EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence

Abstract: We consider the fair division problem of indivisible chores and resolve the long-standing open problem for the existence of EFX allocations with additive cost functions. We show that, even for tri-valued additive cost functions, for every $n\geq 4$, there exists an instance with $n$ agents where no EFX allocation exists. Our counterexample only uses three types of chores, which is also tight on the number of types, as an EFX allocation is known to exist for two types of chores. We then consider bi-valued instances. We show that, for every $n\geq 4$, there exists an instance with $n$ agents where every EFX allocation is not Pareto-optimal. This is also the first example showing the incompatibility of EFX and Pareto-optimality when the costs of items are positive: existing examples showing the incompatibility of EFX and Pareto-optimal exploit items with $0$ costs. Our result shows such an example exists even for bi-valued instances. The number of agents $n$ is also tight: for $n\leq 3$, it is known that EFX is compatible with Pareto-optimality. Finally, we also show that an EFX allocation is guaranteed to exist for $n=4$.

• The paper proves that for every n ≥ 4, a tri-valued additive chore instance can lack any EFX allocation, using a tight three-cost construction and counting arguments.
• The paper shows that for every n ≥ 4, EFX and Pareto-optimality can conflict in strictly positive bi-valued instances, so fairness and efficiency cannot always be achieved together.
• The paper establishes that every four-agent bi-valued instance has an EFX allocation, sharply separating existence from EFX–Pareto compatibility and leaving three-agent general cases open.

This paper settles the long-standing question of whether envy-freeness up to any item (EFX) allocations always exist for indivisible chores with additive cost functions, answering negatively. He and Tao construct, for every number of agents $n \geq 4$, an additive tri-valued instance admitting no EFX allocation, and show that for bi-valued instances with $n \geq 4$, EFX and Pareto-optimality (PO) are incompatible even when all costs are strictly positive. As a counterweight, they prove that every four-agent bi-valued instance admits an EFX allocation. The results sharply delineate the frontier of EFX existence for chores and expose an asymmetry with the goods setting, where additive EFX existence remains open.

Background and motivation

EFX requires that no agent envies another agent even after removing any single chore from her own bundle. For goods, complete EFX existence with additive valuations is open for four or more agents, with known results for three agents, few valuation types, and partial allocations. For chores, exact EFX existence had been established only under strong restrictions: two agents, at most two types of chores, instances with at most twice as many chores as agents, nearly identical-ordering costs, and three-agent personalized bi-valued instances. Nonexistence results existed only for non-additive domains such as superadditive costs. Two gaps therefore persisted: whether EFX always exists for additive chores, and whether EFX is compatible with efficiency for positive bi-valued costs. Prior EFX–PO incompatibility examples relied on zero marginal costs, and EFX together with fractional Pareto-optimality (fPO) was known only for three bi-valued agents.

Tri-valued nonexistence

The central impossibility result states that for every $n \geq 4$, there exists a tri-valued additive chore instance with no EFX allocation, using only three distinct chore types. The number of types is tight, since EFX allocations are known to exist for two types of chores.

The $n=4$ construction uses 13 chores in three groups: three "large" items valued at 20 by everyone, and two groups $B$ and $C$ of five items each, where agents 1 and 2 value $B$-items at 1 and $C$-items at 7, and agents 3 and 4 the reverse. The proof is a chain of counting constraints. First, no bundle may contain two large items: each agent's total cost is 100, so some bundle costs at most 25, while two large items leave a cost of at least 20 after removing the cheapest item, forcing all other bundles to cost exactly 20 — impossible because the five $C$-items cannot be distributed under that cap. Hence exactly one agent receives no large item. Second, every agent values the no-large-item bundle at least 20, which forces that bundle to contain at least three $B$-items and three $n \geq 4$0-items, yielding an EFX threshold of at least 23 after removing a cost-1 item. But then at most two of the remaining three agents can receive $n \geq 4$1-items, so at least one receives at most two $n \geq 4$2-items plus its large item — a cost of at most 22 — and is strongly envied. Contradiction.

The general-$n \geq 4$3 version partitions agents into groups of sizes $n \geq 4$4 and $n \geq 4$5, with cost levels $n \geq 4$6, $n \geq 4$7, and $n \geq 4$8, where $n \geq 4$9. The proof exploits a modular arithmetic obstruction ($n \geq 4$0) to rule out two-large-item bundles, and then shows the unique no-large-item bundle must contain at least $n \geq 4$1 items from each of $n \geq 4$2 and $n \geq 4$3, forcing every other bundle to contain a $n \geq 4$4-item while too few remain. The implication is decisive: EFX for additive chores is not guaranteed under any bounded number of positive cost levels above two, separating the chore setting from the goods setting, where no additive counterexample is known.

