1 Introduction
Participatory budgeting (PB) provides a grassroots and democratic approach to selecting a set of public projects to fund within a given budget. It has been deployed in several cities all over the globe [Shah, 2007]. In contrast to standard political elections, PB requires consideration of the (heterogeneous) costs of projects and must respect a budget constraint. When examining PB settings formally, standard voting axioms and methods that ignore budget constraints and differences in each project’s cost need to be reconsidered. In particular, it has been discussed in policy circles that the success of PB partly depends on how well it provides representation to minorities [Bhatnaga et al., 2003]. We take an axiomatic approach to the issue of proportional representation in PB.
In this paper, we consider PB with weak ordinal preferences. Ordinal preferences provide a simple and natural input format whereby participants rank candidate projects and are allowed to express indifference. A special class of ordinal preferences are dichotomous preferences (sometimes referred to as approval ballots); this input format is used in most realworld applications of PB. However, in recent years, some PB applications have shifted to requiring linear order inputs. For example, in the New South Wales state of Australia, participants are asked to provide a partial strict ranking over projects. The PB model we consider encompasses both approval ballots and linear order inputs.
In most of the PB settings considered, the participants are assumed to have the same weight. However, in many scenarios, symmetry may be violated. For example, in liquid democracy or proxy voting settings, a voter could be voting on behalf of several voters so may have much more voting weight. Similarly, asymmetric weights may naturally arise if PB is used in settings where voters have contributed different amounts to a collective budget. Therefore, we consider PB where voters may have asymmetric weights.
While there is much discussion on fairness and representation issues in PB, there is a critical need to formalize reasonable axioms to capture these goals. We present two new axioms that relate to the proportional representation axiom, proportionality for solid coalitions (PSC), advocated by Dummett for multiwinner elections [Dummett, 1984]. Our axioms provide yardsticks against which existing and new rules and algorithms can be measured. We also provide several justifications for our new axioms.
Contributions
We formalize the setting of PB with weak ordinal preferences. Previously, restricted versions of the setting such as PB with approval ballots have been axiomatically studied before. We then propose two new axioms Inclusive PSC (IPSC) and Comparative PSC (CPSC) that are meaningful proportional representation and fairness axioms for PB with ordinal preferences. In contrast to previous fairness axioms for PB [Aziz et al., 2018b], both IPSC and CPSC imply exhaustiveness (no additional candidate can be funded without exceeding the budget).
We show that an outcome satisfying Inclusive PSC is always guaranteed to exist and can be computed in polynomial time. The concept appears to be the “right” concept for several reasons. For example, it is stronger than the localPJRL concept proposed for PB when voters have dichotomous preferences. It is also stronger than generalized PSC for multiwinner voting with ordinal preferences. It implies the wellstudied concept PJR for multiwinner voting when voters have dichotomous preferences. Even for this restricted setting, it is of independent interest. To show that there exists a polynomialtime algorithm to compute an outcome satisfying IPSC, we present the PB Expanding Approvals Rule (PBEAR) algorithm.
We also show that the CPSC is equivalent to the generalised PSC axiom for multiwinner voting with weak preferences, to Dummett’s PSC axiom for multiwinner voting with strict preferences, and to PJR for multiwinner voting with dichotomous preferences.
2 Related Work
PB with ordinal preferences can be classified across different axes. One axis concerns the input format. Voters either express dichotomous preferences or general weak or linear orders. Along another axis, either the projects are divisible or indivisible. When the inputs are dichotomous preferences, there has been work both for divisible
[Bogomolnaia et al., 2005, Aziz et al., 2019] as well as indivisible projects [Aziz et al., 2018b, Faliszewski and Talmon, 2019]. When the input concerns rankings, then there is work where the projects are divisible. (see, e.g. Aziz and Stursberg [2014], Aziz et al. [2018c], Airiau et al. [2019]). Some of the work is cast in the context of probabilistic voting but is mathematically equivalent to PB for divisible projects.To the best of our knowledge, fairness axioms for PB for discrete projects have not been studied deeply when input preferences are general ordinal preferences. Therefore, this paper addresses an important gap in the literature. Table 1 provides a classification of the literature.
Approval Ballots  Ordinal Prefs  

Divisible  (e.g. Bogomolnaia et al. [2005])  (e.g. Aziz and Stursberg [2014]) 
Indivisible  (e.g Goel et al. [2019])  This paper 
Aziz et al. [2018b] and Faliszewski and Talmon [2019] focused on PB with discrete projects where the input preference format is approval ballots. We show that our general axioms have connections with proportional representation axioms proposed by Aziz et al. [2018b] for the case approvalballots. We will also show how our approach has additional merit even for the case approvalballots. For example, in contrast to previously proposed axioms in [Aziz et al., 2018b], our axioms imply a natural property called exhaustiveness.
