Best-of-Both-Worlds Fairness in Committee Voting

Our first contribution is to broaden the best-of-both-worlds fairness paradigm, which has so far been limited to resource allocation, and explore it in the context of social choice problems, specifically committee voting. Whereas the relaxed notions of ex-post fairness have been examined at great length for committee voting, the literature on ex-ante fairness is less developed.

In Section 3, we formalize several natural axioms for ex-ante fairness that are careful extensions of similar concepts proposed in the restricted setting of single-winner voting. These include the following concepts in increasing order of strength as well as stronger variants: positive share, individual fair share (IFS), unanimous fair share (UFS), and group fair share (GFS). For instance, positive share simply requires that each voter expects to have non-zero probability of having some approved candidate selected. At the other end of the spectrum, GFS gives a desirable level of ex-ante representation to every coalition of voters. In Figure 1, we present ex-ante and ex-post fairness hierarchies and establish logical relations between them.

In line with the goals of the best-of-both-worlds fairness paradigm, our central research question is to understand which combinations of ex-ante and ex-post fairness properties can be achieved simultaneously. In Section 4, we show that ex-ante GFS, ex-ante Strong UFS and ex-post EJR can be achieved simultaneously by devising a class of randomized algorithms. Our class of algorithms, which we call Best-of-Both-Worlds MES (BW-MES), uses as a subroutine the well-known Method of Equal Shares (MES) [Peters and Skowron, 2020]. Furthermore, a lottery satisfying GFS, Strong UFS and EJR can be computed in polynomial time via an algorithm belonging to BW-MES.

Finally, we examine whether it is possible to achieve a stronger ex-post guarantee while maintaining some ex-ante fairness properties. In Section 5, we answer this question affirmatively by presenting an algorithm (referred to as BW-GCR) which outputs a lottery that is ex-post FJR and ex-ante Strong UFS. Our algorithm combines both the Greedy Cohesive Rule [Peters et al., 2021] and the BW-MES rule.

In this paper, we examine approval-based committee voting, a generalization of the classical voting setting which has been studied at length, particularly from the 19th century to the present. One of the persistent questions within the committee voting setting is how to produce committees which proportionally represent groups of voters. Aziz et al. [2017] initiated an axiomatic study of approval-based committee voting based on the idea of “justified representation” for cohesive groups. The study has led to a hierarchy of axioms and a large body of work focusing on voting rules which produce committees satisfying these axioms and thus give some guarantee of fair representation [Aziz et al., 2018; Elkind et al., 2022; Brill et al., 2023]. For a detailed survey of the recent work on approval-based committee voting, we refer the readers to the book of Lackner and Skowron [2023]. While this paper also targets committees satisfying these properties, we examine outcomes that are randomized committees which specify a probability distribution over integral committees.

In social choice theory, randomization is one of the oldest tools used to achieve stronger fairness properties and to bypass various impossibility results which apply to discrete outcomes [see, e.g., Gibbard, 1977]. For single-winner randomized voting (also known as probabilistic voting) with approval preferences, Bogomolnaia et al. [2005] defined ex-ante fairness notions, the individual fair share (IFS) and unanimous fair share (UFS), and provide rules satisfying them. They also proposed a group fairness property called group fair share (GFS) [see Bogomolnaia et al., 2002], independently proposed by Duddy [2015], which is stronger than UFS and IFS but weaker than core fair share (CFS), a group fairness and stability property inspired by that of core from cooperative game theory [Scarf, 1967]. In a setting which generalizes probabilistic voting by allowing arbitrary endowments, Brandl et al. [2021] studied fair distribution rules and introduce GFS, which they show is equivalent to a notion called decomposability. None of their results imply those presented in this paper since their setting places no restriction on the distribution a single alternative can receive our setting does.

Michorzewski et al. [2020] explored the trade-off between group fairness and utilitarian social welfare by measuring the “price of fairness” with respect to fairness axioms such as IFS and GFS. We formulate these ex-ante fairness properties for the more general committee voting setting for the first time. As mentioned, we search for outcomes which also give fair representation to groups ex-post, a desideratum which has no analogue in the classical voting setting.

Aziz [2019] proposed research directions regarding probabilistic decision making with desirable ex-ante and ex-post stability or fairness properties. Freeman et al. [2020] were the first to coin the term “best-of-both-worlds fairness”. They examined the compatibility of achieving ex-ante envy-freeness and ex-post near envy-freeness in the context of resource allocation. There have been several recent works on best-of-both-worlds fairness in resource allocation [Halpern et al., 2020; Babaioff et al., 2021, 2022; Aziz et al., 2023a, b; Hoefer et al., 2023; Feldman et al., 2023]. Other works that consider the problem of implementing a fractional allocation over deterministic allocations subject to constraints include [Budish et al., 2013; Akbarpour and Nikzad, 2020].

