Envy-Freeness up to One Good (EF1)

• EF1 is a fairness criterion that requires, for every pair of agents, the removal of one good from the envied bundle to eliminate envy.
• EF1 ensures near-envy-free allocations exist for indivisible items and can be computed efficiently using methods like envy-cycle elimination and round-robin.
• EF1 highlights trade-offs with stronger fairness notions by providing robust existence guarantees while balancing welfare loss and algorithmic complexity.

Envy-freeness up to one good (EF1) is a relaxation of envy-freeness for allocations of indivisible items. In the standard indivisible-goods model, an allocation $A=(A_1,\dots,A_n)$ is EF1 if for every ordered pair of agents $i,j$ with $A_j\neq\emptyset$, there exists a good $g\in A_j$ such that $u_i(A_i)\ge u_i(A_j\setminus\{g\})$. The notion is central because exact envy-freeness may fail for indivisible goods even with two agents, whereas EF1 has broad existence and algorithmic guarantees and has become a standard benchmark for almost envy-free allocation (Suksompong, 2020, Li et al., 2024).

1. Definition and neighboring fairness notions

Under additive valuations, EF1 is usually stated as

$$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$

The same condition appears in work on the price of fairness, exchange-based reforming, and mechanism design, with the convention that when the compared bundle is empty no removal is needed (Li et al., 2024, Yuen et al., 2024, Mahara et al., 2024).

The literature also studies exact and approximate variants. In the notation of approximate relaxations, an allocation is $\alpha$-EF1 if for every pair $i,j$ with $A_j\neq\emptyset$ there exists $g\in A_j$ such that

$i,j$0

Exact EF1 is the case $i,j$1 (Amanatidis et al., 2018).

EF1 sits in a hierarchy with stronger envy-based notions. Envy-freeness up to any good (EFX) requires the corresponding inequality to hold for every good in the envied bundle, so $i,j$2 (Amanatidis et al., 2019, Amanatidis et al., 2018). Envy-freeness up to one less-preferred good (EFL) strengthens EF1 by additionally requiring that the removed good not exceed the envier’s own bundle in value, and the implication $i,j$3 is explicit in the additive setting (Akrami et al., 12 Feb 2026).

Several domain-specific generalizations alter the deletion rule. For connected bundles on a graph, the removed item must be an outer good whose deletion preserves connectivity of the compared bundle (Bilò et al., 2018). For budget-feasible allocation, EF1 is tested against every budget-feasible probe $i,j$4, including probes into the unallocated charity bundle, and one requires some $i,j$5 such that $i,j$6 (Gan et al., 2021). For arbitrary, possibly non-monotone and non-additive utilities, EF1 is extended by allowing deletion from either the envier’s or the envied’s bundle: $i,j$7 thereby covering goods, chores, and mixed items in a single definition (Bérczi et al., 2020).

2. Existence and combinatorial abundance

For additive valuations, EF1 always exists, and classical constructive procedures such as envy-cycle elimination and round-robin produce EF1 allocations in polynomial time (Li et al., 2024, Amanatidis et al., 2019). The existence phenomenon is substantially stronger in the two-agent case than mere nonemptiness.

A sharp counting theorem shows that for two agents with arbitrary monotone valuations over $i,j$8 indivisible goods, the number of EF1 allocations is always at least

$i,j$9

The bound is tight for every $A_j\neq\emptyset$0 (Suksompong, 2020). For even $A_j\neq\emptyset$1, identical additive valuations giving every item value $A_j\neq\emptyset$2 realize exactly the allocations that split the items into two bundles of size $A_j\neq\emptyset$3; for odd $A_j\neq\emptyset$4, assigning value $A_j\neq\emptyset$5 to the first $A_j\neq\emptyset$6 items and value $A_j\neq\emptyset$7 to the last item yields exactly $A_j\neq\emptyset$8 EF1 allocations (Suksompong, 2020).

The proof is combinatorial. For a fixed agent, bundles are partitioned into “good,” “too small,” and “too large,” and the too-small and too-large families satisfy Hamming distance at least $A_j\neq\emptyset$9. A vertex-isoperimetric lemma on the hypercube allows replacement by Hamming balls around $g\in A_j$0 and $g\in A_j$1 without decreasing that distance, after which counting yields the lower bound. The argument uses a version of the vertex isoperimetric inequality and a Sperner-type size bound (Suksompong, 2020).

