In approval-based budget division, a budget needs to be distributed to some candidates based on the voters' approval ballots over these candidates. In the pursuit of simple, well-behaved, and approximately fair rules for this setting, we introduce the class of sequential payment rules, where each voter controls a part of the budget and repeatedly spends his share on his approved candidates to determine the final distribution. We show that all sequential payment rules satisfy a demanding population consistency notion and we identify two particularly appealing rules within this class called the maximum payment rule (MP) and the 1/3-multiplicative sequential payment rule (1/3-MSP). More specifically, we prove that (i) MP is, apart from one other rule, the only monotonic sequential payment rule and gives a 2-approximation to a fairness notion called average fair share, and (ii) 1/3-MSP gives a 3/2-approximation to average fair share, which is optimal among sequential payment rules.

Abstract We formalize a framework for coordinating funding and selecting projects, the costs of which are shared among agents with quasi-linear utility functions and individual budgets. Our model contains the discrete participatory budgeting model as a special case, while capturing other useful scenarios. We propose several important axioms and objectives and study how well they can be simultaneously satisfied. We show that whereas welfare maximization admits an FPTAS, welfare maximization subject to a natural and very weak participation requirement leads to a strong inapproximability. This result is bypassed if we consider some natural restricted valuations, namely laminar single-minded valuations and symmetric valuations. Our analysis for the former restriction leads to the discovery of a new class of tractable instances for the Set Union Knapsack problem, a classical problem in combinatorial optimization.

In pursuit of participatory budgeting (PB) outcomes with broader fairness guarantees, we initiate the study of lotteries over discrete PB outcomes. As the projects have heterogeneous costs, the amount spent may not be equal ex ante and ex post. To address this, we develop a technique to bound the amount by which the ex-post spend differs from the ex-ante spend---the property is termed budget balanced up to one project (BB1). With respect to fairness, we take a best-of-both-worlds perspective, seeking outcomes that are both ex-ante and ex-post fair. Towards this goal, we initiate a study of ex-ante fairness properties in PB, including Individual Fair Share (IFS), Unanimous Fair Share (UFS) and their stronger variants, as well as Group Fair Share (GFS). We show several incompatibility results between these ex-ante fairness notions and existing ex-post concepts based on justified representation. One of our main contributions is a randomized algorithm which simultaneously satisfies ex-ante Strong UFS, ex-post full justified representation (FJR) and ex-post BB1 for PB with binary utilities.

In the committee voting setting, a subset of k alternatives is selected based on the preferences of voters. In this paper, our goal is to efficiently compute ex-ante fair probability distributions (or lotteries) over committees. Since it is not known whether a lottery satisfying the desirable fairness property of fractional core is polynomial-time computable, we introduce a new axiom called group resource proportionality (GRP), which strengthens other fairness notions in the literature. We characterize our fairness axiom by a correspondence with max flows on a network formulation of committee voting. Using the connection to flow networks revealed by this characterization, we then introduce voting rules which achieve fairness in conjunction with other desirable properties. The redistributive utilitarian rule satisfies ex-ante efficiency in addition to our fairness axiom. We also give a voting rule which maximizes social welfare subject to fairness by reducing to a minimum-cost maximum-flow problem. Lastly, we show our fairness property can be obtained in tandem with strong ex-post fairness properties - an approach known as best-of-both-worlds fairness. We strengthen existing best-or-both-worlds fairness results in committee voting and resolve an open question posed by Aziz et al. [2023a]. These findings follow from an auxiliary result which may prove useful in obtaining best-of-both-worlds type results in future research on committee voting.

Participatory budgeting (PB) is an implementation of direct democracy that allows members of a community to make collective budgeting decisions. However, existing PB processes rely upon a pre-defined central budget. We introduce a framework for pooling resources in addition to selecting projects, which we call PB with Resource Pooling. We motivate the key characteristics of this model and the basic properties we would like a mechanism to satisfy. We summarize results and discuss interesting questions related to our framework.