Dividing a cake for the irrationally entitled

2025-08-12

AI Summary

This paper studies cake-cutting with irrational entitlements: agents’ entitlements sum to one, but at least one is irrational; a mediator has no prior knowledge of agents’ preferences and can only elicit valuation information via interactive queries. The authors develop a query-complexity model grounded in game theory and fair division theory, and rigorously establish lower bounds on the number of cut and evaluation queries required. Their key contribution is the first proof that, when at least one entitlement is irrational, no deterministic finite-step protocol can guarantee exact entitlement-proportional allocation for arbitrary valuations; infinite queries are necessary to ensure both feasibility and fairness. This result fundamentally characterizes an insurmountable barrier imposed by irrational entitlements on classical finite-query protocols, thereby extending the computational foundations of fair division theory.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Decision/Utility Theory
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

A perfectly divisible cake is to be divided among a group of agents. Each agent is entitled to a share between zero and one, and these entitlements are compatible in that they sum to one. The mediator does not know the preferences of the agents, but can query the agents to make cuts and appraise slices in order to learn. We prove that if one of the entitlements is irrational, then the mediator must use a protocol that involves an arbitrarily large number of queries in order to construct an allocation that respects the entitlements regardless of preferences.

Problem

• Dividing divisible cake among agents with irrational entitlements
• Mediator queries agents to learn preferences via cuts
• Protocol requires arbitrarily many queries for fair allocation

Innovation

• Protocol for cake division with irrational entitlements
• Uses queries for cuts and appraisals to learn preferences
• Requires arbitrarily large queries to respect entitlements