Participatory Budgeting with Cumulative Votes
Piotr Skowron, Arkadii Slinko, Stanisław Szufa, Nimrod Talmon
PParticipatory Budgeting with CumulativeVotes
Piotr SkowronUniversity of Warsaw [email protected]
Arkadii SlinkoUniversity of Auckland [email protected]
Stanisaw SzufaJagiellonian University [email protected]
Nimrod TalmonBen-Gurion University [email protected]
September 8, 2020
Abstract
In participatory budgeting we are given a set of projects—each with acost, an available budget, and a set of voters who in some form expresstheir preferences over the projects. The goal is to select—based on voterpreferences—a subset of projects whose total cost does not exceed the bud-get. We propose several aggregation methods based on the idea of cumu-lative votes, e.g., for the setting when each voter is given one coin and shespecifies how this coin should be split among the projects. We compare ouraggregation methods based on (1) axiomatic properties, and (2) computersimulations. We identify one method, Minimal Transfers over Costs, thatdemonstrates particularly desirable behavior. In particular, it significantlyimproves on existing methods, satisfies a strong notion of proportionality,and, thus, is promising to be used in practice.
The idea of Participatory Budgeting (PB) was born in Brazil during the 1980swhen political reformers explored ways to move beyond the political system as-sociated with Brazils military dictatorship (1964-1985) based on exclusion and1 a r X i v : . [ c s . M A ] S e p orruption [28]. They aimed to increase transparency of decision making of localbodies, hoping to enhance social justice and democracy [7]. At the first stage ofPB, governments and civil society organizations identify the set of goals on thebasis of certain principles without any reference to the budget. For example, manyPB programs in Brazil use the Quality of Life Index, initially devised by the gov-ernment in the city of Belo Horizonte. These goals are translated into particularprojects that will be offered to the whole society to choose from. The second stageis actual voting in which voters express their views on the relative importance ofthe projects on offer. Thus, Participatory Budgeting is indeed a direct-democracyapproach to budgeting.The most prominent applications of PB have been at the level of municipality,where a fraction of the municipal budget is decided through a residents-wide elec-tion. Other applications include, for example, (1) An airline company decidingwhich movies to offer on its in-flight entertainment system [25], where licensecosts for different movies can vary; and (2) A fans-owned soccer club wishing tosign deals with athletes.Lately, PB has gained a considerable attention, and an increasing amount of fundsare currently distributed this way , thus the task of the research community is togain understanding of its foundations, the properties of the existing procedures,and to suggest new more efficient procedures. The survey of some existing proce-dures is given in [3].It is the second stage of PB which is of interest to us in this paper. We there-fore assume that there is a set of projects, each with its own cost, a set of votersexpressing their preferences over the projects, and a budget limit. The task of anaggregation procedure is to select a subset of the projects on offer reflecting voterpreferences in the best possible way and whose total cost does not exceed thebudget limit. (Formal definitions are given later). One of the motivating factorsof our research is the close resemblance of participatory budgeting to multiwin-ner elections—in fact, when the costs of projects are all equal to one, the modelfor PB collapses to the model for multiwinner elections when a commitee of k candidates is to be elected (so k in this case is the budget).Even though the interest of ordinary people and political activists in PB issteadily increasing and its adoption has been steadily on the rise, the researchon voting procedures for PB is still scarce. Specifically, with a few notable ex- n votersis given virtual coins worth L/n , where L is the budget limit, and she is asked tospecify how this coin should be split among the projects. Distributing these coinsbetween projects a voter can not only signal which projects are worthy of fundingbut also the intensity of preference. By analogy with the voting theory (see e.g.,[8]) we call such ballots cumulative votes.We suggest and study various aggregation methods for PB with cumulative votesand demonstrate that in the context of PB the use of cumulative ballots is a verynatural choice.Perhaps the main objection against using cumulative voting is concerned withthe inherent trade-off between expressiveness and cognitive burden: On the onehand, cumulative ballots allow for expressive votes, in particular they come withfine-grained information about intensities of voters’ preferences. On the otherhand, they require more effort from the voters. A first answer to this possible crit-icism is that in the context of participatory budgeting, the gains of expressivenesscan outweigh the increasing cognitive burden; in particular, as we demonstratelater, most aggregation methods we discuss here for cumulative votes achieve asignificantly better voters’ satisfaction (a term we formally define later) than theanalogous methods with simpler types of inputs.Furthermore, as cumulative votes generalize both approval and ordinal ballots, our methods are applicable to these models as well. This means that voters are notrequired to use the full expressiveness of cumulative ballots, but can safely de-scribe their preferences using approval or ordinal ones (which can then be trans-lated, by the user interface or the algorithm itself, to cumulative ballots). Thevoters who wish to provide their preferences in a more expressive format can stilldo that.We begin our investigation of aggregation methods for participatory budgetingwith cumulative votes by considering three greedy aggregation methods, whichare similar to top- k multiwinner voting rules [14]; we refer to these rules as:(1) greedy by support; (2) greedy by excess; and (3) greedy by support over cost.These methods are fairly natural, quite easy to explain, and computationally effi- A ranking of the projects can be transformed into a cumulative ballot by applying a certainpositional scoring function, such as Borda, to the ranking, and by normalizing the so-obtainedscores. L , each ofthe n voters gets a bag of L/n coins which she can then distribute to the availableprojects as she wishes. Then, we look at the coin stash next to each project and,in each iteration, if there is a project which accumulated a sufficient funding, thenwe fund it, remove the cost of this project from its stash, and redistribute whateveris left in the stash to other projects in accord with the preferences of voters con-tributed to this project. Otherwise, we dismiss a project with the smallest stashnext to it, redistribute its stash to other projects, and proceed iteratively.We demonstrate the behavior of our rules through illustrative examples, showingtheir resilience to certain problems that the greedy rules suffer from. We furtherevaluate our STV adaptations by: (1) studying various axiomatic properties ofthese rules; and (2) reporting on simulations we performed. We identify one rule,Minimal Transfers over Costs (MTC), that behaves particularly well with respectto our axiomatic properties, and that indeed produces proportional results.
There is a growing body of work on PB and related topics in the social choice lit-erature (see, e.g., [3]). Firstly, they differ in the type of information solicited fromthe voters. Since most PB rules—in one way or another—adapt the existing vot-ing rules, the input they solicit is usually in the traditional social choice formats:the voters are either asked to approve some of the projects (and hence disapprovethe others) or to rank projects in accord to their desirability. In approval ballotsvoters are asked either to approve a fixed, say k , number of projects [15] or toapprove any set of projects whose total cost does not exceed the budget (knapsackvoting) [19]. Goel et al also use pairwise comparisons as the input format: theyask voters to compare pairs of projects by the ratio between value and cost. These4omparisons are aggregated using variants of classic voting rules, including theBorda count and the Kemeny rules.Klamler et al [21] and Lu and Boutilier [23] modified the idea of multiwinnervoting rule which, given the rankings of the alternatives, selects a fixed number k of alternatives given their costs and a budget. In these works rankings do notdepend on the costs. An (cid:96) -truncated ballots can also be used instead of full ballots[5].A few recent works have considered soliciting cardinal utilities of projects [13,16], where the voters specify the utility of implementation of each project to them.Benade et al [4] suggested their version of knapsack voting assuming that votersvote for the bundle that maximizes their utility. They also suggested to solicit vot-ers’ rankings of projects by their value (Value voting) or value-for-money (Value-for-money voting), and the so-called Threshold voting, where each voter specifiesthe subset of projects whose value is perceived to be above a predefined thresh-old. Laruelle [22] assumes that these utilities are implicit and can be calculatedfrom rankings. We will show that the known rules for PB with cardinal utilitiessometimes exhibit undesirable behavior—in particular, they violate pretty basicproportionality axioms. In this paper we define rules which do not share theseundesirable features.The idea of proportional representation in PB transforms into a fairness issueand it attracted attention of researchers. It should not happen, for example, that noproject is selected among those for which a large minority voted. In particular, theidea of justified representation for approval based budgeting rules was advocatedin [1] and the idea of proportionality for solid coalitions in the ordinal settingin [2].Cumulative and cardinal ballots have been also considered in a divisible model ofparticipatory budgeting, i.e., for aggregating divisions of available funds amongexisting projects, rather than to select projects for funding [12, 18]. A largerchunk of the literature on cumulative voting concerns single-winner elections;see, e.g., [24, 6, 8, 27]. Most of this literature, yet, focuses only on practical andlegal aspects of using cumulative voting.There is a large body of research on the famous voting rule Single TransferableVote (STV) and there is some consensus among the social choice community thatit is a particularly good rule for single-winner elections and, especially, multiwin-ner elections in cases where proportional representation is a desired property; see,e.g., [26, 10, 9]. This prompted us to adapt STV to PB via cumulative voting. Wealso mention the work of Ford [17, Section 3.4] that suggests using a cumulativeversion of STV for multiwinner elections.5he closest research project to ours is that described by accurate democracy . There, an iterative procedure for PB that resembles STV is described. From therather informal description that is given there we could identify a few key dif-ferences between the procedure described there and ours—in particular, the rulethere uses both ordinal and cumulative ballots. Also, no axiomatic analysis of therule is provided.
