Proportional Participatory Budgeting with Additive Utilities

1 Introduction . Consider an abstract model where there is a group of agents who have preferences over a set of options . Each option has a cost , and the goal is to select a subset of options whose total cost does not exceed a predefined budget . This model provides a formal framework for a number of real-life scenarios . Perhaps the most natural example is Participatory Budgeting ( PB ) . Through a voting system , PB allows residents of a city ( the agents ) to decide which projects ( the options ) will be funded by the government . In recent years , PB has been started in many cities around the world [ Cabannes , 2004 , Aziz and Shah , 2020 ] , and in some cases is used to decide a significant fraction of the city budget . For example , in Paris , PB has been run every year since 2014 , and since 2016 the total amount of funding for PB in Paris has been more than 100 million euros annually . Besides PB , the formal model captures the problem of electing a representative committee for a group of voters , say a faculty board or a parliament [ Faliszewski et al. , 2017 , Lackner and Skowron , 2020 ] , but also appears useful in situations that do not involve humans . For example , our framework describes the problem of selecting validators in consensus protocols , such as the blockchain [ Cevallos and Stewart , 2020 ] , the problem of selecting web pages that should be displayed in response to user queries , where the selected set of web pages should be useful for different types of user profiles [ Skowron et al. , 2017 ] or the problem of locating public facilities [ Skowron et al. , 2016 , Byrka et al. , 2018 ] . Algorithms for PB can even be used for improving the quality of genetic algorithms [ Faliszewski et al. , 2016 ] . While there are numerous applications of the model that we consider , for concreteness we will use terminology referring to participatory budgeting . In this paper , we focus on designing algorithms ( which we call aggregation rules ) for selecting projects . We are interested in rules that are fair in the sense that each agent has roughly equal influence on the outcome.This implies that the selected subset of projects must proportionally represent the views of 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . the voters , and so every group of voters with similar preferences should have an appropriate portion of the available budget allocated to fulfilling those preferences . Our results contribute to the study of fairness in algorithmic decision making . Notably , while many works on this topic aim to be fair to agents described by feature vectors ( that for example contain demographic information ) , we only use agents ’ preferences . Fair algorithms will need to identify cohesive groups of agents on their own . To understand the constraints that fairness and proportionality place on the decision procedure , let us start by discussing the way cities implement PB today . To count the votes , most cities use a variant of a simple protocol : Each voter is allowed to vote for a certain number of project proposals . Then , the projects with the highest number of votes are funded , until the budget limit is reached . While simple and intuitive , this is a bad voting rule . To see this , consider Circleville , a fictional city divided into four districts . A map of the city is shown in Figure 1 . The districts all have similar sizes , but Northside has the largest population . Suppose $ 400k have been allocated to PB , and suppose that all the project proposals are of a local character ( such as school renovations ) , and so residents only vote for projects that concern their own district . For example , every Northside resident will cast votes for projects A , B , C , and D , but no one else votes for these . Because Northside is the most populous district , the Northside projects will all receive the highest number of votes , and the voting rule described will spend the entire budget on Northside projects . The 280k residents of the other districts are left empty-handed . To circumvent this obvious issue , many cities have opted to hold separate elections for each district . The budget is divided in advance between the districts ( e.g. , in proportion to their number of residents ) , each project is assigned to a district , and voters only vote in their local election . While this avoids the issue of spending the entire budget in Northside , this fix introduces many other problems . For example , projects on the boundary of two districts ( such as A and P ) need to be assigned to one of them . Residents of the other district may be in favor of the boundary project , but can not vote for it . Thus boundary projects are less likely to be funded , even if they would be more valuable overall . Similarly , projects without a specific location that benefit the entire city can not be handled . Also , interest groups that are not geographic in nature will be underserved ; for instance , parents across the city might favor construction of a large playground ( project C ) , but with separate district elections , parents can not form a voting block . Similarly , bike riders across the city can not express their joint interest in the construction of a bike trail along Example River ( projects R , S , H , and G ) . To solve these problems , it seems desirable to hold a single city-wide election , but use a voting system that ensures that money is spent proportionally . The voting system should automatically and endogenously identify groups of voters who share common