The Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI-21) Proportionally Representative Participatory Budgeting with Ordinal Preferences Haris Aziz and Barton E. Lee UNSW Sydney and Data61 CSIRO [email protected], [email protected] Abstract years, some PB applications have shifted to requiring linear order inputs. For example, in the New South Wales state of Participatory budgeting (PB) is a democratic paradigm Australia, participants are asked to provide a partial strict whereby voters decide on a set of projects to fund with a lim- 1 ited budget. We consider PB in a setting where voters report ranking over projects. The PB model we consider encom- ordinal preferences over projects and have (possibly) asym- passes both approval ballots and linear order inputs. metric weights. We propose proportional representation ax- In most of the PB settings considered, the participants are ioms and clarify how they fit into other preference aggre- assumed to have the same weight. However, in many sce- gation settings, such as multi-winner voting and approval- narios, symmetry may be violated. For example, in liquid based multi-winner voting. As a result of our study, we also democracy or proxy voting settings, a voter could be voting discover a new solution concept for approval-based multi- on behalf of several voters so may have much more voting winner voting, which we call Inclusion PSC (IPSC). IPSC weight. Similarly, asymmetric weights may naturally arise if is stronger than proportional justified representation (PJR), PB is used in settings where voters have contributed differ- incomparable to extended justified representation (EJR), and ent amounts to a collective budget or voters are affected by yet compatible with EJR. The well-studied Proportional Ap- proval Voting (PAV) rule produces a committee that satisfies the PB outcome to different extents. Therefore, we consider both EJR and IPSC; however, both these axioms can also be PB where voters may have asymmetric weights. satisfied by an algorithm that runs in polynomial-time. While there is much discussion on fairness and repre- sentation issues in PB, there is a critical need to formal- ize reasonable axioms to capture these goals. We present 1 Introduction two new axioms that relate to the proportional representa- Participatory budgeting (PB) provides a grassroots and tion axiom, proportionality for solid coalitions (PSC), ad- democratic approach to selecting a set of public projects to vocated by Dummett for multi-winner elections (Dummett fund within a given budget (Aziz and Shah 2020). It has been 1984). PSC has been referred to as “a sine qua non for a deployed in several cities all over the globe (Shah 2007). In fair election rule”(Woodall 1994) and the essential feature contrast to standard political elections, PB requires consid- of a voting rule that makes it a system of proportional rep- eration of the (heterogeneous) costs of projects and must re- resentation (Tideman 1995). We use the key ideas underly- spect a budget constraint. When examining PB settings for- ing PSC to design new axioms for PB settings. Our axioms mally, standard voting axioms and methods that ignore bud- provide yardsticks against which existing and new rules and get constraints and differences in each project’s cost need to algorithms can be measured. We also provide several justifi- be reconsidered. In particular, it has been discussed in policy cations for our new axioms. circles that the success of PB partly depends on how well it provides representation to minorities (Bhatnaga et al. 2003). Contributions We formalize the setting of PB with weak We take an axiomatic approach to the issue of proportional ordinal preferences. Previously, only restricted versions of representation in PB. the setting, such as PB with approval ballots, have been ax- In this paper, we consider PB with weak ordinal prefer- iomatically studied (Aziz, Lee, and Talmon 2018). We then ences. Ordinal preferences provide a simple and natural in- propose two new axioms Inclusive PSC (IPSC) and Com- put format whereby participants rank candidate projects and parative PSC (CPSC) that are meaningful proportional rep- are allowed to express indifference. A special class of or- resentation and fairness axioms for PB with ordinal prefer- dinal preferences are dichotomous preferences (sometimes ences. In contrast to previous fairness axioms for PB with referred to as approval ballots); this input format is used approval ballots (see, e.g., Aziz, Lee, and Talmon 2018), in most real-world applications of PB. However, in recent both IPSC and CPSC imply exhaustiveness (i.e., no addi- Copyright © 2021, Association for the Advancement of Artificial 1https://mycommunityproject.service.nsw.gov.au Intelligence (www.aaai.org). All rights reserved. 