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. Funding: A. Bogomolnaia and H. Moulin acknowledge the support from the Basic Research Program of the National Research University Higher School of Economics. H. Moulin’s research was also supported by a Chaire Blaise Pascal of the Region Ile-de-France, 2020-21.