The goal of fair division is to distribute resources among competing players in a “fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.


  1. fair division
  2. envy-freeness
  3. indivisible goods

MSC codes

  1. 91

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Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 1039 - 1068
ISSN (online): 1095-7146


Submitted: 11 February 2019
Accepted: 6 February 2020
Published online: 8 April 2020


  1. fair division
  2. envy-freeness
  3. indivisible goods

MSC codes

  1. 91



Funding Information

Google https://doi.org/10.13039/100006785
John Simon Guggenheim Memorial Foundation https://doi.org/10.13039/100005851
National Science Foundation https://doi.org/10.13039/100000001 : CCF-1524062

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