EF1: Envy-Freeness Up to One Item

Updated 10 July 2026

EF1 is a fairness concept that relaxes exact envy-freeness, ensuring any envy can be neutralized by removing one item from the envied bundle.
It applies to various domains including goods, chores, mixed items, and externality-laden settings, and is supported by algorithms like envy cycle elimination.
Although EF1 guarantees broad existence and tractability, it can entail significant welfare trade-offs and requires adaptations in nonmonotone or personalized models.

Envy-freeness up to one item, usually abbreviated EF1 and sometimes written EF-1, is a relaxation of envy-freeness for the allocation of indivisible items. In the standard goods model, an allocation $A=(A_1,\ldots,A_n)$ is EF1 if for every pair of agents $i,j$ with $A_j\neq \emptyset$ , there exists an item $g\in A_j$ such that $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ . EF1 is central because exact envy-freeness often fails once items are indivisible, whereas EF1 admits broad existence, strong algorithmic support, and a large body of refinements, separations, and domain-specific extensions (Farhadi et al., 2020).

1. Standard formalization

The canonical setting consists of a set of agents $[n]$ , a set of indivisible items $[m]$ , and valuation functions $v_i:2^{[m]}\to \mathbb{R}_{\ge 0}$ , often assumed additive: $v_i(S)=\sum_{g\in S} v_i(g).$ An allocation $A=(A_1,\dots,A_n)$ partitions the item set. Exact envy-freeness requires

$i,j$ 0

EF1 weakens this to

$i,j$ 1

Under monotonicity, this is equivalent to the formulation

$i,j$ 2

which makes the “up to one item” relaxation explicit (Farhadi et al., 2020).

EF1 is pairwise and asymmetric. For each ordered pair $i,j$ 3, the witnessing item may differ. That feature is mathematically mild but conceptually important: EF1 does not claim that there is a single globally removable item for each envied bundle, only that every envy relation can be neutralized by some one-item deletion. This local character underlies both its tractability and many later refinements (Hosseini et al., 2019).

The two-agent case admits a particularly simple complement formulation. If $i,j$ 4 is one bundle and $i,j$ 5 the other, then $i,j$ 6 is EF1 for agent $i,j$ 7 iff either $i,j$ 8, or there exists $i,j$ 9 such that

$A_j\neq \emptyset$ 0

This viewpoint is useful in counting and geometric arguments on the hypercube of subsets (Suksompong, 2020).

2. Position in the fairness hierarchy

In the standard goods setting, EF1 sits strictly below envy-freeness up to any item (EFX), which requires envy to disappear after removing any item from the envied bundle. The standard implication chain is

$A_j\neq \emptyset$ 1

and recent work places envy-freeness up to a random good (EFR) strictly between EFX and EF1: $A_j\neq \emptyset$ 2 where EFR requires

$A_j\neq \emptyset$ 3

Here $A_j\neq \emptyset$ 4 is the uniform distribution over $A_j\neq \emptyset$ 5 (Farhadi et al., 2020).

A different strengthening is strong EF1 (sEF1), where for each envied agent $A_j\neq \emptyset$ 6, the same item $A_j\neq \emptyset$ 7 must work against all other agents. This notion is equivalent to uniformly hiding at most one good from each bundle in the “hidden goods” framework. More generally, HEF- $A_j\neq \emptyset$ 8 asks for a single global hidden set $A_j\neq \emptyset$ 9 of at most $g\in A_j$ 0 goods such that

$g\in A_j$ 1

This reframes EF1’s pairwise removals as a global consistency problem and makes precise why standard EF1 can be weak in aggregate (Hosseini et al., 2019).

Several domain-specific one-item variants modify what is compared. For mixed manna, introspective envy-freeness up to one item (IEF1) requires each agent to remove or add at most one item to her own bundle: $g\in A_j$ 2 IEF1 coincides with standard EF1 for chores and is implied by EF1 for goods (Barman et al., 23 Sep 2025). In collaborative settings with endogenous entitlements, average envy-freeness up to one item (AEF-1) replaces total bundle values by average bundle values: $g\in A_j$ 3 for some $g\in A_j$ 4 (Han et al., 2023).

