Envy-Free Up to One Item (EF1) Fairness

Updated 11 November 2025

EF1 is a fairness concept for indivisible items where any envy between agents is eliminated by removing a single item from the envied bundle.
It guarantees the existence of allocations under additive utilities and supports polynomial-time algorithms like cut-and-choose that balance fairness and efficiency.
EF1 plays a foundational role in fair division theory by offering robust, flexible guarantees even in dynamic or perturbed allocation settings.

Envy-Free Up to One Item (EF1) is a central relaxation of envy-freeness in the allocation of indivisible items. In contrast to strict envy-freeness—which is unattainable in the majority of discrete-resource scenarios—EF1 admits robust, algorithmically tractable, and broadly-applicable guarantees. The concept is defined such that any agent’s pairwise envy towards another can always be eliminated by hypothetically removing a single item from the envied bundle. This relaxation has become a foundational fairness notion in economics, computer science, and operations research, and it underpins both classical fair-division algorithms and a wide range of recent generalizations. The following sections present the rigorous mathematical properties of EF1, its combinatorial and structural implications, algorithmic methods, generalizations for non-additive and mixed settings, connections to efficiency, and its limitations as a fairness guarantee.

1. Formal Definition and Structural Properties

EF1 is defined for a finite ground set $M$ of $m$ indivisible items and %%%%2%%%% agents with monotonic valuations $u_i:2^M\rightarrow \mathbb{R}_{\ge 0}$ (assuming $u_i(\emptyset)=0$ ).

An allocation is a partition $(A_1, \ldots, A_n)$ of $M$ , with $A_i$ assigned to agent $i$ . The allocation is called envy-free up to one item (EF1) if, for every pair of agents $i, j$ , with $A_j \neq \emptyset$ , there exists an item $g \in A_j$ such that: $u_i(A_i) \geq u_i(A_j \setminus \{g\})$ In words, even if $i$ envies $j$ , removing some single item from $j$ 's bundle suffices to eliminate $i$ 's envy.

For $n=2$ agents with arbitrary monotonic valuations, the definition can equivalently be stated on ordered partitions $(M_1, M_2)$ : $M_1$ is EF1 for agent 1 if $u_1(M_1) \geq u_1(M_2)$ , or there exists $j \in M_2$ such that $u_1(M_1) \geq u_1(M_2 \setminus \{j\})$ , and vice versa.

EF1 always exists for additive or monotonic utility functions (see below for nuanced cases).

A salient property—demonstrated in (Suksompong, 2020)—is that the number of EF1 allocations is always exponential in the number of items $m$ for two agents, in contrast to stronger relaxations like EFX, for which only $2$ such allocations may exist. This combinatorial richness underpins the robustness of EF1.

2. Existence, Counting, and Robustness Results

EF1 is strictly weaker than envy-freeness (EF), which is unachievable with indivisible items except in degenerate cases. In the two-agent case, Suksompong establishes that the number of EF1 allocations is bounded below as: $f_{\mathrm{EF1}}(m) = \begin{cases} \binom{m}{m/2} & \text{if %%%%24%%%% even} \ 2 \cdot \binom{m-1}{(m-1)/2} & \text{if %%%%25%%%% odd} \end{cases}$ with tightness for uniform additive valuations. Asymptotically, $f_{\mathrm{EF1}}(m) = \Theta(2^m / \sqrt{m})$ .

Sketch of argument (Suksompong, 2020):

The EF1-good subsets are middle layers of the $m$ -cube, separated from too-small or too-large subsets by Hamming distance at least 2 (vertex-isoperimetric inequality).
EF1 allocations persist under small perturbations to the instance (valuation changes, addition/deletion of items), unlike EFX, where there can be only two allocations regardless of $m$ .
This abundance provides robustness: as the combinatorial space of EF1 allocations is large, the probability of losing all EF1 allocations due to a minor change is negligible.

The result rigorously explains why EF1 is combinatorially and computationally much more tractable than EFX, and helps justify its central role in fair-division theory and algorithms.

3. Algorithmic Approaches and Computational Aspects

For $n=2$ agents and arbitrary monotonic, possibly non-additive, utility functions, a polynomial-time algorithm always finds an EF1 allocation. The standard approach is a prefix-suffix (cut-and-choose) procedure:

For items s_1, ..., s_m in a fixed order:
    For j = 0 to m:
        F_j = {s_1, ..., s_j}, L_j = {s_{j+1}, ..., s_m}
        If u_1(F_j) = u_1(L_j): (perfect balance)
            Assign to agents based on u_2(F_j) vs u_2(L_j), return allocation
        If u_1(F_j) >= u_1(L_j) and u_2(F_j) <= u_2(L_j):
            Assign as above, return allocation
    [Handle cases where u_1(S) ≠ 0 by finding sign flip crossing and assigning accordingly]

(Bérczi et al., 2020)

For $n \geq 3$ , existence of EF1 is guaranteed under additive utilities and can be obtained by round-robin or envy-graph algorithms, but for non-additive or non-monotonic utilities, the existence question remains open (Bérczi et al., 2020).

Additionally, there is a deep connection to algorithmic social choice:

Under buyer utility functions (additive, but each agent values goods at certain fixed prices or at zero), a greedy algorithm gives both utilitarian optimality (maximum sum-utility) and EF1 in $O(nm)$ time (Camacho et al., 2021).