Fairness properties for integral committees are well-studied in committee voting. A desideratum that has received significant attention is justified representation (JR) [Aziz et al., 2017]. In order to reason about JR and its strengthenings, an important concept is that of a cohesive group. For any positive integer $\ell$, a group of voters $N^{\prime}\subseteq N$ is said to be $\ell$-cohesive if $|N^{\prime}|\geq\ell\cdot n/k$ and $\left|\bigcap_{i\in N^{\prime}}A_{i}\right|\geq\ell$.

Two important strengthenings of JR have been proposed.

It follows directly from the definitions that EJR implies PJR, which in turn implies JR. A committee providing EJR (and therefore PJR and JR) always exists and can be computed in polynomial time [Aziz et al., 2017; Peters and Skowron, 2020].

Whereas the literature on fairness concepts for integral committees is very well-developed, fairness properties for fractional committees are largely unexplored except for the special case of single-winner voting [Bogomolnaia et al., 2005; Duddy, 2015; Aziz et al., 2020]. We introduce a hierarchy of fairness notions for fractional committees in the committee voting setting by generalizing axioms from the single-winner context based on fair share. The weakest in the hierarchy of axioms is individual fair share (IFS), the idea behind which is that “each one of the $n$ voters ‘owns’ a $1/n$-th share of decision power, so she can ensure an outcome she likes with probability at least $1/n$”, as Aziz et al. [2020, page 18:2] put it. This idea suggests at least two distinct interpretations of the utility lower bound guaranteed by IFS:

1. (a)

   each voter is given $1/n$ probability to choose their favourite integral outcome, or

2. (b)

   each voter can select $1/n$ of the (fractional) outcome.

In probabilistic voting, as we will see shortly, both interpretations coincide. Critically, this is not the case in the committee voting setting. Instead, these interpretations diverge and lead to two alternative hierarchies of fair share axioms for committee voting, which we term fair share and strong fair share, respectively.

We begin by defining both generalizations of individual fair share (IFS). Both impose a natural lower bound on individual utilities stronger than that of positive share, which requires that $u_{i}(\vec{p})>0$. In the single-winner setting, IFS requires that the probability that the (single) alternative selected is approved by any individual voter is no less than $1/n$. It is thus tempting to require $u_{i}(\vec{p})=\sum_{c\in A_{i}}p_{c}\geq\frac{k}{n}$, which turns out to be too strong in our setting as a fractional committee satisfying it may not exist.²²2For instance, let $k>n$ and consider the case where voter $i$ only approves a single candidate. Then, the above inequality cannot hold for $i$ as the left-hand side is upper bounded by $|A_{i}|=1$ while the right-hand side is greater than one and can be arbitrarily large. Intuitively speaking, this is because our only restriction on the voters’ approval sets is that each voter approves of at least one candidate, just as is standard in the single-winner literature. However, whereas in the $k=1$ special case this assumption is sufficient to ensure that a uniform cut-off utility lower bound for each voter is feasible, the same is not true for general $k$.

While IFS captures interpretation (a) of fair share, Strong IFS reflects interpretation (b) which says that each voter should control $1/n$ of the fractional outcome.

Next, we strengthen IFS (resp., Strong IFS) to unanimous fair share (UFS) (resp., Strong UFS), which guarantees any group of like-minded voters an influence proportional to its size.

Our primary focus in this paper is a stronger notion—group fair share (GFS)—which gives a non-trivial ex-ante representation guarantee to every coalition of voters.

We note that a GFS fractional committee always exists and can be computed by a very natural algorithm called Random Dictator, which selects each voter’s favourite integral committee (breaking ties arbitrarily) with probability $1/n$.

However, Random Dictator does not satisfy Strong IFS.³³3To see this, consider an instance with $k=2$, three candidates $\{c_{1},c_{2},c_{3}\}$, and two voters with $A_{1}=\{c_{1}\}$ and $A_{2}=\{c_{2},c_{3}\}$. Since each voter must select an integral committee, voter $1$ allocates some of her probability to a candidate she does not approve, and thus $\sum_{c\in A_{1}}p_{c}=p_{c_{1}}=1/2<\min\left\{\frac{k}{n},|A_{1}|\right\}=1$. Indeed, this is the principal reason we chose to name the respective axiom hierarchies as we did. There is a significant precedent of treating the Random Dictator mechanism as the utility lower bound for fair share axioms, including by the authors [Bogomolnaia et al., 2005, page 167] who introduced fair share:

Fair welfare share uses the random dictator mechanism as the disagreement option that each participant is entitled to enforce.

