1 Introduction
In multimember elections, a central concern is proportional representation of voters. When voters elicit ranked preferences over candidates, one particular axiom for proportional representation is Proportionality of Solid Coalitions (PSC). This axiom was advocated by Dummett (1984) and has been referred to as the most important requirement for proportional representation (Tideman, 1995; Tideman and Richardson, 2000; Woodall, 1994, 1997).
PSC is the subject of many theoretical and empirical studies. Theoretical studies have focused on designing voting rules that satisfy PSC; these include single transferable vote (STV) (Tideman, 1995), Quota Borda System (QBS) (Dummett, 1984), SchulzSTV (Schulze, 2011), and the Expanding Approvals Rule (EAR) (Aziz and Lee, 2020). STV is the most prominent of these rules^{2}^{2}2STV is used for elections in several countries including Australia, Ireland, India, and Pakistan. and has attracted significant attention in the literature from both a theoretical (Geller, 2005; Miller, 2007; Ray, 1986; Van Deemen, 1993) and empirical perspective (Gallagher, 1986; Latner and McGann, 2005). However, few studies consider the structure imposed by the PSC axiom.^{3}^{3}3A notable exception includes Peters (2018). In a setting where voters are assumed to have dichotomous preferences, Peters’ (2018) results imply that an analogous PSC axiom is incompatible with strategyproofness. Note that in this setting the GibbardSatterthwaite theorem (Gibbard, 1977; Satterthwaite, 1975) does not apply.
In this note, we present a characterization of PSC committees as the range of outcomes from a certain procedure; we formalize and call this rule the Minimal Demand (MD) rule. MD generalizes an approach pioneered by Michael Dummett. We also present an alternative way of viewing MD in terms of a “Dummett Tree,” which represents a decision tree with decisions taken at each branching of a node. Our main result is the following:
A committee satisfies PSC if and only it is a possible outcome of MD if and only if it is an outcome of a branch of a Dummett tree.This contribution is important because it provides an intuitive and tractable method for researchers to analyze the demands of PSC on committee outcomes. We hope that this characterization will be useful for understanding the interaction between PSC and other axioms.
2 Preliminaries
We consider the standard social choice setting with a set of voters , a set of candidates and a preference profile such that each is linear order over . If then we say that voter prefers candidate to candidate or, equivalently, voter ranks candidate ahead of candidate . Based on the preference profile, the goal is to select a committee of predetermined size .
The focus of this paper is on committees that satisfy the Proportionality of Solid Coalitions (PSC) axiom. Before formally defining PSC, we introduce the notion of a solid coalition which is central to the PSC axiom. Intuitively, a set of voters forms a solid coalition for a set of candidates if every voter in prefers every candidate in to any candidate outside of . Importantly, voters that form a solid coalition for a candidateset are not required to have identical preference orderings over candidates within nor .
Definition 1 (Solid coalition).
A set of voters is a solid coalition for a set of candidates if every voter in ranks (strictly prefers) every candidates in ahead of every candidate in . That is, for all and for any
The candidates in are said to be supported by the voter set , and, conversely, the voter set is said to be supporting the candidate set .
We now state the PSC definition. Informally, PSC requires that a committee “adequately” represents the preferences of solid coalitions that are “sufficiently large” by including some candidates that they support in relation to the size of the solid coalition. Here “adequately” and “sufficiently large” are defined according to a parameter and leads to a hierarchy of PSC definitions denoted by PSC. In slightly more detail, a committee satisfies PSC if, for every positive integer , every solid coalition of voters for a candidateset with size has: at least candidates included in from the candidateset or, if is smaller than , has all candidates in included in .
Definition 2 (Psc).
Let . We say a committee satisfies PSC if for every positive integer , and for every solid coalition supporting a candidate subset with size , the following holds
The PSC axiom captures intuitive features of proportional representation. The axiom ensures representation of minority voters so long as they share similar preferences over candidates, i.e., they form a solid coalition, and the amount of representation given to a group of voters that form a solid coalition is (approximately) in proportion to their size.
