PROP1: Fair Allocation of Indivisible Goods

• PROP1 is an approximate fairness concept that relaxes exact proportionality by allowing each agent to achieve a proportional share with the addition of one good.
• It guarantees the existence of fair allocations under additive valuations and serves as a baseline for stronger variants such as PROPm and PROPavg.
• Research on PROP1 investigates efficient computation, fairness hierarchies, and diverse applications in mixed, public, online, and ordinal allocation settings.

Proportionality up to one good (PROP1) is an approximate fairness notion for allocations of indivisible goods. In the standard private-goods model with agents $N=\{1,\dots,n\}$, goods $O$, and additive non-negative utilities $u_i$, exact proportionality requires $u_i(p(i)) \ge \frac{1}{n}u_i(O)$ for every agent $i$. PROP1 relaxes this by requiring that each agent can reach her proportional benchmark after the hypothetical addition of one good not currently in her bundle: $\forall i\in N\;\exists o\in O\setminus p(i)$ such that $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$. Exact proportionality may fail because goods are indivisible, whereas PROP1 allocations always exist under additive valuations (Aziz et al., 2020). Subsequent work has treated PROP1 as a central benchmark across mixed utilities and weights, public goods, completion and online models, query-based elicitation, and group allocation (Aziz et al., 2019, Garg et al., 2021, HV et al., 2024, Choo et al., 5 Aug 2025).

1. Standard definition and formal model

In the classical model, an allocation is a function $p:N\to 2^O$ such that bundles are disjoint and $\bigcup_{i\in N} p(i)=O$. Utilities are additive: $u_i(O')=\sum_{o\in O'}u_i(o)$ for every $O$0. Exact proportionality is the requirement

$O$1

Because exact proportionality can be impossible for indivisible goods, the standard relaxation is proportionality up to $O$2 items, denoted PROP$O$3: for each agent $O$4, there exists $O$5 with $O$6 such that

$O$7

PROP1 is the special case $O$8, and proportionality up to any item, PROPx, strengthens it by requiring the inequality to hold for every single item outside the bundle rather than for one suitable item (Aziz et al., 2020).

The same relaxation reappears in several adjacent models with adapted benchmarks. In the completion model, where a frozen partial allocation $O$9 is extended by allocating the unallocated goods, Prop1 is defined on the final allocation $u_i$0 by requiring that for each agent $u_i$1, there exists a good $u_i$2 such that $u_i$3 (HV et al., 2024). In public goods with a cardinality constraint $u_i$4, the proportional benchmark becomes

$u_i$5

and Prop1 is defined by a one-good swap rather than a one-good addition: for each agent $u_i$6, there exist $u_i$7 and $u_i$8 such that

$u_i$9

because the budget constraint forbids simply adding a good (Garg et al., 2021).

2. Position in the fairness hierarchy

Under additive valuations, the hierarchy recorded in the welfare-maximization literature is

$u_i(p(i)) \ge \frac{1}{n}u_i(O)$0

where EF is envy-freeness, EF1 is envy-freeness up to one item, and PROPx is proportionality up to any item (Aziz et al., 2020). PROP1 is therefore weaker than EF1. The separation is strict even for two agents: with one item $u_i(p(i)) \ge \frac{1}{n}u_i(O)$1 of value $u_i(p(i)) \ge \frac{1}{n}u_i(O)$2 and six items $u_i(p(i)) \ge \frac{1}{n}u_i(O)$3 of value $u_i(p(i)) \ge \frac{1}{n}u_i(O)$4 to both agents, the allocation in which Alice receives $u_i(p(i)) \ge \frac{1}{n}u_i(O)$5 and Bob receives the six $u_i(p(i)) \ge \frac{1}{n}u_i(O)$6's is PROP1 for Alice, since $u_i(p(i)) \ge \frac{1}{n}u_i(O)$7, but it is not EF1 because Bob’s bundle remains valued at $u_i(p(i)) \ge \frac{1}{n}u_i(O)$8 after removal of any single $u_i(p(i)) \ge \frac{1}{n}u_i(O)$9, still exceeding Alice’s $i$0 (Aziz et al., 2020).

A substantial part of the recent literature sharpens PROP1 by replacing the compensating good with more demanding surrogates. PROPm, proportionality up to the maximin item, was introduced as a middle-ground between PROP1 and PROPx (Baklanov et al., 2020). It was later shown that PROPm allocations exist for all instances and can be computed in polynomial time (Baklanov et al., 2021). PROPavg, proportionality up to the least valued good on average, was introduced as a stronger notion than PROPm and was likewise shown to always exist with a polynomial-time algorithm (Kobayashi et al., 2022). This sequence of refinements makes clear that PROP1 is not the endpoint of approximate proportionality, but rather the weakest universally available member of a broader family.

