PROP1: Fairness in Indivisible Goods Allocation

• PROP1 is a fairness concept that ensures each agent nearly achieves a proportional share by allowing the hypothetical addition (or removal) of one good.
• It bridges traditional proportionality and EF1, extending fairness to weighted, ordinal, and mixed goods settings with robust computational guarantees.
• Algorithms like Round-Robin, Envy-Cycle Elimination, and bag-filling provide efficient PROP1 allocations, though maximizing welfare under PROP1 is NP-hard.

Proportionality Up to One Good (PROP1) is a central fairness relaxation in the allocation of indivisible goods (and, more generally, goods and chores) among agents with potentially diverse preferences. Born from the recognition that exact proportionality is often unattainable with indivisibilities, PROP1 weakens the classic proportionality benchmark by permitting the hypothetical addition or removal of a single item. This concept bridges the tractable and the infeasible in fair division, aligning closely with related relaxations such as envy-freeness up to one good (EF1), possesses robust polynomial-time computability in many cases, and extends—albeit with care—to ordinal, weighted, public, and mixed (divisible + indivisible) settings.

1. Formal Definition and Core Properties

Let $M$ be a set of $m$ indivisible goods, $N = [n]$ the set of $n$ agents, and $v_i:2^M \to \mathbb{R}_{\geq 0}$ agent $i$'s valuation. An integral allocation $A = (A_1, \dots, A_n)$—a partition of $M$—is said to satisfy proportionality up to one good (PROP1) if for every agent $i \in N$, there exists $g \in M \setminus A_i$ such that

$m$0

This condition ensures that even if direct proportionality ($m$1) fails, the shortfall can be compensated by the hypothetical addition of one remaining good. For chores (disutilities), or public goods settings, analogous definitions apply via bundle removal or one-swap arguments (Aziz et al., 2020, V. et al., 2023, Garg et al., 2021).

In weighted settings, if each agent $m$2 has entitlement $m$3 (with $m$4), the PROP1 requirement becomes

$m$5

(V. et al., 2023, Aziz et al., 2019). For mixed-goods extensions (divisible + indivisible), PROP1 is smoothly interpolated through the indivisibility ratio $m$6, and proportionality is required up to an $m$7-fraction of a missing indivisible good (Li et al., 2024).

PROP1 sits in a strict hierarchy of fairness relaxations: $m$8 where EF1 implies PROP1 in additive and submodular domains, but the reverse implication does not hold in general (Aziz et al., 2020, Andersen et al., 17 Aug 2025).

2. Existence and Algorithmic Guarantees

A crucial property of PROP1 is universal existence—every fair division instance with indivisible goods and additive valuations admits such an allocation. This existence follows from generalizations such as the PROP$m$9 theorem: for every $N = [n]$0, there exists an allocation where each agent can achieve their proportional share up to $N = [n]$1 excluded goods. For $N = [n]$2, this is precisely PROP1. The constructive argument proceeds via recursive divider algorithms and dynamic decompositions (Baklanov et al., 2021).

Setting	Existence	Polynomial-time algorithm?	Reference
Additive	Always	Yes (Round-Robin, ECE, Bag-filling)	(Andersen et al., 17 Aug 2025, Baklanov et al., 2021)
Monotone submodular	Always	Yes (ECE, Round-Robin, local moves)	(Andersen et al., 17 Aug 2025)
Satiating subadditive	Always	Yes (local-improvement + ECE)	(Andersen et al., 17 Aug 2025)
Weighted, ordinal	Always	Yes (reduction to perfect matching)	(V. et al., 2023)
Public goods	Always (MNW)	Pseudo-polytime (MNW); $N = [n]$3 approx	(Garg et al., 2021)

Pseudocode for the classic algorithm in additive valuations: $n$9

Notably, in comparison-based models (agents respond only to bundle comparisons), a PROP1 allocation can be computed in $N = [n]$4 queries when the number of agents is constant. The algorithm achieves simultaneous PROP1 and $N = [n]$5-MMS guarantees (Bu et al., 2024).

3. Computational Complexity and Extensions

While basic PROP1 allocations admit efficient computation, several extensions introduce computational hardness:

• Welfare Maximization: Finding a PROP1 allocation that also maximizes utilitarian welfare is strongly NP-hard when $N = [n]$6 is part of the input, and remains NP-hard for fixed $N = [n]$7 (Aziz et al., 2020).
• PROP1 in Completion Problems: Given a partial ("frozen") allocation, deciding whether a PROP1 completion exists is polynomial-time solvable for binary/lexicographic valuations, but NP-complete for unrestricted additive valuations with just two or three agents (HV et al., 2024).
• Online Models: Greedy online allocation fails to guarantee even a constant approximation to PROP1, but uniformly random allocation achieves an $N = [n]$8-PROP1 guarantee against nonadaptive adversaries; minimal side information (maximum item value) enables a deterministic $N = [n]$9-PROP1 (Choo et al., 5 Aug 2025).
• Mixed Goods: For settings with both divisible and indivisible goods, the notion extends to proportionality up to an $n$0-fraction of a single indivisible good; sharp upper and lower bounds on $n$1 ensure tightness (Li et al., 2024).