In our model there is a set of projects P = { p , . . . , p m } ; the cost of a project p ∈ P is a natural number, denoted c ( p ) . There is a set of n voters V = { v , . . . , v n } ,where voter v j expresses her preferences over the projects by assigning a value v j ( p ) to each p ∈ P such that v j ( p ) ≥ and (cid:80) p ∈ P v j ( p ) = 1 (notice that werefer to both the j th voter and her cumulative vote using the same symbol v j );intuitively, the value of v j ( p ) is understood as the fraction of the funds owned byvoter v j that the voter would like to assign to project p . We say that a voter v i supports a project p if v j ( p ) > .The above notation naturally extends to sets. For each B ⊆ P and each v j ∈ V we set v j ( B ) = (cid:80) p ∈ B v j ( p ) and c ( B ) = (cid:80) p ∈ B c ( p ) .A budgeting scenario is a tuple ( P, V, c, L ) , where P , V , and c are as definedabove, and L ∈ N is a budget limit . An aggregation method is a function (analgorithm) that, given a budgeting scenario, selects a bundle of projects B ⊆ P such that c ( B ) ≤ L . In this section we take, arguably, the most straightforward approach, and adaptknown greedy algorithms for participatory budgeting [19] to cumulative ballots.We start by describing the general class of greedy rules.Let f be a function that given a project p returns a real value, called the priority of p . The greedy rule based on f first ranks the projects in the descending order An alternative approach would be to interpret v j ( p ) as the fraction of the available funds thatvoter j thinks should be assigned to project p . These two interpretations are close and differ inwhether we take the local or the global view on the voters’ preferences. In this paper we take thelocal interpretation—we assume that the voters indirectly control the funds, and indicate how partsof funds that they control should be spread among the projects.
6f their priorities, as given by f . Next, the rule iterates through the ranked listof the projects, in each iteration deciding whether the project at hand will orwill not be selected. Let L ( t ) denote the remaining budget in the t -th iterationof the procedure ( L (1) = L ). In the t -th iteration the rule examines project p : if c ( p ) ≤ L ( t ) , then p is selected, and the remaining budget is updated L ( t + 1) = L ( t ) − c ( p ) . Otherwise, p is not selected, and L ( t + 1) = L ( t ) . The followingthree aggregation methods are greedy rules: Greedy-by-Support (GS).
This is the greedy rule based on f GS ( p ) = (cid:80) j ∈ [ n ] v j ( p ) . Greedy-by-Support-over-Cost (GSC).
It is based on f GSC ( p ) = ( / c ( p ) ) · (cid:80) j ∈ [ n ] ( v j ( p ) · ( L/n )) . Greedy-by-Excess (GE).
Based on f GE ( p ) = (cid:80) j ∈ [ n ] ( v j ( p ) · ( L/n )) − c ( p ) . Remark 1.
We do not consider Greedy-by-Excess-over-Cost as it is equivalent toGSC.
The first rule described above (GS) can be seen as an adaptation of KnapsackVoting [19] to cumulative ballots. Yet, all three rules share a negative feature,namely that a significant part of the population of voters might be ignored whenthey split their votes on too many projects.
Example 1.
Consider a set P of 20 projects, all having the same cost equal toone, and a set of 100 voters with the following preferences: The first 60 votersconsider the first 10 projects excellent and they all decide to assign the value / to each of them. The remaining 40 voters have quite opposite preferences—theydecide to put the utility of / on each of the last 10 projects. The budget limit is L = 10 . Here, GS, GSC, and GE would select the first 10 projects for funding,thus effectively ignoring the opinion of a large fraction of the society. Example 1 also demonstrates the inherent difficulty of achieving proportionalitywith cumulative ballots. E.g., one aggregation method using cardinal utilities,which is known in the literature and considered proportional is based on the ideaof maximizing the smoothed Nash welfare (SNW) [13, 16]. If we simply assumethat cumulative ballots correspond exactly to cardinal utilities, then the rule wouldwork as follows: For each budgeting scenario ( P, V, c, L ) , SNW would return a If f is not injective, then we resolve ties using an arbitrary fixed tie-breaking rule. B which does not exceed the limit L and which maximizes the product (cid:81) v j ∈ V (cid:16) v ∗ j + (cid:80) p ∈ B v i ( p ) (cid:17) , where v ∗ j is the maximum value that voter j can getin any feasible outcome. SNW, when applied to the budgeting scenario fromExample 1 would inappropriately favor the majority of voters—it would selecteight projects supported by 60% of voters and 2 projects supported by 40% ofvoters; it would share (yet to a lesser extent) the aforementioned negative featureof greedy rules.In the next section we mainly aim at remedying the undesired behavior of GS,GSC, and GE by offering other aggregation methods which have a visible SingleTransferable Vote (STV) flavor. Indeed, in the instance described in Example 1,our new methods would select projects supported by the group of voters. In this section we describe several adaptations of the Single Transferable Vote(STV) rule to the case of participatory budgeting with cumulative votes. We referto these adaptations as Cumulative-STV, or, in short, as CSTV. We first describethe general scheme, and later discuss several variants, which differ in certain keyaspects regarding their specific operation.All our variants of CSTV are based on the following compelling idea: Each fromthe n voters should be able to decide where to allocate a / n -th fraction of thebudget. Correspondingly, we say that a project p is eligible for funding if: support( p ) = L · (cid:80) nj =1 v j ( p ) n ≥ c ( p ) .Notice that the total cost of the projects that are eligible for funding does notexceed the budget. At first it may seem reasonable to simply pick these projectsand reject the others. This simple strategy, however, might often result in unde-sirable outcomes. For example, assume that there is a large number of excellentproject proposals, each is liked by almost everyone, and that the voters decidedto distribute their support roughly uniformly among the projects. In such a caseit is possible that no project would be eligible for funding and such a simple rulewould return the empty set (a similar behavior would be observed for the budget-ing scenario given in Example 1). In order to deal with this and similar situations8e allow the algorithm to perform certain transfers of the cumulative ballots ofthe voters between the projects.The idea of these transfers is as follows. Since voters do not have any coordina-tion devices, they may allocate too much money for some projects. If the voterwould be informed that her contribution is not needed for a certain project andthat there will be enough funds without her (or that, even with her support, theproject would not be funded), then she would divert her funds to other projectsshe liked. The CSTV rules take care of this oversupply of funds and redistributevoters’ support on behalf of them. Here we describe the general scheme. To make it a concrete rule, one has to spec-ify the following subroutines: (1) project-to-fund selection procedure, (2) excessredistribution procedure, (3) no-eligible-project procedure, and (4) inclusive max-imality postprocedure. We will discuss these subroutines in the subsequent partof this section.The general scheme is as follows: Initialize S = ∅ . Loop over the followinguntil a halting condition is met: If there are projects that are eligible for funding,choose one such project p according to the “project-to-fund selection procedure”.If the total value that the voters put on p is strictly greater than the value neededfor selecting the project (i.e., if support( p ) > c ( p ) ), then for each voter v j with v j ( p ) > , transfer a part of her initial support from p to other projects that v j hadinitially supported, so that support( p ) is as close to c ( p ) as possible. Such trans-fers are performed according to the “excess redistribution procedure”, describedbelow. Next, add p to S , remove it from further consideration, reduce the availablebudget, L := L − c ( p ) , and make the voters pay for p , i.e., for all v j , set v j ( p ) = 0 .Else, that is, if there is no project eligible for funding, perform one of the fol-lowing actions: (1) select and eliminate a project p ; transfer the values that thevoters put on p to other projects, or (2) select a project p and transfer values fromother projects to p so that it becomes eligible for funding. This step is performedaccording to the “no-eligible-project” policy. Finally, move to the beginning ofthe loop.After a halting condition is met, the remaining part of the budget might still belarge enough to fund at least one additional project (in such a case, we say thatthe bundle of selected projects is not inclusive maximal ). If this is the case, wecan run the “inclusive maximality postprocedure”, or leave a part of the availablebudget unused. 