1 Introduction
Social choice is a general framework of preference aggregation in which voters express preferences over outcomes and a desirable outcome is selected based on the preferences of the voters (Aziz et al., 2017a; Conitzer, 2010). The most classic model of social choice is (single winner) voting in which voters express preferences over a set of alternatives and exactly one alternative is selected (Brams and Fishburn, 2002). A natural generalization of the model is mutiwinner voting or committee voting in which a set of alternatives is selected (Faliszewski et al., 2017). Another model is multiple referenda in which voters vote over multiple but independent binary decisions (Brams et al., 1997). Probabilistic versions of singlewinner voting have also been examined (Gibbard, 1977).
In this paper, we study a natural model of social choice that simultaneously generalizes all the social choice settings mentioned above. The advantage of considering a more general combinatorial model (Lang and Xia, 2016) is that instead of coming up with desirable axioms, rules, and algorithms in a piecemeal manner for different settings, one can design or apply general principles and approaches that may be compelling for a wide range of settings. Of course certain axioms may only be meaningful for a certain subsetting but as we show in this paper, a positive algorithmic or axiomatic result for welljustified axioms can be viewed favourably for all relevant subsettings as well. Another advantage of formalising a general model is that it provides an opportunity to unify different strands of work in social choice. Our model also helps approach the committee voting problem in which there are additional diversity constraints possibly relating to gender, race or skill. Finally, our model applies to general participatory budgeting scenarios Cabannes (2004) where multiple decisions needs to be made and the minority representation needs to be protected.
After formalizing the SCV (subcommittee voting) setting, we focus on a particular restriction of SCV in which agents or voters only express approvals over some of the alternatives or candidates. The restriction to approvals is desirable because approvals capture dichotomous/binary preferences that are prevalent in many natural settings. Secondly, ordinal and cardinal preferences coincide when preferences are dichotomous. This is desirable since elicitation of cardinal utilities has been considered controversial in decisions concerning public goods.
SCV with approvals can be viewed as a multidimensional generalization of approvalbased committee voting (Kilgour, 2010). For approvalbased committee voting, a particularly appealing axiom that captures representation is justified representation () that requires that a set of voters that is large enough and cohesive enough in their preferences should get at least one approved candidate in the selected committee. The axiom has received considerable attention (Brill et al., 2017; SánchezFernández et al., 2017, 2016). For SCV with approvals, we extend the justified representation axiom (Aziz et al., 2017b) that has only been studied in the context of committee voting.
One interesting application captured by this SCV framework, which is not possible under standard models, is committee voting in the presence of diversity constraints or quotas. Considering this application highlights the conflict between diversity constraints and the original axioms of fair, or justified, representation. As will be shown this conflict leads to conceptual issues of what is the ‘appropriate’ generalisation of the axiom for SCV instances and also technical issues such as existence and computational intractability of achieving certain axioms whilst diversity constraints are enforced.
Contributions
Our contributions are threefold with the first two being conceptual contributions.
Firstly, we study a natural model of social choice called SCV (subcommittee voting) that simultaneously generalizes several previously studied settings.
Secondly, we focus on approvalbased SCV and present new notions of justified representation () concepts including Intrawise JR (IWJR) and Spanwise JR (SWJR). These distinct notions lead to ‘local’ and ‘global’ approaches to representation, respectively.
Thirdly, we present technical results concerning the extent to which these properties can be satisfied. We show that although SWJR is a natural extension of JR to the SCV setting, a committee satisfying SWJR may not exist even under severe restrictions. Furthermore, checking whether there exists a committee satisfying SWJR is NPcomplete. The results always show that the more general setting SCV is considerably more challenging than approvalbased committee voting. We then formalize a weakening of SWJR called weakSWJR and present a polynomialtime algorithm that finds a committee that simultaneously satisfies weakSWJR and IWJR. We also propose two natural generalizations of PAV (Proportional Approval Voting), a wellknown rule for committee voting under approvals. However we show that neither of these two extensions satisfies both weakSWJR and IWJR.
