EFX-with-Charity: Partial Fair Division

• EFX-with-Charity is a relaxation of envy-freeness that permits some goods to remain unallocated while ensuring that every allocated bundle fulfills the EFX condition and that no agent envies the charity set.
• It employs structured frameworks like the minimal-envied-subset and champion-graph approaches to construct partial allocations across various valuation models including additive, monotone, subadditive, and nice cancelable valuations.
• The method not only guarantees fairness with bounded charity (e.g., |U| < n, |U| ≤ n-2, or |U| ≤ 1) but also enhances efficiency metrics such as Nash welfare and MMS guarantees, with recent work refining approximation bounds using probabilistic and combinatorial techniques.

EFX-with-Charity is a partial-allocation relaxation of envy-freeness up to any item for the fair division of indivisible goods. Instead of insisting that every good be assigned, one permits a set of goods to remain unallocated—typically denoted by $U$ or $P$—while requiring that the allocated bundles satisfy EFX and, in the strongest formulations, that no agent values the unallocated set more than her own bundle. The notion arose as a way to obtain robust fairness guarantees in settings where complete EFX is unresolved or structurally difficult, and it has since developed into a distinct line of work spanning exact existence theorems, approximate EFX guarantees, structural proof frameworks, efficiency analyses, and algorithmic variants under general monotone, additive, subadditive, and typed valuation models (Chaudhury et al., 2019, Berger et al., 2021).

1. Formal definition and variants

The standard setting consists of $n$ agents and a finite set $M$ of indivisible goods. Each agent $i$ has a valuation function $v_i:2^M \to \mathbb{R}_{\ge 0}$, typically assumed to be normalized and monotone, meaning $v_i(\varnothing)=0$ and $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$. In the additive case,

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

An allocation $A=(A_1,\dots,A_n)$ is EFX if, for all agents $P$0 and all goods $P$1,

$P$2

Equivalently, no agent strongly envies another after the removal of any single good from the other agent’s bundle (Berger et al., 2021).

EFX-with-Charity replaces complete allocations by partial allocations. A partial allocation $P$3 leaves a charity set

$P$4

with the EFX condition checked only over the allocated bundles: $P$5 A recurrent additional guarantee is that the charity set is itself not envied: $P$6 This “no envy of charity” condition is explicit in the general monotone-valuation framework of Chaudhury et al. and in the later exact and approximate existence results (Chaudhury et al., 2019, Berger et al., 2021).

Two further variants are standard. First, $P$7-EFX relaxes the comparison multiplicatively: $P$8 Second, bounded-charity forms of the concept constrain the size of the unallocated set, for example $P$9, $n$0, or $n$1 in special cases. In the four-agent setting, Berger et al. call the $n$2 guarantee “almost full EFX” (Akrami et al., 2022, Berger et al., 2021).

2. Existence guarantees and the quantitative role of charity

The modern literature on EFX-with-Charity is organized around progressively stronger upper bounds on the number of unallocated goods. A basic chronology is as follows.

Setting	Guarantee	Source
Normalized, monotone valuations	EFX, no agent values charity more than own bundle, $n$3	(Chaudhury et al., 2019)
Four additive agents; extended to nice cancelable valuations	EFX with $n$4, and $n$5 is not envied	(Berger et al., 2021)
$n$6 additive agents; extended to nice cancelable valuations	EFX with $n$7, and $n$8 is not envied	(Berger et al., 2021)
Monotone valuations, $n$9-EFX	$M$0 unallocated goods	(Akrami et al., 2022)
Transformation via rainbow path degree	$M$1 discarded goods	(Jahan et al., 2022)
Additive valuations with at most $M$2 distinct valuation types	$M$3 charity	(HV et al., 21 Aug 2025)

The first general theorem shows that for normalized and monotone valuations there always exists a partition $M$4 satisfying EFX, the non-envy-of-charity condition $M$5, and the size bound $M$6. This theorem is constructive and already treats charity as a controlled resource rather than an arbitrary remainder (Chaudhury et al., 2019).

Berger et al. substantially sharpened the exact landscape. For four additive agents, and more generally for nice cancelable valuations, they proved that an EFX allocation exists with at most one unallocated good, and that the unallocated set is not envied. In the general $M$7-agent additive case they established the bound $M$8, again extending beyond additivity to nice cancelable valuations (Berger et al., 2021). These results improve the earlier $M$9 guarantee and are the standard exact benchmarks for bounded charity in the additive regime.

Charity is not merely a technical device for existence. For three additive agents, Chaudhury, Garg, and Mehlhorn proved complete EFX existence, but they also exhibited instances in which a partial EFX allocation has higher Nash welfare than any complete EFX allocation, thereby falsifying the monotonicity conjecture of Caragiannis et al. In that sense, charity can enlarge the efficiency frontier even when existence without charity is already known (Chaudhury et al., 2020).

