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Achieving Envy-Freeness Through Items Sale

Authors Vittorio Bilò , Evangelos Markakis , Cosimo Vinci

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Author Details

Vittorio Bilò
  • Department of Mathematics and Physics "Ennio De Giorgi", University of Salento, Italy
Evangelos Markakis
  • Department of Informatics, Athens University of Economics and Business, Greece
  • Input Output Global (IOG)
Cosimo Vinci
  • Department of Mathematics and Physics "Ennio De Giorgi", University of Salento, Italy

Funding

This work is partially supported by: GNCS-INdAM - Programma Professori Visitatori; the PNRR MIUR project FAIR - Future AI Research (PE00000013), Spoke 9 - Green-aware AI; MUR - PNRR IF Agro@intesa; the Project SERICS (PE00000014) under the NRRP MUR program funded by the EU – NGEU; the Italian MIUR PRIN 2017 project ALGADIMAR - Algorithms, Games, and Digital Markets (2017R9FHSR_002); the framework of H.F.R.I call "Basic research Financing (Horizontal support of all Sciences)" under the National Recovery and Resilience Plan "Greece 2.0" funded by the European Union – NextGenerationEU (H.F.R.I. Project Number: 15877).

Cite AsGet BibTex

Vittorio Bilò, Evangelos Markakis, and Cosimo Vinci. Achieving Envy-Freeness Through Items Sale. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.26

Abstract

We consider a fair division setting of allocating indivisible items to a set of agents. In order to cope with the well-known impossibility results related to the non-existence of envy-free allocations, we allow the option of selling some of the items so as to compensate envious agents with monetary rewards. In fact, this approach is not new in practice, as it is applied in some countries in inheritance or divorce cases. A drawback of this approach is that it may create a value loss, since the market value derived by selling an item can be less than the value perceived by the agents. Therefore, given the market values of all items, a natural goal is to identify which items to sell so as to arrive at an envy-free allocation, while at the same time maximizing the overall social welfare. Our work is focused on the algorithmic study of this problem, and we provide both positive and negative results on its approximability. When the agents have a commonly accepted value for each item, our results show a sharp separation between the cases of two or more agents. In particular, we establish a PTAS for two agents, and we complement this with a hardness result, that for three or more agents, the best approximation guarantee is provided by essentially selling all items. This hardness barrier, however, is relieved when the number of distinct item values is constant, as we provide an efficient algorithm for any number of agents. We also explore the generalization to heterogeneous valuations, where the hardness result continues to hold, and where we provide positive results for certain special cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Algorithmic game theory and mechanism design
  • Applied computing → Economics
Keywords
  • Fair Item Allocation
  • Approximation Algorithms
  • Envy-freeness
  • Markets

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