3. Existence and computation in classical goods models

A foundational fact is that EF1 always exists for agents with subadditive valuations, and in the additive setting it can be obtained by simple procedures such as round robin; envy-graph methods provide polynomial-time guarantees in broader classes (Farhadi et al., 2020). In random assignment, the probabilistic serial outcome can be implemented as a lottery over deterministic EF1 allocations in polynomial time, reconciling ex-ante envy-freeness with ex-post EF1 fairness (Aziz, 2020).

A central constructive paradigm is the envy cycle elimination (ECE) algorithm: repeatedly allocate an unassigned good to an unenvied agent and then eliminate any envy cycle by rotating bundles. ECE always preserves EF1. Recent work shows that the freedom in choosing both the next good and the unenvied recipient has substantial welfare consequences: vanilla ECE has strong utilitarian price $g\in A_j$ 5, whereas jointly choosing the good–agent pair that maximizes utility improves the guarantee to $g\in A_j$ 6 while still preserving EF1 (Celine et al., 1 Jun 2026).

Connectivity constraints sharply change the picture. When items form a path and each agent must receive a connected bundle, EF1 is guaranteed for $g\in A_j$ 7 agents with arbitrary monotone valuations; for any number of agents on a path the paper establishes EF2 rather than EF1; and for identical valuations it gives a polynomial-time EF1 algorithm for all $g\in A_j$ 8 (Bilò et al., 2018). For two agents on a general graph, EF1 is guaranteed exactly when the graph’s biconnected components are arranged in a path, equivalently when the block tree is a path (Bilò et al., 2018).

Budget constraints create another nonstandard model. In budget-feasible allocation with identical additive valuations, fairness is defined against any subbundle $g\in A_j$ 9 of another agent’s or the charity’s bundle satisfying the envying agent’s budget bound. In that model, a polynomial-time $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 0-EF1 allocation exists in general, exact EF1 is computable in polynomial time for uniform budgets and for two agents, and in the large-budget regime the guarantee improves to $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 1-EF1 for the constructive algorithm (Gan et al., 2021).

4. Beyond monotone goods

Once valuations become nonmonotone, the standard “remove one item from the envied bundle” formulation ceases to be canonical. A natural generalization allows the witness item to come from either bundle: $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 2 Under this definition, EF1 always exists for two agents with arbitrary utility functions, even when utilities are neither monotone nor additive, and such an allocation can be found in polynomial time (Bérczi et al., 2020).

Additional structured nonmonotone domains also support EF1. For identical trilean valuations, where every bundle value lies in $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 3, EF1 exists for any number of agents. For separable single-peaked valuations, EF1 exists for any number of agents when each type has a common threshold across agents, and for three agents under arbitrary thresholds. In both valuation classes, stronger EFX-type guarantees fail to exist (Bhaskar et al., 2024).

For chores and mixed items, the one-item relaxation must often be interpreted through the envying agent’s own bundle. In the chore analogue, envy can be removed by deleting one chore from the envying agent’s bundle. For mixed manna, the existence of Pareto-optimal EF1 remains open, but every instance admits a Pareto-optimal IEF1 allocation; the result is exact for chores and weaker than EF1 for goods (Barman et al., 23 Sep 2025).

Externalities require a deeper reformulation. When agent $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 4’s utility depends on who receives every item,

$v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 5

envy is defined via bundle swaps rather than bundle comparison: $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 6 The externalities-aware EF1 notion deletes at most one item from the whole allocation and then compares the original and swapped outcomes. This extension coincides with standard EF1 when externalities vanish, yields an $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 7-time two-agent algorithm, and gives a polynomial-time three-agent existence theorem under binary valuations and the no-chore assumption (Aziz et al., 2021). Subsequent work settled the general existence question negatively: EF1 does not always exist with externalities, and the optimal universal relaxation is EF- $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 8, with a polynomial-time EF- $v_i(A_i)\ge v_i(A_j\setminus\{g\})$ 9 algorithm and a matching $[n]$ 0 lower bound even for binary no-chore valuations (Connor et al., 19 Jan 2026).