Note that the PSC axiom forms a hierarchy of axioms: if, for some , an outcome satisfies PSC, then satisfies PSC for all (Aziz and Lee, 2020, Lemma 2).
The PSC definition assumes that .^{4}^{4}4See (Aziz and Lee, 2020, Footnote 10) for justification. If , then the PSC property is referred to as HarePSC. If , for sufficiently small , then the PSC property is referred to as DroopPSC.^{5}^{5}5Droop PSC is also referred to as Droop’s proportionality criterion (DPC). Technically speaking the Droop quota is . The exact value is referred to as the HagenbachBischoff quota.^{6}^{6}6Formally, is required to be positive and small enough so that that for any ,
Remark 1.
There are some reasons to prefer the Droop quota. First, for the Droop quota satisfies the majority principle (Woodall, 1997): A candidate that is most preferred by more than half of the voters is selected whenever such a candidate exists.^{8}^{8}8A related property for general is the fixed majority principle (Debord, 1993). However, for , the fixed majority principle is incomparable with PSC and is not aligned with proportional representation. For discussion on other properties in multiwinner voting that are related to majority comparisons, we refer readers to Aziz et al. (2017); Gehrlein (1985). Second, DroopPSC implies HarePSC, which was stated as an essential property that a rule designed for PR should satisfy (Dummett, 1984).
3 Minimal Demand Rule and the Dummett Tree
Dummett proposed the Quota Borda System (QBS) rule as follows. It examines the prefixes (of increasing sizes) of the preference lists of voters and checking whether there exists a corresponding solid coalition for a set of voters. If there is such a solid set of voters, then an appropriate number of candidates with the highest Borda count are selected so as to satisfy the corresponding PSC demand.
We view this approach as a more general class of rules that we refer to as the Minimal Demand (MD) rule. Just like in QBS, we examine prefixes (of increasing sizes) of the preference lists of voters and check whether there exists a corresponding solid coalition for a set of voters. If there is such a solid set of voters, then a minimal subset of candidates is selected to satisfy the corresponding PSC requirement.^{9}^{9}9In our formalization of the rule candidates from the candidate subset are selected in a sequential manner. The MD algorithm is formalized as Algorithm 1.
Since MD specifies a candidate subset from which a candidate should be selected, rather than specifying a single candidate, there is a great deal of flexibility in how these “ties” are resolved, i.e., which candidate from the subset is selected. By considering different tiebreaking decisions, we can attain different outcomes of MD. These outcomes can be represented in the form of a decision tree, which we will refer to as the Dummett Tree.
The Dummett tree can be viewed as a (possibly) nondeterministic tree corresponding to how the MD rule can be run depending on the selection of candidates in each stage. Each node along a path of Dummett tree corresponds to a stage. The Dummett tree has depth . If candidates have already been selected by stage , we still go over all the stages. At each stage of the tree the rule only considers prefixes of voter preferences lists up to their th most preferred candidate. Accordingly, only those PSC demands that pertain to the most preferred candidates of each voter are considered. Each path of the tree reflects the selection of candidates. Each node in the path represents the selection of candidates at that point keeping in view the candidates already selected at nodes higher up in the path.^{10}^{10}10Equivalently, one may also include “null” nodes, which represent stages where no candidate was selected, in the tree. For a preference profile , we will denote the corresponding Dummett tree by DT.
Note that each preference profile gives rise to PSC demands. We can partition the PSC demands among demands pertaining to rank from to . For any given , we only consider candidates in : the set of candidates involved in preferences of voters up to their first positions.
A reformulation of the MD rule
Consider the following voting rule, which we call wMD.
For each voter initialize , initialize and . Consider every voters top preferred candidates (denote the union of all voters top preferences by ), if there exists a set of candidates supported by a solid coalition of voters with weight
(2) and there exists a positive integer such that
(3) then select an unelected candidate from and update to , and rescale the weight of all voters in to . Repeat until no such set of candidates exists, or all candidates in any such set have already been elected. If , update to ; otherwise terminate the algorithm.
Some observations:

