# 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} }

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.

## 66 Citations

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

- Economics, Computer ScienceIJCAI
- 2020

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

- EconomicsAutonomous Agents and Multi-Agent Systems
- 2015

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.

### On Fair and Efficient Allocations of Indivisible Goods

- Computer Science, EconomicsAAAI
- 2021

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.

### Beyond identical utilities: buyer utility functions and fair allocations

- EconomicsArXiv
- 2021

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.

### Computing Pareto-Optimal and Almost Envy-Free Allocations of Indivisible Goods

- Computer Science, EconomicsArXiv
- 2022

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.

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

- EconomicsECAI
- 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

- Economics
- 2010

We study the problem of fairly dividing a set of goods amongst a group of agents, when those agents have preferences that are ordinal relations over alternative bundles of goods (rather than utility…

### On Fair Division of Indivisible Items

- EconomicsFSTTCS
- 2018

A polynomial time approximation algorithm is given that maximizes Nash social welfare up to a factor of e^{1/e} \approx 1.445$.

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

- EconomicsArXiv
- 2016

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.

### Finding Fair and Efficient Allocations

- EconomicsEC
- 2018

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.

## References

SHOWING 1-10 OF 14 REFERENCES

### Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity

- EconomicsIJCAI
- 2005

This work considers the problem of allocating fairly a set of indivisible goods among agents from the point of view of compact representation and computational complexity, and identifies the complexity of determining whether there exists an efficient and envy-free allocation.

### On approximately fair allocations of indivisible goods

- EconomicsEC '04
- 2004

In the presence of indivisibilities, it is shown that there exist allocations in which the envy is bounded by the maximum marginal utility, and an algorithm for computing such allocations is presented.

### Combinatorial Auctions with k-wise Dependent Valuations

- Computer Science, EconomicsAAAI
- 2005

It is shown that the minimum non-trivial degree of interdependency (2-wise dependency) is sufficient to render NP-hard the problem of computing the optimal allocation (but the class of valuations displaying up to k-wise dependencies is exhibited, which exhibits a restricted class of such valuations for which computing the ideal allocation is easy.

### Multiagent Resource Allocation with K -additive Utility Functions

- Computer Science, Economics
- 2004

This work gives particular consideration to scenarios where the preferences of agents are modelled in terms of k-additive utility functions, i.e. scenarios where synergies between different resources are restricted to bundles of at most k items.

### Fair division - from cake-cutting to dispute resolution

- Mathematics
- 1996

This chapter discusses jealousy-freeness and equitability for n=2, the divisible case, and applications for the point-allocation procedures.

### Issues in Multiagent Resource Allocation

- EconomicsInformatica
- 2006

A survey of some of the most salient issues in Multiagent Resource Allocation, including various languages to represent the pref-erences of agents over alternative allocations of resources as well as different measures of social welfare to assess the overall quality of an allocation.

### Distributed task allocation in social networks

- Computer ScienceAAMAS '07
- 2007

This paper proposes a new variant of the task allocation problem, where the agents are connected in a social network and tasks arrive at the agents distributed over the network through an algorithm based on the contract-net protocol.