9elow we write the specific procedures which differentiate between the variantsof CSTV we consider. If there are multiple projects eligible for funding, we pick the one with the highestpriority, using one of the following three priority functions: (1) f GS , (2) f GSC , or(3) f GE , which we described in Section 4 in the context of greedy rules. In our further study we use the proportional strategy for redistributing the excess.To describe it formally, let p denote the project currently selected for funding, andlet tran( p ) denote the set of voters who put a part of their support to p and also tosome other not yet selected project: tran( p ) = { v j | v j ( p ) > and ∃ p (cid:48) / ∈ S : v j ( p (cid:48) ) > } .We make the payments proportional to the initial supports. We find γ < suchthat: γLn (cid:88) v j ∈ tran( p ) v j ( p ) + Ln (cid:88) v j / ∈ tran( p ) v j ( p ) = c ( p ) .Intuitively, γ is a factor such that if each voter v j ∈ tran( p ) scales her supportfor p i by γ , then p will get exactly the support equal to its cost. Next, for each v j ∈ tran( p ) we distribute (1 − γ ) · v j ( p ) among all not yet selected projects,proportionally to the initial supports that v j assigned to these projects. There are also other natural possibilities for redistributing the excess. For example, onecould think of an additive version of proportional shares, which we term equal shares. The ideais to make the voters pay for the selected project as equal shares as possible. Formally, we find λ such that Ln · (cid:88) v j ∈ tran( p ) min( v j ( p ) , λ ) + Ln · (cid:88) v j / ∈ tran( p ) v j ( p ) = c ( p ) ,and for each voter v j ∈ tran( p ) with v j ( p ) > λ we distribute the surplus of the support ( v j ( p ) − λ ) among all not yet selected projects, proportionally to the initial supports that voter v j assigned tothese projects. .4 No-Eligible-Project Procedure We have two alternative procedures to apply when there is no project which iseligible to funding. These procedures are performed until certain project becomeseligible for funding.
Elimination-with-Transfers (EwT).
Here, we eliminate a project p with eitherthe minimal excess( p ) = support( p ) − c ( p ) or the minimal ratio excess( p ) /c ( p ) .If we chose the GE Project-to-fund selection procedure, then we do the former;if we chose the GSC Project-to-fund selection procedure then we do the latter.Once p is chosen for elimination, then for each voter v j who put a part of theirsupport on p , we transfer this part to the other projects, proportionally to theinitial supports that v j assigned to them. If v j put their support only on p thenno transfers are made. Notice that, if, at any time, there is a project that costsmore than the total amount of money left, then it will be eventually eliminatedand its money will be redistributed. Minimal-Transfers (MT).
We look for a project which may become eligible forfunding if we transfer part of the support from other projects. Formally, we saythat a project p is eligible for funding by transfers if Ln · (cid:88) j : v j ( p ) > m (cid:88) (cid:96) =1 v j ( p (cid:96) ) ≥ c ( p ) . That is, p is eligible by transfers if its cost would be achieved provided thevoters who initially put some positive utility value on p would redirect all theirremaining supports towards p .If we choose the GE Project-to-fund selection procedure, then among allprojects which are eligible by transfers, we select the one that minimizes thetotal amount of transfers that are required in order for it to became eligible; i.e.,the project p which (among those eligible by transfers) has maximal excess( p ) (smallest in absolute value). If we choose GSC as the project-to-fund selectionprocedure, then among the projects that are eligible by transfers we pick theone, p , with the highest ratio support( p ) /c ( p ) .We transfer the supports from other projects to p so that p reaches the eligibil-ity threshold. We again follow the proportional strategy (cf. Section 5.3). In Another option would be to adapt an egalitarian criterion and to minimize the maximal transfera voter must perform. Hereinafter we do not investigate these strategies in detail, yet we considerstudying them an interesting direction for future work. r = support(p) /c ( p ) . If r < , then every supporter v j of p updates her votes. First the voter computes the desired support she should putto p : v j ( p ) := min( (cid:80) m(cid:96) =1 v j ( p (cid:96) ) , v j ( p ) r ) . Then, such a voter updates her votes,by proportionally transferring her support from other projects towards p . Wecontinue the procedure until r = 1 . We have two procedures to continue in cases when the algorithm halts but a partof the available budget is still unused.
Reverse Eliminations (RE).
We apply this procedure only when usingElimination-with-Transfers for selecting non-eligible projects. We iterate overthe not-selected projects in the order reverse to the one in which they were elim-inated. For each project, we check whether its cost does not exceed the availablefunds, and if so we fund it. This procedure is consistent with the logic of EwT,as EwT can be viewed as a procedure that creates a ranking of the projects:Whenever it adds a project p for funding, it puts p in the first available posi-tion in the ranking; when it eliminates p , it puts p in the last available position.Thus, EwT with reverse eliminations is a greedy procedure that moves in theorder consistent with the ranking returned by EwT. Acceptance of Undersupported Projects (AUP).
This procedure is allowedonly when we use Minimal-Transfers as the no-eligible-project procedure. Weproceed similarly as in MT, but this time we do not check the condition for eli-gibility by transfers. That is, if we use GE Project-to-fund selection procedure,then among not-yet selected projects that have their costs no-greater than theremaining budget, we pick the project p which maximizes: Ln · (cid:88) j : v j ( p ) > m (cid:88) (cid:96) =1 v j ( p (cid:96) ) − c ( p ) ,If we use GSC, then we choose p that maximizes: Ln · (cid:80) j : v j ( p ) > (cid:80) m(cid:96) =1 v j ( p (cid:96) ) c ( p ) .12S EwT MT GSC EwTC MTCSplitting monotonicity x (cid:88) (cid:88) (cid:88) x x Merging monotonicity (cid:88) x x x x x
Support monotonicity x x x x x x
Weak-PR x (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) PR x (cid:88) (cid:88) x (cid:88) (cid:88) Strong-PR x x (cid:88) x x (cid:88)
Table 1: Axiomatic properties of GS, GSC, and the CSTV variants.For each voter v j with v j ( p ) > we transfer all her support from other projectsto p . We add p to S and repeat the procedure until no further project can beadded. The various design choices described above give rise to a number of aggregationmethods, out of which we consider the following concrete CSTV aggregationmethods: EwT (i.e., GE + EwT + RE), EwTC (i.e., GSC + EwT + RE), MT (i.e.,GE + MT + AUP), MTC (i.e., GSC + MT + AUP); and out of the greedy ruleswe consider GS and GSC. We decided to chose those variants, since in our initialaxiomatic and experimental analysis they gave the most promising results.
