Rental Harmony: Sperner's Lemma in Fair Division

The Simplex Reminiscent of Sperner's Lemma

“…This implies in particular that the hardness result for Hypergraph Labeling with Color Lists holds even for hypergraphs with a feasible labeling such that each hyperedge uses at most 4 colors. We also give an interpretation of this result in the setting of fair division [8]. Our result shows that for a certain restricted variant of fair division, not only is it impossible to satisfy all players but in fact all players except four will be always unsatisfied.…”

Constrainedly fair job assignments under minimum wages

“…In fact, we cannot adjust wages for over-demanded jobs, because jobs may always be over-demanded due to the lower compensation limits. It should also be noted that the current algorithm is different from those proposed by Su (1999), Klijn (2000), Brams and Kilgour (2001), and Haake et al (2002), because they all require that a fixed (possibly non-negative) amount of money should be completely allocated among the agents. Hence, no such minimal compensation scheme exists in their models.…”

“…A crucial difference between this algorithm and the procedures in, e.g., Crawford and Knoer (1981), Demange et al (1986) and Su (1999) is that we do not, and cannot, work with over-demanded jobs, because it may well be the case that a job is always over-demanded (e.g., job 1 in Example 1 when s 1 > 2, see footnote 4).…”

Dividing connected chores fairly

“…The result of Steinhaus [16] can be applied to chores too, so proportional divisions always exist. By Su [18], it is proven that envy-free divisions of chores with connected pieces also always exist. For the existence of equitable divisions with connected pieces, as far as we know no constructive proof was given so far.…”