2 Subcommittee Voting
We propose a new setting called SCV that generalizes a number of voting models. The setting is a tuple

is the set of voters/agents.

is the set of candidates.

is a partitioning of the candidates. Each is referred to as a candidate subset from which a subcommittee is to be chosen.

is the quota function that specifies the number of candidates to be selected from each subset . We denote by .

specifies for each agent , her preferences/utilities over . We allow the possibility that an agent does not compare candidates across candidate subsets. At a minimum it is required that each is transitive and complete within each subset , however additional restrictions can be introduced, as befitting the setting; they might even be replaced with cardinal utilities
An SCV outcome specifies a real number for each with the following constraints:
In this paper we restrict our attention to discrete outcomes so that but in general SCV can allow for probabilistic outcomes where
is the probability of selecting candidate
. For discrete outcomes, an outcome will be a committee of size that consists of subcommittees where each and .If , for all and , we are in the voting setting. If , for all , we are in the committee/multiwinner voting setting (Faliszewski et al., 2017). If and , we are in the probabilistic voting setting (Gibbard, 1977). If for all and for all , we are in the public decision making setting (Conitzer et al., 2017). Note that public decision making setting is equivalent to the “voting on combinatorial domain” setting studied by Lang and Xia (2016). The latter setting allows for more complex preferences over the set of combinatorial outcomes but the preferences may not be polynomial in the number of candidates and voters. If for all and for all and for all , we are in the multiplereferenda setting (Brams et al., 1997; Lacy and Niou, 2000; Çuhadaroğlu and Lainé, 2012).
At a very abstract level, even a bicameral legislature can be viewed as an SCV setting in which . Even if there are no explicit multiple committees, there can be diversity constraints imposed on the committee that can be easily modelled as an SCV problem. For example, the problem of selecting five people with 3 women and 2 men can be viewed as an SCV problem with two candidate subsets.
In this paper, we will focus exclusively on approvalbased voting in the SCV setting. In approvalbased voting we replace each agent ’s preference with an approval ballot which represents the subset of candidates that she approves. The list of approval ballots is referred to as the ballot profile. As per the general SCV setting introduced at the start of the section, the goal is to select a target number of candidates from which satisfy the quota function for each candidate subset.
3 Justified Representation in Approvalbased Subcommittee Voting
We now focus on the SCV setting in which each agent approves a subset of the candidates. Based on the approvals, the goal is to identify a fair or representative outcome. Note that if , we are back in the committee voting setting. The approvalbased SCV setting can be seen as capturing independent committee voting settings.
For the approvalbased committee voting setting, justified representation () is a desirable property.
Definition 1 (Justified representation (JR)).
Given a ballot profile over a candidate set and a target committee size , we say that a set of candidates of size satisfies justified representation for if :
One natural extension of to the case of SCV is to treat each candidate subset as an independent committee voting problem. Then an SCV outcome satisfies Intrawise JR (IWJR) if each subcommittee satisfies .
Definition 2 (Intrawise JR (IWJR)).
An SCV outcome satisfies Intrawise JR (IWJR) if and :
We note that since a committee satisfying can always be attained by a polynomialtime algorithm (Aziz et al., 2017b), IWJR is easy to achieve by treating each subcommittee voting as a separate committee voting problem.
The limitation of this approach is that it could be that each time the same voters are unrepresented in each subcommittee and they may ask for some representation in at least some subcommittee. Thus IWJR can be considered as a ‘local’ axiom which ignores whether or not a given voter has already been represented in some other subcommittee.
In view of this limitation, another extension of to the case of SCV is to impose a type condition across all subcommittees.^{1}^{1}1Imposing representation requirements across all subcommittees implicitly assumes that the selections of all subcommittees are of comparable significance to the voters. The definition of SWJR is identical to the definition of for the committee voting setting.
Definition 3 (Spanwise JR (SWJR)).
An SCV outcome satisfies Spanwise JR (SWJR) if :
This approach to representation leads to a ‘global’ axiom which aims to represent large, cohesive groups of voters (i.e. and ) in some subcommittee, but not necessarily a subcommittee where they are cohesive.
