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Maximum Flow is Fair: A Network Flow Approach to Committee Voting

Mashbat Suzuki, Jeremy Vollen

TL;DR

This paper addresses fair committee voting under approval ballots by introducing Group Resource Proportionality (GRP) and establishing a max-flow characterization that connects fairness to network flows. It then develops two algorithms, Redistributive Utilitarian Rule (RUT) and Generalized CUT (GCUT), that achieve GRP with efficiency and, in GCUT’s case, excludable strategyproofness, while enabling best-of-both-worlds fairness results. The work shows fundamental trade-offs: no rule can be GRP-efficient and strategyproof, but GCUT attains GRP with excludable strategyproofness and maximizes social welfare within GRP, supported by a rich flow-network framework. Moreover, the paper strengthens best-of-both-worlds guarantees by combining ex-ante GRP with ex-post FJR/EJR+ via affordability, resolving open questions and offering polynomial-time constructions. Overall, the study advances flow-based methods for fair probabilistic committee voting and demonstrates practical, scalable approaches to coalition representation and welfare maximization under fairness constraints.

Abstract

In the committee voting setting, a subset of $k$ alternatives is selected based on the preferences of voters. In this paper, our goal is to efficiently compute $\textit{ex-ante}$ fair probability distributions over committees. We introduce a new axiom called $\textit{group resource proportionality}$, which strengthens other fairness notions in the literature. We characterize our fairness axiom by a correspondence with max flows on a network formulation of committee voting. Using the connection to flow networks revealed by this characterization, we introduce two voting rules which achieve fairness in conjunction with other desiderata. The first rule - the $\textit{redistributive utilitarian rule}$ - satisfies ex-ante efficiency in addition to our fairness axiom. The second rule - Generalized CUT - reduces instances of our problem to instances of the minimum-cost maximum flow problem. We show that Generalized CUT maximizes social welfare subject to our fairness axiom and additionally satisfies an incentive compatibility property known as $\textit{excludable strategyproofness}$. Lastly, we show our fairness property can be obtained in tandem with strong $\textit{ex-post}$ fairness properties - an approach known as $\textit{best-of-both-worlds}$ fairness. We strengthen existing best-or-both-worlds fairness results in committee voting and resolve an open question posed by Aziz et al. [2023].

Maximum Flow is Fair: A Network Flow Approach to Committee Voting

TL;DR

This paper addresses fair committee voting under approval ballots by introducing Group Resource Proportionality (GRP) and establishing a max-flow characterization that connects fairness to network flows. It then develops two algorithms, Redistributive Utilitarian Rule (RUT) and Generalized CUT (GCUT), that achieve GRP with efficiency and, in GCUT’s case, excludable strategyproofness, while enabling best-of-both-worlds fairness results. The work shows fundamental trade-offs: no rule can be GRP-efficient and strategyproof, but GCUT attains GRP with excludable strategyproofness and maximizes social welfare within GRP, supported by a rich flow-network framework. Moreover, the paper strengthens best-of-both-worlds guarantees by combining ex-ante GRP with ex-post FJR/EJR+ via affordability, resolving open questions and offering polynomial-time constructions. Overall, the study advances flow-based methods for fair probabilistic committee voting and demonstrates practical, scalable approaches to coalition representation and welfare maximization under fairness constraints.

Abstract

In the committee voting setting, a subset of alternatives is selected based on the preferences of voters. In this paper, our goal is to efficiently compute fair probability distributions over committees. We introduce a new axiom called , which strengthens other fairness notions in the literature. We characterize our fairness axiom by a correspondence with max flows on a network formulation of committee voting. Using the connection to flow networks revealed by this characterization, we introduce two voting rules which achieve fairness in conjunction with other desiderata. The first rule - the - satisfies ex-ante efficiency in addition to our fairness axiom. The second rule - Generalized CUT - reduces instances of our problem to instances of the minimum-cost maximum flow problem. We show that Generalized CUT maximizes social welfare subject to our fairness axiom and additionally satisfies an incentive compatibility property known as . Lastly, we show our fairness property can be obtained in tandem with strong fairness properties - an approach known as fairness. We strengthen existing best-or-both-worlds fairness results in committee voting and resolve an open question posed by Aziz et al. [2023].
Paper Structure (12 sections, 24 theorems, 34 equations, 3 figures, 2 algorithms)

This paper contains 12 sections, 24 theorems, 34 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Characterization of Fair Committees
  4. Fairness and Efficiency: Redistributive Utilitarian Rule
  5. GRP
  6. Efficiency
  7. Polynomial time computation
  8. On Strategyproofness, Fairness, and Efficiency
  9. Structural Observations on Generalized CUT
  10. GRP-efficiency of Generalized CUT
  11. Excludable Strategyproofness of Generalized CUT
  12. Best-of-both-worlds Fairness

Key Result

Theorem 3.3

A fractional committee $\vec{p}$ satisfies group resource proportionality if and only if there exists a max flow $f$ on the network representation $\mathcal{N}$ such that $p_c \geq f(c , t)$ for each $c \in C$.

Figures (3)

  • Figure 1: Illustration of the network representation of an instance of committee voting.
  • Figure 2: Visualization of hierarchy of fairness properties mentioned in this paper. An arrow from (A) to (B) denotes that (A) implies (B). Properties highlighted in green are polynomial time computable, following from \ref{['thm:gr_characterization']}. (*) Strict fractional core is not guaranteed to exist.
  • Figure 3: Illustration of a flow update along a cycle $Q_i=(s,i_1,c_1,\ldots,i_r,c_r,i,s)$ with bottleneck $b>0$. Labels in black denote flow bounds sufficient to trigger the flow update. Green arcs show the path of flow redistribution.

Theorems & Definitions (58)

  • Example 1.1
  • Definition 2.1: Efficiency
  • Definition 2.2: Strategyproofness
  • Definition 2.3: Excludable Strategyproofness
  • Definition 2.4
  • Definition 3.1: Fractional core
  • Definition 3.2: Group Resource Proportionality
  • Theorem 3.3
  • proof
  • Definition 3.4: Strong UFS ALS+23
  • ...and 48 more