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Allocating Mixed Goods with Customized Fairness and Indivisibility Ratio

Bo Li, Zihao Li, Shengxin Liu, Zekai Wu

TL;DR

This paper studies fair division of mixed divisible and indivisible goods by introducing agent-specific indivisibility ratios and new fairness notions EFα and PROPα, along with their fractional-relaxation variants EFf(α) and PROPf(α). It proves PROPα allocations always exist and can be computed in polynomial time, while EFα allocations may fail for more than two agents, establishing tight bounds and identifying cases where EF$f(α)$ is near-optimal. It also shows that PROP$f(α)$ cannot always be guaranteed below a certain threshold, and that MNW allocations satisfy PROPα and Pareto optimality. The work connects these notions to EFM, MMS, and PROPmM, clarifying when relaxations strengthen or weaken fairness and efficiency in mixed goods. Overall, the results quantify how the mix of divisible and indivisible items shapes the extent of achievable fairness and efficiency, with implications for customized fair division in heterogeneous settings.

Abstract

We consider the problem of fairly allocating a combination of divisible and indivisible goods. While fairness criteria like envy-freeness (EF) and proportionality (PROP) can always be achieved for divisible goods, only their relaxed versions, such as the ''up to one'' relaxations EF1 and PROP1, can be satisfied when the goods are indivisible. The ''up to one'' relaxations require the fairness conditions to be satisfied provided that one good can be completely eliminated or added in the comparison. In this work, we bridge the gap between the two extremes and propose ''up to a fraction'' relaxations for the allocation of mixed divisible and indivisible goods. The fraction is determined based on the proportion of indivisible goods, which we call the indivisibility ratio. The new concepts also introduce asymmetric conditions that are customized for individuals with varying indivisibility ratios. We provide both upper and lower bounds on the fractions of the modified item in order to satisfy the fairness criterion. Our results are tight up to a constant for EF and asymptotically tight for PROP.

Allocating Mixed Goods with Customized Fairness and Indivisibility Ratio

TL;DR

This paper studies fair division of mixed divisible and indivisible goods by introducing agent-specific indivisibility ratios and new fairness notions EFα and PROPα, along with their fractional-relaxation variants EFf(α) and PROPf(α). It proves PROPα allocations always exist and can be computed in polynomial time, while EFα allocations may fail for more than two agents, establishing tight bounds and identifying cases where EF$f(α)$ is near-optimal. It also shows that PROP$f(α)$ cannot always be guaranteed below a certain threshold, and that MNW allocations satisfy PROPα and Pareto optimality. The work connects these notions to EFM, MMS, and PROPmM, clarifying when relaxations strengthen or weaken fairness and efficiency in mixed goods. Overall, the results quantify how the mix of divisible and indivisible items shapes the extent of achievable fairness and efficiency, with implications for customized fair division in heterogeneous settings.

Abstract

We consider the problem of fairly allocating a combination of divisible and indivisible goods. While fairness criteria like envy-freeness (EF) and proportionality (PROP) can always be achieved for divisible goods, only their relaxed versions, such as the ''up to one'' relaxations EF1 and PROP1, can be satisfied when the goods are indivisible. The ''up to one'' relaxations require the fairness conditions to be satisfied provided that one good can be completely eliminated or added in the comparison. In this work, we bridge the gap between the two extremes and propose ''up to a fraction'' relaxations for the allocation of mixed divisible and indivisible goods. The fraction is determined based on the proportion of indivisible goods, which we call the indivisibility ratio. The new concepts also introduce asymmetric conditions that are customized for individuals with varying indivisibility ratios. We provide both upper and lower bounds on the fractions of the modified item in order to satisfy the fairness criterion. Our results are tight up to a constant for EF and asymptotically tight for PROP.
Paper Structure (18 sections, 19 theorems, 42 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 18 sections, 19 theorems, 42 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Example
  3. Main Results
  4. Related Work
  5. Preliminaries
  6. Envy-freeness up to a Fractional Good
  7. Two Agents
  8. General Number of Agents
  9. Existence and Computation of PROP$f(\alpha)$
  10. The Algorithm
  11. Analysis
  12. Impossibility Result
  13. Compatibility of PROP$\alpha$ and PO
  14. Relation with Other Fairness Notions
  15. Connections to EFM
  16. ...and 3 more sections

Key Result

Theorem 3.1

When $n=2$, an EF$\alpha$ allocation exists and can be found in polynomial time.

Figures (1)

  • Figure 1: Relations among different fairness notions, where $X \rightarrow Y$ means that an allocation satisfying $X$ must also satisfy $Y$.

Theorems & Definitions (52)

  • Definition 2.1: EF & PROP
  • Definition 2.2: EF1 & PROP1
  • Definition 2.3: Indivisibility Ratio
  • Definition 2.4: EF$\alpha$ & PROP$\alpha$
  • Definition 2.5: EF$f(\alpha)$ & PROP$f(\alpha)$
  • Definition 2.6: PO
  • Definition 2.7: MNW
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 42 more