Note that both SWJR and IWJR concern representation that are not at the level of single individual but at the level of large enough cohesive groups.^{2}^{2}2In a related paper, Conitzer et al. (2017) proposed fairness concepts for Public Decision Making that is equivalent to SCV in which for each . They considered different fairness notions that are based on proportional or envyfree allocations. The concepts involve viewing agents independently and are different from proportional representation concerns. When the number of subcommittees is less than the number of voters, the concepts they consider are trivially satisfied.
Next we show that a SWJR committee may not exist and is NPhard to compute.
4 (Non)existence and complexity of SWJR committees
We show that a committee satisfying SWJR may not exist under either of the two restriction (1) there are exactly two candidate subsets, and (2) for all .
Proposition 1.
A committee satisfying SWJR may not exist even if there are exactly two candidate subsets and for .
Proof.
Consider an SCV instance where , with and , and . Note that .
If the approval ballots are and , then there is no SCV outcome (i.e. a committee) which satisfies SWJR. This can be immediately observed since SWJR requires both voters to be represented, however the quota prevents this from being possible. ∎
The reader may note that above proof utilises an example where the voters have ballots which do not approve of any voter in some candidate subset (i.e. ). This feature is not required to show the nonexistence of an SWJR committee however, it greatly simplifies the example.
Above we proved that a committee satisfying SWJR may not exist. One could still aim to find such a committee whenever it exists. Next we prove that the problem of checking whether a SWJR committee exists or not is NPcomplete.
Proposition 2.
Checking whether an SWJR committee exists or not is NPcomplete.
To show that checking whether a SWJR committee exists is NPcomplete we will reduce a given instance of the Set Cover problem, a known NPcomplete problem Garey and Johnson (1979), to an SCV instance  such that an SWJR committee exists if and only if the Set Cover instance has a yes answer.
Below is a statement of the Set Cover problem.
0.98 Set Cover
[5pt] Input: Ground set of elements, a collection of subsets of such that and an integer . Question: Does there exist a such that and
To build intuition for the formal proof, which follows, we provide an overview of the reduction.
The Set Cover problem involves answering whether or not there exists a collection of at most subsets which cover another set . This problem can be embedded into an SCV instance by letting the set represent the set of voters and considering a candidate subset such that each element denotes an element of ; that is,
We then let each element of , say , be approved by voters if and only if . By appropriately defining quota values and voter approval ballots on the remaining candidate subset it is shown that an SWJR committee exists if and only if every voter is represented via a candidate in – this of course possible if and only if we have a yesinstance of the Set Cover problem
A formal proof is presented below.
Proof.
We reduce a given Set Cover instance to an SCV instance as follows: Let
denote the set of voters, candidate set and partition into two candidate subsets. Let voter approval ballots be
and . Without loss of generality we may assume that , also note that and so .
Since every voter has a nonempty approval ballot (i.e. ) and , a committee satisfies SWJR if and only if every voter is represented.
If is a yesinstance of the Set Cover problem then these exists a subset with such that – it follows that the committee
is a solution to the SCV problem and satisfies SWJR. Conversely, if is an SWJR committee then the set
provides a yesinstance to the Set Cover problem.
∎
5 WeakSWJR
In the previous section we showed that a committee satisfying SWJR need not exist, and checking whether it does is NPcomplete. Naturally and in the pursuit of a computationally tractable representation axiom we weaken the concept of SWJR. In this section we present a weak version of SWJR, appropriately referred to as weakSWJR. An SCV outcome satisfying weakSWJR is guaranteed to exist and is attainable via a polynomialtime algorithm.
Definition 4 (WeakSWJR).
An SCV outcome satisfies weakSpanwise Justified Representation (weakSWJR) if
Informally speaking, the weakSWJR axiom captures the idea that if a “large”, cohesive set of voters unanimously support at least one candidate in each candidate subset then they require representation in some subcommittee.
First observe that weakSWJR is indeed a (strict) weakening of the SWJR concept.
Proposition 3.