3. Existence, efficiency, and computation in the classical model

For additive valuations, PROP1 allocations are known to exist and to be computable in polynomial time (Aziz et al., 2020). The same paper studies the interaction between PROP1 and utilitarian social welfare $i$1. It defines two optimization problems: deciding whether some utilitarian-maximal allocation is also PROP1, and computing a utilitarian-maximal allocation among all PROP1 allocations. The decision problem ExistsUMandPROP1 is strongly NP-complete when the number of agents is part of the input, NP-complete for three agents, and polynomial-time solvable for two agents. The optimization problem ComputeUMwithinPROP1 is strongly NP-hard when the number of agents is unbounded, NP-hard already for three agents, and remains NP-hard even for two agents. For fixed $i$2 and integer valuations bounded by $i$3, however, a pseudopolynomial dynamic program computes a utilitarian-maximal PROP1 allocation in time $i$4 by tracking each agent’s accumulated utility together with the most valuable external item for that agent (Aziz et al., 2020).

PROP1 also remains compatible with strong efficiency in settings that are not limited to non-negative additive goods. With mixed utilities, where an item may be a good for one agent and a chore for another, and with asymmetric entitlements $i$5 summing to $i$6, weighted PROP1 requires that agent $i$7 either already gets at least $i$8, or can reach that benchmark by adding one good outside her bundle, or can reach it by removing one chore from her bundle. In that model, there always exists an integral allocation satisfying weighted PROP1 and fractional Pareto optimality, and a strongly polynomial-time algorithm computes one (Aziz et al., 2019).

Beyond additivity, the recent non-additive analysis shows that EF1 implies PROP1 for satiating submodular valuations, that Round-Robin computes a complete PROP1 allocation for monotone submodular valuations and a partial PROP1 allocation after the second-to-last round for satiating submodular valuations, that PROP1 allocations for satiating subadditive goods can be computed in polynomial time, and that maximum Nash welfare allocations are PROP1 for monotone submodular goods (Andersen et al., 17 Aug 2025). The same work also shows that these extensions are sharp: for monotone XOS goods, there is an EF1 allocation that is not PROP1, and Round-Robin may fail to be PROP1 (Andersen et al., 17 Aug 2025).

4. Completion, elicitation, and online allocation

In the completion problem, part of the allocation is frozen and only the remaining goods can be assigned. This turns PROP1 into a constrained extension problem rather than a from-scratch construction problem. For binary additive valuations, Prop1-COMPLETION is polynomial-time solvable by a flow-network method, and Prop1+PO-COMPLETION is polynomial-time as well. Under lexicographic valuations, every allocation is Prop1, so Prop1-COMPLETION is trivial, and Prop1+PO-COMPLETION is polynomial-time via sequencibility. In contrast, for general additive valuations, Prop1-COMPLETION is NP-complete even for two agents and even for three agents with identical additive valuations, while for two agents with identical additive valuations there is again a polynomial-time algorithm (HV et al., 2024).

When valuations are not directly available and the algorithm can only ask comparison queries of the form $i$9, PROP1 remains computable with logarithmic dependence on the number of goods. For identical additive valuations, a PROP1 allocation can be found in $\forall i\in N\;\exists o\in O\setminus p(i)$0 queries; for non-identical additive valuations, the query complexity is $\forall i\in N\;\exists o\in O\setminus p(i)$1. The same comparison model also supports computation of allocations satisfying both PROP1 and $\forall i\in N\;\exists o\in O\setminus p(i)$2-MMS within $\forall i\in N\;\exists o\in O\setminus p(i)$3 queries. A matching lower bound of $\forall i\in N\;\exists o\in O\setminus p(i)$4 holds even for a constant number of agents with identical valuations (Bu et al., 2024).

Online fair division changes the problem more radically because goods arrive sequentially and must be allocated immediately and irrevocably. In that setting, three natural greedy algorithms fail to guarantee any positive approximation to PROP1 against an adaptive adversary. Against a non-adaptive adversary, however, uniformly random allocation achieves an $\forall i\in N\;\exists o\in O\setminus p(i)$5-PROP1 guarantee with high probability for

$\forall i\in N\;\exists o\in O\setminus p(i)$6

With side-information in the form of perfect predictions of each agent’s maximum item value, an online algorithm achieves $\forall i\in N\;\exists o\in O\setminus p(i)$7-PROP1; with one-sided prediction error $\forall i\in N\;\exists o\in O\setminus p(i)$8, the guarantee becomes

$\forall i\in N\;\exists o\in O\setminus p(i)$9

The same work shows that stronger notions such as EF1, MMS, and PROPX remain inapproximable even with perfect maximum-item-value predictions (Choo et al., 5 Aug 2025).

