EFX: Envy-Freeness up to Any Good

• EFX is a fairness criterion for allocating indivisible goods, stipulating that any agent’s envy vanishes once any single good is removed from another’s bundle.
• It extends envy-freeness concepts by strictly strengthening EF1 and lies between full envy-freeness and EF1, posing significant algorithmic and theoretical challenges.
• Research on EFX focuses on existence proofs, approximation techniques, welfare trade-offs, and applications in structured domains like graph-restricted allocations.

Envy-freeness up to any good (EFX) is a fairness criterion for allocating indivisible goods under which an agent’s envy toward another bundle must disappear after the removal of any single good from that bundle. In additive settings, the standard formulation requires that for all agents $i,j$ and all goods $g \in A_j$ with $v_i(g) > 0$, $v_i(A_i) \ge v_i(A_j \setminus \{g\})$; a stronger zero-inclusive version is commonly denoted EFX0. EFX lies strictly between envy-freeness (EF) and envy-freeness up to one good (EF1): EF implies EFX, and EFX strictly strengthens EF1. Its general existence for additive valuations remains open, and the literature correspondingly studies exact existence in structured domains, approximate and partial variants, welfare and efficiency trade-offs, algorithmic methods such as leximin and local search, and epistemic or graph-restricted relaxations (Plaut et al., 2017, Lim et al., 19 Jan 2026).

1. Formal definitions and variant notions

For agents $N=[n]$, goods $G$, and additive nonnegative valuations $v_i$, an allocation $A=(A_1,\dots,A_n)$ is EF if $v_i(A_i)\ge v_i(A_j)$ for all $i,j$. It is EF1 if for every pair $g \in A_j$0 there exists some good $g \in A_j$1 such that $g \in A_j$2. It is EFX if for every pair $g \in A_j$3 and every good $g \in A_j$4 with $g \in A_j$5, one has $g \in A_j$6. In the stronger EFX0 variant, the same inequality must hold for all $g \in A_j$7, including goods of zero value to the evaluating agent (Lim et al., 19 Jan 2026). Some recent work adopts this zero-inclusive form as the default “any-good” definition, so the literature contains both conventions (Akrami et al., 12 Feb 2026).

The basic goods formulation has several extensions. For chores, one removes a chore from the envious agent’s own bundle rather than from the envied bundle; for mixed goods-and-chores, strong envy is defined by taking the maximum of the goods-side and chores-side forms. In graph-restricted EFX, the requirement is imposed only along edges of an undirected graph on the agents, yielding $g \in A_j$8-EFX rather than global EFX (Payan et al., 2022). In mixed divisible-and-indivisible settings, EFXM generalizes EFX by requiring exact envy-freeness whenever the compared bundle contains divisible goods, and EFX-type comparisons only for all-indivisible bundles (Li et al., 2024).

Approximate and related notions are central in the area. An allocation is $g \in A_j$9-EFX if for all relevant $v_i(g) > 0$0, $v_i(g) > 0$1. Weighted EFX (WEFX) normalizes utilities by entitlements $v_i(g) > 0$2, requiring $v_i(g) > 0$3 for all pairs and goods; a weaker variant is WWEFX (Neoh et al., 4 Apr 2025). EFX+ replaces removal from the other bundle by addition to the evaluator’s bundle, requiring $v_i(g) > 0$4 for all $v_i(g) > 0$5; under additive valuations, EFX+ is equivalent to EFX, while under general monotone valuations the two notions are incomparable (Neoh et al., 4 Apr 2025).

A distinct epistemic line replaces actual pairwise comparisons by certificate-based feasibility. In epistemic EFX (EEFX), agent $v_i(g) > 0$6’s bundle need only admit a partition of the remaining goods that certifies EFX from $v_i(g) > 0$7’s perspective. This leads to the minimum EFX share (MXS) and, more recently, to the strong EEFX share $v_i(g) > 0$8, which satisfies $v_i(g) > 0$9 and underpins simultaneous EFL+EEFX existence for additive valuations (Caragiannis et al., 2022, Akrami et al., 12 Feb 2026).