This abundance sharply contrasts with EFX. In the same two-agent setting, the number of EFX allocations can be as small as $g\in A_j$2, independent of $g\in A_j$3, whereas the number of EF1 allocations is always exponential in $g\in A_j$4 (Suksompong, 2020). A common misconception is that EF1 and EFX differ mainly in logical strength; the counting result shows a large gap in robustness as well.

3. Core algorithms and heuristic refinements

The classical algorithmic template for EF1 under additive valuations is envy cycle elimination (ECE). One maintains the directed envy-graph on agents, where there is an arc $g\in A_j$5 if $g\in A_j$6. In each round, ECE picks an unallocated good and an unenvied agent, assigns the good to that agent, and then breaks any newly formed envy cycle by rotating entire bundles along the cycle. The standard inductive proof shows that assigning to an unenvied agent and rotating along envy cycles preserves EF1 after every step, so the final allocation is EF1 (Celine et al., 1 Jun 2026, Amanatidis et al., 2019).

Because ECE leaves freedom in choosing both the next good and the receiving agent, recent work studies welfare-oriented heuristics. If no heuristic is used and arbitrary valid choices are allowed, the worst-case utilitarian guarantee is $g\in A_j$7. Restricting the heuristic to only the receiving agent does not improve this bound: the greedy-agent rule still has $g\in A_j$8. Restricting it to only the good improves the guarantee to $g\in A_j$9. A single greedy joint choice in the first round yields $u_i(A_i)\ge u_i(A_j\setminus\{g\})$0 for $u_i(A_i)\ge u_i(A_j\setminus\{g\})$1 and $u_i(A_i)\ge u_i(A_j\setminus\{g\})$2 for $u_i(A_i)\ge u_i(A_j\setminus\{g\})$3. Greedy joint choice in every round,

$u_i(A_i)\ge u_i(A_j\setminus\{g\})$4

achieves a worst-case guarantee of approximately $u_i(A_i)\ge u_i(A_j\setminus\{g\})$5, hence $u_i(A_i)\ge u_i(A_j\setminus\{g\})$6 for large $u_i(A_i)\ge u_i(A_j\setminus\{g\})$7 (Celine et al., 1 Jun 2026).

The same study reports average-case behavior substantially better than worst-case theory. For $u_i(A_i)\ge u_i(A_j\setminus\{g\})$8 and $u_i(A_i)\ge u_i(A_j\setminus\{g\})$9 from $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$0 up to $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$1, with i.i.d. Uniform$$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$2 utilities normalized to sum to $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$3, the joint-every-round heuristic has average $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$4 near $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$5–$$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$6, while greedy-agent and greedy-good stay around $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$7–$$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$8 (Celine et al., 1 Jun 2026). This suggests that the flexibility of ECE is practically consequential even when its baseline correctness proof is purely fairness-based.

Round-robin remains a second canonical EF1 procedure. In additive settings it lets agents repeatedly pick their most-valued remaining good in a fixed order and yields an EF1 allocation (Mahara et al., 2024, Amanatidis et al., 2018). Later work uses EF1-preserving combinations of round-robin and envy-cycle elimination as a backbone for approximation guarantees to stronger notions such as EFX, GMMS, and PMMS (Amanatidis et al., 2019).

4. Efficiency loss and relations to MMS, PMMS, and EFX

The welfare cost of imposing EF1 can be quantified through the price of fairness. For additive indivisible goods, with scaled utilities $$\forall i,j\in N,\ \exists g\in M_j\ \text{such that}\ u_i(M_i)\ge u_i(M_j\setminus\{g\}).$$9, the price of EF1 is

$\alpha$0

The complete characterization gives $\alpha$1 for two agents, $\alpha$2 for general $\alpha$3 under scaled utilities, and $\alpha$4 for general $\alpha$5 under unscaled utilities (Li et al., 2024).

The tight two-agent proof partitions goods into

$\alpha$6

defines the surplus $\alpha$7, and analyzes three regimes: $\alpha$8, $\alpha$9, and $i,j$0. In the middle regime the worst-case ratio is attained at $i,j$1, yielding the tight constant $i,j$2 (Li et al., 2024).

Comparison with EFX is explicit. For two agents and scaled utilities, $i,j$3 whereas $i,j$4 (Li et al., 2024). The welfare comparison therefore does not merely reflect logical implication; EF1 can be strictly less costly than EFX in the worst case.