Fluschnik et al. [2017] consider the discrete PB model and study the computational complexity of maximizing various notions of social welfare, including Nash social welfare. Benadè et al. [2017] study issues surrounding preference elicitation in PB with the goal of maximizing utilitarian welfare. In their model, they also consider input formats in which voters express ordinal rankings. However, their focus is not on proportional representation. Fain et al. [2016] considered PB both for divisible settings as well as discrete settings. However, their focus was on cardinal utilities. In particular, they focus on a demanding but cardinalutility centric concept of core fairness. Our ordinal approach caters for many settings in which voters only express rankings over projects. Other works on cardinal utilities include [Fain et al., 2018, Bhaskar et al., 2018]. Both ordinal and cardinal utilities have their own merits. Whereas cardinal utilities allow users to express an intensity of preferences, ordinal preferences are typically easier to elicit and put less cognitive burden on the voters.
The paper is also related to a rapidly growing literature on multiwinner voting [Aziz et al., 2017a, Faliszewski et al., 2017, Aziz et al., 2017b, Elkind et al., 2017, Janson, 2016, Schulze, 2002, Tideman, 2006]. PB is a strict generalization of multiwinner voting. Our axiomatic approach is inspired by the PSC axiom in multiwinner voting. The axiom was advocated by Dummett [1984]. PSC has been referred to as the most important requirement for proportional representation in multiwinner voting [Woodall, 1994, 1997, Tideman and Richardson, 2000, Woodall, 1994, Tideman, 1995]. We dedicate a separate section to multiwinner voting because one of our axioms gives rise to a new and interesting axiom for the restricted setting of multiwinner voting.
3 Preliminaries
A PB setting is a tuple where is the set of voters, is the set of candidate projects (candidates), and is the total budget limit. The function specifies the cost of each candidate . The function specifies a voter’s budget for each . We assume that the . For any set of voters , we will denote by . Therefore . A set of candidates is feasible with respect to if . The preference profile specifies for each voter , her ordinal preference relation over . In the terminology of Benadè et al. [2017], the input format can be viewed as ‘rank by value’ so that voters rank projects according to how they value them without taking costs into account.
We write to denote that voter values candidate at least as much as candidate and use for the strict part of , i.e., if and only if but not . Finally, denotes ’s indifference relation, i.e., if and only if both and . The relation results in (nonempty) equivalence classes for some such that if and only if and for some . Often, we will use these equivalence classes to represent the preference relation of a voter as a preference list If each equivalence is of size , then the preference will be a called strict preference. If for each voter, the number of equivalence classes is at most two, the preferences are referred to as dichotomous preferences. When the preferences of the voters are dichotomous, the voters can be seen as approving a subset of voters. In this case for each voter , the first equivalence class is also referred to as an approval ballot, and is denoted by . Note that in this special case, where a voter has dichotomous preferences, the approval set contains all information about voter
’s preference. The vector
is referred to as the approval ballot profile. If a voter is indifferent between all candidates, then voter ’s approval ballot could be interpreted to be either or ; our results and axioms are independent of this interpretation.Multiwinner voting can be viewed a special kind of PB in which for all and for all . The budget limit is typically denoted by committee size . Any setting that allows for weak preferences can be viewed as encapsulating the corresponding setting with approval ballots. The reason is that approval ballots can be viewed as dichotomous preferences. Figure 1 provides an overview of which model reduces to which other model.
Definition 1 (Exhaustive outcomes).
An outcome is said to be exhaustive w.r.t. if and for all .
Definition 2 (Maximal cost outcomes).
An outcome is said to be a maximal cost outcome w.r.t. if and for all .
Note that a maximal cost outcome is always exhaustive but an exhaustive outcome need not be maximal cost. In multiwinner voting, since we only consider outcomes that use up the budget of , it means that all feasible outcomes are by default both exhaustive and maximal cost.
4 Proportional Representation in PB with Ordinal Preferences
Let denote voter ’s th most preferred candidate or one such candidate if indifferences are present. To attain such a candidate in the presence of indifferences the following procedure can be used: (1) break all ties in voter ’s preferences temporarily to get an artificial strict order and (2) identify the th candidate in the artificial strict order.
Definition 3 (Generalised solid coalition).
Suppose voters have weak preferences. A set of voters is a generalised solid coalition for a set of candidates if every voter in weakly prefers every candidate in at least as high as 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 .
