# On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences

```@inproceedings{Keijzer2009OnTC,
title={On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences},
author={Bart de Keijzer and Sylvain Bouveret and Tomas Klos and Yingqian Zhang},
booktitle={Algorithmic Decision Theory},
year={2009}
}```
• Published in Algorithmic Decision Theory 13 October 2009
• Economics
We study the problem of allocating a set of indivisible goods to a set of agents having additive preferences. We introduce two new important complexity results concerning efficiency and fairness in resource allocation problems: we prove that the problem of deciding whether a given allocation is Pareto-optimal is coNP-complete, and that the problem of deciding whether there is a Pareto-efficient and envy-free allocation is \$\Sigma_2^p\$-complete.

### Almost Group Envy-free Allocation of Indivisible Goods and Chores

• Economics, Computer Science
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This work takes the group envy-freeness concept that is well-established in the literature and presents stronger and relaxed versions that are especially suitable for the allocation of indivisible items, and presents a clear taxonomy of the fairness concepts.

### Characterizing conflicts in fair division of indivisible goods using a scale of criteria

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Five different fairness criteria in a simple model of fair resource allocation of indivisible goods based on additive preferences are investigated, forming an ordered scale that can be used to characterize how conflicting the agents’ preferences are.

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A polynomial-time algorithm is designed that computes Nash welfare maximizing allocations when there are constantly many agents with constant many different values for the goods.

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• Economics
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This work proposes a polynomial time algorithm which maximizes the utilitarian social welfare and at the same time produces an allocation which is EF1 and PO in a special case of additive utility functions called buyer utility functions.

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A polynomial-time algorithm is designed that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many diﬀerent values for the goods.

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• 2010
This work considers the algorithmic problem of deciding whether there exists an allocation that is possibly (or necessarily) envy-free, given the incomplete preference information available, if in addition some mild economic efficiency criteria need to be satisfied.

### Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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• 2010
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### On Fair Division of Indivisible Items

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### Efficiency and Sequenceability in Fair Division of Indivisible Goods with Additive Preferences

• Economics
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It is shown that any Pareto-optimal allocation (under additive preferences) is sequenceable, but that the converse is not true anymore, and the links between these efficiency properties and the "scale of fairness" are investigated.

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A pseudopolynomial time algorithm for finding allocations that are EF1 and Pareto efficient; in particular, when the valuations are bounded, the algorithm finds such an allocation in polynomial time.

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