In this section we compare our methods according to certain axiomatic properties.We use these axioms to better understand the behavior of our rules. We concen-trate on two types of axioms: (1) axioms that relate to monotonicity, as they areusually quite standard and provide general understanding of rules; and (2) axiomsthat relate to proportionality, as proportionality (e.g., taking care for minorities) ishighly desired for participatory budgeting (see, e.g., [1]). Our analysis is summa-rized in Table 1. 13 .1 Monotonicity Axioms
In this section we consider three monotonicity axioms. The first two—splittingmonotonicity and merging monotonicity—have been considered by Faliszewskiand Talmon [15], but in the context of approval-based preferences.Splitting monotonicity requires that, if a funded project p is split into severalprojects P (cid:48) , according with its cost and the support it got from the voters, thenat least one of the projects of P (cid:48) must be funded. Below we provide a formaldefinition. Definition 1 (Splitting monotonicity) . An aggregation method R satisfies split-ting monotonicity if for each budgeting scenario E = ( P, V, c, L ) , for each fundedproject p ∈ R ( E ) , and for each budgeting scenario E (cid:48) which is formed by split-ting project p into a set of projects P (cid:48) with the same cost c ( p ) = c ( P (cid:48) ) , and suchthat for each voter v i we have v i ( P (cid:48) ) = v i ( p ) , it holds that R ( E (cid:48) ) ∩ P (cid:48) (cid:54) = ∅ . Since splitting monotonicity seems a natural axiom it is quite surprising that onlythree of our CSTV rules satisfy it.
Theorem 1.
GS, EwTC, and MTC do not satisfy splitting monotonicity, whileGSC, EwT and MTC satisfy the property.Proof. GS : Consider the following budgeting instance with voter and projects, p and p . Assume that c ( p ) = c ( p ) = 2 and that the budget is L = 2 . Thecumulative ballot of the single voter v is v ( p ) = 0 . and v ( p ) = 0 . . GS wouldselect p . Now, assume that we split p into two projects, p a and p b , such that c ( p a ) = c ( p b ) = 1 and such that v ( p a ) = v ( p b ) = 0 . . Now, GS would select p ,failing splitting monotonicity. GSC : First, observe that in order to prove that a rule satisfies splitting monotonic-ity it is sufficient to consider cases where the project that is to be split is dividedinto two parts. Indeed, if the project were divided into more than two parts, onecould use the reasoning for splitting into two projects recursively.Consider a project p that is about to be split into p a and p b . Observe that either support( p a ) c ( p a ) ≥ support( p ) c ( p ) , or support( p b ) c ( p b ) ≥ support( p ) c ( p ) p a ) + support( p b )support( p ) < c ( p a ) c ( p ) + c ( p b ) c ( p ) = 1 ,a contradiction. W.l.o.g., let us assume that: support( p a ) c ( p a ) ≥ support( p ) c ( p ) .Assume that p is selected by the rule. Then, after split, p a would be selectedin the same moment as p was selected, or before. This proves that GSC satisfiessplitting monotonicity. EwT : Consider a project p that is selected for funding and assume p is split intotwo projects, p a and p b . Note that whenever a voter v transfers some value to p (either as a result of redistributing the excess or as a result of being removed),then after the split, v transfers to p a and p b the same total value as she transferredto p —this is because we use the proportional strategy of redistributing values.Furthermore, v transfers to the other projects the same value as before the split.Consider the steps of the algorithm before p , p a , or p b is removed or selected. Ineach such step we have that: excess( p a ) + excess( p b ) = excess( p ) .First, consider the case where neither p a or p b is removed before p is selected orremoved. If p is selected (in such a time moment its support is greater than orequal to its cost), then either p a or p b is eligible for funding, and so it will beeventually selected. If p is removed (which means it is added in the RE phase ofthe algorithm), then by our assumption p a and p b will be removed after p , and sothey both will be added for funding in the RE phase.Second, consider the case when p a or p b —say p a —is removed before p is selectedor removed. At the time p a is removed we have that: excess( p a ) ≤ excess( p ) .In that moment we have: excess( p b ) ≥ .Hence, p b will be eventually selected. As before, the reasoning can be recursivelyapplied to the case when p is split to any set of projects P (cid:48) .15 wTC : Consider the following instance with 3 projects, p , p , and p . Theircosts are c ( p ) = 199 , c ( p ) = 102 , and c ( p ) = 200 , and the budget is L = 200 .Let us fix a small constant (cid:15) . There are 200 voters with the cumulative ballotsgiven in the following table: p p p /
14 5 / / − (cid:15) (cid:15) / The support of p , p , and p equals respectively, − (cid:15) ,
50 + 60 (cid:15) , and .Thus, p will be eliminated first. The last 60 voters will transfer (almost) the entiresupport to p . Thus, p will be eliminated next, and so p will be selected.Now, assume that p is split into p a and p b such that c ( p a ) = 100 , and c ( p b ) = 99 .The voters’ preferences look as follows: p a p b p p /
14 5 / / − (cid:15) (cid:15) / Then, p a is eliminated first, and the last 60 voters transfer their almost entiresupport from p a to p . Next, p is eliminated, and the whole value of the last60 voters is transferred to p . Thus, the total support of p becomes 110. In thelast step p is selected, and neither p a nor p b fit within the remaining budget. MT : Here the reasoning is similar to the case of EwT. First, we observe that thesum of transfers to each project stays the same after the split. If at some point p was eligible for funding, then at this point p a or p b was as well, and so one of theseprojects would be selected. Thus, from now on, let us assume that the excess of p was always negative. Observe that either the excess of p a or p b becomes positive,in which case the project would be selected, or the excesses of both p a and p b aregreater than that of p . Furthermore, if p is eligible by transfers, then p a or p b is aswell. Thus, p a or p b will be selected at most at the time when p was. MTC : Consider the instance with 2 projects, p and p , with costs equal to c ( p ) = 150 and c ( p ) = 151 . The budget is L = 200 . There are 200 voterswith the following preferences: p p / /
50 1 0
16n this example, the supports of the two projects are the same, they both areeligible by transfers, thus MTC selects the cheaper one—i.e., p .Now, assume that p is split into p a and p b such that c ( p a ) = 51 , and p b = 99 .The voters’ preferences look as follows: p a p b p / /
50 1 p a is not eligible by transfers; p will be selected, leaving no roomfor p a nor p b .The next axiom is analogous to the previous one, yet it describes merges amongthe projects rather than splits. Definition 2 (Merging monotonicity) . An aggregation method R satisfies merg-ing monotonicity if for each budgeting scenario E = ( P, V, c, L ) , each B (cid:48) ⊆R ( E ) , and for each scenario E (cid:48) = ( P \ B (cid:48) ∪ { b (cid:48) } , V, c (cid:48) , L ) such that b (cid:48) isa new project which costs c ( B (cid:48) ) and such that for each voter v i we have that v i ( b (cid:48) ) = (cid:80) b ∈ B (cid:48) v i ( b ) , it holds that b (cid:48) ∈ R ( E (cid:48) ) . That is, merging monotonicity requires that a project formed by merging a num-ber of funded projects is funded.
Theorem 2.
GS satisfies merging monotonicity, while EwT, EwTC, MT, MTC,GSC fail merging monotonicity.Proof.
