Maximin share and minimax envy in fair-division problems

Abstract

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envyminimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.

On approximately fair allocations of indivisible goods

“…In the case of additive utilities, [10] consider a more general model, where the object to be partitioned is a measurable space (Ω, F) (think as an example that (Ω, F) = ([0, 1], Borel sets)).…”

“…In that case, the marginal utility of any good is clearly the maximum value of an atom, which is α. It was shown in [10] that there exist allocations with envy at most O(αn 3/2 ).…”

“…However, in most of the models considered so far, either all goods are infinitely divisible or there is at least one divisible good such as "money" to let the players compensate each other in order to achieve envy-freeness [1,29]. Under these assumptions, it has been proved that envy-free allocations always exist (see among others [28,10,1,29,3]). …”

“…Furthermore, we give a polynomial time algorithm for producing an allocation with maximum envy at most α. The problem of finding allocations with bounded envy in the presence of indivisible goods was introduced in [10]. A bound of O(αn 3/2 ) was obtained, where n is the number of players.…”

Finding maxmin allocations in cooperative and competitive fair division

“…In particular, its relationship with other important notions such as efficiency (or Pareto optimality) and, above all, envy-freeness has been investigated with alternating success: each maxmin partition is efficient, but while for the two children case Brams and Taylor [6] showed that it is also envy-free, the same may not hold when three or more children are to be served, as shown in Dall'Aglio and Hill [9].…”