A separate line of work maps EF1 against maximin-share-type notions. Exact EF1 implies $i,j$5-MMS for $i,j$6 and $i,j$7-MMS for $i,j$8. More generally, $i,j$9-EF1 implies both

$A_j\neq\emptyset$0

with the latter recovering the classic $A_j\neq\emptyset$1 bound at $A_j\neq\emptyset$2. For PMMS, $A_j\neq\emptyset$3-EF1 implies $A_j\neq\emptyset$4-PMMS, and exact EF1 implies $A_j\neq\emptyset$5-PMMS for $A_j\neq\emptyset$6 (Amanatidis et al., 2018). The same source emphasizes that EFX implies EF1 exactly, but EF1 may fail to imply any constant-factor EFX guarantee (Amanatidis et al., 2018).

5. Connected bundles and group allocation

When goods must form connected bundles in an underlying graph, EF1 changes both combinatorially and algorithmically. For a path and monotonic valuations on connected bundles, connected EF1 allocations exist for at most four agents, and connected EF2 allocations exist for any number of agents (Bilò et al., 2018).

For two agents on a path, there is a discrete cut-and-choose protocol based on a “lumpy tie.” One agent identifies the smallest vertex $A_j\neq\emptyset$7 such that

$A_j\neq\emptyset$8

the other agent chooses between the two sides, and the first agent receives the remainder together with $A_j\neq\emptyset$9. The resulting allocation is connected and EF1 (Bilò et al., 2018).

For three agents on a path, a discrete moving-knife protocol yields a connected EF1 allocation in $g\in A_j$0 time. For four agents, a Sperner-lemma construction yields existence of connected EF1. For identical monotonic valuations on a path, a polynomial-time algorithm computes a leximin allocation and then performs local moves of boundary items; with suitable dynamic programming for leximin, the runtime is $g\in A_j$1 (Bilò et al., 2018).

For general connected graphs and two agents, the characterization is exact: a connected graph guarantees a connected EF1 allocation for all two-agent instances if and only if its block-tree is a path, equivalently if and only if it admits a bipolar numbering. The block-tree can be computed by DFS in $g\in A_j$2 time, and the path property can be checked in linear time (Bilò et al., 2018).

EF1 has also been generalized to allocations between groups. If goods are allocated to bundles $g\in A_j$3 for groups of agents, EF1 requires that for every agent $g\in A_j$4 in group $g\in A_j$5 and every other group $g\in A_j$6, there exists $g\in A_j$7 such that

$g\in A_j$8

For two fixed groups with responsive valuations of sizes $g\in A_j$9, a balanced EF1 allocation always exists. Under arbitrary monotonic valuations, existence is obtained for bounded total group size through chromatic properties of generalized Kneser graphs (Kyropoulou et al., 2019).

Binary valuations admit a complete two-group characterization. Up to symmetry, EF1 always exists for group-size pairs $i,j$00 and $i,j$01, while there are counterexamples for $i,j$02, $i,j$03, and $i,j$04. Moreover, deciding whether an EF1 allocation exists for two fixed groups with binary valuations is NP-complete (Kyropoulou et al., 2019).

A distinct “partition-selection” model allows the partition of agents into groups to be chosen together with the allocation. For arbitrary monotonic valuations and any target sizes $i,j$05, there always exists a partition into two groups of sizes $i,j$06 together with an EF1 allocation of goods. A stronger statement guarantees balanced group sizes and goods bundles differing in size by at most one (Kyropoulou et al., 2019).

6. Budgets, chores, mixed items, and non-monotone utilities

In the budget-feasible allocation model, each item has a size and each agent has a capacity $i,j$07. An allocation includes a charity bundle $i,j$08 of unallocated items, and EF1 is tested against every subset $i,j$09 that fits the envier’s budget. Formally, for every agent $i,j$10, every $i,j$11, and every probe $i,j$12 with $i,j$13, there must exist $i,j$14 such that

$i,j$15

For identical additive valuations, a polynomial-time algorithm computes a $i,j$16-approximate EF1 allocation. For uniform budgets and for two agents, exact EF1 can be computed efficiently. In the large-budget regime $i,j$17, the same algorithm yields $i,j$18-approximate EF1, and any Nash-social-welfare maximizing allocation is $i,j$19-EF1 when valuations are identical (Gan et al., 2021).