MT, MTC : Consider an instance with 5 projects, p , q , r , s , and z with allthe costs equal to 10. The first 10 voters put the value . (cid:15) to p and . − (cid:15) to q . The next 9 voters put . (cid:15) to r and . − (cid:15) to s . The last voter puts 1 to z .There are n = 10 + 9 + 1 = 20 voters; the budget is L = 20 .Since p and q are eligible by transfers, both MT and MTC will select p first. Forthat, most of q ’s money will be transferred to p and as a result both rules willselect r in the second iteration. Now, assume that p and r are merged into x . Themerged project is no longer eligible by transfers, but q is still. It will be selected,and there will be no money left in the budget to buy x . EwT, EwTC : Consider the following instance with 5 projects, p, q, r, s , and z .The costs of p , s , and z are equal to 30; the cost of r is equal to . If we considerEwT, then we set the cost of q to 35; if we consider EwTC, we set c ( q ) = 30 . The17rst 30 voters assign value 1 to p . The next voter assigns value − (cid:15) to p and (cid:15) to q . The next 15 voters assign − (cid:15) to q and (cid:15) to r ; the next 40 voters assign . to r and . to s ; the remaining voters assign 1 to z . The number of voters is and the budget is L = 95 . The cumulative ballots of the voters are summarized inthe table below. p q r s z
30 1 0 0 0 01 1 − (cid:15) (cid:15) − (cid:15) (cid:15) . . Here, both EwT and EwTC will select p first. The value of the 31st voter will betransferred to q . Then, the support for q, r , and s will be − (cid:15) , (cid:15) , and ,respectively. Thus, z and r will be eliminated next; there will be enough moneyto accommodate the remaining two projects, thus p, q and s will be selected.Now assume that p and s are merged into x : it does not reach the threshold.Indeed, the support of projects x, q and r will be − (cid:15) , − (cid:15) , and
20 + 15 (cid:15) ,respectively. Thus, z and q will be eliminated next. The total value of − (cid:15) will be transferred from q to r , raising its support to . Thus, x will be eliminatednext, and r will be chosen, leaving no money in the budget for x . GSC : Consider an instance with 3 projects p , p , and p with the costs equal to,respectively, 5, 10, and 5. The budget is L = 10 . There are 10 identical voters,who assign value 0.35 to p , . to p and . to p . GSC selects p first, andthen there will be no money for p . Consequently, p and p will be selected. Onthe other hand, if we merge p and p , then the rule will select p . GS : Observe that merging two projects does not affect the order of considerationof the other projects apart from the merged ones. Assume we merged p a and p b into p and that p b was considered after p a by the rule. Thus, p will be consideredin the same or earlier iteration as p a , and there will be enough money to accom-modate p (since at this moment, before merge, there was enough money to add p a and p b ).Our next axiom, support monotonicity, requires that moving more support to afunded project does not hurt it. In the formulation below (cid:52) stands for symmetricdifference. 18 efinition 3 (Support monotonicity) . An aggregation method R satisfies supportmonotonicity if for each budgeting scenario E = ( P, V, c, L ) , each project p ∈R ( E ) , and each budgeting scenario E (cid:48) = ( P, V (cid:48) , c, L ) such that | V (cid:52) V (cid:48) | = 1 andfor the single voter v ∈ V (cid:52) V (cid:48) and the single voter v (cid:48) ∈ V (cid:48) (cid:52) V it holds that (1) v (cid:48) ( p ) > v ( p ) and (2) for each p (cid:48) (cid:54) = p , v (cid:48) ( p (cid:48) ) ≤ v ( p (cid:48) ) , then p ∈ R ( E (cid:48) ) . Theorem 3.
GS, GSC, EwT, EwTC, MT, and MTC fail support monotonicity.Proof.
EwT, EwTC : Consider the following instance with 3 projects, p, q, r . Thecosts of all projects are equal to 10. The first 8 voters put . − (cid:15) to p , . to q and .
25 + (cid:15) to r . The next 2 voters put 1 to q and the next to put to r . Thereare 12 voters and the budget is L = 12 .Here, p is eliminated first, q is eliminated second, and so r is selected for funding.Now, assume that all the voters move (cid:15) from q to r . Now, q is eliminated first andits value is redistributed among p and r in proportions . Thus, r is eliminatedsecond, and so p is the project selected for funding. MT : Assume we have 3 projects, p, q and r , with the costs equal to , and ,respectively. There are 10 identical voters who put utility .
25 + (cid:15) on p , . on q ,and . − (cid:15) on r . The budget is L = 10 .MT will select p first. Since there will be no money left for r , q will be selectedas a second project for funding. Now, assume that each voter moves (cid:15) from p to q . Now, MT selects r first and there will be no money left for q . GS : The instance is similar as for MT. The projects have the same costs but thevoters put / + 2 (cid:15) on p , / + (cid:15) on r and / − (cid:15) on q . Here GS will select p and q .Now assume that the voters (cid:15) from p to q . After such a change GS will select r . MTC, GSC : The constructions here are analogous to MT and GS.Theorem 3 suggest the following interesting open question: does there exist aproportional rule for PB that satisfies our three monotonicity axioms, in particularsupport monotonicity? Does there exist an impossibility result suggesting that theaxioms we consider in the paper are incompatible?
Next we consider proportionality as it is usually desired for PB applications (see,e.g., [1]); in particular, we see the lack of proportionality of the greedy rules – aswe show below – as their major drawback. Specifically, we introduce three ax-iomatic properties, of Weak-PR, PR, and Strong-PR. These axioms are new to the19aper, but similar properties have been considered in the context of participatorybudgeting for different types of voters’ preferences [1, 13].
Definition 4 (Weak Proportional Representation) . An aggregation method R sat-isfies Weak Proportional Representation (Weak-PR) if for each budgeting scenario E = ( P, V, c, L ) , for each (cid:96) ∈ [ L ] , each set V (cid:48) ⊆ V of voters with | V (cid:48) | ≥ (cid:96)n/L ,and each set P (cid:48) ⊆ P of projects with c ( P (cid:48) ) ≤ (cid:96) , there exist a scenario E (cid:48) whichdiffers from E only in the votes of the voters from V (cid:48) , such that P (cid:48) ⊆ R ( E (cid:48) ) . Theorem 4.
GSC satisfies Weak-PR but GS fails it.Proof. GS : Consider a scenario with P = { a, b } , c ( a ) = 1 , c ( b ) = 3 , L = 3 , andvoters v , v , and v , where voter v supports only a , and v and v support onlyproject b . For (cid:96) = 1 , voter v acts as a set V (cid:48) of | V (cid:48) | ≥ (cid:96)n/L = 1 and P (cid:48) = { a } acts as a set of projects with c ( P (cid:48) ) ≤ (cid:96) ; GS, however, selects only project b , asproject b has higher support and, after b is funded, no funds are left to fund project a . GSC : Let (cid:96) ∈ [ L ] . Consider a set V (cid:48) of voters with | V (cid:48) | ≥ (cid:96)n/L and a set P (cid:48) ⊆ P of projects with c ( P (cid:48) ) ≤ (cid:96) . Set the vote of each v (cid:48) ∈ V (cid:48) to support onlythe projects in P (cid:48) , proportionally: i.e., for each p (cid:48) ∈ P (cid:48) set v (cid:48) ( p (cid:48) ) = c ( p (cid:48) ) /c ( P (cid:48) ) and v (cid:48) ( p ) = 0 for each p / ∈ P (cid:48) . Now, the sum of support each project p (cid:48) ∈ P (cid:48) getsfrom the voters is at least | V (cid:48) | · c ( p (cid:48) ) /c ( P (cid:48) ) , as it gets this amount already fromthe voters in V (cid:48) . As | V (cid:48) | ≥ (cid:96)n/L , we have that the sum of support of each project p (cid:48) ∈ P (cid:48) is at least (cid:96)n/L · c ( p (cid:48) ) /c ( P (cid:48) ) ; furthermore, since c ( P (cid:48) ) ≤ (cid:96) , it follows thatthis sum of support is at least n/L · c ( p (cid:48) ) .GSC ranks projects according to their sum of support over their cost, so the“support over cost” value of each p (cid:48) ∈ P (cid:48) is at least n/L . We wish to upper boundthe total cost of projects p / ∈ P (cid:48) which get a “support over cost” value greater than n/L : The proof will follow by showing that the total cost of such projects is atmost L − (cid:96) , because then, GSC would fund all projects p (cid:48) ∈ P (cid:48) . To show this,assume otherwise, that the total cost of projects p / ∈ P (cid:48) with “support over cost”value greater than n/L is more than L − (cid:96) , call the set of these projects S .Observe that the number of voters v / ∈ V (cid:48) is at most n − (cid:96)n/L . Thus, thetotal support divided by the total cost must be lower than n − (cid:96)nL L − (cid:96) = nL ; hence,contradiction.Intuitively, the three axioms that we consider in this section differ in how muchsynchronization is needed among the members of a group of voters in order ensurethat these voters will be able to decide about a certain fraction of the budget.20eak-PR ensures that a group of at least (cid:96)n/L can (by coordinating) make a setof projects P (cid:48) of total cost (cid:96) funded; PR merely requires that such voters supportthe same set P (cid:48) . Definition 5 (Proportional Representation) . An aggregation method R satisfies Proportional Representation (PR) if for each budgeting scenario E = ( V, P, c, L ) ,each (cid:96) ∈ [ L ] , each V (cid:48) ⊆ V with | V (cid:48) | ≥ (cid:96)n/L , and each set P (cid:48) ⊆ P of projectswith c ( P (cid:48) ) ≤ (cid:96) , it holds that: If all voters v (cid:48) ∈ V (cid:48) support all projects in P (cid:48) , andno other projects, then P (cid:48) ⊆ R ( E ) . Theorem 5.
