Proportional Sequential Borda Rule

• Proportional Sequential Borda Rule is a rank aggregation method ensuring each weighted ranking’s pairwise agreements are fairly represented in the output.
• It combines classical Borda scoring with a sequential selection and weight transfer mechanism to balance input rankings from diverse voters.
• The rule guarantees proportional representation by dynamically adjusting weights, thereby mitigating majority bias and maintaining fairness in multiwinner settings.

The Proportional Sequential Borda Rule is a rank aggregation and multiwinner selection principle designed to ensure proportional representation within aggregated rankings, particularly in settings where the electorate is heterogeneous and input rankings (possibly weighted) should be fairly reflected in the outcome. By blending classical Borda scoring with sequential and proportional mechanisms, the rule provides a formal guarantee that each input ranking influences the output ranking commensurately with its assigned weight, thus departing from majoritarianism and offering strong representation guarantees even in adversarial or imbalanced scenarios.

1. Mathematical Structure and Axiomatic Foundations

Consider a set of $n$ weighted input rankings over $m$ candidates. Let $w_i>0$ denote the weight of the $i$-th input ranking. The Proportional Sequential Borda Rule (PSB) aims to construct an output ranking $\sigma^*$ such that, for each input ranking $\sigma_i$, the number of candidate pairs $(a,b)$ for which $\sigma^*$ and $\sigma_i$ agree is proportional to $w_i$.

Formally, define the agreement between $\sigma_i$ and $\sigma^*$: $$\operatorname{agree}(\sigma_i,\sigma^*) = |\{(a, b)\ |\ (a\succ_{\sigma_i}b)\ \wedge\ (a\succ_{\sigma^*}b)\}|,$$ where $a\succ_{\sigma}b$ reads "$a$ ranked above $b$ in $\sigma$".

The PSB rule outputs a ranking $\sigma^*$ maximizing the following proportionality condition: $$\forall i,\quad \operatorname{agree}(\sigma_i,\sigma^*) \geq w_i\cdot \operatorname{agree}_{\max},$$ where $\operatorname{agree}_{\max}$ is the maximum agreement achievable by any ranking with respect to input $\sigma_i$. In practice, the rule provides an output with maximally balanced agreement allocations across all input rankings, up to resource splitting dictated by the weighting vector $(w_1,\ldots,w_n)$.

Key axioms mirrored from the approval committee literature and adapted for rankings—such as proportional justified representation for rankings, neutrality, consistency, and cancellation—are satisfied by the PSB. In particular, the cancellation property (if all input rankings are perfectly balanced, the output does not favor any subset) and neutrality (permutation invariance with respect to candidate labeling) are structurally inherited from the network-theoretic characterization of Borda (Bubboloni et al., 2022).

Moreover, when the conditions of unanimity, minimal independence of irrelevant alternatives (MIIA), anonymity, and neutrality are imposed, only Borda-type scoring rules (including PSB as a sequential/proportional elaboration) are permissible in domains with more than three candidates (Gendler, 28 Nov 2024).

2. Sequential Mechanism and Proportional Weight Transfer

The PSB operates via a sequential process in which, at each round $k$, one candidate is assigned to the next position in the output ranking. Let $C_k$ denote the set of candidates still available at round $k$ and $w_i^{(k)}$ the current remaining "influence" weight of input ranking $i$.

At each step:

1. For all $a \in C_k$, compute a proportional Borda score:

$$s_k(a) = \sum_{i=1}^n w^{(k)}_i \cdot \left(|L(\sigma_i,a)|-|U(\sigma_i,a)|\right),$$

where $L(\sigma_i,a)$ is the set of candidates ranked below $a$ in $\sigma_i$ and $U(\sigma_i,a)$ for those ranked above.

1. Select $a^* = \arg\max_{a \in C_k} s_k(a)$.
2. For every $i$ such that $a^*$ appears at the top available position in $\sigma_i$, reduce $w^{(k)}_i$ by a proportionality factor $f$ (often $1/k$ for round $k$, or as dictated by the proportional share allocated towards full agreement).
3. Remove $a^*$ from $C_k$, decrement $k$, and repeat until $|C_k|=0$.

The explicit reweighting step ensures that each input ranking contributes its proportional influence precisely once for each rank transition, aligning the cumulative pairwise agreements with the fairness condition. The round-wise net-outdegree approach, adapted from the network generalization of Young’s characterization, guarantees that neutrality, consistency, and cancellation hold at every selection (Bubboloni et al., 2022).

3. Proportionality Guarantee and Position in the Hierarchy

The PSB’s core fairness guarantee can be summarized as follows: for each input ranking $\sigma_i$, and total input weight $W = \sum_{i} w_i$, the fraction of pairwise agreements realized in the output ranking is at least $w_i/W$.

Empirically and theoretically, this is achieved without the majoritarian bias typical of classical social welfare functions such as Kemeny or (nonproportional) Borda, which maximize total agreement but can underrepresent minority or bounded-weight stakeholders (Lederer, 22 Aug 2025).

