Computational Complexity of Envy-Free Allocations under Weakly Lexicographic Preferences

Core Concepts

Determining the existence of envy-free allocations is computationally hard for weakly lexicographic preferences, even with at most two indifference classes per agent. However, an algorithm can be developed to find allocations that are Pareto optimal and satisfy fairness notions like envy-freeness up to one item (EF1), maximin share (MMS), or envy-freeness up to any item (EFX), depending on the chosen criteria.

Abstract

The paper investigates fair division of indivisible items under weakly lexicographic preferences, where agents can express indifferences between sets of items.
Key highlights:

Deciding the existence of envy-free (EF) allocations is NP-complete, even when agents have at most two indifference classes.
An algorithm is developed that can find Pareto optimal (PO) allocations satisfying fairness notions like EF1, MMS, or EFX, depending on the chosen criteria.
The algorithm utilizes techniques like preference graphs and potential envy to handle indifferences when guaranteeing fairness and efficiency.
For chores-only instances, the paper shows that an EF1 and PO allocation always exists and can be computed efficiently, in contrast to the challenges in achieving EFX.
It is proven that EFX implies MMS for chores-only instances, which stands in contrast to the goods-only case.

The paper provides insights into the computational and axiomatic boundaries of fair division under weakly lexicographic preferences, highlighting the differences between goods and chores.

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Key Insights Distilled From

Almost Envy-Freeness under Weakly Lexicographic Preferences

by Hadi... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19740.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to handle more complex preference structures beyond weakly lexicographic orderings

The techniques developed in the paper can be extended to handle more complex preference structures beyond weakly lexicographic orderings by incorporating additional constraints and considerations into the algorithm. One approach could be to introduce a more nuanced scoring system that takes into account not just the order of preferences but also the intensity of preferences. This could involve assigning weights to different items within each indifference class based on the agent's relative preference for those items. By incorporating these weighted preferences into the allocation algorithm, it would be possible to capture a more detailed and nuanced representation of the agents' preferences.
Another extension could involve incorporating probabilistic or fuzzy preferences into the model. This would allow for a more flexible representation of preferences, where agents can express not just strict rankings but also degrees of uncertainty or ambiguity in their preferences. By adapting the algorithm to handle probabilistic or fuzzy preferences, it would be possible to accommodate a wider range of preference structures and provide more personalized and tailored allocations that better reflect the agents' true preferences.
Furthermore, exploring the use of machine learning techniques to analyze and predict agents' preferences based on historical data or behavioral patterns could also enhance the algorithm's ability to handle more complex preference structures. By leveraging machine learning algorithms, the system could learn from past allocations and feedback to make more accurate and personalized allocations in the future, even in the presence of intricate and multifaceted preference structures.

What are the implications of the finding that EFX implies MMS for chores-only instances, but not for goods-only instances

The finding that EFX implies MMS for chores-only instances but not for goods-only instances has significant implications for the understanding of fairness and efficiency in resource allocation problems. In the context of chores, the implication that EFX implies MMS suggests that achieving envy-freeness up to any item automatically ensures that each agent receives a bundle that is at least as preferable as the best possible bundle they can get in a partition where they receive the worst bundle. This implies a strong connection between envy-freeness and the maximin share fairness notion in the context of chores.
On the other hand, the fact that this implication does not hold for goods-only instances indicates a fundamental difference in the nature of fairness considerations between goods and chores. It suggests that the relationship between envy-freeness and maximin share fairness is more intricate and nuanced in the context of goods, potentially due to the different characteristics and properties of goods compared to chores. This distinction highlights the importance of considering the specific context and characteristics of the resources being allocated when analyzing fairness notions in resource allocation problems.

Can the challenges in achieving EFX allocations for chores be overcome by considering alternative fairness notions or relaxations

The challenges in achieving EFX allocations for chores can potentially be overcome by considering alternative fairness notions or relaxations that provide a balance between fairness and efficiency. One approach could be to relax the strict requirement of EFX and consider a more relaxed notion of envy-freeness that allows for minor instances of envy under certain conditions. By introducing a more flexible definition of envy-freeness, it may be possible to find allocations that are closer to EFX while still ensuring a high level of fairness and efficiency.
Additionally, exploring alternative fairness notions such as envy-freeness with compensation or envy-freeness with relaxation could provide alternative paths to achieving fair allocations in the presence of complex preference structures. These alternative notions allow for a more nuanced understanding of fairness and can accommodate situations where strict envy-freeness may be difficult to achieve. By considering a range of fairness notions and relaxations, it may be possible to find a suitable balance between fairness and efficiency in allocating chores, even in the presence of challenging preference structures.