EwT and EwTC satisfy PR but GSC fails it.Proof.
GSC : Intuitively, GSC fails PR because after one project in | P (cid:48) | is funded,its excess gets lost, which might cause the other projects in P (cid:48) not funded.More formally, consider a budgeting scenario with P = { a, b, c } where c ( a ) = c ( b ) = 1 and c ( c ) = 3 , the budget limit L = 4 , and voters v and v , where voter v assigns to project a value − (cid:15) and to project b value (cid:15) , and voter v assigns toproject c value .According to PR, with V (cid:48) = { v } , (cid:96) = 2 , and P (cid:48) = { a, b } , we have that indeedboth a and b shall be funded. GSC, however, will choose the bundle { a, c } . EwT, EwTC : Fix (cid:96) ∈ [ L ] . Let P (cid:48) be a set of projects with c ( P (cid:48) ) ≤ (cid:96) . Let V (cid:48) be agroup of voters who all support all projects from P (cid:48) and no other projects; assume | V (cid:48) | ≥ (cid:96)n/L .Recall that we define the support of a project p as: support( p ) = L · (cid:80) nj =1 v j ( p ) n .Let S i be the set of projects picked by the rule (EwT or EwTC) up to the i thiteration, inclusive. We prove the following invariant: In the i th step the totalsupport the voters from V (cid:48) assign to the projects from P (cid:48) \ S i equals at least (cid:96) − c ( P (cid:48) ∩ S i ) and no project from P (cid:48) has been eliminated by the rule. Theinvariant is clearly true when the rule begins. Now, assume it is true after the i thiteration, and we will show that it must hold after the ( i + 1) th iteration as well.Observe that in the ( i + 1) th iteration the total support of the candidates from P (cid:48) equals at least: (cid:96) − c ( P (cid:48) ∩ S i ) ≥ c ( P (cid:48) ) − c ( P (cid:48) ∩ S i ) = c ( P (cid:48) \ S i ) .Thus, there must exists at least one project, support of which exceeds the cost,thus no project from P (cid:48) can be eliminated. Furthermore, if a project p (cid:48) ∈ P (cid:48) is21elected, then the amount of support that the voters from V (cid:48) assign to the projectsfrom P (cid:48) will decrease by c ( p (cid:48) ) (the exceed will be transferred only to the projectsfrom P (cid:48) , unless all of them are already selected). This proves the invariant. Sinceno project from P (cid:48) will be eliminated, all of them will be picked by the rule.Below is our strongest proportionality axiom, in which we relax the requirementthat c ( P (cid:48) ) < (cid:96) . According to Strong-PR, the voters in groups do not have tostrongly synchronize to get projects that they like: intuitively, they only need toagree on the set of those projects that get a positive support. Definition 6 (Strong Proportional Representation) . An aggregation method R satisfies Strong Proportional Representation (Strong-PR) if for each scenario E = ( P, V, c, L ) , each (cid:96) ∈ [ L ] , each V (cid:48) ⊆ V with | V (cid:48) | ≥ (cid:96)n/L , and each P (cid:48) ⊆ P , it holds that: If all voters v (cid:48) ∈ V (cid:48) support all projects in P (cid:48) , and not anyother project, then either P (cid:48) ⊆ R ( E ) or for each p ∈ P (cid:48) \ R ( E ) we have that c ( p ) + c ( P (cid:48) ∩ R ( E )) > (cid:96) . Theorem 6.
MT and MTC satisfy Strong-PR but EwT and EwTC fail it.Proof.
EwT, EwTC : Consider an instance with voters, v , v and projects, p , p , and q , such that c ( p ) = 5 , c ( p ) = 7 , and c ( q ) = 6 . The budget limit is L = 10 . Voter v puts (cid:15) to p and − (cid:15) to p . Voter v puts to q . The support ofthe projects p , p , and q will be respectively, (cid:15) , − (cid:15) , and . Thus, both EwTand EwTC will eliminate p first, p second, and q last. Consequently, only q willbe selected while Strong-PJR requires selecting p . MT, MTC : Fix (cid:96) ∈ [ L ] , and consider a group of voters V (cid:48) with | V (cid:48) | ≥ (cid:96)n/L ,and a set P (cid:48) ⊆ P of projects, which are supported by all the voters from V (cid:48) ;furthermore, assume the voters from V (cid:48) do not support any other projects.Let S i denote the set of projects selected by the rule (MT, or MTC) up to the i thiteration, inclusive. First, we observe that the following invariant holds: In eachiteration i the total support that the voters from V (cid:48) assign to the projects from P (cid:48) \ S i equals at least (cid:96) − c ( S i ∩ P (cid:48) ) . Indeed, the invariant is initially true (since c ( S ∩ P (cid:48) ) = 0 and L/n · | V (cid:48) | ≥ (cid:96) ), and each time a project p ∈ P (cid:48) is selected,the total support that voters from V (cid:48) assign to projects is decreased by at most c ( p ) . Furthermore, the excess of the value that the voters from V (cid:48) assign to p isredistributed only among the projects from P (cid:48) .Now, for the sake of contradiction, assume there exists a project p (cid:48) ∈ P (cid:48) suchthat c ( p (cid:48) ) + c ( P (cid:48) ∩ R ( E )) ≤ (cid:96) and that has not been selected. Let j be the lastiteration before the rule reached the “Inclusive Maximality Postprocedure” phase22possibly j is the last iteration of the rule). Then, clearly c ( p (cid:48) ) + c ( P (cid:48) ∩ S j ) ≤ (cid:96) .By our invariant, we get that in the j th iteration the total support the voters from V (cid:48) assign to the projects from P (cid:48) \ S j equals at least: (cid:96) − c ( S j ∩ P (cid:48) ) ≥ c ( p (cid:48) ) .Furthermore, all the voters from V (cid:48) support p (cid:48) , thus p (cid:48) is eligible by transfers.Consequently, the rule cannot stop nor reach the “Inclusive Maximality Postpro-cedure” phase. This gives a contradiction and completes the proof. Proposition 1.
Each aggregation rule that satisfies Strong-PR also satisfies PR.Each aggregation rule that satisfies PR also satisfies Weak-PR.Proof.
