Maximum Nash welfare and other stories about EFX

Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, Alexandros Hollender, Alexandros A. Voudouris

Research output: Contribution to journalArticlepeer-review


We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations.
Original languageEnglish
Pages (from-to)69-85
Number of pages17
JournalTheoretical Computer Science
Early online date9 Feb 2021
Publication statusPublished - 8 Apr 2021


  • Fair division
  • Nash welfare
  • EFX
  • Approximation


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