2. Exact existence in the classical goods setting

The general existence question for EFX under additive valuations remains unresolved, but a substantial exact-existence frontier has been established. Early results showed that for general identical valuations the leximin++ solution is EFX, and for two players a cut-and-choose protocol based on leximin++ yields an EFX allocation; with nonzero marginal utility, standard leximin gives EFX and Pareto optimality for identical valuations and also for two players with additive valuations (Plaut et al., 2017).

For additive valuations with heterogeneous agents, the known exact cases have progressively expanded. EFX is known for three agents, for any number of agents when there are at most two distinct valuation types, and for any number of agents when there are at most three distinct additive valuation types. The three-type theorem simultaneously generalizes the three-agent case and the two-type case (HV et al., 2024). A related earlier result established complete EFX when all but two agents share identical valuations, yielding in particular complete EFX for four agents with at most three distinct types (Ghosal et al., 2023).

Another exact regime is defined by few surplus items. Writing $v_i(A_i) \ge v_i(A_j \setminus \{g\})$0, complete EFX allocations are guaranteed when $v_i(A_i) \ge v_i(A_j \setminus \{g\})$1 via envy-cycle elimination and when $v_i(A_i) \ge v_i(A_j \setminus \{g\})$2 via a potential-function-based augmentation; in this regime, complete EFX allocations, indeed EFX0 allocations, exist even for general monotone valuations (Lim et al., 19 Jan 2026). This threshold is structurally significant: beyond $v_i(A_i) \ge v_i(A_j \setminus \{g\})$3, no unconditional existence guarantee is currently known.

Counting arguments sharpen the importance of the near-balanced regime. For $v_i(A_i) \ge v_i(A_j \setminus \{g\})$4, every instance has at least $v_i(A_i) \ge v_i(A_j \setminus \{g\})$5 EFX allocations, and this bound is tight; for $v_i(A_i) \ge v_i(A_j \setminus \{g\})$6, every instance has at least $v_i(A_i) \ge v_i(A_j \setminus \{g\})$7 EFX allocations, again tightly. More conceptually, if there exists a function $v_i(A_i) \ge v_i(A_j \setminus \{g\})$8 such that EFX always exists for $v_i(A_i) \ge v_i(A_j \setminus \{g\})$9, then EFX exists for every $N=[n]$0 (Neoh et al., 4 Apr 2025). This suggests that understanding instances with only slightly more goods than agents is not a peripheral case but a possible route to the full existence problem.

3. Approximate, almost-full, and partial EFX

Because exact EFX is unresolved in general, approximation has been a major line of attack. A foundational result proved existence of $N=[n]$1-EFX for subadditive valuations (Plaut et al., 2017). A stronger guarantee, $N=[n]$2-EFX with $N=[n]$3 the golden ratio, was obtained by the Draft-and-Eliminate algorithm, which simultaneously guarantees EF1, a $N=[n]$4-approximation of PMMS, and a $N=[n]$5-approximation of GMMS (Amanatidis et al., 2019). This established the long-standing benchmark $N=[n]$6.

More recent work pushed the approximation frontier to $N=[n]$7 in structured domains. Polynomial-time $N=[n]$8-EFX allocations exist when there are at most seven agents, when each agent’s singleton values come from at most three values, and when valuations can be represented by a multigraph (Amanatidis et al., 2024). In the few-types direction, a $N=[n]$9-EFX allocation exists for any number of agents when there are at most four distinct additive valuations (HV et al., 21 Aug 2025). These results are significant because they strictly generalize settings where exact EFX was previously known while also exceeding the general $G$0 barrier.

A complementary approach allows unallocated goods. For every $G$1, there exists a $G$2-EFX partial allocation with a sublinear number of unallocated goods, specifically at most $G$3, while preserving a $G$4-approximation to Nash social welfare (Chaudhury et al., 2021). Later type-sensitive results show that with at most $G$5 distinct additive valuations there exists a $G$6-EFX allocation with charity using only $G$7 unallocated goods (HV et al., 21 Aug 2025). These bounds connect approximate EFX to extremal graph parameters such as the rainbow cycle number.