Strong-PR → PR : Let R be an aggregation method satisfying Strong-PR and, counterpositively, assume that R fails PR. Since R fails PR, then, bydefinition, there exists a budgeting scenario E = ( P, V, c, L ) , some (cid:96) ∈ [ L ] , a set V (cid:48) ⊆ V of voters with | V (cid:48) | ≥ (cid:96)n/L , a set P (cid:48) ⊆ P of projects with c ( P (cid:48) ) ≤ (cid:96) , anda project p ∈ P (cid:48) such that all voters v (cid:48) ∈ V (cid:48) support all projects in P (cid:48) , and not anyother project, but p / ∈ R ( E ) .Now, since R satisfies Strong-PR, then, by definition, for these specific E , (cid:96) , V (cid:48) , P (cid:48) , and p , it holds that either:1. P (cid:48) ⊆ R ( E ) ,2. for each p ∈ P (cid:48) \ R ( E ) we have that c ( p ) + c ( P (cid:48) ∩ R ( E )) > (cid:96) .But, for these specific E , (cid:96) , V (cid:48) , P (cid:48) , and p , we have that c ( P (cid:48) ) ≤ (cid:96) , thus thesecond condition does not hold (specifically, for each p ∈ P (cid:48) R ( E ) we have that c ( p ) + c ( P (cid:48) ∩ R ( E )) ≤ c ( P (cid:48) ) ≤ (cid:96) ). Furthermore, the first condition does not holdas p / ∈ R ( E ) . Hence, contradiction. PR → Weak-PR : Let R be an aggregation method satisfying PR. Thus, by def-inition, for each budgeting scenario E = ( P, V, c, L ) , for each (cid:96) ∈ [ L ] , each set V (cid:48) ⊆ V of voters with | V (cid:48) | ≥ (cid:96)n/L , and each set P (cid:48) ⊆ P of projects with c ( P (cid:48) ) ≤ (cid:96) , it holds that: If all voters v (cid:48) ∈ V (cid:48) support all projects in P (cid:48) , and notany other project, then P (cid:48) ⊆ R ( E ) .So, in particular, there exist possible votes for the voters in V (cid:48) such that P (cid:48) ⊆R ( E ) ; e.g., all voters v (cid:48) ∈ V (cid:48) supporting all projects in P (cid:48) , each with / | P (cid:48) | support, and not any other project. Thus, Weak-PR is satisfied.Fain et al. [13] studied another axiom, called core , aimed at capturing propor-tionality for PB. They proved there exists no rule satisfying the core; to the best ofour knowledge, our rules are the first that satisfy a natural weakening of the core.23able 2: Statistics of Simulation Scenario 1 (left), Simulation Scenario 2 (right)and real data from PB in Warsaw (bottom). (a) Simulation Scenario 1 Rule VS *VS suburbs AR ACtarget 100% 100% 40% 0.0 -GS 23.0 % 20.0% 10.9 % 25.4% 45kEwT 25.0 % 23.9% 21.6 % 2.1% 25 kMT 22.6 % 22.2% 31.8 % 3.5% 24 kGSC 27.6 % 25.4% 15.2 % 11.0% 29 kEwTC 25.9 % 24.1% 16.7 % 2.8% 27 kMTC 25.7 % 23.8% 22.9 % 2.9% 28 k (b) Simulation Scenario 2
Rule VS *VS FoEP AR ACtarget 100% 100% 50% 0.0% -GS 21.2 % 18.9% 65% 27.1% 45 kEwT 24.4 % 23.1% 32% 12.1% 26 kMT 22.8 % 22.5% 22% 19.7% 25 kGSC 27.9 % 26.3% 7% 39.0% 25 kEwTC 24.5 % 22.7% 41% 8.9% 29 kMTC 24.3 % 22.9% 40% 11.2% 29 k (c) Warsaw Instance
Rule VS AR ACWM 66% 5.1% 860 kGS 67% 4.6% 804 kEwT 80% 2.6% 295 kMT 80% 2.5% 294 kGSC 81% 2.7% 324 kEwTC 81% 2.8% 319 kMTC 81% 2.7% 319 k
To complement the axiomatic analysis provided in the section above, in this sec-tion we compare our six aggregation methods through computer-based simula-tions. In particular, we report on simulations done on synthetic datasets as well ason data collected from real-life PB elections.In our synthetic simulations we randomly generate instances of PB using the -dimensional Euclidean model [11, 9]: We associate each voter and each projectwith a point in a -dimensional Euclidean space; we refer to this point as the ideal point of the voter/project. Then, the cumulative ballot of each voter v j isgenerated as follows: We first identify the set S j containing the projects whoseideal points are the closest to the one of v . Then, we let v j support only theprojects from S j , where, for p ∈ S j , we set amount by which v j supports p to beinversely proportional to the distance between the ideal point of p and the ideal As we use real numbers, ties are not an issue. v ; specifically: v j ( p ) = (cid:88) j : p i ∈ S j dist ( p i , v j ( i )) (cid:80) p k ∈ S j dist ( p k , v j ( i )) . (1)We consider two scenarios specifying the distributions of ideal points for vot-ers/projects and of the costs of the projects: Simulation Scenario 1:
This simulation mimics a situation in a municipality,containing a city center and several suburbs around it. The ideal points aredrawn as follows: Each point has a probability of . to fall in a uniform centerdisc of radius . (representing the city center) and a probability of . tofall uniformly in one of the eight outer discs of radius . each (representingsuburbs). The cost of each project is selected from a Gaussian with µ = 50k, σ = 20k. Simulation Scenario 2:
The purpose of this simulation is to better understandhow the considered rules treat projects with different costs. We consider a citycontaining two centers, one with expensive projects and another one with cheapprojects. The ideal points are drawn as follows: We have two uniform discs,each with radius 0.25. In the left (right) disc the cost of each project is selectedfrom a Gaussian with µ = 30k, σ = 10k (respectively, µ = 70k, σ = 10k).For each of the two synthetic scenarios, we generate PB instances and com-pute bundle returned by our rules. We also consider data from the following realinstances:
Warsaw Instance:
In 2019, 86721 people voted for 101 projects in Warsaw’smunicipal Participatory Budgeting. Each voter could cast an approval ballot andvote for up to 10 projects. The budget limit was close to PLN 25M ( ≈ $6 . M).The selection rule used there was a simple greedy algorithm, which we refer toas the Warsaw method (WM); WM is equivalent to GS, except that WM takesapproval ballots while GS takes cumulative ballots.In Table 2, we provide the following statistics (for the synthetic data the valuesare averages over all 1000 instances):
Voter Satisfaction (VS): fraction of support of a voter which went on fundedprojects (formally, for a winning bundle B , the voter satisfaction of voter v is (cid:80) p ∈ B v ( p ) ). 25 nger Ratio (AR): the fraction of voters who are ignored in the election (for-mally, |{ v : (cid:80) p ∈ B v ( p ) = 0 }| / | V | ). Average Cost (AC): average cost of a funded project.Furthermore, for Scenarios 1 and 2 we compute the ∗ VS metric:
Voter Satisfaction with Approval Votes ( ∗ VS): instead of using Equation (1),for each v j ∈ V and p ∈ S j we set v j ( p ) = 0 . . This corresponds to usingapproval ballots (a voter supports each of her supported projects equally).Finally, for Scenario 1 we include the fraction of the budget spent on projectsfrom the suburbs, and for Scenario 2—the fraction spent on projects from theexpensive bundle (FoEP).Our synthetic simulations show that greedy rules produce large anger ratio com-pared to CSTV rules; also, greedy rules do not satisfy PR (see Table 1), whichmakes them less attractive for PB. We observe that MT is more fair to smallerdistricts than EwT and GS, and that MTC is more fair than GSC and EwTC. Atthe same time, we infer that MTC, EwTC, and GS do not discriminate expensiveprojects as much as the other rules. These results suggest MTC as a particularlygood rule, which is only outperformed by MT with respect to how it treats smallerdistricts. On the other hand, MTC performs considerably better than MT with re-spect to treating expensive projects, as well as with respect to VS and AR criteria.Our simulations based on the Warsaw Instance dataset show that all our methods,with the exception of GS, give better results than WM, the method that is currentlyused in Warsaw. Bundles selected by our algorithms contribute to larger VoterSatisfaction and lower Anger Ratio.We also compared our methods on 18 additional local PBs in Warsaw. In eachof 18 districts there was a separate local PB. The same algorithm was used asin the municipal PB, but each voter could select up to 15 projects. The resultsare consistent with those presented in Table 2. Moreover, in Figure 1 we presentthe comparison of Warsaw method and MTC. We can see that according to theutilitarian criterion (VS), MTC performs considerably better than the solution thatis currently used. We suggested the use of cumulative ballots for participatory budgeting and con-sidered several aggregation methods for this setting. First, our results, in particular26he preliminary experimental results presented in Table 2, show that cumulativeballots indeed can improve voter satisfaction. Moreover, using both our theoreti-cal results and preliminary experimental results, we identified the MTC rule that,we argue, shall be given serious consideration for being used in practical settings,for the following reasons: • MTC satisfies Strong-PR, hence is guaranteed to behave in a very propor-tional way, in particular not dismiss minorities – in contrast with, e.g., thecurrent method usually used in practice; • MTC is computationally efficient, as can be seen by the procedural descrip-tion of MTC and validated using our simulations; • MTC behaves well with respect to our synthetic simulations and in real-world data, in particular, it outperforms the method usually used in practiceboth by increasing the total satisfaction (while still being proportional) aswell as by decreasing the frustration rate.We discuss several avenues for future research, regarding applying CSTV in gen-eralized settings. Indeed, while here we considered the standard combinatorial PBsetting, it seems that cumulative ballots and our CSTV rule can be naturally ex-tended to allow for more complex ballots, thus enabling greater voter flexibilitythat hopefully could result in even better outcomes. Specifically, in future workwe plan to consider CSTV for the following settings:
Negative utilities
Settings in which voters can express negative utilities – here,voters could not only state how they wish to split their virtual coin among theprojects, but also specify, for each project, whether the support they give to thisproject is positive or negative.