5. Domain-specific variants

In indivisible public goods, an allocation is a subset $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$0 with $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$1, and the relevant proportional benchmark is agent-specific: $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$2 Because the budget constraint prevents adding one more public good, Prop1 is formulated as a single swap. In that model, all maximum Nash welfare allocations satisfy Prop1, Pareto optimality, and $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$3-RRS; all leximin-optimal allocations satisfy RRS, Prop1, and Pareto optimality; exact computation of MNW and leximin is NP-hard; and an $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$4-factor approximation algorithm for MNW also satisfies RRS, $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$5-Prop, and Prop1 for additive valuations (Garg et al., 2021).

For mixed divisible and indivisible goods, PROP1 appears as the extreme indivisible endpoint of a family of fractional relaxations indexed by the indivisibility ratio

$u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$6

The paper defines PROP$u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$7 by requiring that for each agent $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$8, there exists an indivisible good $u_i(p(i)\cup\{o\}) \ge \frac{1}{n}u_i(O)$9 such that

$p:N\to 2^O$0

When $p:N\to 2^O$1, this reduces to PROP1; when $p:N\to 2^O$2, it reduces to exact proportionality. PROP$p:N\to 2^O$3 allocations exist for any number of agents and are computable in polynomial time, and any maximum Nash welfare allocation satisfies PROP$p:N\to 2^O$4 (Li et al., 2024).

Group allocation introduces another variant. If goods are allocated to groups and each agent fully enjoys her group’s bundle, PROPk for an agent $p:N\to 2^O$5 means that adding at most $p:N\to 2^O$6 goods outside the group bundle suffices to reach $p:N\to 2^O$7. For groups of size at most $p:N\to 2^O$8, a PROPk allocation exists and can be found efficiently; more precisely, the algorithm guarantees PROP1 to the first agent in each group, PROP2 to the second, and so on. For couples, the same work identifies special cases in which full PROP1 exists for any number of couples (Gölz et al., 19 Aug 2025).

Ordinal and weighted settings admit yet another reformulation. Under ordinal preferences with unequal entitlements $p:N\to 2^O$9, the notion of weighted necessarily proportional up to one item, WSD-PROP1, requires the PROP1 condition to hold under every additive valuation consistent with each agent’s ranking. WSD-PROP1 allocations always exist for both goods and chores and can be computed in polynomial time via perfect matchings in a bipartite graph. The perfect matching polytope then allows optimization over all WSD-PROP1 allocations, including min-cost allocations of goods and most efficient allocations of chores. The same paper shows existence of sequencible WSD-PROP1 allocations and also shows incompatibility between Pareto optimality under all compatible valuations and WSD-PROP1 (V. et al., 2023).

6. Ordinal repeated allocation and the current boundary of the notion

Repeated assignment problems impose a temporal strengthening of PROP1: fairness is required not only at the end, but after every day. In the model of balanced sequences of permutations, there are $\bigcup_{i\in N} p(i)=O$0 players and $\bigcup_{i\in N} p(i)=O$1 items, each day assigns one item to each player, and fairness is interpreted ordinally, meaning that the cumulative allocation after day $\bigcup_{i\in N} p(i)=O$2 must satisfy PROP1 for every valuation consistent with the common ranking of items. A sufficient condition is weak balance: $\bigcup_{i\in N} p(i)=O$3 where $\bigcup_{i\in N} p(i)=O$4 is the $\bigcup_{i\in N} p(i)=O$5-th best item in player $\bigcup_{i\in N} p(i)=O$6’s cumulative bundle after day $\bigcup_{i\in N} p(i)=O$7. Weakly balanced sequences guarantee perpetual ordinal PROP1; such sequences exist for all $\bigcup_{i\in N} p(i)=O$8, while for $\bigcup_{i\in N} p(i)=O$9 with $u_i(O')=\sum_{o\in O'}u_i(o)$0 no weakly balanced sequence exists (Adams et al., 25 Feb 2026).

Across these models, a consistent picture emerges. PROP1 is weaker than EF1 and exact proportionality, and it is not interchangeable with them even in very small instances (Aziz et al., 2020). At the same time, it is substantially more robust than stronger proportional relaxations such as PROPx, which may fail to exist, and it remains meaningful in settings where exact fairness or stronger envy-based guarantees are computationally or structurally unattainable. The literature therefore treats PROP1 both as a baseline notion with strong existence properties and as a reference point for sharper variants such as PROPm, PROPavg, and weighted or ordinal formulations (Baklanov et al., 2021, Kobayashi et al., 2022).