Relaxing the number of removed items yields another hierarchy. For restricted additive valuations, complete EF2X allocations always exist, and an EFX allocation can be found while discarding at most $G$8 goods, improving the prior restricted-setting bound by a factor of $G$9 (Akrami et al., 2022). This shows that exact EFX is not the only robust relaxation: EF$v_i$0X can preserve completeness even when EFX itself remains difficult.

4. Structured domains: graphs, multigraphs, and lexicographic preferences

Graph structure has produced several exact or restricted EFX theories. In graph-restricted EFX, the fairness condition is imposed only along the edges of an undirected graph on the agents. For goods, $v_i$1-EFX allocations exist for the star $v_i$2, for broader star-like graphs with identical or consistent core valuations, and for $v_i$3-type path structures; many of these results extend to chores. For mixed goods and chores under lexicographic preferences, a $v_i$4-EFX allocation exists in polynomial time whenever the graph has diameter at least $v_i$5 (Payan et al., 2022). The same paper introduced graph-restricted hidden envy-freeness, tied existence to vertex cover, and reported that a sweeping algorithm on Spliddit path instances terminated with a $v_i$6-EFX allocation in every one of the $v_i$7 tested instances (Payan et al., 2022).

A more structural orientation-based program studies when a graph guarantees EFX under every assignment of graphical valuations. A graph is strongly EFX orientable if every edge-weight assignment admits an EFX orientation. Bipartite graphs, equivalently graphs with $v_i$8, are strongly EFX orientable, whereas graphs with $v_i$9 are not. The binary case admits a complete characterization in terms of forests and independent neighborhoods, and the $A=(A_1,\dots,A_n)$0 frontier contains both positive and negative examples (Zeng et al., 2024). This chromatic-number connection isolates a precise graph-theoretic obstruction to universal orientation-based EFX.

The multigraph case was long open because parallel edges defeat the simple-graph exchange arguments. That problem is now resolved: EFX allocations exist on multigraph instances under cancelable valuations, a strict superclass of additive valuations, and can be computed in polynomial time. The proof uses precomputed per-pair unit bundles, a height-one resent-forest reduction, and a dumping phase that safely allocates the remaining bundles (Afshinmehr et al., 17 Jun 2026).

Lexicographic preferences form another domain where exact EFX is unusually well behaved. Under strict lexicographic preferences, EFX and Pareto optimal allocations always exist, and there is an algorithmic characterization: every EFX+PO allocation arises from a two-phase procedure consisting of one round of serial dictatorship followed by allocating the leftovers only among unenvied agents. If one also requires strategyproofness, non-bossiness, and neutrality, the admissible mechanisms collapse to the IQSD rule. By contrast, EFX combined with rank-maximality may fail to exist, and deciding existence of an EFX and rank-maximal allocation is NP-complete (Hosseini et al., 2020).

5. Welfare, price of fairness, and Pareto compatibility

A large recent development studies EFX not only as a feasibility condition but as a welfare constraint. In the few-surplus regime $A=(A_1,\dots,A_n)$1 with $A=(A_1,\dots,A_n)$2, generalized-mean welfare $A=(A_1,\dots,A_n)$3 exhibits a sharp complexity dichotomy at $A=(A_1,\dots,A_n)$4. For any fixed $A=(A_1,\dots,A_n)$5, deciding whether EFX can attain the global $A=(A_1,\dots,A_n)$6-mean optimum and computing a $A=(A_1,\dots,A_n)$7-mean-optimal EFX allocation are NP-hard even with at most three surplus goods. For any fixed $A=(A_1,\dots,A_n)$8, by contrast, one can optimize $A=(A_1,\dots,A_n)$9-mean welfare over the EFX or EFX0 domain in polynomial time and can certify whether EFX attains the global optimum (Lim et al., 19 Jan 2026). This separates utilitarian or near-utilitarian objectives from inequality-averse objectives such as Nash welfare and egalitarian welfare.

