The paper studies the problem of implementing fractional participatory budgeting (PB) outcomes as lotteries over integral outcomes. It first shows that not all fractional outcomes can be implemented by lotteries that satisfy a natural "up to any" budget feasibility property. The paper then introduces a weaker budget feasibility axiom called "budget balanced up to one project" (BB1) and demonstrates an approach to implement any fractional outcome using a lottery that satisfies BB1.
The paper then defines a hierarchy of ex-ante fairness axioms for PB, extending the "fair share" and "strong fair share" axioms from the committee voting literature. It investigates the compatibility of these ex-ante fairness notions with various ex-post fairness properties, such as justified representation (JR), extended justified representation (EJR), and full justified representation (FJR).
In the setting of PB with binary utilities, the paper shows that ex-ante group fair share (GFS) and ex-post JR are incompatible. However, it provides a randomized algorithm that simultaneously achieves ex-ante strong unanimous fair share (Strong UFS) and ex-post FJR, though the algorithm is not polynomial-time. The paper also gives a polynomial-time algorithm that achieves ex-ante Strong UFS and ex-post EJR.
Finally, the paper establishes that in the general PB setting with cardinal utilities, ex-ante and ex-post fairness are not compatible, even for the weakest pair of axioms and even in the restricted case where projects have unit costs.
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by Haris Aziz,X... at arxiv.org 04-09-2024
https://arxiv.org/pdf/2404.05198.pdfDeeper Inquiries