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

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.

[1]  Yann Chevaleyre,et al.  Multiagent Resource Allocation with K -additive Utility Functions , 2004 .

[2]  Steven J. Brams,et al.  Fair division - from cake-cutting to dispute resolution , 1998 .

[3]  Toby Walsh,et al.  Principles and Practice of Constraint Programming — CP 2001: 7th International Conference, CP 2001 Paphos, Cyprus, November 26 – December 1, 2001 Proceedings , 2001, Lecture Notes in Computer Science.

[4]  Vincent Conitzer,et al.  Combinatorial Auctions with k-wise Dependent Valuations , 2005, AAAI.

[5]  Yann Chevaleyre,et al.  Issues in Multiagent Resource Allocation , 2006, Informatica.

[6]  Theodore P. Hill,et al.  Equitable distribution of indivisible objects , 1988 .

[7]  Yingqian Zhang,et al.  Distributed task allocation in social networks , 2007, AAMAS '07.

[8]  Jérôme Lang,et al.  Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity , 2005, IJCAI.

[9]  Michael Wooldridge,et al.  The complexity of contract negotiation , 2005, Artif. Intell..

[10]  Elchanan Mossel,et al.  On approximately fair allocations of indivisible goods , 2004, EC '04.

[11]  Sarit Kraus,et al.  Methods for Task Allocation via Agent Coalition Formation , 1998, Artif. Intell..

[12]  S. Brams,et al.  Efficient Fair Division , 2005 .

[13]  M. Schaefer,et al.  Completeness in the Polynomial-Time Hierarchy A Compendium ∗ , 2008 .

[14]  Wei-Min Shen,et al.  A Dynamic Distributed Constraint Satisfaction Approach to Resource Allocation , 2001, CP.