If a set of voters supports a set of candidates , we will refer to as the periphery of the set of candidates with respect to voter set .
4.1 Main New Concepts
We first present a concept that is similar in spirit to the PSC concept that was proposed by Dummett [1984] for multiwinner voting for strict preferences.
Definition 4 (Comparative PSC (CPSC) for PB with general preferences).
A budget satisfies Comparative PSC (CPSC) if there exists no set of voters such that solidly supports a set of candidates such that and there is a subset of candidates such that
The intuition for CPSC is that if a set of voters solidly supports a subset then it may start to think that a weight should be selected from or its periphery especially if there is enough weight present. At the very least it should not be the case that there is a feasible subset of of weight at most but the weight of is strictly less.
Definition 5 (Inclusion PSC for PB with general preferences).
A budget satisfies Inclusion PSC (IPSC) if there exists no set of voters who have a solidly supported set of candidates such that and there exists some candidate such that
The intuition for IPSC is that a set of voters solidly supports a subset then it may start to think that a weight should be selected from or its periphery especially if there is enough weight present. At the very least it should not be the case that weight of does not exceed even if some unselected candidate in can be added to .
For both IPSC and CPSC, we avoid violation if for solidly supporting candidates in , the weight of is large enough. That is, we only impose representation requirements for sets of voters who solidly support a set of candidates. If, instead, representation requirements were enforced for all sets of voters, regardless of whether they solidly supported a set of candidates, then it may not be possible to satisfy either axiom. This observation has already been made in the context of multiwinner voting (see, e.g. Aziz et al. [2017a]). Similarly, both axioms focus on whether the weight is large enough. If we only care about the weight of , then, again, it is not possible to define a fairness concept that is guaranteed to exist for all instances.
Both IPSC and CPSC imply exhaustiveness.
Proposition 1 (CPSC and IPSC are exhaustive).
Any outcome that satisfies CPSC or IPSC is exhaustive.
Proof.
Let be a nonexhaustive outcome. That is, there exists a candidate such that .
Later, in Proposition 2, we prove the stronger result that a CPSC outcome is always a maximal cost outcome. Thus, we omit the proof that a CPSC outcome is exhaustive.
For the sake of a contradiction, suppose that satisfies IPSC. The set of all voters solidly supports the entire candidate set , , and
Definition 5 is violated since and This is the desired contradiction. ∎
CPSC implies the stronger maximal cost property. As will be shown within the proof of Proposition 3, an IPSC outcome need not be a maximal cost outcome.
Proposition 2 (CPSC implies maximal cost).
Any outcome that satisfies CPSC is a maximal cost outcome.
Proof.
Suppose that and are two distinct budgets that satisfy CPSC and assume . We prove that cannot satisfy CPSC. The set of all voters is a solid coalition for the entire candidate set . Take . We have and
Thus, does not satisfy CPSC. ∎
Proposition 3.
For PB with ordinal preferences, IPSC does not imply CPSC and CPSC does not imply IPSC.
Proof.
First, we show that IPSC does not imply CPSC. Let with and , and suppose that voters have dichotomous preferences:
Consider the outcome . This does not satisfy CPSC since the set of voters is a generalized solid coalition for with
and, yet, such that . On the other hand, satisfies IPSC. For example, take and as above, there is a single candidate and . Thus, IPSC is not violated by the set of voters and solid coalition . It can similarly be shown that for all other subsets of voters and sets of solidly supported candidates that IPSC is not violated.
Second, we show that CPSC does not imply IPSC. Let with , and suppose that the voters’ preferences are
Consider the outcome . This does not satisfy IPSC. The set of voters forms a generalized solid coalition for and
However, the candidate and . Thus, IPSC is violated. On the other hand, satisfies CPSC. For example, take and as above, there is only one subset that does not exceed . However,
Thus, CPSC is not violated by the set of voters and solid coalition . It can similarly be shown that for all other subsets of voters and sets of solidly supported candidates that CPSC is not violated. ∎
4.2 Concepts with Approval Ballots
Proposition 4 (Comparative PSC (CPSC) for PB with approval preferences).
Suppose voters have dichotomous preferences. An outcome satisfies Comparative PSC (CPSC) if and only if
 (i)

there exists no set of voters such that there is a subset of candidates such that but .
 (ii)

the outcome is a maximal cost outcome.
Proof.
() We prove the result via the contrapositive. Suppose that an outcome does not (simultaneously) satisfy (i) and (ii). If (ii) does not hold, then, by Proposition 2, CPSC does not hold. Now, suppose that (ii) holds but (i) does not. That is, is maximal cost and there exists such that there exists with and
(1) 
Since , the set of voters forms a generalized solid coalition for and
(2) 
Further, the (trivial) subset is such that and
by (1). Thus, CPSC is violated.
