Envy-Free up to One Item (EF1)

• EF1 is a fairness concept for indivisible item allocations that permits neutralizing envy by removing a single item from the envied bundle.
• It extends to various settings including goods, chores, mixed manna, budgets, and externalities by adapting deletion rules to the context.
• Algorithmic strategies such as sequential picking, envy-cycle elimination, and welfare maximization balance fairness and efficiency under EF1.

Searching arXiv for recent and foundational papers on EF1 and closely related variants. Envy-free up to one item (EF1) is a relaxation of envy-freeness for allocations of indivisible items. In the standard goods model, with agents $N$, items $M$, and valuations $v_i$, an allocation $\mathbf{A}=(A_i)_{i\in N}$ is EF1 if for every pair of agents $i,j$ with $A_j\neq \varnothing$, there exists some 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 may fail even in trivial indivisible instances, whereas EF1 often exists and supports a large theory spanning existence, computation, efficiency loss, dynamic restoration, and extensions to chores, mixed manna, budgets, graphs, and externalities (Kyropoulou et al., 13 Aug 2025, Aziz, 2020).

1. Formal definition and principal variants

In the standard additive setting, each agent $i$ has

$v_i(S)=\sum_{g\in S} v_i(g)$

for every bundle $M$0, and EF1 asks whether each envy relation can be neutralized by removing one item from the envied bundle. This is the canonical formulation for indivisible goods and remains the reference point for most algorithmic and welfare analyses (Kyropoulou et al., 13 Aug 2025).

Once items may be goods, chores, or mixed, the one-item relaxation is no longer unique. For arbitrary, possibly non-monotone utilities, a generalized EF1 condition requires that for every pair $M$1, either $M$2 does not envy $M$3, or there exists an item $M$4 such that

$M$5

This “either-side deletion” is necessary when removing an item from the envied bundle need not be the correct repair operation (Bérczi et al., 2020). In the goods-and-chores formulation used in neural allocation work, the same idea appears as: envy can be removed either by deleting a good from the envied bundle or by deleting a chore from the envying agent’s own bundle (Mishra et al., 2021).

Budget constraints induce another variant. If items have costs $M$6, agents have budgets $M$7, and only affordable subsets of another bundle are relevant, then $M$8-EF1 requires that for every pair $M$9 and every affordable $v_i$0, there exists an item $v_i$1 such that

$v_i$2

When $v_i$3, this is the budget-feasible EF1 notion (Wu et al., 2020). Closely related work with item sizes and a charity bundle adopts the same affordable-subbundle logic and compares agents not only to one another but also to unallocated items (Gan et al., 2021).

For mixed manna, standard EF1 is difficult to extend. An allocation $v_i$4 is called introspective envy free up to one item (IEF) if for each agent $v_i$5 there exists a set $v_i$6 with $v_i$7 such that

$v_i$8

Here the symmetric difference allows agent $v_i$9 either to add one item to or remove one item from its own bundle. This notion coincides with EF for indivisible chores, while for indivisible goods EF implies IEF (Barman et al., 23 Sep 2025).

With externalities, agents evaluate whole allocations rather than only their own bundles. One formulation defines envy via bundle swaps: agent $\mathbf{A}=(A_i)_{i\in N}$0 envies $\mathbf{A}=(A_i)_{i\in N}$1 if $\mathbf{A}=(A_i)_{i\in N}$2. The associated EF-$\mathbf{A}=(A_i)_{i\in N}$3 relaxation permits deletion of at most $\mathbf{A}=(A_i)_{i\in N}$4 items from the allocation before comparing $\mathbf{A}=(A_i)_{i\in N}$5 and $\mathbf{A}=(A_i)_{i\in N}$6, and standard EF1 is recovered when externalities vanish (Connor et al., 19 Jan 2026).

2. Existence and computation in the classical model

A strong baseline is available for two agents. For arbitrary utility functions, not necessarily monotone or additive, an EF1 allocation always exists for two agents and can be computed in polynomial time via a constructive prefix/suffix argument on an arbitrary item ordering (Bérczi et al., 2020). This result settles the first genuinely difficult non-monotone case and separates EF1 from stronger EFX-type notions, which can fail even for two agents with identical non-monotone utilities (Bérczi et al., 2020).

Connectivity constraints substantially complicate existence. When items form a path and bundles must be connected, EF1 exists for two, three, and four agents under monotone valuations; EF2 exists for any number of agents; and for identical valuations there is a polynomial-time algorithm computing a connected EF1 allocation for any number of agents (Bilò et al., 2018). For two agents on general graphs, the graphs that guarantee connected EF1 are exactly those whose block tree is a path, equivalently those admitting a bipolar numbering (Bilò et al., 2018).

Structured valuation classes extend the EF1 frontier beyond standard additive monotonicity. For identical trilean valuations, where every set has value in $\mathbf{A}=(A_i)_{i\in N}$7, 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 even with agent-specific thresholds (Bhaskar et al., 2024). These existence results are accompanied by explicit impossibility results for stronger EFX-type guarantees in the same classes (Bhaskar et al., 2024).

