Guarantees in Fair Division: General or Monotone Preferences

Abstract

When dividing a “manna” Ω of private items (commodities, workloads, land, time slots) between n agents, the individual guarantee is the welfare each agent can secure in the worst case of other agents’ preferences and actions. If the manna is nonatomic and utilities are continuous (not necessarily monotone or convex) the minmax utility, that of our agent’s best share in the agent’s worst partition of the manna, is guaranteed by Kuhn’s generalization of divide and choose. The larger maxmin utility—of the agent’s worst share in the agent’s best partition—cannot be guaranteed even for two agents. If, for all agents, more manna is better than less (or less is better than more), the new bid and choose rules offer guarantees between minmax and maxmin by letting agents bid for the smallest (or largest) size of a share they find acceptable.

On guarantees, vetoes, and random dictators

“…The main focus of the subsequent literature is envy‐ free divisions: how to achieve one by simple cuts and queries (Brams and Taylor (1995), Robertson and Webb (1998), Aziz and McKenzie (2016)) and proving its existence under preferences more general than additive utilities (Stromquist (1980), Woodall (1980)). An exception is the recent paper of Bogomolnaia and Moulin (2020) returning to the worst case approach under very general preferences and identifying the MinMaxShare (my best share in the worst partition of the cake I can be offered) as a feasible guarantee, though not a maximal one.…”

Mind the gap: Cake cutting with separation

“…We are aware of two recent studies of the maximin share in cake cutting. Bogomolnaia and Moulin [15] considered agents with general continuous valuations-not necessarily additive or even monotonic. They showed that the maximin share is not always attainable, but the minimax share (i.e., the worst-case share of an agent when the items are partitioned into n bundles and the agent gets the best bundle) can always be guaranteed.…”

Ordinal Maximin Share Approximation for Goods

“…The implication relations between -out-of-d MMSfairness guarantees for different values of and d were characterized by Segal-Halevi (2019). Recently, the maximin share and its ordinal approximations have also been applied to some variants of the cake-cutting problem (Elkind, Segal-Halevi, & Suksompong, 2021c, 2021b, 2021aBogomolnaia & Moulin, 2022).…”