() We prove the result via the contrapositive. Suppose that does not satisfy CPSC. If is not maximal cost, then (ii) is violated and we are done. Assume that is maximal cost but does not satisfy CPSC. That is, is maximal cost and there exists a set of voters that solidly supports such that and but
(3) 
Now, suppose that, for some ,
This can only occur if or . In both cases, this implies that and
But this is a contradiction since, combined with (3), this shows that cannot be a maximal cost outcome. Thus, it must be that, for all ,
and the solidly supported candidate set is a subset of . It follows that is also a subset of such that and
by (3). Thus, (ii) is violated; this completes the proof. ∎
Proposition 5 (Inclusion PSC for PB with approval preferences).
Suppose voters have dichotomous preferences. An outcome satisfies Inclusion PSC (IPSC) if and only if
 (i)

there exists no set of voters such that and there exists some such that .
 (ii)

the outcome is exhaustive.
Proof.
() We prove the result using the contrapositive. If (ii) does not hold, then, by Proposition 1, we see that IPSC is violated. Now, assume that (ii) holds but (i) does not hold. That is, is an exhaustive outcome, and there exists a set of voters with
(4) 
and some such that
(5) 
Let . The set is solidly supported by the set of voters and, since for all , we have
for all . It then follows from (4) that and, yet, by (5) there exists a candidate
such that That is, IPSC is violated.
() We prove the result via the contrapositive. Suppose that is an outcome such that IPSC does not hold. If is not exhaustive, then (ii) is violated and we are done. Now, suppose the is exhaustive and does not satisfy IPSC. That is, is exhaustive, and there exists a set of voters who solidly support a set of candidates with
(6) 
and there exists some candidate such that
(7) 
First, suppose that, for some ,
This can only occur if or . In either case, this implies that and
for some ; but this contradicts the assumption that is exhaustive. Thus, it must be that
for all . It then follows that
and, by (6),
Further, the candidate subset must correspond to a subset of . Thus, the candidate and
Thus, condition (i) is violated. ∎
PB with approval ballots has been considered by Aziz et al. [2018b]. For example, they proposed the following concept.
Definition 6 (BpjrL [Aziz et al., 2018b]).
Assume that . A budget satisfies BPJRL if for all there exists no set of voters with such that and .
In the restricted setting studied by Aziz et al. [2018b], CPSC for PB with approval preferences is equivalent to the combination of the BPJRL and the maximal cost concepts.^{1}^{1}1The setting studied by Aziz et al. [2018b] normalizes the minimalcost candidate to have unit cost and the budget limit, , is required to be a positive integer. BPJRL is weaker than CPSC because BPJRL does not imply maximal cost.
Remark 1.
IPSC for PB with approval preferences is stronger than the LocalBPJRL proposed by Aziz et al. [2018b].
Definition 7 (LocalBPJRL [Aziz et al., 2018b].).
A budget satisfies LocalBPJRL if for all there exists no set of voters such that , and there exists some such that
4.3 Concepts with Strict Preferences
Below we rewrite the definitions for strict preferences. The definitions turn out to be simpler compared to preferences that allow for ties.
Definition 8 (Comparative PSC (CPSC) for PB with strict preferences).
Under strict preferences, a candidate set satisfies Comparative PSC (CPSC) if for all solid coalitions supporting candidate subset , if there exists some such that , then .
Definition 9 (Inclusion PSC for PB with strict preferences).
Under strict preferences, a candidate set satisfies local PSC if for all solid coalitions supporting candidate subset , there exists no subset such that and .
5 Computing proportionally representative outcomes
Our first observation is that computing a CPSC outcome is computationally hard, even for one voter.
Proposition 6.
Computing a CPSC outcome is NPhard even for the case of one voter.
Proof.
Take an instance concerning one voter who approves of all candidates. Then, due to Proposition 2, we know that outcome must contain the set of projects that maximizes the total weight constrained to budget limit . The problem is equivalent to a knapsack problem and hence NPhard. ∎
Next, we show that even for one voter with strict preferences, a CPSC outcome may not exist.
Example 1.
Consider the following PB instance with one voter and 4 candidate projects. The voter’s preferences are as follows.
The limit is 4 and the weight are: . CPSC requires that must be selected. It also requires that should be selected. Therefore a CPSC outcome does not exist.