Several computational frameworks combine EF1 with stronger desiderata. In buyer utility functions, a greedy $\mathbf{A}=(A_i)_{i\in N}$8-time algorithm simultaneously maximizes utilitarian social welfare, achieves Pareto optimality, and returns an EF1 allocation; a slight modification yields EFX (Camacho et al., 2021). In random assignment, the probabilistic serial outcome can be implemented in polynomial time as a lottery over deterministic EF1 allocations, and the same holds for an SD-efficient EPS variant (Aziz, 2020). Thus EF1 is compatible not only with deterministic constructive procedures but also with ex-ante fairness mechanisms (Aziz, 2020).

3. Welfare loss and welfare optimization under EF1

The efficiency cost of imposing EF1 is commonly measured by the price of EF1, namely the worst-case ratio between maximum unconstrained social welfare and maximum social welfare among EF1 allocations. For additive ternary valuations, where each item has value $\mathbf{A}=(A_i)_{i\in N}$9, $i,j$0, or $i,j$1 for each agent with $i,j$2, the price of EF1 remains asymptotically large: for $i,j$3,

$i,j$4

Thus even a highly restricted ternary domain does not improve the asymptotic $i,j$5 behavior known for broader classes (Kyropoulou et al., 13 Aug 2025).

For few agents, the picture is sharper. With two agents and additive ternary valuations, the exact price of EF1 is

$i,j$6

With three agents, the paper establishes

$i,j$7

The two-agent upper bound is achieved by a modified round-robin procedure, M2RR, and the three-agent upper bound by Repeated-Max-Matching (RMM) with non-wasteful tie-breaking (Kyropoulou et al., 13 Aug 2025).

Optimizing welfare subject to EF1 is itself computationally difficult. In the additive goods/chores/mixed setting, maximizing utilitarian social welfare among EF1 allocations is NP-hard, even though unconstrained maximum utilitarian welfare is easy and EF1 allocations themselves are easy to find (Mishra et al., 2021). This hardness motivates approximation algorithms in restricted domains. For two utility types, there is a $i,j$8-approximation for normalized utilities for any number of agents, a tight $i,j$9-approximation for three normalized agents, and a tight $A_j\neq \varnothing$0-approximation for three unnormalized agents (Ma et al., 11 Sep 2025).

The envy-cycle-elimination (ECE) paradigm illustrates how welfare depends on the choice of EF1 algorithm rather than only on the existence theorem. Vanilla ECE has strong utilitarian price $A_j\neq \varnothing$1 and infinite strong egalitarian price. Jointly optimizing both the selected good and the receiving agent substantially improves the strong utilitarian bound, whereas optimizing only one of these two choices does not yield comparable gains (Celine et al., 1 Jun 2026).

4. Budgets, chores, mixed manna, and externalities

Budget feasibility degrades exact EF1 guarantees but preserves approximate analogues. In the model with costs, budgets, and a charity agent for unallocated goods, a feasible allocation maximizing Nash social welfare is always $A_j\neq \varnothing$2-EF1 and Pareto optimal, and the $A_j\neq \varnothing$3 factor is tight (Wu et al., 2020). When item costs are small relative to budgets, the guarantee improves toward $A_j\neq \varnothing$4; specifically, if

$A_j\neq \varnothing$5

then Max-NSW is

$A_j\neq \varnothing$6

and this asymptotic $A_j\neq \varnothing$7 limit is itself tight (Wu et al., 2020).

Under identical additive valuations with sizes and budgets, stronger algorithmic results are available. A polynomial-time algorithm computes a $A_j\neq \varnothing$8-EF1 allocation in general, an exact EF1 allocation when budgets are uniform, and an exact EF1 allocation for two agents. In the large-budget regime the same algorithm achieves $A_j\neq \varnothing$9-EF1, while an NSW-maximizing allocation is $g\in A_j$0-EF1 (Gan et al., 2021).

For mixed manna, the existence of PO and EF1 allocations remains open, but there is a full existence theorem for the introspective variant: an IEF and PO allocation always exists for any mixed-manna instance (Barman et al., 23 Sep 2025). This generalizes the known chores existence result because IEF coincides with EF for indivisible chores (Barman et al., 23 Sep 2025).

Externalities fundamentally weaken the standard benchmark. One line of work extends EF1 to additive externalities and proves that EF1 always exists for two agents, and for three agents under binary valuations with a no-chore assumption (Aziz et al., 2021). A later asymptotic result resolves the general question negatively: EF1 does not always exist with externalities, and the correct universal guarantee is EF-$g\in A_j$1. More precisely, every instance admits an EF-$g\in A_j$2 allocation in deterministic polynomial time, and there are binary no-chore instances requiring $g\in A_j$3 deletions (Connor et al., 19 Jan 2026).

