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Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings

Vishwa Prakash H. V., Prajakta Nimbhorkar

TL;DR

This paper studies fair allocation of indivisible goods and chores under ordinal valuations with unequal entitlements. It shows that weighted necessarily proportional up to one item (WSD-PROP1) allocations exist and can be computed in polynomial time by reducing to perfect matchings, and provides a full characterization as corner points of a perfect matching polytope. It also connects WSD-PROP1 to rank-maximal matchings to obtain sequencible allocations and establishes Best-of-Both-Worlds ex-ante ex-post guarantees via a fractional-perfect-matching decomposition. The framework unifies goods and chores, enables objective-based optimization, and highlights limits such as PO incompatibility, while offering extensions to incorporate agent-specific competence and broader category quotas.

Abstract

We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.

Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings

TL;DR

This paper studies fair allocation of indivisible goods and chores under ordinal valuations with unequal entitlements. It shows that weighted necessarily proportional up to one item (WSD-PROP1) allocations exist and can be computed in polynomial time by reducing to perfect matchings, and provides a full characterization as corner points of a perfect matching polytope. It also connects WSD-PROP1 to rank-maximal matchings to obtain sequencible allocations and establishes Best-of-Both-Worlds ex-ante ex-post guarantees via a fractional-perfect-matching decomposition. The framework unifies goods and chores, enables objective-based optimization, and highlights limits such as PO incompatibility, while offering extensions to incorporate agent-specific competence and broader category quotas.

Abstract

We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.
Paper Structure (27 sections, 21 theorems, 15 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 27 sections, 21 theorems, 15 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Fractional and Randomized Allocations:
  6. Cardinal Valuations:
  7. The Interval Representation of Items:
  8. Stochastic Dominance(SD):
  9. Fairness and Efficiency Notions
  10. Matchings
  11. Rank-Maximal Matchings irving2003greedyirving2006rank:
  12. Existence and Computation of WSD-PROP1 Allocations for Chores via Matchings
  13. Optimizing Over Allocations
  14. Extending the Allocation Graph:
  15. Considering Agent Competence
  16. ...and 12 more sections

Key Result

lemma 1

Let $T\subseteq B$ be a set of $m_i$ chores, and let $r_1<r_2<\cdots<r_{m_i}$ be the ranks of the chores in $T$ in the ranking $\pi_i$ of agent $a_i$ (i.e, this set consists of the $r_1$-least favorite chore, $r_2$-least favorite chore,$\cdots$, and the $r_{m_i}$-least favorite chore for agent $a_i$

Figures (1)

  • Figure 2: The edges in red form a perfect matching - but it is not sequencible. The blue edges correspond to a rank-maximal matching, but it is not perfect. The squiggly edges corresponds to a rank-maximal $A$-perfect matching. This is sequencible, and the picking sequence is $\langle a_1,a_2,a_4,a_3\rangle$

Theorems & Definitions (30)

  • definition 1
  • definition 2: WSD-EF
  • definition 3: WPROP1 propx-doesnt-exist
  • definition 4: fractional WPROP1
  • definition 5: WSD-PROP1
  • definition 6: Pareto Optimailty (PO)
  • definition 7: Sequencibility (SEQ)
  • definition 8: Rank-Maximal Matching
  • definition 9: Rank-Maximal Perfect Matching
  • lemma 1
  • ...and 20 more