EF1 is also robust to generalizations for mixed-manna (goods and chores): introspective EF1 (IEF1) allocations always exist for any additive instance, matching EF1 on pure goods and on chores (Barman et al., 23 Sep 2025). While the existence is established via a geometric KKM-type fixed-point argument, the development of efficient algorithms for PO+EF1 in mixed manna remains open.

4. Relation to Other Fairness Notions and Extensions

EF1 is part of a hierarchy of relaxations:

EF (Envy-freeness): $u_i(A_i) \geq u_i(A_j)$ always; may not exist for indivisible goods.
EF1: as above; always exists for additive utilities.
EFX (Envy-free up to any good): even after removing any single good from $A_j$ , $i$ no longer envies $j$ ; may not exist even with two agents in the non-additive setting (Bérczi et al., 2020).

For utility models with negative utilities (“chores”), the analogous definition is: $\forall i,j\in[n], \exists c\in A_i:\; u_i(A_i \setminus \{c\}) \geq u_i(A_j)$ (Chandramouleeswaran et al., 18 Oct 2024).

Extensions to more structured or restricted valuation classes have been shown:

Identical trilean valuations ( $\{0,a,b\}$ -valued): EF1 existence for any $n$ (Bhaskar et al., 29 Nov 2024).
Separable single-peaked: EF1 for three agents (Bhaskar et al., 29 Nov 2024).
Mixed goods/chores and externality-rich settings: suitable relaxations of EF1 coincide or generalize previous results (Barman et al., 23 Sep 2025, Aziz et al., 2021).

In information-theoretic terms, related notions (e.g., HEF- $k$ ) quantify aggregate envy-freeness by demanding global hiding of up to $k$ goods; this is strictly stronger than EF1 and results in sharp complexity distinctions (Hosseini et al., 2019).

5. Efficiency Trade-offs and Hardness

Although EF1 is always achievable in polynomial time for standard settings, maximizing utilitarian social welfare (USW) within the set of EF1 allocations is NP-hard, even for additive utilities and two agents (Mishra et al., 2021). Approximation results include:

For two agent types with normalized utilities, a $2$-approximation for USW under EF1 is achievable in polynomial time; for $n=3$ , a $5/3$-approximation is tight (Ma et al., 11 Sep 2025).
The “price of EF1” (the efficiency loss due to imposing EF1) with additive ternary valuations is $\Omega(\sqrt{n})$ , matching that of subadditive valuations for $n\gg 1$ , but for $n=2$ and $n=3$ , nearly tight constants ($12/11$, $[1.2,1.256]$ ) are established (Kyropoulou et al., 13 Aug 2025).

Neural-net–based methods (EEF1-NN) can find allocations empirically satisfying EF1 in $>99\%$ of random instances and yielding near-optimal welfare, vastly outperforming standard combinatorial methods in scalability and speed, albeit with no deterministic guarantees (Mishra et al., 2021).

6. Limitations, Contextual Failures, and Open Problems

EF1 is not sufficient to guarantee statistical or group-level fairness in personalized or highly heterogeneous settings. In recommendation systems, allocations may be EF1 but severely group-unfair or display disparate impact across protected groups. Pathological cases (such as empty allocations or majority domination) illustrate that EF1 does not inherently address group or individual parity, nor does it generalize to multi-stakeholder or dynamic settings (Aird et al., 10 Sep 2025).

Robustness comes at the price of potentially weak guarantees: the minimal aggregate number of items needing to be hidden or “excused” to restore actual envy-freeness may be large (up to $n-1$ ), and minimizing this number is NP-complete (Hosseini et al., 2019).

Several algorithmic and structural questions remain open:

Existence and efficient computation of EF1 (or PO+EF1) for general non-additive and mixed-valuation settings with $n\geq 3$ .
Stronger guarantees under connectivity/graph constraints or budget-feasibility.
Generalization to stronger notions such as EFX remains subtle and largely intractable in many domains (Bérczi et al., 2020, Bhaskar et al., 29 Nov 2024).
Dynamic restoration (maintaining EF1 under agent/item churn) can be PSPACE-complete even for binary, monotone utilities (Chandramouleeswaran et al., 18 Oct 2024).
Fair randomization over EF1 allocations (simultaneous ex-ante (stochastic-dominance) and ex-post EF1 fairness) is achievable via the probabilistic serial lottery mechanism in polynomial time, but decomposition into EF1 + Pareto optimal allocations is NP-hard (Aziz, 2020).

Summary Table: Key Features of EF1

Property	EF1	EFX	EF
Existence (additive)	Always exists	Open	Rare
Number of allocations	Exponential in items (2 agents)	Constant ( $\leq2$ for 2 agents)	$0$ or few
Polynomial algorithm	Yes (additive $n$ ), $n=2$ for non-additive	Rare/No	No
Price of fairness	$\Theta(\sqrt{n})$ (worst-case)	Higher/unknown	High
Robustness	High	Low	N/A
Group/aggregate fairness	Not guaranteed	Not guaranteed	Not guaranteed

EF1 is thus a uniquely robust and combinatorially rich notion that, while not a panacea for all fairness desiderata—especially in contexts with heterogeneous, group-based, or dynamic fairness requirements—remains the principal workhorse for polynomial-time fair division with indivisible items. Its combinatorial abundance, algorithmic tractability, and resistance to perturbation explain its dominance in both foundational and applied fair-division research.