5. Quantitative and efficiency perspectives

For two agents with arbitrary monotone valuations, EF1 allocations are not merely guaranteed to exist; they are combinatorially abundant. The number of EF1 allocations is always at least

$[n]$ 1

and

$[n]$ 2

and these bounds are tight. By contrast, the number of EFX allocations can be as small as $[n]$ 3, also tightly. This abundance-versus-fragility gap gives a combinatorial explanation for the practical robustness of EF1 relative to EFX (Suksompong, 2020).

The welfare cost of imposing EF1 can still be substantial. For additive ternary valuations, the price of EF1 is $[n]$ 4 for large $[n]$ 5, matching the order known for general subadditive valuations. For two agents, the price is exactly $[n]$ 6; for three agents it lies between $[n]$ 7 and $[n]$ 8 (Kyropoulou et al., 13 Aug 2025).

Optimizing welfare subject to EF1 becomes easier under type structure. When agents have only two utility types, maximizing utilitarian social welfare over EF1 allocations admits a $[n]$ 9-approximation for normalized utilities, improving the earlier $[m]$ 0 guarantee for general normalized utilities. For $[m]$ 1, the ratio sharpens to a tight $[m]$ 2 in the normalized case and a tight $[m]$ 3 in the unnormalized case (Ma et al., 11 Sep 2025).

These quantitative results suggest a stable pattern. EF1 is usually easy to satisfy as a feasibility notion, often flexible in the space of allocations, yet potentially expensive when combined with secondary objectives such as utilitarian welfare, connectivity, or externalities. A plausible implication is that EF1’s algorithmic tractability and its efficiency loss must be analyzed separately rather than inferred from one another.

Because EF1 is inherently pairwise, several later notions aim to control its global or dynamic behavior. The hidden-goods framework replaces pairwise hypothetical deletions by a single global hidden set. In that language, sEF1 is equivalent to uniformly hiding at most one good per bundle, and every additive instance admits a $[m]$ 4 allocation in polynomial time. Finding the minimum hidden set is computationally hard, but empirical results show that standard EF1 algorithms often require hiding no more than three goods on synthetic and Spliddit data (Hosseini et al., 2019).

EF1 also supports asymmetric strengthening. EFPrior requires both global EF1 and exact envy-freeness from a prioritized subset $[m]$ 5 toward the remaining agents $[m]$ 6. With additive valuations, round robin with all agents in $[m]$ 7 ordered before those in $[m]$ 8 computes a complete EFPrior allocation; for general monotone valuations, a polynomial-time algorithm computes a partial EFPrior allocation leaving fewer than $[m]$ 9 items unallocated (Bu et al., 2022).

A dynamic viewpoint appears in EF1-Restoration. Starting from an EF1 allocation disrupted by the loss of an item or the arrival of a new agent, the goal is to recover EF1 via valid transfers while passing only through near-EF1 states. For identical monotone goods and chores, restoration is always possible; for identical additive valuations, the number of transfers is optimally bounded by $v_i:2^{[m]}\to \mathbb{R}_{\ge 0}$ 0; for additive binary graphical valuations, an optimal restoration algorithm is given; and for monotone binary valuations, deciding restorability is PSPACE-complete (Bhaskar et al., 2024).

Random assignment contributes another refinement: the probabilistic serial allocation can be decomposed into a lottery over deterministic EF1 allocations, so ex-ante envy-freeness and ex-post EF1 can be achieved simultaneously in polynomial time (Aziz, 2020).

Outside classical fair division, EF1 has also been criticized. In personalized recommendation, the core objection is that envy tracks cross-bundle desirability under heterogeneous preferences rather than fairness of treatment. The position paper “Envy-Free but Still Unfair: Envy-Freeness Up To One Item (EF-1) in Personalized Recommendation” argues that there is no inherent relationship between fairness, utility, and envy in personalized systems, so allocations may be envy-free or EF1 yet still be unfair to protected groups, or highly envious despite equal achieved utility (Aird et al., 10 Sep 2025).

EF1 therefore occupies a distinctive position in the theory of fair division. It is the most durable one-item relaxation of envy-freeness across indivisible goods, nonmonotone domains, and several constrained environments; it is often algorithmically accessible where stronger notions fail; but its meaning depends strongly on the model, and in settings such as externalities or personalization its standard interpretation can require substantial reformulation or may cease to be the right fairness proxy altogether.