We initiate the study of the communication complexity of fair division with indivisible goods. We focus on some of the most well studied fairness notions (envy-freeness, proportionality, and approximations thereof) and valuation classes (submodular, subadditive, and unrestricted). We show that for more than two players (and any combination of other parameters), determining whether a fair allocation exists requires exponential communication (in the number of goods). For two players, tractability depends heavily on the specific combination of parameters, and most of the paper is focused on the two-player setting. Taken together, our results completely resolve whether the communication complexity of computing a fair allocation (or determining that none exists) is polynomial or exponential, for every combination of fairness notion, valuation class, and number of players, for both deterministic and randomized protocols.


  1. fair division
  2. communication complexity
  3. envy-freeness

MSC codes

  1. 91

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Information & Authors


Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 206 - 243
ISSN (online): 1095-7111


Submitted: 13 February 2019
Accepted: 6 January 2020
Published online: 27 February 2020


  1. fair division
  2. communication complexity
  3. envy-freeness

MSC codes

  1. 91



Funding Information

Guggenheim Fellowship

Funding Information

Google https://doi.org/10.13039/100006785

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : CCF-1524062

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