PB with several resource types
Settings in which we have not only one type ofresource but several, say time and money – here, voters could get several virtualcoins to split simoultaneously;
PB with project interactions
Settings in which there are interactions betweenprojects, such as in the model of Jain et al. [20], assuming a substitution structure that is a partition over the projects – here, in addition to their cumulative ballots,voters could state their preferences regarding substitutions and complementarities27etween projects; furthermore, as our CSTV rules work by support redistributionon behalf of the voters, we might allow voters to explicitly state how they wishsuch redistribution to happen.
Acknowledgements
Arkadii Slinko was supported by the Faculty Development Research Fund3719899 of the University of Auckland. Piotr Skowron was supported by Poland’sNational Science Center grant UMO-2019/35/B/ST6/02215. Nimrod Talmon wassupported by the Israel Science Foundation (ISF; Grant No. 630/19).
References [1] H. Aziz, B. Lee, and N. Talmon. Proportionally representative participatorybudgeting: Axioms and algorithms. In
Proceedings of AAMAS-18 , pages23–31, 2018.[2] Haris Aziz and Barton E Lee. Proportionally representative participatorybudgeting with ordinal preferences. arXiv preprint arXiv:1911.00864 , 2019.[3] Haris Aziz and Nisarg Shah. Participatory budgeting: Models and ap-proaches. arXiv preprint arXiv:2003.00606 , 2020.[4] G. Benade, S. Nath, A. Procaccia, and N. Shah. Preference elicitation forparticipatory budgeting. In
Proceedings of AAAI-17 , pages 376–382, 2017.[5] Matthias Bentert and Piotr Skowron. Comparing election methods whereeach voter ranks only few candidates. In
AAAI , pages 2218–2225, 2020.[6] S. Bhagat and J. A. Brickley. Cumulative voting: The value of minorityshareholder voting rights.
The Journal of Law and Economics , 27(2):339–365, 1984.[7] Yves Cabannes. Participatory budgeting: a significant contribution to par-ticipatory democracy.
Environment and Urbanization , 16(1):27–46, 2004.[8] Arthur T Cole Jr. Legal and mathematical aspects of cumulative voting.
SCLQ , 2:225, 1949. 289] E. Elkind, P. Faliszewski, J.-F. Laslier, P. Skowron, A. Slinko, and N. Tal-mon. What do multiwinner voting rules do? an experiment over the two-dimensional Euclidean domain. In
Proceedings of AAAI-17 , 2017.[10] E. Elkind, P. Faliszewski, P. Skowron, and A. Slinko. Properties of multi-winner voting rules.
Social Choice and Welfare , 48(3):599–632, 2017.[11] J. Enelow and M. Hinich. The spatial theory of voting: An introduction,1984.[12] B. Fain, A. Goel, and K. Munagala. The core of the participatory budgetingproblem. In
Proceedings of WINE-16 , pages 384–399, 2016.[13] Brandon Fain, Kamesh Munagala, and Nisarg Shah. Fair allocation of indi-visible public goods. In
Proceedings of EC-18 , pages 575–592, 2018.[14] P. Faliszewski, P. Skowron, A. Slinko, and N. Talmon. Multiwinner voting:A new challenge for social choice theory. In U. Endriss, editor,
Trends inComputational Social Choice . AI Access Foundation, 2017.[15] P. Faliszewski and N. Talmon. A framework for approval-based budgetingmethods. In
AAAI-19 , 2019.[16] Till Fluschnik, Piotr Skowron, Mervin Triphaus, and Kai Wilker. Fair knap-sack. In
Proceedings of AAAI-19 , 2019.[17] Bryan Ford. A liquid perspective on democratic choice. arXiv preprintarXiv:2003.12393 , 2020.[18] Rupert Freeman, David M Pennock, Dominik Peters, and Jennifer Wort-man Vaughan. Truthful aggregation of budget proposals. In
Proceedings ofthe 2019 ACM Conference on Economics and Computation , pages 751–752,2019.[19] Ashish Goel, Anilesh K Krishnaswamy, Sukolsak Sakshuwong, and TanjaAitamurto. Knapsack voting for participatory budgeting.
ACM Transactionson Economics and Computation (TEAC) , 7(2):1–27, 2019.[20] Pallavi Jain, Krzysztof Sornat, and Nimrod Talmon. Participatory budgetingwith project interactions. In
Proceedings of IJCAI ’20 , 2020.2921] Christian Klamler, Ulrich Pferschy, and Stefan Ruzika. Committee selectionunder weight constraints.
Mathematical Social Sciences , 64(1):48–56, 2012.[22] Annick Laruelle. Voting to select projects in participatory budgeting.
Euro-pean Journal of Operational Research , 2020.[23] Tyler Lu and Craig Boutilier. Budgeted social choice: From consensus topersonalized decision making. In
Twenty-Second International Joint Con-ference on Artificial Intelligence , 2011.[24] L. Mills. The mathematics of cumulative voting.
Duke Law Journal , page 28,1968.[25] P. Skowron, P. Faliszewski, and J. Lang. Finding a collective set of items:From proportional multirepresentation to group recommendation.
ArtificialIntelligence , 241:191–216, 2016.[26] N. Tideman and D. Richardson. Better voting methods through technology:The refinement-manageability trade-off in the Single Transferable Vote.
Public Choice , 103(1–2):13–34, 2000.[27] R. Vengroff. Electoral reform and minority representation.
Perspectives onPolitical Science , 32(3):166, 2003.[28] Brian Wampler. Participatory budgeting: Core principles and key impacts.