4. Relationship with Envy-Freeness, Pareto Optimality, and Public Goods

• Relation to EF1: For additive/more generally submodular valuations, every EF1 allocation is PROP1, but not vice versa. This relationship breaks down for valuations beyond submodular (e.g., monotone XOS), where there exist EF1 allocations that do not satisfy PROP1 and vice versa (Andersen et al., 17 Aug 2025).
• Compatibility with Efficiency: PROP1 is compatible with Pareto optimality (PO), and, under monotone submodular/additive utilities, maximum Nash welfare (MNW) allocations are both PROP1 and PO. This does not extend seamlessly to EF1 (Garg et al., 2021, Aziz et al., 2019).
• Public Goods: In the selection of $n$2 out of $n$3 public goods, MNW allocations guarantee PROP1 for all agents. However, PROP1 does not guarantee classical proportionality or round-robin share for public goods (Garg et al., 2021).

Examples illustrate that PROP1 balances tractability and fairness; it is sometimes the strongest efficiency-compatible fairness guarantee attainable (Aziz et al., 2019, Garg et al., 2021).

5. Ordinal, Weighted, and Perpetual Fairness Variants

PROP1 extends to ordinal and weighted settings by stipulating that the fairness guarantee must hold under all consistent additive representations or for agent-specific entitlements:

• Ordinal (SD-PROP1, WSD-PROP1): Given strict orderings, an allocation is PROP1 under stochastic dominance if PROP1 holds for every consistent additive utility. Existence and computation reduce to perfect matching in bipartite graphs, and the set of such allocations coincides with perfect matching polytope vertices (V. et al., 2023).
• Weighted: When agents have weights or entitlements (e.g., $n$4), a weighted PROP1 allocation exists and can be computed in polynomial time (Aziz et al., 2019, V. et al., 2023).
• Perpetual/Online Fairness: In multi-round settings, it is possible to define and achieve sequential PROP1 (i.e., after every round), provided certain combinatorial balance conditions are met, but for large $n$5 there are impossibility barriers (Adams et al., 25 Feb 2026). In online allocation with irrevocable assignments, minimal predictions suffice for deterministic PROP1, while EF1 and MMS remain unachievable (Choo et al., 5 Aug 2025).

6. Methodological Frameworks and Practical Algorithms

PROP1 allocations in additive domains are efficiently computed by diverse mechanisms:

• Round-Robin: Agents pick goods in turn; always outputs PROP1 (and EF1) for additive, and PROP1 for monotone submodular valuations (Andersen et al., 17 Aug 2025).
• Envy-Cycle Elimination (ECE): Processes envy-graphs and cycles as goods are allocated, guaranteeing EF1 and thus PROP1 for monotone submodular valuations (Andersen et al., 17 Aug 2025).
• Bag-filling and Divider Algorithms: Used to establish universal PROP1 existence and to construct allocations recursively, crucially in the PROPm framework (Baklanov et al., 2021).
• Dynamic Programming: For welfare-maximizing PROP1 with small $n$6, pseudo-polynomial dynamic programming is employed (Aziz et al., 2020).
• Matching and Polyhedral Optimization: Weighted and ordinal PROP1 reduces to finding perfect matchings in allocation graphs, making use of integrality of the matching polytope (V. et al., 2023).

Each of these approaches is rigorously analyzed for runtime, correctness, and invariant maintenance, ensuring that the properties of PROP1 are preserved throughout the allocation process.

7. Limits, Variants, and Open Questions

While PROP1 is widely attainable, its boundaries are sharply delineated by hardness and impossibility results:

• Stronger notions such as EF1+PO, or (unrelaxed) proportionality, do not always coexist with PO or may not be computationally feasible in the indivisible domain (Aziz et al., 2019, Garg et al., 2021, Aziz et al., 2020).
• In mixed (divisible + indivisible) goods, PROP1 interpolates gracefully through the indivisibility ratio; attempts to strengthen the guarantee beyond this context encounter hard impossibility thresholds (Li et al., 2024).
• Online and streaming models reveal that PROP1 is the precise fairness notion allowing nontrivial (and robust) guarantees even under minimal information or adversarial conditions (Choo et al., 5 Aug 2025).
• Perpetual PROP1 under strong ordinal models can be constructed via "weakly balanced" permutations for $n$7, but not for arbitrary $n$8; perpetual PROP2 remains open (Adams et al., 25 Feb 2026).

Open research directions include extending comparison-based efficient algorithms for PROP1 + stronger guarantees to monotone non-additive domains, characterizing the price of fairness for PROP1 in new settings, and refining the understanding of PROP1’s compatibility with other fairness and efficiency benchmarks.

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References:

• "Fair Division of Indivisible Goods with Comparison-Based Queries" (Bu et al., 2024)
• "Computing Welfare-Maximizing Fair Allocations of Indivisible Goods" (Aziz et al., 2020)
• "PROPm Allocations of Indivisible Goods to Multiple Agents" (Baklanov et al., 2021)
• "Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings" (V. et al., 2023)
• "A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation" (Aziz et al., 2019)
• "Fair and Efficient Completion of Indivisible Goods" (HV et al., 2024)
• "Allocating Mixed Goods with Customized Fairness and Indivisibility Ratio" (Li et al., 2024)
• "Computing Approximately Proportional Allocations of Indivisible Goods: Beyond Additive and Monotone Valuations" (Andersen et al., 17 Aug 2025)
• "Perpetually Fair Assignments Via Balanced Sequences of Permutations" (Adams et al., 25 Feb 2026)
• "Approximate Proportionality in Online Fair Division" (Choo et al., 5 Aug 2025)
• "On Fair and Efficient Allocations of Indivisible Public Goods" (Garg et al., 2021)