Furthermore, the natural extensions of Strong UFS to Strong GFS are not guaranteed to exist. For instance, following [Bogomolnaia et al., 2005; Brandl et al., 2021] and our own Definition 3.5, we may be tempted to formulate the RHS of Strong GFS as the sum of the Strong IFS guarantees, i.e., $\sum_{i\in S}\min\left\{\frac{k}{n},|A_{i}|\right\}$. However, Example 3.7 will show that a fractional committee satisfying this notion may not always exist.⁴⁴4This demonstrates that the setting of Brandl et al. [2021] deviates from our setting as Algorithm 1 outputs a decomposable outcome satisfying Strong IFS, but Strong GFS is not satisfied. Another natural generalization would be the following:

Equation 1 captures the spirit of strong fair share well by affording each coalition of voters control over the outcome proportional to their size, upper bounded by the number of candidates they collectively approve. Example 3.7 below shows the formulation of Strong GFS given by Equation 1 is also impossible to satisfy.

It follows directly from the definitions that GFS implies UFS, which in turn implies IFS, and that each of our generalizations of IFS, UFS, and GFS correspond to their definitions in the single-winner voting scenarios. The relations between these axioms are pictured in Figure 1. We would also like to remark that any pair of ex-ante fairness notions not given explicit logical relation in Figure 1 are logically independent (i.e., neither implies the other). The details are deferred to Appendix A.

Before describing and proving our approach to best-of-both-worlds fairness in this setting, we first investigate the logical relations between our ex-ante and ex-post properties for integral committees. In doing so, we rule out some naive approaches to our problem of interest and illustrate the usefulness of our ex-ante properties. We begin by remarking that, as mentioned in Section 1, there may not exist an integral committee satisfying any of our ex-ante fairness properties.

As mentioned, this fact is the principal motivation for studying randomized committees. However, we would also like to understand what our fairness concepts for fractional committees can tell us about the space of integral committees satisfying our ex-post fairness properties. The following example and summarizing remark show that our fractional fairness concepts can help in reasoning about which outcomes satisfying our ex-post properties are more desirable.

From this, we conclude that a successful algorithm must select carefully from the space of outcomes satisfying our ex-post properties. We next explore to what extent our ex-ante properties imply our ex-post properties in the integral case.

While Proposition 3.11 hints at a synergy between our ex-ante and ex-post properties, Proposition 3.12 below shows that even the strongest ex-ante property in our hierarchy does not imply the next strongest ex-post property.

Despite this negative finding, in the following section, we will present a class of algorithms computing randomized committees which simultaneously satisfy ex-ante GFS and ex-post EJR.

We start by providing an intuition behind our family of rules BW-MES, whose pseudocode can be found in Algorithm 1. At a high level, BW-MES follows in spirit the idea of the Method of Equal Shares (MES) of Peters and Skowron [2020]. To be more precise, we follow the MES algorithm description of Lackner and Skowron [2023, Rule 11], and make use of its first phase. For ease of exposition, we simply refer to this first phase as “MES”.

Algorithm 1 returns an integral EJR committee $W_{\text{MES}}$ in algorithm 1. Denote by $(b_{i})_{i\in N}$ the remaining budget of the voters after executing MES (algorithm 1). Our next step is to extend $W_{\text{MES}}$ to a fractional GFS committee of size $k$ using voters’ remaining budget. We first initialize a fractional committee $\vec{p}$ using $W_{\text{MES}}$ in algorithm 1. It is worth noting that for any $c\in C\setminus W_{\text{MES}}$, $\sum_{i\in N_{c}}b_{i}<1$; otherwise candidate $c$ would have been included in $W_{\text{MES}}$ in algorithm 1. The key idea behind our completion method for the fractional committee in this family of rules is the following:

• •

  We first let each $i\in N$ such that $A_{i}\setminus W_{\text{MES}}\neq\emptyset$ spend her remaining budget $b_{i}$ on candidates $A_{i}\setminus W_{\text{MES}}$, in an arbitrary way.

• •

  Next, for any other voter, her remaining budget can be spent on any candidate $c\in C$ provided $p_{c}\leq 1$.

Finally, for the implementation step, we can use any rounding method that implements the fractional committee $\vec{p}$ by randomizing over integral committees of the same size; see, e.g., the AllocationFromShares method of Aziz et al. [2019], the stochastic method of Grimmet [2004], or the dependent randomized rounding scheme of Gandhi et al. [2006].

