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Nearly Equitable Allocations beyond Additivity and Monotonicity
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Studies existence and efficient computation of EQx (equitability up to any item) allocations of indivisible goods and chores, giving existence for monotone valuations, polynomial algorithms for two agents with additive objective valuations, and hardness results for nonmonotone subjective valuations.
Problem
Fair division of indivisible items: whether EQx (equitability up to any item) allocations exist and can be computed efficiently beyond monotone additive valuations, including when items are goods and chores.
Approach
Extends the Add-and-Fix greedy procedure to monotone non-additive valuations (choosing poorest agent and greedy goods), proves existence via leximin++ for identical chores, gives a two-way greedy algorithm for two agents with additive objective valuations, and uses reductions from Partition/3-Partition for nonmonotone subjective cases.
Key results
- For monotone valuations, EQx allocations always exist and Greedy Add-and-Fix computes one in pseudo-polynomial time; weakly well-layered valuations admit polynomial-time computation.
- For two agents with additive objective valuations, a polynomial-time two-way greedy algorithm computes an EQx allocation, which was previously unknown.
- For nonmonotone subjective additive valuations, EQx existence is weakly NP-hard to decide for two agents and strongly NP-hard for an arbitrary number of agents.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| allow monotone valuations beyond additive valuations | EQx allocation existence · Indivisible items with monotone valuations | EQx existence | Helped |
| restrict monotone valuations to weakly well-layered valuations | Greedy Add-and-Fix · Indivisible goods; weakly well-layered valuations | computation time | Helped |
| use the approximate Greedy Add-and-Fix algorithm with ε > 0 | Monotone valuations · Indivisible goods; general monotone valuations | computation time | Helped |
| allow nonmonotone subjective additive valuations, including mixed items | EQx allocation existence · Two agents with normalized valuations | EQx existence | Hurt |
| use objective additive valuations with two agents | Two-Way Greedy Algorithm · Two agents; goods and chores | EQx existence and computation | Helped |
| require chores to have identical values across agents | Leximin++ allocation · Additive objective valuations | EQx existence | Helped |
| restrict objective additive valuations to a single chore | EQx allocation existence · Additive goods and one chore | EQx existence | Helped |
Surprising findings
- EQx allocations may fail to exist for two agents with nonmonotone additive subjective valuations, even with normalized valuations.
- Leximin++ allocations need not be EQx when there is a single non-identically valued chore.
Limitations
- Existence of EQx under general objective additive valuations with multiple agents and mixed chores/goods remains open
- Subjective nonmonotone case admits no EQx in general; hardness results
- Pseudo-polynomial algorithm for constant agents with subjective valuations only
Experimental setup
Builds on: Add-and-Fix leximin++ EFx for identical valuations
Closest papers by meaning
- Epistemic EFX Allocations Exist for Monotone Valuations (AAAI 2025 · similarity 0.93)
- Understanding EFX Allocations: Counting and Variants (AAAI 2025 · similarity 0.92)
- The Complexity of Fair Division of Indivisible Items with Externalities (AAAI 2024 · similarity 0.92)
- (Almost Full) EFX for Three (and More) Types of Agents (AAAI 2025 · similarity 0.91)
- On Optimal Tradeoffs between EFX and Nash Welfare (AAAI 2024 · similarity 0.91)
- The Complexity of Extending Fair Allocations of Indivisible Goods (AAAI 2025 · similarity 0.90)
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