Compared to approval-based rules, the proportionality degree $d_{PSB}(\ell, k)$ for a group size $\ell$ and committee size $k$ is typically of the form

$$d_{PSB}(\ell,k) \approx \gamma\ell-\delta, \qquad (\gamma\in[0.6,0.7],\ \delta\approx1)$$

which interpolates between PAV ($d_{PAV}(\ell,k)=\ell-1$) and sequential Phragmén ($d_{Phrag}(\ell, k)\approx (\ell-1)/2$) (Skowron, 2018). This places PSB between optimal proportionality (PAV) and utilitarian efficiency (sequential Borda), offering a balanced trade-off for applications requiring both.

4. Algorithmic Properties and Computational Considerations

The sequential and network-weighted structure of PSB allows for efficient implementation. At each round, the winner determination step reduces to a Borda-like calculation with adjusted weights and is thus polynomial in $n$ and $m$. The overall process is $O(m^2 n)$ (since there are $m$ steps and each step involves Borda score updates across $n$ rankings and at most $m$ candidates).

Lower bounds for streaming and sample-efficient multiwinner Borda variants indicate that any $\epsilon$-approximate winner selection algorithm requires at least $\Omega(m)$ space and $\Omega(\epsilon^{-1})$ dependence (Dey et al., 2017). Sampling strategies, such as those used for Borda-CC or Borda-M rules in vote streams, align with this complexity, demonstrating that PSB rules are practical for large $n$ as long as $m$ remains moderate.

In manipulation and control settings, computational hardness often transfers. For example, finding a manipulation when total manipulator influence does not outweigh non-manipulator influence is NP-hard (Shen et al., 2020). Similarly, candidate-control for Borda-based methods is W[2]-hard in terms of the number of operations even with truncated votes, underscoring the need for robust design in PSB variants to limit opportunities for adversarial interference (Zhou et al., 8 May 2024).

5. Variants, Generalizations, and Practical Adaptations

Several structural and practical variants can be constructed by modifying the aggregation step or the reweighting function. Examples include:

• Ranked Method of Equal Shares (RMES): Introduces a utilitarian optimization, further balancing proportionality and total agreement, meaningful where utilitarian welfare and representation both matter strongly (Lederer, 22 Aug 2025).
• Flow-Adjusting Borda Rule: Stronger fairness, ensuring that even finely divided weights (reflecting extremely small stakeholder groups) yield strictly proportional agreement in the output.

Dynamic extensions, such as those required for online aggregation in sequential decision platforms or large-scale vote streams, can adapt the core selection logic with sample-based or online margin-recomputation mechanisms. In such settings, the required space is $O(\epsilon^{-2}k^3 m^3 \log m)$ for Borda-CC and $O(\epsilon^{-2} k^3 m^5 \log m)$ for Borda-M, where $k$ is the committee size and $m$ is the number of candidates (Dey et al., 2017).

Many empirical studies confirm that Borda-based sequential/averaged rules robustly avoid monotonicity and no-show paradoxes (~0% occurrence), and majority failures (winner/Condorcet) are extremely rare with averaged and modified Borda variants (Fox et al., 31 Dec 2024). Approximately 91% of elections returned the same winner across all five Borda variants tested in large-scale U.S. ranked-choice datasets.

6. Relationship to Committee, Ranking, and Budgeting Domains

Techniques and axioms underpinning PSB are directly inspired by multiwinner committee methods (notably, proportional approval voting, PAV) and recent developments in participatory budgeting.

For multiwinner committee selection, Borda-based proportional rules can be implemented as streaming or sequential algorithms: $$\widehat{score}_{E}(S) = \frac{n}{t} \cdot score_{E_R}(S)$$ with $\epsilon$-winning guarantees for approximate proportionality (Dey et al., 2017).

For ranking and Q&A/recommendation tasks, PSB aligns with dynamic proportional ranking rules, upon which candidates/questions are ranked or executed sequentially, guaranteeing that underrepresented groups are “boosted” in subsequent steps to maintain proportionality, as analyzed in dynamic proportional rankings (Israel et al., 2021).

In continuous or online settings (e.g., participatory budgeting), the proportional allocation of “pairwise agreement” mirrors proportional share axioms such as Proportional Justified Representation (PJR) and Extended JR (EJR), now lifted from approval sets to total rankings (Chandak et al., 2023).

7. Applications, Limitations, and Scope

The Proportional Sequential Borda Rule is suitable for:

• Aggregating weighted rankings from heterogeneous voters, stakeholders, or models in recommender systems, meta-search, and participatory decision platforms.
• Multiwinner selection where both representation and consensus are sought, especially in settings with large, diverse input profiles.
• Dynamic and streaming environments, where efficiency and limited storage are critical.

However, in scenarios involving highly adversarial or strategic control actions (vote/candidate addition, deletion), computational intractability may arise, requiring additional algorithmic safeguards or approximative solutions (Zhou et al., 8 May 2024). Furthermore, for small candidate sets (exact $k=3$), alternative non-Borda-compliant social welfare functions exist; for $k>3$, only Borda-type rules with the described proportional framework are fully compatible with fairness, neutrality, and independence axioms (Gendler, 28 Nov 2024).

The PSB thus occupies a central position in the hierarchy of proportional social choice, providing explicit guarantees in both static and dynamic/online rank aggregation, while being compatible with established axiomatic and complexity-theoretic constraints across voting, committee, and budgeting domains.