques.

Fairness concepts and axioms such as proportional justified representation (PJR, EJR, uPJR), priceability, and lexicographic maximin fairness underpin nearly all RMES-inspired rules.

6. Practical Applications and Empirical Insights

RMES and its relatives have demonstrated practical impact across application areas:

• Hiring and Committee Selection: The need for proportionality in “shortlists” of uncertain length leads directly to ranking rules with justified group representation in every prefix (Skowron et al., 2016).
• Collaborative Filtering and Recommender Systems: Proportionalistic ranking is relevant where diverse user groups must see their preferences reflected in top-k recommendations or result lists.
• PB and Crowd-sourced Decision Making: Equal shares-based rules balance total utility, representation, and computational tractability, with empirical studies confirming better proportional fairness and lower exclusion of minorities than majoritarian baselines (Nelissen, 2023).
• Assignment and Resource Sharing: Probabilistic and maximin-equal shares approaches enable fair division even with arbitrary ordinal preferences or multi-round offline scenarios (Chen et al., 2021, Li et al., 2021).

Empirical findings consistently show that methods rooted in equal shares—especially those allowing minor adaptivity or efficiency-oriented modifications (e.g., 2-Geometric RAV, RMES, AMES)—achieve strong proportional guarantees at a negligible cost to overall social welfare. In participatory budgeting, RMES variants reliably achieve lower exclusion rates and better “bottom end” satisfaction, with only a small loss in total welfare relative to utilitarian methods (Nelissen, 2023). Completion heuristics (e.g., Add1U, BOS Equal Shares) deal with non-exhaustiveness and are necessary for practical deployment (Papasotiropoulos et al., 23 Sep 2024, Kraiczy et al., 17 Feb 2025).

7. Axiomatic and Algorithmic Significance

RMES serves as a bridge between approval- and budget-based methods and classical ranking aggregation, enabling the transference of justified representation, proportional share, priceability, and network flow techniques into ranking and assignment domains. Notably:

• It demonstrates that proportionality can be satisfied not only in collective selection but also in ordering and ranking settings, challenging the primacy of majoritarian rules for rank aggregation.
• The explicit tying of budget splitting and candidate pricing to pairwise and prefixwise fairness guarantees enables unified formal analysis across domains.
• Theoretical and empirical work confirms that equal shares-inspired mechanisms can reach utilitarian efficiency in initial rounds, with a “controlled” trade-off for proportionality as the ranking or allocation proceeds (Lederer, 22 Aug 2025, Skowron et al., 2016).

The ranked method of equal shares represents a convergence of algorithmic social choice, fair division, and resource allocation, providing scalable, principled solutions to proportionality in ranked collective outcomes.