5110 Approval Ballots Ordinal Prefs Divisible (e.g. Bogomolnaia, Moulin, and Stong 2005) (e.g. Aziz and Stursberg 2014) Indivisible (e.g. Goel et al. 2019) This paper Table 1: Classification of the literature on fair participatory budgeting with ordinal preferences. tional candidate can be funded without exceeding the budget mat is approval ballots. We show that our general axioms limit). have connections with proportional representation axioms We show that an outcome satisfying Inclusive PSC is proposed by Aziz, Lee, and Talmon (2018) for the case of always guaranteed to exist and can be computed in poly- approval-ballots. We will also show how our approach has nomial time. The concept appears to be the “right” con- additional merit even for the case of approval-ballots. For cept for several reasons. First, it is stronger than the example, in contrast to previously proposed axioms in Aziz, local-BPJR-L concept proposed for PB when voters have Lee, and Talmon (2018), our axioms imply a natural prop- dichotomous preferences (Aziz, Lee, and Talmon 2018). erty called exhaustiveness. Second, it is also stronger than generalised PSC for multi- Fluschnik et al. (2017) consider the discrete PB model winner voting with ordinal preferences (Aziz and Lee 2020). and study the computational complexity of maximizing var- Third, when voters have dichotomous preferences, it implies ious notions of social welfare, including Nash social wel- the well-studied concept PJR for multi-winner voting, is in- fare. Benade` et al. (2017) study issues surrounding prefer- comparable to the EJR axiom (Aziz et al. 2018), and yet ence elicitation in PB with the goal of maximizing utilitarian is compatible with EJR. In particular, the well-studied pro- welfare. In their model, they also consider input formats in portional approval voting rule (PAV) computes an outcome which voters express ordinal rankings. However, their focus that satisfies both IPSC and EJR; however, there also exists is not on proportional representation. Fain, Goel, and Mu- polynomial-time algorithms that can achieve this. Even for nagala (2016) considered PB both for divisible settings as this restricted setting, it is of independent interest. To show well as discrete settings. However, their focus was on car- that there exists a polynomial-time algorithm to compute an dinal utilities. In particular, they focus on a demanding but outcome satisfying IPSC, we present the PB Expanding Ap- cardinal-utility centric concept of core fairness. Our ordinal provals Rule (PB-EAR) algorithm. approach caters to many settings in which voters only ex- We also show that the CPSC is equivalent to the gener- press rankings over projects. Other works on cardinal utili- alised PSC axiom for multi-winner voting with weak pref- ties include Fain, Munagala, and Shah (2018) and Bhaskar, erences, to Dummett’s PSC axiom for multi-winner voting Dani, and Ghosh (2018). In recent work, Rey, Endriss, and with strict preferences, and to PJR for multi-winner voting de Haan (2020) study an end-to-end model of participatory with dichotomous preferences. budgeting and focus primarily on strategic behaviour. The paper is also related to a rapidly growing literature 2 Related Work on multi-winner voting (Aziz et al. 2017a; Faliszewski et al. PB with ordinal preferences can be classified across differ- 2017; Aziz et al. 2017b; Elkind et al. 2017; Janson 2016; ent axes. One axis concerns the input format. Voters either Schulze 2002; Tideman 2006). PB is a strict generalization express dichotomous preferences or general weak or linear of multi-winner voting. Our axiomatic approach is inspired orders. Along another axis, either the projects are divisi- by the PSC axiom in multi-winner voting. The axiom was ble or indivisible. When the inputs are dichotomous prefer- advocated by Dummett (1984). PSC has been referred to as ences, there has been work both for divisible (Bogomolnaia, the most important requirement for proportional representa- Moulin, and Stong 2005; Aziz, Bogomolnaia, and Moulin tion in multi-winner voting (Woodall 1994, 1997; Tideman 2019) as well as indivisible projects (Aziz, Lee, and Tal- and Richardson 2000; Woodall 1994; Tideman 1995). Fig- mon 2018; Faliszewski and Talmon 2019). When the input ure 1 provides an overview of which model reduces to which concerns rankings, then there is work where the projects are other model. We dedicate a separate section to multi-winner divisible (see, e.g., Aziz and Stursberg 2014; Airiau et al. voting because one of our axioms gives rise to a new and 2019). Some of the work is cast in the context of probabilis- interesting axiom for the restricted setting of multi-winner tic voting but is mathematically equivalent to PB for divisi- voting. ble projects. To the best of our knowledge, fairness axioms for PB for 3 Preliminaries discrete projects have not been studied deeply when the in- A PB setting is a tuple (N; C; %; b; w; L) where N is the put preferences are general ordinal preferences. Therefore, set of n voters, C is the set of candidate projects (candi- this paper addresses an important gap in the literature.