Later, we will show that in a more restrictive setting (multiwinner approval voting) a CPSC outcome always exists, can be computed in computed in polynomialtime, and coincides with a wellestablished proportional representation axiom, called PJR. In fact, a CPSC outcome always exists in the PB setting if voters have dichotomous preferences. This follows from the observation that, when voters have dichotomous preferences, CPSC is equivalent to requiring that an outcome satisfy BPJRL and maximal cost, which is guaranteed to exist [Aziz et al., 2018b, Proposition 3.7].^{2}^{2}2It should be noted that the proof of Proposition 3.7 in Aziz et al. [2018b] requires a slight modification to fit the more general PB setting studied in the present paper. The lack of guarantee of existence of CPSC in the general PB settings versus multiwinner voting demonstrates the challenges posed when moving from multiwinner voting to PB.
In contrast to CPSC, we show that an IPSC outcome is not only guaranteed to exist but it can be computed in polynomial time.
Proposition 7.
PBEAR satisfies Inclusion PSC for PB.
Proof.
Let be an outcome of PBEAR. For sake of a contradiction, suppose that does not satisfy IncPSC. That is, there exists a set of voters who solidly support a candidate set such that
where , and there exists a candidate such that
(8) 
We will denote by .
First, suppose the PBEAR terminated at some iteration. At the end of the iteration, the sum of voter weights in is at least
This follows because when each candidate is added to a total weight of is subtracted from the set of voters supporting this candidate. Our lower bound is attained by assuming that every candidate that can possibly reduce the weight of voters in (i.e., those candidates for ) subtracts the entire weight from the voter set .
But (8) implies that , and so
Thus, at the end of the iteration
which implies that and . This is contradiction since no other candidates from are contained in besides those already accounted for in , and so PBEAR could not have iterated to the th stage.
Second, suppose that PBEAR terminated at some iteration. This can only occur if and no candidate can be added without exceeding the budget or . The total voter weight that has been subtracted (from all voters) via the algorithm is exactly
(9) 
As noted in the above paragraphs, the voter weights of is at least . This gives an upper bound on the total voter weight that has been decreased from all voters
(10) 
That is, . This is a contradiction since the set does not equal and contains at least one candidate, namely , such that does not exceed . ∎
Next we show that not all IPSC outcomes are possible outcomes of PBEAR even for the restricted setting of multiwinner voting.
Example 2.
Suppose and approval ballots
The outcome satisfies IPSC. However, it will never be an outcome of PBEAR. This follows because in the first iteration the support for candidate and are equal to 2, candidate receives support and candidate zero. The threshold for consideration to be added into is . Thus, only candidate and can be considered. They are supported by disjoint sets of voters. Each must be added. The unique PBEAR outcome is .
6 Special Focus on Multiwinner Voting
In the section, we dive into the wellstudied setting of multiwinner voting which is also referred to as committee voting. In this setting, candidates are to be selected from the set of candidates. We uncover some unexpected relations between fairness concepts for this particular setting. We also show that whereas PSCS does not give rise to a new fairness concept, IPSC gives to a new fairness concept even for the setting concerning approval ballots.
Let us first introduce generalised PSC, which was proposed by Aziz and Lee [2019] and applies to multiwinner settings with ordinal preferences.
Definition 10 (Generalised PSC [Aziz and Lee, 2019]).
A committee satisfies generalised PSC if for every positive integer , and for all generalised solid coalitions supporting candidate subset with size , there exists a set with size at least such that for all
Aziz and Lee [2019] showed that generalised PSC extends the PJR concept for multiwinner voting with approval ballots.
Definition 11 (Pjr).
Suppose all voters have dichotomous preferences. A committee with satisfies PJR for an approval ballot profile over a candidate set if for every positive integer there does not exists a set of voters with such that
Let us first rewrite CPSC for the multiwinner voting setting.
Definition 12 (Comparative PSC (CPSC) for multiwinner voting ).
A budget satisfies Comparative PSC (CPSC) if there exists no set of voters such that solidly supports a set of candidates such that there is a subset of candidates such that but .
Proposition 8.
For multiwinner voting, CPSC is equivalent to Generalised PSC.
Proof.
Suppose does not satisfy CPSC. Then, there exists a set of candidates solidly supported by , for which there is some subset of candidates such that but
(11) 
To show that generalised PSC does not hold, take , and notice that is a generalized solid coalition for with . We wish to show that there is no subset of size at least such that for all
If such a set did exist, then it must be that
which contradicts (11). Therefore, no such set can exist and generalized PSC is violated.
Suppose does not satisfy generalised PSC. Then, for some positive integer , there exists a generalised solid coalition supporting candidate subset such and there does not exist any subset such that for all
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