3. Structural proof frameworks and constructive algorithms

Two proof paradigms dominate the exact theory. The first is the minimal-envied-subset framework of Chaudhury et al. for general monotone valuations. Their algorithm maintains a partial EFX allocation and a charity pool $i$0, together with an acyclic envy graph. It uses three update rules: $i$1, which greedily assigns a good when this preserves EFX; $i$2, which replaces an agent’s bundle by a minimal envied subset of the pool when some agent values $i$3 more than her own bundle; and $i$4, a cyclic reallocation rule that operates from sources of the envy graph using fresh goods from the pool. The algorithm repeatedly applies these rules, decycles the envy graph when necessary, and terminates in pseudo-polynomial time with EFX, $i$5, and $i$6 (Chaudhury et al., 2019).

The second paradigm is the champion-graph framework of Berger et al. For a partial EFX allocation $i$7, they build a directed multigraph $i$8 whose vertices are agents and whose edges include envy edges, champion edges $i$9 for an unallocated item $v_i:2^M \to \mathbb{R}_{\ge 0}$0, and generalized champion edges $v_i:2^M \to \mathbb{R}_{\ge 0}$1, where items $v_i:2^M \to \mathbb{R}_{\ge 0}$2 are donated and items $v_i:2^M \to \mathbb{R}_{\ge 0}$3 are released. The framework introduces good $v_i:2^M \to \mathbb{R}_{\ge 0}$4-cycles, good $v_i:2^M \to \mathbb{R}_{\ge 0}$5-edges, bottom-half bundles, Pareto-improvable cycles, and a lexicographic domination order. A central lemma states that if $v_i:2^M \to \mathbb{R}_{\ge 0}$6 contains a Pareto-improvable cycle, then there exists a partial EFX allocation $v_i:2^M \to \mathbb{R}_{\ge 0}$7 that Pareto-dominates $v_i:2^M \to \mathbb{R}_{\ge 0}$8, and agents on the cycle strictly improve. Another key lemma shows that domination can be converted into a reduction in the charity count. In the general $v_i:2^M \to \mathbb{R}_{\ge 0}$9-agent case, the “parallel rings” lemma isolates the structure that arises when no Pareto-improvable cycle exists and there are at least $v_i(\varnothing)=0$0 unallocated items; in the four-agent case, a detailed case analysis combines generalized champion edges, good cycles, and lexicographic improvements to reduce charity to at most one item (Berger et al., 2021).

Both frameworks are constructive, but they differ algorithmically. The monotone-valuation algorithm of Chaudhury et al. is explicitly pseudo-polynomial. Berger et al. emphasize existence and structured reallocation steps rather than polynomial-time guarantees; their paper states that finding Pareto-improvable cycles and generalized champion edges may require case-specific reasoning (Chaudhury et al., 2019, Berger et al., 2021).

4. Valuation classes, typed agents, and domain extensions

A major extension of the exact theory is the class of nice cancelable valuations. A valuation $v_i(\varnothing)=0$1 is cancelable if

$v_i(\varnothing)=0$2

for all bundles $v_i(\varnothing)=0$3 and all items $v_i(\varnothing)=0$4. A valuation is nice cancelable if there exists a non-degenerate cancelable valuation $v_i(\varnothing)=0$5 that respects $v_i(\varnothing)=0$6, meaning that every strict inequality of $v_i(\varnothing)=0$7 is preserved by $v_i(\varnothing)=0$8. Berger et al. show that additive, unit-demand, budget-additive, and multiplicative valuations are all nice cancelable, and they extend their $v_i(\varnothing)=0$9 and $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$0 results, as well as previous additive EFX existence results, to this larger class. They also note that not all cancelable valuations are nice (Berger et al., 2021).

A distinct direction replaces dependence on the number of agents by dependence on the number of valuation types. In the additive $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$1-type setting, where agents are partitioned into at most $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$2 groups sharing common valuations, the 2025 work “Almost and Approximate EFX for Few Types of Agents” proves that for any $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$3 there exists a $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$4-EFX allocation with charity of size

$S\subseteq T\Rightarrow v_i(S)\le v_i(T)$5

The proof restricts attention to $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$6 leading agents, one per type, and uses heavy envy, valuable goods, heavy champions, and a group champion graph. This replaces the earlier dependence on $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$7 by dependence on $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$8, which is significant when the agent population is large but preference heterogeneity is low (HV et al., 21 Aug 2025).

Type structure can also remove the need for charity entirely. A 2023 three-type result proves complete EFX when at least $S\subseteq T\Rightarrow v_i(S)\le v_i(T)$9 agents have identical valuations and the remaining two agents may differ; the result extends from additive to MMS-feasible valuations. In this regime, EFX-with-Charity becomes a comparison point rather than the final guarantee: the typed structure is strong enough to upgrade “almost full” EFX to full EFX (Ghosal et al., 2023).