The price of fairness literature quantifies the efficiency loss from imposing EFX. In the general indivisible-goods model, the price of EFX for two agents is $v_i(A_i)\ge v_i(A_j)$0 under scaled utilities and $v_i(A_i)\ge v_i(A_j)$1 under unscaled utilities; for $v_i(A_i)\ge v_i(A_j)$2 agents it is $v_i(A_i)\ge v_i(A_j)$3 in the scaled case and $v_i(A_i)\ge v_i(A_j)$4 in the unscaled case. The same asymptotic picture persists for the mixed-goods generalization EFXM (Li et al., 2024). In the few-surplus regime, the welfare picture is more refined: for $v_i(A_i)\ge v_i(A_j)$5, the price of EFX can grow linearly with $v_i(A_i)\ge v_i(A_j)$6, while for $v_i(A_i)\ge v_i(A_j)$7 it is bounded by a constant depending only on the surplus, and for Nash welfare it is at most $v_i(A_i)\ge v_i(A_j)$8, which vanishes asymptotically (Lim et al., 19 Jan 2026).

Pareto optimality is not uniformly compatible with EFX. In the few-surplus regime with $v_i(A_i)\ge v_i(A_j)$9, deciding whether an allocation is both EFX and Pareto optimal is NP-hard, and the EFX0+PO decision problem is $i,j$0-complete (Lim et al., 19 Jan 2026). Earlier work already showed impossibility of EFX+PO in natural additive and general settings when zero marginal utility is allowed, and even under nonzero marginal utility EFX+PO can fail for two players with general distinct valuations (Plaut et al., 2017). The lexicographic domain therefore stands out sharply: there EFX+PO is always achievable and exactly characterized (Hosseini et al., 2020).

6. Computation, counting, epistemic relaxations, and open directions

The algorithmic status of EFX is shaped by both positive structure and strong lower bounds. In the value-query model, identifying an EFX allocation can require exponentially many queries even for two players with identical submodular valuations: the deterministic lower bound is $i,j$1 and the randomized lower bound is $i,j$2 (Plaut et al., 2017). These lower bounds explain why existence proofs in special domains often rely on strong structural invariants rather than generic search.

At the same time, simple heuristics can be surprisingly effective. A recent local-search algorithm based on simulated annealing uses the total number of EFX violations as an objective function and a single-transfer neighborhood. It found an EFX allocation in all the instances tested, including thousands of randomly generated inputs, and scaled to settings with hundreds of agents and/or thousands of items. In the identical additive case, the paper gives a potential function $i,j$3 ensuring that any strict-descent procedure under the single-transfer neighborhood terminates at an EFX allocation (Brânzei, 6 Oct 2025). This yields an alternative proof of existence for identical additive valuations.

Counting EFX allocations is itself computationally hard: computing the number of EFX allocations is #P-complete for general monotone valuations (Neoh et al., 4 Apr 2025). Nonetheless, the same work derives exact or near-exact lower bounds on the minimum number of EFX allocations in several sparse regimes, studies WEFX under binary additive valuations, and proves that under additive valuations EFX+ and EFX are equivalent. This counting perspective suggests that extremal multiplicity, not only existence, may be informative for the global EFX question (Neoh et al., 4 Apr 2025).

Epistemic relaxations offer a different response to the open existence problem. EEFX allocations always exist and are polynomial-time computable for additive valuations (Caragiannis et al., 2022). More strongly, there always exists an allocation that is simultaneously EFL and EEFX for additive valuations, obtained via the strong EEFX share and the lone-divider/RMMS framework, although the current proof is existential and only yields an exponential-time method; polynomial-time computation remains open even for four agents (Akrami et al., 12 Feb 2026). This line suggests that certificate-based fairness may retain much of EFX’s conceptual force while avoiding some of its existential fragility.

Open directions remain numerous and sharply defined. The central problem is still exact EFX existence for arbitrary additive valuations. Beyond that, the literature isolates several concrete frontiers: extending exact or $i,j$4-approximate results beyond three or four valuation types, understanding the regime beyond three surplus goods, characterizing the $i,j$5 boundary for strong graph orientability, tightening price-of-fairness bounds under structural restrictions, and developing polynomial-time algorithms for simultaneous EFL+EEFX. A plausible implication is that progress will continue to come from domains where combinatorial structure and fairness interact cleanly, rather than from a single uniform proof strategy.