SWJR implies WeakSWJR. But weakSWJR does not imply SWJR.
Proof.
We prove the proposition via the contrapositive, suppose weakSWJR does not hold. Then there exists a set such that with for all and for all But
and so SWJR does not hold.
The second claim can be easily observed from the definition and simple counter examples can be constructed (for an example see within the proof of Proposition 4 in the supplement material). ∎
The next proposition states that for a given SCV instance there may be three distinct committees satisfying, respectively, weakSWJR but not IWJR, IWJR but not weakSWJR, and both weakSWJR and IWJR simultaneously. That is, the weakSWJR and IWJR representation axioms are distinct but are not mutually exclusive (the proof can be found in the supplement material).
Proposition 4.
WeakSWJR does not imply IWJR and IWJR does not imply weakSWJR. Also weakSWJR and IWJR are not mutually exclusive concepts.
Proof.
We provide an example of an SCV instance which admits three distinct SCV outcomes which satisfy

weakSWJR but not IWJR

IWJR but not weakSWJR

both weakSWJR and IWJR.
Consider the SCV with ,
where , , and . Let the approval ballots of each voter be as follows
First observe that for an SCV outcome to satisfy weakSWJR the only set of voters which must be represented is since they are a the only group of size greater or equal to who unanimously support a voter in each of the candidate subsets i.e. and are approved by every voter .
Whilst, for an SCV outcome to satisfy IWJR it is required that the group are represented in by candidate . This is because is the only group of size greater or equal to who unanimously support a candidate in .
Thus it follows immediately that there exists three distinct SCV outcomes which satisfy the three properties stated at the beginning of this proof, namely;
satisfies weakSWJR but not IWJR,
satisfies IWJR but not weakSWJR and,
satisfies both IWJR and weakSWJR. ∎
6 An Algorithm for weakSWJR and IWJR
The previous sections have introduced two appealing representation axioms, weakSWJR and IWJR, which capture distinct notions of representation or fairness. The former axiom considers the structure of approvals across all candidate subsets, whilst the latter axiom considers each candidate subset as an independent event. This section will combine these two axioms and consider SCV outcomes which satisfy both weakSWJR and IWJR. We prove that a committee satisfying both weakSWJR and IWJR is guaranteed to exists for every SCV setting and such a committee can be computed in polynomialtime.
We begin by presenting the following intermediate fact before providing a constructive existence proof of a committee which satisfies weakSWJR and IWJR.
Lemma 1.
Let be a sequence of positive numbers and be a sequence of nonnegative numbers. Then,
Proof.
Let and let , then
∎
Proposition 5.
A committee which satisfies both weakSWJR and IWJR always exists and can be attained via Algorithm 1.
Proof.
Consider Algorithm 1. We argue that the committee returned by Algorithm 1 satisfies both weakSWJR and IWJR.
First we show that IWJR is satisfied. Suppose not, then during step 2 for some we allocated winning spots but failed to represent a group, say , of size at least who unanimously supports some candidate(s) in . However, at each stage at least additional voters are represented and so it must be the case that at least voters were represented in . That is, all voters have been represented which contradicts the existence of the set . Thus IWJR is always satisfied.
Now we show that weakSWJR is also satisfied. Suppose that after step 2, for all – if this were not the case then weakSWJR is trivially satisfied since all voters would then be represented in . The proof now reduces to showing that there are enough ‘places’ left in after allocating the places in Step 2.
In the ‘worst case’ every allocation for each represents the same subset of voters – in this case represents voters with elected candidates. But then there are
possible mutually exclusive groups of size which are unrepresented in . In the worst case, each of these groups would unanimously support a different candidate in every candidate subset and so correspond to a problem set with respect to weakSWJR. Recall that we have winning spots left and so suffices to show that
(1) 
Note that we use the property that if problem set exists they must unanimously support some candidate in every subcommittee and so we can ignore the quota issues. Finally, (1) follows immediately from dividing by and applying Lemma 1. ∎
7 Testing Representation
Testing for SWJR and IWJR are easy given the polynomialtime testing of JR. Testing for weakSWJR is more involved, we conjecture the complexity is coNPcomplete. Before we proceed, we outline the standard approvalbased voting setting and an algorithm identified by Aziz et al. (2017b) to test JR.