5. Algorithmic paradigms and learned allocation rules

EF1 computation has developed through several distinct paradigms. Sequential picking remains a core mechanism. Standard round robin already suffices for additive valuations in the prioritized-agent model, where the allocation must be EF1 globally and additionally envy-free from prioritized agents to non-prioritized agents; placing prioritized agents before the others in each round yields the stronger property (Bu et al., 2022). More specialized sequential algorithms include M2RR for two-agent ternary welfare guarantees and divide-and-choose-style algorithms for exact budget-feasible EF1 with two agents (Kyropoulou et al., 13 Aug 2025, Gan et al., 2021).

Envy-graph methods supply another major family. ECE iteratively allocates a good to an unenvied agent and eliminates any envy cycle, preserving EF1 regardless of how the next good and recipient are chosen (Celine et al., 1 Jun 2026). In prioritized fair division, envy-graph update rules $g\in A_j$4 maintain EF1 while preserving the no-envy requirement from prioritized to non-prioritized agents, leading to partial allocations with a small unallocated pool for general valuations (Bu et al., 2022).

Market and welfare-based procedures give EF1 indirectly through optimization. NSW maximization yields exact EF1 in the unconstrained indivisible-goods setting as recalled in later budget-feasible work, and approximate EF1 under budget constraints (Wu et al., 2020). In buyer utility functions, utilitarian maximization and Pareto optimality coincide, enabling a direct greedy algorithm for EF1+PO+MSW rather than an NSW computation (Camacho et al., 2021). For ex-ante fair random allocation, Birkhoff decompositions of PS or EPS outcomes produce lotteries over recursively balanced sequential allocations, each of which is EF1 (Aziz, 2020).

Learning-based approaches treat EF1-constrained welfare optimization as a prediction problem. EEF1-NN is a fully convolutional, U-Net-inspired architecture trained with a Lagrangian loss combining utilitarian welfare and EF1-style envy penalties. The method targets allocations that are EF1 and approximately maximum-USW, and empirically achieves $g\in A_j$5 and $g\in A_j$6 for several distributions when $g\in A_j$7, while being much faster at inference than solving the constrained optimization problem directly (Mishra et al., 2021).

6. Structural limitations, robustness, and interpretation

Although EF1 is weaker than envy-freeness, it is far from fragile. For two agents with arbitrary monotonic valuations over $g\in A_j$8 goods, the number of EF1 allocations is always exponential in $g\in A_j$9. The tight lower bound is

$v_i(A_i)\ge v_i(A_j\setminus\{g\})$0

for even $v_i(A_i)\ge v_i(A_j\setminus\{g\})$1, and

$v_i(A_i)\ge v_i(A_j\setminus\{g\})$2

for odd $v_i(A_i)\ge v_i(A_j\setminus\{g\})$3 (Suksompong, 2020). This contrasts sharply with EFX, for which as few as two allocations may exist (Suksompong, 2020).

At the same time, EF1 is only a pairwise notion. The information-withholding framework formalizes this by asking how many goods must be hidden globally to make an allocation envy-free. An allocation is $v_i(A_i)\ge v_i(A_j\setminus\{g\})$4 if there exists a set $v_i(A_i)\ge v_i(A_j\setminus\{g\})$5 of at most $v_i(A_i)\ge v_i(A_j\setminus\{g\})$6 goods such that for every pair $v_i(A_i)\ge v_i(A_j\setminus\{g\})$7,

$v_i(A_i)\ge v_i(A_j\setminus\{g\})$8

Here $v_i(A_i)\ge v_i(A_j\setminus\{g\})$9 is exactly envy-freeness, while $i$0. Deciding whether a $i$1 allocation exists is NP-complete for every fixed $i$2, even for identical valuations (Hosseini et al., 2019).

Connectivity requirements create additional structural obstructions. In graph fair division with connected bundles, cutsets of gap $i$3 and suitable valence block connected EF1 guarantees for specific agent counts; if a graph guarantees connected EF1 for $i$4 agents under common additive or common monotone valuations, then it contains no cutset in the corresponding forbidden interval (Chen et al., 2024). Yet cutset-freeness is not sufficient: the paper gives an eight-vertex graph with no cutsets of gap at least $i$5 that still fails to guarantee connected EF1 for three agents under common additive valuations (Chen et al., 2024).

Finally, EF1 is not automatically the right fairness metric outside resource-allocation settings. In personalized recommendation, a position paper argues that envy and EF-1 often track preference mismatch rather than unfair treatment: a recommendation system can be envy-free and still unfair, or highly envious and yet equalize user utility. The critique is directed especially at consumer-side personalization, where heterogeneous preferences are definitional rather than pathological (Aird et al., 10 Sep 2025). This suggests that EF1 is best interpreted as a fairness criterion for rival or allocative environments, not as a universal proxy for fairness across all personalized decision systems (Aird et al., 10 Sep 2025).