5. Approximate EFX with charity and the rainbow-cycle line of work

Approximate EFX with charity is tied to a combinatorial object called the rainbow cycle number. Akrami et al. study the reduction of Chaudhury et al., in which constructing $v_i(S)=\sum_{g\in S} v_i(\{g\}).$0-EFX allocations with few unallocated goods is reduced to finding rainbow cycles in multipartite digraphs. If $v_i(S)=\sum_{g\in S} v_i(\{g\}).$1 is an upper bound on the rainbow cycle number and $v_i(S)=\sum_{g\in S} v_i(\{g\}).$2, then the reduction yields a polynomial-time algorithm producing a partial $v_i(S)=\sum_{g\in S} v_i(\{g\}).$3-EFX allocation with

$v_i(S)=\sum_{g\in S} v_i(\{g\}).$4

unallocated goods and no envy of the unallocated set. Their main improvement is the near-tight bound

$v_i(S)=\sum_{g\in S} v_i(\{g\}).$5

obtained by a direct probabilistic argument and derandomized by the method of conditional expectations. Substituting this into the reduction gives the bound

$v_i(S)=\sum_{g\in S} v_i(\{g\}).$6

for the number of unallocated goods, improving the earlier $v_i(S)=\sum_{g\in S} v_i(\{g\}).$7 guarantee (Akrami et al., 2022).

The subsequent paper “Rainbow Cycle Number and EFX Allocations: (Almost) Closing the Gap” introduces the rainbow path degree $v_i(S)=\sum_{g\in S} v_i(\{g\}).$8, proves

$v_i(S)=\sum_{g\in S} v_i(\{g\}).$9

and derives again the upper bound $A=(A_1,\dots,A_n)$0. Combined with the existing transformation theorem, this yields a $A=(A_1,\dots,A_n)$1-EFX allocation with

$A=(A_1,\dots,A_n)$2

discarded goods. The same paper conjectures

$A=(A_1,\dots,A_n)$3

which would imply $A=(A_1,\dots,A_n)$4 and tighten the fair-division bound to $A=(A_1,\dots,A_n)$5 discarded goods (Jahan et al., 2022).

A different approximation regime appears in the best-of-both-worlds literature. For monotone valuations, a 2025 paper proves the existence of randomized allocations that are ex-post EFX-with-charity and ex-ante $A=(A_1,\dots,A_n)$6-EF; more precisely, they satisfy a $A=(A_1,\dots,A_n)$7 approximate stochastic-dominance envy-freeness guarantee. The algorithm, “Random Charity Swaps,” initializes the pool as all goods, repeatedly chooses an inclusion-wise minimal envied subset of the pool, selects uniformly at random an agent who envies it, and swaps that subset with the agent’s current bundle. For subadditive valuations, the same work combines this with the “little charity” procedure to obtain ex-post EFX-with-bounded-charity with $A=(A_1,\dots,A_n)$8 and ex-ante $A=(A_1,\dots,A_n)$9-proportionality (Kavitha et al., 22 Jul 2025).

6. Efficiency consequences, specialized models, and open questions

The efficiency implications of charity are explicit in both exact and approximate settings. In the additive case, the general monotone-valuation algorithm of Chaudhury et al. yields an MMS guarantee

$P$00

and a minor variant produces a complete $P$01-GMMS allocation by assigning the charity set to a source of the envy graph. The bound improves as the charity set becomes smaller, so bounded charity is directly linked to stronger efficiency guarantees (Chaudhury et al., 2019).

For three additive agents, the counterexamples of Chaudhury, Garg, and Mehlhorn show that partial EFX can strictly dominate complete EFX in Nash welfare. This result rules out a naive interpretation of charity as a merely temporary defect: in some instances, unallocated goods are what permit the fairer-and-more-efficient allocation (Chaudhury et al., 2020).

The concept has also migrated to specialized models. In the fair-orientation framework, where agents are vertices and goods are edges that must be oriented to one endpoint, charity means deleting a minimum number of edges so that an EFX orientation exists. The 2025 paper on EF(X) orientations gives a dynamic program over nice tree decompositions running in time

$P$02

where $P$03 is the maximum shared weight and $P$04 is treewidth; for polynomially bounded weights this becomes $P$05. This is a specialized but conceptually faithful analogue of EFX-with-Charity: fairness is restored by bounded exclusion of contested resources (Kanellopoulos et al., 31 Dec 2025).

Several open problems remain central. Berger et al. explicitly ask whether the last unallocated item can always be allocated in the four-agent case, and whether the general $P$06 bound can be improved to $P$07 or further (Berger et al., 2021). The rainbow-cycle line asks whether the conjectured exact growth of $P$08 can be proved, which would remove the $P$09 factor from current approximate charity bounds (Jahan et al., 2022). The randomized best-of-both-worlds line asks whether the $P$10 ex-ante guarantees for monotone and subadditive valuations can be improved, and whether pseudopolynomial-time procedures can be made polynomial-time without losing ex-post EFX-with-charity (Kavitha et al., 22 Jul 2025). More broadly, no impossibility instance for full EFX with four or more additive agents is known, so EFX-with-Charity continues to function both as a practical fairness notion and as a proxy for the unresolved complete-allocability question (Berger et al., 2021).