In the following, we use an illustrative example to demonstrate our algorithm.

Before proving Theorem 4.1, we first show a lower bound on voters’ utilities provided by BW-MES.

We are now ready to establish our main result.

Because MES may return an EJR committee $W_{\text{MES}}$ of size less than $k$, several ways of extending $W_{\text{MES}}$ to an integral committee of size exactly $k$ have been discussed [see, e.g., Peters and Skowron, 2020; Lackner and Skowron, 2023]. As we have seen previously (Remark 3.8), an EJR committee may not provide positive share, let alone GFS. We provide a novel perspective on the completion of MES. Specifically, we define a family of rules which extends an integral committee returned by MES to a randomized committee providing GFS.

Due to the flexibility in the definition of the BW-MES family, the number of BW-MES rules obtaining distinct outcomes can be quite large. For example, one BW-MES rule that seems quite natural is that which continues in the spirit of MES: for the candidate whose supporters have the most collective budget leftover, this budget is spent on the candidate, and we continue in this fashion sequentially. It is an interesting future direction to further identify specific algorithms in the BW-MES family which provide additional desiderata such as high social welfare or additional ex-ante properties.

We give an algorithm, referred to as BW-GCR, which obtains ex-ante Strong UFS in addition to ex-post FJR. The pseudocode of the algorithm is presented as Algorithm 2. At a high level, our algorithm is an interesting synthesis of the Greedy Cohesive Rule and our BW-MES Rule (Algorithm 1). A key ingredient here is how we set voters’ budgets for the BW-MES phase in order to obtain a feasible Strong UFS fractional committee in the end.

More specifically, our algorithm begins by calling GCR, denoting by $r$ the number of steps it executes before terminating. For each $j\in[r]$, we refer to $N_{j}$, $T_{j}$, and $\beta_{j}$ as the values of $N^{\prime}$, $T$, and $\beta$ for the weakly cohesive group selected in the $j$-th step of GCR. Let $W_{\text{GCR}}=\bigcup_{j\in[r]}T_{j}$ be the returned integral FJR committee, $N_{\text{GCR}}=\bigcup_{j\in[r]}N_{j}$ the set of voters who are inactivated (put differently, represented), and $N_{\text{MES}}=N\setminus N_{\text{GCR}}$ the set of active voters. For ease of exposition, we sometimes discuss a specific unanimous group within a given weakly cohesive group. Given any set of voters $S$, we can partition the voters into a collection of (maximal) unanimous groups $\{S^{1},S^{2},\dots,S^{\eta}\}$ such that for each $z\in[\eta]$, voters $S^{z}$ have an identical preference (i.e., they are unanimous) and for any $i\in S^{z}$ and $i^{\prime}\in S^{z^{\prime}}$ with $z\neq z^{\prime}$, $A_{i}\neq A_{i^{\prime}}$ (i.e., each unanimous group is maximal). We will denote by $N_{j}^{z}$ the $z$-th unanimous group of the $j$-th weakly cohesive group $N_{j}$ encountered by Algorithm 2 and will refer to the number of distinct unanimous groups in $N_{j}$ as $\eta_{j}$.

Next, Algorithm 2 proceeds by carefully setting an individual budget $b_{i}$ for each voter $i\in N_{\text{GCR}}$ in algorithm 2, and next for voters in $N_{\text{MES}}$ in algorithm 2 in a different way. We then call BW-MES (Algorithm 1) as a subroutine with budgets $(b_{i})_{i\in N}$, candidates $C\setminus W_{\text{GCR}}$ and committee size $k_{\text{MES}}=k-|W_{\text{GCR}}|$, and get a randomized outcome $\{(\lambda_{j},W_{j})\}_{j\in[s]}$. We will show shortly that the randomized committee $\{(\lambda_{j},W_{j}\cup W_{\text{GCR}})\}_{j\in[s]}$ is feasible, ex-post FJR, and ex-ante Strong UFS.

We now state the main result of this section, which shows that Algorithm 2 provides best-of-both-worlds fairness.

We begin by proving some auxiliary statements related to the budgets $b_{i}$ for each $i\in N$. These statements also provide a rationale for the particular way in which budgets are set in algorithm 2 and algorithm 2. The first two claims concern voters in $N_{\text{GCR}}$, for whom the budget is set in algorithm 2. Our first claim suggests that if a unanimous group is given zero budget, this unanimous group has already received representation amounting to its fair share.

Our next claim establishes a connection between $\beta_{j}$ and $|T_{j}|$ for any weakly cohesive group such that some voter is assigned positive budget.