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Randomized Algorithms for Fair Participatory Budgeting Outcomes


Core Concepts
This paper introduces randomized algorithms for participatory budgeting that simultaneously achieve strong ex-ante fairness guarantees and desirable ex-post fairness properties, such as justified representation and full justified representation.
Abstract
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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Key Insights Distilled From

by Haris Aziz,X... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05198.pdf
Fair Lotteries for Participatory Budgeting

Deeper Inquiries

How can the randomized algorithms presented in this paper be extended to handle additional constraints or objectives beyond fairness, such as diversity or representation of different demographic groups

The randomized algorithms presented in the paper can be extended to handle additional constraints or objectives beyond fairness by incorporating those constraints into the optimization framework. For example, to address diversity, the algorithm could be modified to ensure that the selected projects represent a diverse range of categories or demographics. This could be achieved by introducing constraints that promote diversity in the selection process, such as requiring a minimum number of projects from different categories or ensuring representation from various demographic groups. Similarly, to enhance the representation of different demographic groups, the algorithm could be adapted to prioritize projects that benefit underrepresented communities or allocate a certain percentage of the budget to projects that address specific needs of those groups. By incorporating these additional constraints into the algorithm, it can be tailored to optimize for a combination of fairness, diversity, and representation in the participatory budgeting process.

What are the computational complexity implications of achieving stronger ex-ante fairness guarantees in participatory budgeting

Achieving stronger ex-ante fairness guarantees in participatory budgeting can have significant computational complexity implications. As the fairness constraints become more stringent, the search space for feasible solutions may increase, leading to higher computational complexity. This can result in algorithms requiring more time and resources to find optimal or near-optimal solutions that satisfy the fairness criteria. In cases where ex-ante and ex-post fairness are shown to be incompatible, efficient algorithms can still be developed by balancing the trade-off between the two types of fairness. While it may not be possible to achieve both types of fairness simultaneously in all scenarios, algorithms can be designed to prioritize one type of fairness over the other based on the specific requirements of the participatory budgeting process. Efficient algorithms for settings where ex-ante and ex-post fairness are incompatible may involve heuristic approaches, approximation algorithms, or optimization techniques that aim to find solutions that balance fairness considerations while respecting the constraints of the problem. By carefully designing algorithms that strike a balance between ex-ante and ex-post fairness, it is possible to achieve satisfactory outcomes within reasonable computational limits.

Are there efficient algorithms for the settings where ex-ante and ex-post fairness are shown to be incompatible

The incompatibility results shown in the paper have practical implications for the design and implementation of participatory budgeting processes in real-world applications. Understanding the limitations and trade-offs between ex-ante and ex-post fairness can help policymakers and decision-makers make informed choices when designing participatory budgeting mechanisms. These insights can inform the design of participatory budgeting processes by highlighting the need to carefully consider the trade-offs between different fairness criteria. Decision-makers can use this knowledge to tailor the participatory budgeting process to prioritize certain fairness principles over others based on the specific goals and objectives of the budget allocation. Additionally, the incompatibility results can guide the development of decision-support tools and algorithms that assist in the selection and prioritization of projects in participatory budgeting. By taking into account the constraints and implications of ex-ante and ex-post fairness, these tools can help streamline the decision-making process and ensure that the outcomes are aligned with the desired fairness objectives.
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