Polynomialtime algorithm to verify JR
The standard setting of approvalbased voting (AV) is a special case of SCV. In particular, an AV instance is a tuple where is a set of voters, is the set of candidates, is a positive integer and is an approval ballot profile. An AV outcome (or committee) is a subset such that . Note, that this is simply a special case of SCV when , and .
The algorithm proposed by Aziz et al. (2017b) to test JR is as follows: given an Approval Voting instance and outcome , for each candidate compute
The set fails to provide JR for if and only if there exists a candidate with .
With minor modifications, the above algorithm provides a polynomialtime algorithm to test whether an SCV outcome satisfies SWJR and IWJR.
Proposition 6.
It can be checked in polynomial time whether a given committee satisfies SWJR or not.
Proof.
Same as the polynomialtime algorithm for testing JR. ∎
Proposition 7.
It can be checked in polynomial time whether a given committee satisfies IWJR or not.
Proof.
First note that an SCV outcome satisfies IWJR if and only if it satisfies JR for every candidate subset when approvals ballots are restricted to . That is, satisfies IWJR if and only if the Approval Voting instance satisfies JR for all .
It follows immediately that applying the polynomialtime algorithm to verify that JR is satisfied in each of these Approval Voting instances is also a polynomialtime algorithm. ∎
8 Generalizing PAV to SCV
In the setting of approvalbased multiwinner voting, the Proportional Approval Voting (PAV) rule has been extensively studied and shown to satisfy many desirable representation properties. It has been shown in (Aziz et al., 2017b) that PAV committees satisfy JR, though computing a PAV committee is NPhard.^{3}^{3}3In fact PAV is viewed as one of the most compelling rules for approvalbased committee voting because it satisfies EJR a property stronger than JR (Aziz et al., 2017b).
Under PAV in the standard approval voting setting (AV), each voter who has of their approved candidates in the committee is assumed to derive utility of if and zero otherwise. The total utility of a committee is then defined as
this is known as the PAVscore. The PAV rule outputs the committee of size which maximizes the PAVscore among all committees of size .
In this section we consider generalizing the PAV rule to the SCV setting. This leads to two distinct PAV rules for the SCV setting which are both natural generalizations.
Spanwise PAV (SWPAV) is a generalization of the PAV rule which assumes voters gain utility solely from the number of their approved candidates in . Thus, each voter derives utility and the SWPAV score of a committee is
Intrawise PAV (IWPAV) is a generalization of the PAV rule which assumes voters gain utility from both the number of their approved candidates in and also the diversity across subcommittees. Thus each voter derives utility
and the IWPAV score of a committee is
In both generalizations, a SWPAV (IWPAV) committee is defined to be a committee satisfying the SCV quota conditions and maximizing the SWPAV (IWPAV) score.
To illustrate the distinction between SWPAV and IWPAV the following example is provided.
Example 1.
Consider a voter with approved and elected candidates such that and . Then voter ’s contribution to the SWPAV score is
Whilst her contribution to the IWPAV score is
Generalized Justified Representation under SWPAV and IWPAV
We consider the representation properties of SCV committee outcomes from the IWPAV and SWPAV rules. We show that IWPAV satisfies IWJR and SWPAV satisfies weakSWJR, whilst neither satisfies both. In addition, we show that both rules can fail to output an SWJR committee when such a committee exists.
Proposition 8.
IWPAV satisfies IWJR.
Proof.
Any IWPAV maximizing committee must be such that for all the set is a PAV maximizing committee in the standard AV setting. Hence JR must be satisfied in each , as shown in (Aziz et al., 2017b), thus IWJR is satisfied. ∎
We now show that SWPAV satisfies weakSWJR, however first we introduce some notation and a lemma.