interests , and make sure that those groups are appropriately represented . This aim has been identified by several researchers [ Aziz et al. , 2018b ] , but no convincing proposal for a proportional voting rule has emerged so far . Indeed , no good formalization of “ proportionality ” for the PB context has been identified in the literature , except for the concept of the core . However , the core is a very demanding requirement , and there are situations where it fails to exist [ Fain et al. , 2018 ] . In this paper , we formalize proportionality for participatory budgeting as an axiom called extended justified representation ( EJR ) . The axiom requires that no group of voters with common interests is underserved . We construct a simple and attractive voting rule ( the Method of Equal Shares ) that satisfies EJR for approval preferences , and that satisfies EJR up to one project for general additive valuations . We then discuss a strengthening of EJR—which we call fully justified representation ( FJR ) —and show that this strengthening is still satisfiable , albeit by a different voting rule . Both our proportionality axiom and our voting rule are generalizations of concepts that have been introduced in the literature on multi-winner voting [ Faliszewski et al. , 2017 ] . That literature can be seen as handling a special case of PB , where all projects cost the same amount of money . This is often called the unit cost assumption . It turns out that the unit cost assumption substantially simplifies the problem . Further , much of the relevant literature studies rules that work with approval ballots , where voters are allowed to approve or disapprove each project . In our paper , we allow any additive valuations ( not just 0/1 ) , which is more expressive . The proportionality axioms and voting rules that we introduce all work for general additive valuations . This is notable , since allowing additive valuations introduces significant conceptual difficulty . Indeed , most prominent multi-winner voting rules do not naturally extend to additive valuations ( or at least not gracefully ) . Allowing additive valuations gives voters a way to more precisely describe their preference intensities . This can be valuable information in a PB setting where there are often projects which differ significantly in their cost . In practice , PB elections typically only elicit approval information . Using rules such as ours ( defined for general additive valuations ) , we can interpret this approval information in two different ways : either as 0/1 utilities ( so that a voter ’ s utility for a selected outcome is the number of approved funded projects ) or as cost-based utilities ( so the utility for an outcome is the total cost of approved funded projects ) . These two interpretations lead to interestingly different rules , with cost-based utilities favoring more expensive projects and leading to outcomes that are more similar to the outcomes that are selected by the greedy rule typically used by cities today . 2 Preliminaries . For each t 2 N , write [ t ] = { 1 , 2 , . . . , t } . An election is a tuple ( N , C , cost , { ui } i2N ) , where : • N = [ n ] and C = { c1 , . . . , cm } are the sets of voters and candidates ( or projects ) . • cost : C ! Q+ is a function that for each c 2 C assigns the cost that needs to be paid if c is selected . For each T ✓ C , we write cost ( T ) = P c2T cost ( c ) for the total cost of T . • For each voter i 2 N , the function ui : C ! [ 0 , 1 ] defines i ’ s additive utility function . If a set T ✓ C of candidates is implemented , i ’ s overall utility is ui ( T ) = P c2T ui ( c ) . For a subset S ✓ N of voters , we further write uS ( T ) = P i2S P c2T ui ( c ) for the total utility enjoyed by S if T is implemented . We assume that uN ( c ) > 0 for each c 2 C , so that every candidate is assigned positive utility by at least one voter . The voters have a fixed common budget which we normalize to 1 . A subset of candidates W ✓ C is feasible if cost ( W ) 1 . Our goal is to choose a feasible subset of candidates , which we call an outcome , based on voters ’ utilities . An aggregation rule ( or , in short , a rule ) is a function R that for each election E returns a feasible outcome W = R ( E ) called the winning outcome.1 There are two interesting special cases of our model : Committee elections . In this case , there exists k 2 N ( the committee size ) such that each candidate costs 1/k . Then W is an outcome if and only if |W | k. In this special case we also refer to outcomes as committees , and we say that the election satisfies the unit cost assumption . Approval utilities . In this case , for each i 2 N and c 2 C it holds that ui ( c ) 2 { 0 , 1 } . The approval set of voter i is A ( i ) : = { c 2 C : ui ( c ) = 1 } , and we say that i approves candidate c if c 2 A ( i ) . If c 2 A ( i ) \W , we say that c is a representative of i . Often we combine of these special cases , and study approval-based committee elections . 1Sometimes there are ties . For the results of this paper it does not matter how these ties are broken .

Summary: The paper deals with a setting of participatory budgeting in which voters's preferences over projects are aggregated to fund a subset of the candidate projects with a limited budget. Based on justified representation in committee elections (a related setting without costs) a fairness concept is introduced as well as an aggregation rule (named Rule X) which is shown to satisfy the concept, while other commonly used rules fail to do so. Several variants, restrictions and extensions are discussed.