The selection of different candidates, and the different choices of solid coalitions leads to different outcomes. Thus, the set of all possible outcomes (represented by the tree) forms an irresolute voting rule.

At the start of the algorithm, the total weight of all voters is . At the election of a new candidate this (total weight) decreases by precisely . Furthermore, since
and the entire set of voters is a solid coalition for the voting rule is guaranteed to have elected precisely candidates by the end of the th stage.
Lemma 1.
MD and wMD are equivalent. That is, an outcome is attainable under MD if and only if it is attainable under wMD.
Proof.
Fix the preferences of voters, and let be some outcome from MD such that candidate is elected before candidate for all .
Now suppose that there is no outcome of wMD that coincides with . Let be an outcome of wMD, and denote by the first index such that and for all . Note that is the index of the ‘pointoffirstdifference’ between the election outcomes and . Furthermore, suppose that among all wMD outcomes maximizes this ‘pointoffirstdifferences’ index .
Suppose that is elected in MD at the th stage (i.e., when considering the set of candidates). Consider the stage where the candidates are considered under wMD. Since is required for PSC (due to minimal demand election procedure of MD), it must be that is an element of a candidate which is solidly supported by a group of voters for some , and
(4) 
Now consider the wMD voting rule, the group of voters clearly satisfies the second condition (3). Thus, it only remains to prove that (2) also holds. The original total weight of voters in was (since we intialize ). Given that (4) holds, and the group of voters only solidly support candidates in , it must be that the remaining weight is at least . We conclude that (2) holds. Thus, is a valid choice for wMD and at least one of the outcomes of wMD must include which has ‘pointoffirstdifference’ index of at least . This contradicts the maximality of ; we conclude that is attainable under wMD.
Now, conversely, let be some outcome from wMD such that candidate is elected before candidate for all .
Suppose that there is no outcome of MD that coincides with . Let be an outcome of MD, and again denote by the index such that and for all . Furthermore, suppose that among all MD outcomes maximizes this ‘pointoffirstdifferences’ index .
Suppose that is elected in wMD at the th stage (i.e., when considering the set of candidates). Consider the stage where the candidates are considered under MD. Since was elected under wMD it must be the case that there exists a group of voters who solidly supporting a candidate subset with such that the weight of is . That is, both (2) and (3) hold. But the MD voting rule only requires (3) to hold to make a valid choice. This leads to a new outcome which has a strictly larger ‘pointoffirstdifference’ index of . This is a contradiction and so it must be that is attainable under MD. ∎
4 MD, Dummett Trees and PSC
We present some connections between MD, Dummett Trees and PSC. Lemma 2 states that each possible outcome of MD satisfies PSC. To the best of our knowledge, it is the first formal argument that MD returns a PSC committee and the outcome being PSC does not depend on the specific choices made during the algorithm.
Lemma 2.
Each path along the Dummett Tree gives rise to a selection of a committee satisfying PSC. Equivalently, each possible outcome of MD satisfies PSC.
Proof.
Equivalently, we consider the wMD formulation of MD (recall Lemma 1).
We denote by the set of candidate involved in the preferential election, and fix .
Let be an arbitrary outcome from the wMD voting rule. For sake of a contradiction suppose that does not satisfy PSC; that is, there exists a positive integer and a group of voters with solidly supporting a candidate subset such that
Assume that and are minimal such that
This can always be achieved by considering smaller subsets of and smaller values .
Now consider preferential election for . Note that , hence voters in only vote for candidates in in this election, and also all previous preferential elections for , Now, let denote the subset of elected candidates at the termination of this preferential election. Since , it follows that and so condition (3) in the wMD rule is satisfied. Furthermore, it must be the case that the total weight of voters in exceeds ; that is, condition (2) is also satisfied. This follows since the group starts with weight and the election of each candidate ( such candidates) reduces this weight by at most (strictly less when there is overlap of coalitions), and so
The wMD voting rule cannot terminate without either an additional candidates from being elected, or the total weight of voters in being reduced below . Voters in can only have their weight reduced below in this preferential election if an additional candidates in are elected, and so the former is required before the voting rule can proceed to the preferential election. But, by assumption, these candidates were not elected, and so it must be that the preferential election is never attained and so . That is, the additional candidates were not elected because it would exceed the size limit of the election outcome . However, this cannot be the case. For if it were, then the total weight of all voters at the termination of the preferential election would be (note that ), which contradicts the existence of the set of voters with weight at the termination of this preferential election.∎
Notice that Lemma 2 implies the following. Any voting rule that proceeds by sequentially electing candidates is guaranteed to satisfy PSC so long as: (1) candidates are only added if they resolve a PSC violation, and (2) candidates that resolve violation of PSC for smaller prefixes of voters preferences are elected before candidates that resolve violations for larger prefixes.
Lemma 3.
Take any committee satisfying PSC. Then the candidates in are selected along some path of the Dummett tree.
Proof.
Consider a committee satisfying PSC. We simulate MD that makes makes decisions along the tree and selects candidates from . The proof is by induction on the number of stages. At each stage, we can select candidates only from to fulfill the PSC demands for the first positions. Since satisfies PSC and hence all PSC demands, if we have selected by the th stage, there exists a set such that that satisfies the PSC demands up to the first positions. Hence there always exists a path in the Dummett tree that selects . ∎
Combining the two lemmas gives us the following equivalence theorem.
Theorem 1.
The following four statements are equivalent.

A committee satisfies PSC.

A committee is an outcome of some path of the Dummett tree.

A committee is a possible outcome of MD.

A committee is a possible outcome of wMD.
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