EF1 also survives beyond monotonicity and additivity. For arbitrary utility functions over indivisible items, a polynomial-time algorithm for two agents orders the items $i,j$20, studies the prefix-suffix pairs $i,j$21, and locates a sign change in $i,j$22. The resulting allocation is EF1 in $i,j$23 utility evaluations, assuming each set-utility query is polynomial-time (Bérczi et al., 2020). By contrast, stronger EFX-type relaxations can fail even for two agents with identical non-monotone, non-additive utilities (Bérczi et al., 2020).

The chores setting requires a different orientation of the one-item removal. For indivisible chores, EF1 means that for every pair $i,j$24 with $i,j$25, there exists a chore $i,j$26 such that

$i,j$27

A polynomial-time algorithm computes such allocations by assigning chores to sinks of the envy graph and resolving only cycles in the top-trading envy graph. The restriction to top-trading cycles is essential: arbitrary envy-cycle swaps can destroy EF1 for chores. The algorithm runs in polynomial time and returns an EF1 allocation (Bhaskar et al., 2020).

The same paper extends the framework to doubly monotone instances and to mixed resources consisting of indivisible items and a divisible bad cake. In that setting, an allocation satisfying envy-freeness for mixed resources (EFM) always exists, and the proposed algorithm runs in $i,j$28 rounds assuming an oracle for perfect cake partitions (Bhaskar et al., 2020).

7. Exchanges, information, mechanisms, and simultaneous guarantees

Beyond static existence, EF1 has a substantial reconfiguration theory. If one fixes a size vector and allows pairwise exchanges of one good for one good, the EF1-exchange graph is the subgraph induced by EF1 allocations. Two EF1 allocations need not be reachable from one another even for two agents with general utilities, and in general reachability is PSPACE-complete. For two agents with identical utilities or with binary utilities, however, the EF1-exchange graph is connected and optimal EF1 paths always exist and can be found in polynomial time. For identical binary utilities and arbitrary $i,j$29, connectivity is restored, but optimal paths may still fail for $i,j$30 (Igarashi et al., 2023).

A related reforming problem starts from an arbitrary allocation and asks whether some EF1 allocation with the same size vector can be reached by exchanges. The complexity depends sharply on the number of agents and on the utility class. Reformability is in P for two agents with identical utilities, for binary utilities with two agents, for constant $i,j$31 with binary utilities, and for arbitrary $i,j$32 with identical binary utilities; it is weakly or strongly NP-complete in the complementary regimes described in the paper (Yuen et al., 2024). When all agents start with exactly $i,j$33 goods, the worst-case number of exchanges needed to reach EF1 satisfies essentially tight bounds. For general utilities,

$i,j$34

where $i,j$35 and $i,j$36, and for $i,j$37 one has the exact formula $i,j$38 (Yuen et al., 2024).

EF1 is also only a pairwise condition, which motivates aggregate refinements. In the information-withholding model, an allocation is HEF-$i,j$39 if there exists a set $i,j$40 of at most $i,j$41 hidden goods such that the revealed allocation is envy-free. Every EF1 allocation is automatically uHEF-$i,j$42, but finding an allocation that hides an optimal number of goods is computationally hard even for restricted valuations. Given a fixed allocation $i,j$43, minimizing the number of hidden goods is NP-hard to approximate within $i,j$44, where

$i,j$45

yet there is a greedy polynomial-time algorithm returning a hiding set of size at most $i,j$46 (Hosseini et al., 2019).

Mechanism design raises a different issue: many EF1 mechanisms depend on an exogenous order of agents. Position envy-freeness up to $i,j$47 goods (PEF$i,j$48) compares an agent’s outcome across different orderings of the same mechanism. Round-robin and envy-cycle are not PEF1 in general, but a deterministic matching-based mechanism, PEF1-Matching, is polynomial-time, scale-invariant, always outputs an EF1 allocation, and is PEF1. In the two-agent case, any mechanism that always returns a maximum Nash social welfare allocation is PEF1, and a modified adjusted winner mechanism achieves EF1, Pareto-optimality, and PEF1 simultaneously (Mahara et al., 2024).

Recent work also studies coexistence with stronger epistemic fairness guarantees. For additive valuations, there always exists an allocation that is simultaneously EFL, hence EF1, and epistemic EFX (EEFX). The construction introduces the strong EEFX share

$i,j$49

proves $i,j$50, and then applies a lone-divider framework. The resulting existence theorem is existential and yields an exponential-time procedure by scanning all subsets to compute the shares (Akrami et al., 12 Feb 2026). This places EF1 not only as a relaxation of envy-freeness, but also as a component in composite fairness guarantees that interact with EFL, EEFX, and residual maximin-share ideas.