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The Landscape of Almost Equitable Allocations

Hadi Hosseini, Vishwa Prakash HV, Aditi Sethia, Jatin Yadav

TL;DR

The paper broadens fair division by analyzing equitable up to one item (EQ1) allocations under general, possibly non-monotone valuations, assuming a common sign of the grand bundle. It establishes a nuanced landscape: EQ1 may fail to exist or be NP-hard to decide for three or more agents, while two-agent cases admit polynomial-time EQ1 allocations; several valuation classes (doubly monotone, submodular; nonnegative; identical subadditive) admit polynomial-time EQ1 allocations via a marginal-witness framework. A constructive exponential-time algorithm yields EQ1 for nonnegative valuations, resolving long-standing open questions, and results in EF1 for identical subadditive valuations. The findings significantly extend EQ1 existence/computation beyond monotone/additive settings and connect to broader notions like EF1 under special valuation structures, with implications for nonmonotone fair division in theory and practice.

Abstract

Equitability is a fundamental notion in fair division which requires that all agents derive equal value from their allocated bundles. We study, for general (possibly non-monotone) valuations, a popular relaxation of equitability known as equitability up to one item (EQ1). An EQ1 allocation may fail to exist even with additive non-monotone valuations; for instance, when there are two agents, one valuing every item positively and the other negatively. This motivates a mild and natural assumption: all agents agree on the sign of their value for the grand bundle. Under this assumption, we prove the existence and provide an efficient algorithm for computing EQ1 allocations for two agents with general valuations. When there are more than two agents, we show the existence and polynomial-time computability of EQ1 allocations for valuation classes beyond additivity and monotonicity, in particular for (1) doubly monotone valuations and (2) submodular (resp. supermodular) valuations where the value for the grand bundle is nonnegative (resp. nonpositive) for all agents. Furthermore, we settle an open question of Bil`o et al. by showing that an EQ1 allocation always exists for nonnegative(resp. nonpositive) valuations, i.e., when every agent values each subset of items nonnegatively (resp. nonpositively). Finally, we complete the picture by showing that for general valuations with more than two agents, EQ1 allocations may not exist even when agents agree on the sign of the grand bundle, and that deciding the existence of an EQ1 allocation is computationally intractable.

The Landscape of Almost Equitable Allocations

TL;DR

The paper broadens fair division by analyzing equitable up to one item (EQ1) allocations under general, possibly non-monotone valuations, assuming a common sign of the grand bundle. It establishes a nuanced landscape: EQ1 may fail to exist or be NP-hard to decide for three or more agents, while two-agent cases admit polynomial-time EQ1 allocations; several valuation classes (doubly monotone, submodular; nonnegative; identical subadditive) admit polynomial-time EQ1 allocations via a marginal-witness framework. A constructive exponential-time algorithm yields EQ1 for nonnegative valuations, resolving long-standing open questions, and results in EF1 for identical subadditive valuations. The findings significantly extend EQ1 existence/computation beyond monotone/additive settings and connect to broader notions like EF1 under special valuation structures, with implications for nonmonotone fair division in theory and practice.

Abstract

Equitability is a fundamental notion in fair division which requires that all agents derive equal value from their allocated bundles. We study, for general (possibly non-monotone) valuations, a popular relaxation of equitability known as equitability up to one item (EQ1). An EQ1 allocation may fail to exist even with additive non-monotone valuations; for instance, when there are two agents, one valuing every item positively and the other negatively. This motivates a mild and natural assumption: all agents agree on the sign of their value for the grand bundle. Under this assumption, we prove the existence and provide an efficient algorithm for computing EQ1 allocations for two agents with general valuations. When there are more than two agents, we show the existence and polynomial-time computability of EQ1 allocations for valuation classes beyond additivity and monotonicity, in particular for (1) doubly monotone valuations and (2) submodular (resp. supermodular) valuations where the value for the grand bundle is nonnegative (resp. nonpositive) for all agents. Furthermore, we settle an open question of Bil`o et al. by showing that an EQ1 allocation always exists for nonnegative(resp. nonpositive) valuations, i.e., when every agent values each subset of items nonnegatively (resp. nonpositively). Finally, we complete the picture by showing that for general valuations with more than two agents, EQ1 allocations may not exist even when agents agree on the sign of the grand bundle, and that deciding the existence of an EQ1 allocation is computationally intractable.

Paper Structure

This paper contains 8 sections, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. General valuations.
  4. Doubly monotone and submodular valuations.
  5. Nonnegative valuations.
  6. Identical subadditive valuations.
  7. Related Work
  8. Divisible Items.

Figures (1)

  • Figure 1: A pictorial representation of various valuation classes, their intersections, and an overview of our results. $\checkmark$ implies existence of 1EQEQ$1$ allocations, $\times$ implies non-existence, $?$ implies that the existence is an open question. For two agents, we show the existence and efficient computation of 1EQEQ$1$ allocations under general valuations when the grand bundle is valued nonnegatively (resp. nonpositively) (\ref{['thm:two_agents']}).