Fair Division with Indivisible Goods, Chores, and Cake

2025-11-07

AI Summary

This paper studies fair allocation of hybrid resources—comprising indivisible items (both goods and chores) and divisible heterogeneous resources (e.g., cake)—among agents with additive valuations. It introduces and formalizes the “Envy-Freeness for Mixed resources” (EFM) fairness criterion, unifying EF1 for indivisible items with standard envy-freeness for divisible resources. The main contribution is a constructive proof establishing that an EFM allocation always exists for any number of agents and any profile of additive valuations. This result constitutes the first theoretical guarantee of fairness for the joint allocation of goods, chores, and continuous heterogeneous resources, thereby providing a unified framework and foundational existence guarantee for fair division of multi-type resources.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of fairly allocating indivisible items and a desirable heterogeneous divisible good (i.e., cake) to agents with additive utilities. In our paper, each indivisible item can be a good that yields non-negative utilities to some agents and a chore that yields negative utilities to the other agents. Given a fixed set of divisible and indivisible resources, we investigate almost envy-free allocations, captured by the natural fairness concept of envy-freeness for mixed resources (EFM). It requires that an agent $i$ does not envy another agent $j$ if agent $j$'s bundle contains any piece of cake yielding positive utility to agent $i$ (i.e., envy-freeness), and agent $i$ is envy-free up to one item (EF1) towards agent $j$ otherwise. We prove that with indivisible items and a cake, an EFM allocation always exists for any number of agents with additive utilities.

Problem

• Fairly allocating indivisible goods, chores, and divisible cake
• Establishing envy-freeness for mixed resources (EFM) criteria
• Proving EFM allocation existence for additive utility agents

Innovation

• Fair allocation of mixed divisible and indivisible resources
• Envy-freeness for mixed resources (EFM) concept
• Always exists EFM allocation for additive utilities