N-person cake-cutting: There may be no perfect division

Steven J. Brams, Michael A. Jones, Christian Klamler

    Research output: Contribution to journalArticlepeer-review


    A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.

    Original languageEnglish (US)
    Pages (from-to)35-47
    Number of pages13
    JournalAmerican Mathematical Monthly
    Issue number1
    StatePublished - Jan 2013

    ASJC Scopus subject areas

    • General Mathematics


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