EFX–Pareto incompatibility for bi-valued instances

The second theorem shows that for every $n \geq 4$5 and $n \geq 4$6, there is a bi-valued instance with costs in $n \geq 4$7 such that every EFX allocation is Pareto-dominated. This is the first incompatibility example with strictly positive costs; prior examples exploited zero-cost items. Since EFX+fPO allocations exist for three bi-valued agents, the threshold $n \geq 4$8 is tight.

The construction uses $n \geq 4$9 chores: $n=4$0 large items valued at $n=4$1 by everyone, plus groups $n=4$2 (size $n=4$3) and $n=4$4 (size $n=4$5) that are small for one agent group and large for the other. The proof shows any EFX allocation must give every agent at least one large item (by counting arguments against the $n=4$6 available large items), forcing every agent's cost to be at least $n=4$7, with at least one agent at $n=4$8 because some agent must receive an "across-group" item. A Pareto-dominating allocation is then constructed explicitly: the agent with cost $n=4$9 is rearranged so that one agent in the opposite group receives only two small items of cost $B$0, strictly improving that agent while weakly improving all others.

An important corollary, combined with the paper's third result, is that for four positive bi-valued agents, EFX existence and EFX+PO compatibility diverge: an EFX allocation always exists, yet some instances admit no Pareto-optimal EFX allocation.

EFX existence for four bi-valued agents

The positive result establishes that every four-agent bi-valued instance admits an EFX allocation. For $B$1, a variant of round-robin (with a "first reverse round" when the chore count is not divisible by four) suffices for any number of agents. The main work assumes $B$2 and proceeds by classifying chores by the number of agents for whom they are small.

Insertion of easy items. Chores small for at least three of the four agents ($B$3) can be inserted one at a time into any EFX allocation. The proof manipulates the most-envy graph: cycles are eliminated by bundle rotations (which strictly decrease costs along the cycle), and a new item is assigned to a sink, with a path-rotation argument handling the case where the unique sink is the unique agent for whom the item is large. A remark notes this technique generalizes: under bi-valued costs, items large for at most one agent are never an obstacle.

The $B$4 prefix. Items small for at most one agent ($B$5) are allocated in "canonical" form: writing $B$6, each agent receives $B$7 or $B$8 items, prioritizing uniquely-small items. Canonical allocations are always EFX (envy-free when $B$9), and a super-canonical strengthening gives short agents a cost advantage of at least $C$0 over long agents, with existence guaranteed for any choice of short agents — a flexibility exploited later.

The $C$1 multigraph algorithm. Items small for exactly two agents ($C$2) are modeled as a multigraph on four vertices. Deficiency $C$3 measures whether a set has too few incident small items. When no set is deficient, a balanced orientation (via a Hall-type flow argument) gives every agent only small items. When a singleton or pair is maximally deficient, explicit constructions — including careful integer interval arguments for choosing split counts — produce EFX allocations. The paper further characterizes the only two exceptional residual configurations in which some agent must use a large item as the EFX-removal item, and shows when a prescribed agent can be made residual-favorite (envy-free in the residual).

Concatenation. The final phase combines the prefix and residual allocations case by case over $C$4 and the multiplicity structure of the $C$5 graph. Some $C$6 items are used for "gap filling" to make the prefix envy-free; a composition lemma then ensures the concatenation is EFX provided residual envy is certified by removing small items. Exceptional configurations — where one agent must remove a large item — are handled by choosing the disadvantageous agent so that either her final bundle contains only large items or an $C$7 prefix advantage compensates for the large-item removal. The case analysis is extensive, and its correctness rests on the exact arithmetic structure of bi-valued costs; the paper does not claim the approach extends beyond four agents.

Limitations and open problems

The paper's negative results are tight but leave the natural next cases open. Whether an EFX allocation always exists for three agents with general additive costs remains unresolved, as does existence for bi-valued instances with more than four agents — the insertion and concatenation techniques here are specific to $C$8. For goods, additive EFX existence remains open even though nonexistence is now known for submodular and subadditive valuations, and the compatibility of EFX with PO for personalized bi-valued goods is unknown. The incompatibility result for bi-valued chores concerns plain Pareto-optimality; whether every bi-valued instance with $C$9 admits an EFX allocation that is fractionally Pareto-optimal is not settled by this work.

Conclusion

This paper resolves the EFX existence question for additive chores in the negative, via a tri-valued counterexample that is tight on both the number of agents and the number of chore types, and demonstrates that EFX and Pareto-optimality are incompatible for positive bi-valued costs with four or more agents. The accompanying existence theorem for four bi-valued agents completes the small-$B$0 picture and shows that existence and efficiency compatibility genuinely diverge in this setting. The results establish a structural asymmetry between goods and chores in discrete fair division and leave the three-agent additive case and the beyond-four bi-valued case as the immediate open problems.