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Fair Division Beyond Monotone Valuations

Siddharth Barman, Paritosh Verma

Abstract

This paper studies fair division of divisible and indivisible items among agents whose cardinal preferences are not necessarily monotone. We establish the existence of fair divisions and develop approximation algorithms to compute them. We address two complementary valuation classes, subadditive and nonnegative, which go beyond monotone functions. Considering both the division of cake (divisible resources) and allocation of indivisible items, we obtain fairness guarantees in terms of (approximate) envy-freeness (EF) and equability (EQ) In the context of envy-freeness, we prove that an EF division of a cake always exists under cake valuations that are subadditive and globally nonnegative. This result complements the nonexistence of EF allocations for burnt cakes known for more general valuations. In the indivisible-items setting, we establish the existence of EF3 allocations for subadditive and globally nonnegative valuations. In addition, we obtain universal existence of EF3 allocations under nonnegative valuations. We study equitability under nonnegative valuations. Here, we prove that EQ3 allocations always exist when the agents' valuations are nonnegative. Also, in the indivisible-items setting, we develop an approximation algorithm that, for given nonnegative valuations, finds allocations that are equitable within additive margins. Our results have combinatorial implications. For instance, the developed results imply the universal existence of proximately dense subgraphs: Given any graph $G=(V, E)$ and integer $k$ (at most $|V|$), there always exists a partition $V_1, V_2, \ldots, V_k$ of the vertex set such that the edge densities within the parts, $V_i$, are additively within four of each other. Further, such a partition can be computed efficiently.

Fair Division Beyond Monotone Valuations

Abstract

This paper studies fair division of divisible and indivisible items among agents whose cardinal preferences are not necessarily monotone. We establish the existence of fair divisions and develop approximation algorithms to compute them. We address two complementary valuation classes, subadditive and nonnegative, which go beyond monotone functions. Considering both the division of cake (divisible resources) and allocation of indivisible items, we obtain fairness guarantees in terms of (approximate) envy-freeness (EF) and equability (EQ) In the context of envy-freeness, we prove that an EF division of a cake always exists under cake valuations that are subadditive and globally nonnegative. This result complements the nonexistence of EF allocations for burnt cakes known for more general valuations. In the indivisible-items setting, we establish the existence of EF3 allocations for subadditive and globally nonnegative valuations. In addition, we obtain universal existence of EF3 allocations under nonnegative valuations. We study equitability under nonnegative valuations. Here, we prove that EQ3 allocations always exist when the agents' valuations are nonnegative. Also, in the indivisible-items setting, we develop an approximation algorithm that, for given nonnegative valuations, finds allocations that are equitable within additive margins. Our results have combinatorial implications. For instance, the developed results imply the universal existence of proximately dense subgraphs: Given any graph and integer (at most ), there always exists a partition of the vertex set such that the edge densities within the parts, , are additively within four of each other. Further, such a partition can be computed efficiently.
Paper Structure (29 sections, 25 theorems, 33 equations, 1 figure, 6 algorithms)

This paper contains 29 sections, 25 theorems, 33 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction
  2. Applications Beyond Fair Division
  3. Notation and Preliminaries
  4. Indivisible Item Allocation.
  5. Agents' Valuations.
  6. Multilinear Extensions.
  7. Cake Division.
  8. Envy-Freeness.
  9. Equitability.
  10. Tools and Techniques
  11. Cake Construction
  12. Cake Rounding
  13. Envy-Freeness Under $\sigma$-Subadditive Valuations
  14. Combinatorial to Multilinear Subadditivity
  15. Base case:
  16. ...and 14 more sections

Key Result

Theorem 1

Given any graph $G=(V,E)$ and integer $k \leq |V|$, there always exists a partition $V_1, V_2, \ldots, V_k \neq \emptyset$ of the vertex set such that $| \delta(V_i) - \delta(V_j)| \leq 5 \Delta +1$, for every $i,j \in [k]$; here, $\Delta$ is the maximum degree of $G$. Further, such a $k$-partition

Figures (1)

  • Figure 1: The figure illustrates the fair division settings considered in this work and the associated theorems.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1: $\mathrm{EF}k$
  • Theorem 4: Brouwer's Fixed-Point Theorem border1985fixed
  • Theorem 5: Sperner's Lemma
  • Theorem 6: Envy-Free Cake Division Su1999rental
  • Lemma 7: $\mathrm{EF}$ Rounding Lemma
  • proof
  • Lemma 8: $\mathrm{EQ}$ Rounding Lemma
  • ...and 33 more