Let be an SCV committee and let , define the marginal contribution of as
The following lemma was explicitly presented by Aziz and Huang (2017) for the standard approval voting setting and was implicitly used in (Aziz et al., 2017b). The lemma applies to the SCV setting via an identical argument. We omit the proof and provide a reference.
Lemma 2.
[Aziz and Huang (2017)] For any committee such that , there exists at least one such that
Proposition 9.
SWPAV satisfies weakSWJR.
Proof.
Suppose for the purpose of a contradiction that is a SWPAV committee which does not satisfy weakSWJR. That is, there exists a group of unrepresented voters with such that for all and for all .
First, note that there must exist a candidate such that . Suppose otherwise, then by Lemma 2 it must be that all voters are represented which contradicts the existence of the set .
Now suppose that for some and let such that . Then it is clear that adding to the committee increases the SWPAV score by at least and so
which contradicts being a SWPAV committee. Thus, a SWPAV committee must satisfy weakSWJR. ∎
The following proposition shows that both SWPAV and IWPAV can fail to produce a SWJR committee when such a committee exists. The proof illustrates a tradeoff between maximizing voter utility and pursuing the representation axiom of SWJR.
Proposition 10.
Both SWPAV and IWPAV can fail SWJR when a SWJR committee exists.
Proof.
Consider the following counterexample. Let ,
and let the approval ballots be
with quotas and .
For SWJR to be satisfied we require that at least one voter from each of the groups and are represented in . Thus, the only committees satisfying SWJR are of the following form:
Any committee with the form above has an SWPAV score of 7 and an IWPAV score of 7. However the committee maximizes the SWPAV and IWPAV scores, with both equal to 8, and does not satisfy SWJR. ∎
We now present two propositions showing that SWPAV can fail to produce a committee satisfying IWJR, and IWPAV can fail weakSWJR.
The proof of the following proposition highlights the conflict between SWPAV, which does not incentivize diversity, and IWJR, which may demand representation of a group already represented in another subcommittee.
Proposition 11.
SWPAV can fail to produce a committee satisfying IWJR.
Proof.
Let ,
and let approval ballots be
with quotas and .
The only candidate required to be in a candidate subset for IWJR is candidate since . However, direct computation shows that a maximal SWPAV score is attained from .∎
The following proposition shows that IWPAV can fail weakSWJR. The proof highlights the conflict between IWPAV, which incentives diversity, and weakSWJR, which may require a smaller unrepresented group of voters to be represented.
Proposition 12.
IWPAV can fail to produce a committee satisfying weakSWJR
Proof.
Let , let approval ballots be
with quotas . Note .
For a weakSWJR committee every voter must be represented. However, the IWPAV committee is which gives an IWPAV score of and does not satisfy weakSWJR. ∎
9 Discussion
In this paper we formalized a general social choice model called subcommittee voting. We focussed on natural generalization of JR from the approvalbased committee voting setting to the approvalbased SCV setting. Some of the results are summarized in Table 1. It will be interesting to consider generalizations of stronger versions of justified representation such as PJR and EJR. For example, IWJR can straightforwardly be strengthened to IWPJR or IWEJR.
It will be interesting to consider more general preferences that need not be approvalbased. Several research questions that have been intensely studied in subdomains of SCV apply as well to SCV. For example, it will be interesting to extend axioms and rules for singlewinner or multiwinner voting to that of SCV.
Representative  Complexity  

committee  of  
exists  computing  
SWJR  No  NPc 
IWJR  Yes  in P 
weakSWJR  Yes  in P 
IWJR & weakSWJR  Yes  in P 
In any combinatorial setting, one can view the voting process as either simultaneous voting or sequential voting (Barrot and Lang, 2016; Lang and Xia, 2016; Freeman et al., 2017b, a). We formalized SCV as a static model in which subcommittees are to be selected simultaneously. The representation notions that we formalized can also be considered if voting over each subcommittee is conducted sequentially over time. The axioms that we consider such as SWJR apply as well to understand the quality of an outcome in these online or sequential settings.
Acknowledgments
Haris Aziz is supported by a Julius Career Award. Barton Lee is supported by a Scientia PhD fellowship.
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