Mixed Manna in Fair Division

• Mixed manna is a fair-division framework where items can be goods, chores, or neutral, and their desirability may vary across agents.
• The literature applies key fairness notions such as MMS, envy-based relaxations (EF1, IEF1, EFR-k), and Pareto efficiency to manage both positive and negative utilities.
• Algorithmic methods and market formulations, including competitive equilibria and fixed-parameter approaches, address the computational challenges in mixed manna.

Mixed manna is the fair-division setting in which the endowment contains items that may be goods, chores, or neutral, and the same item may be desirable for one agent and undesirable for another. In the standard indivisible model, agents have additive utilities $v_i(S)=\sum_{j\in S} v_i(j)$ with item values that may be positive, zero, or negative; the literature studies fairness notions such as maximin share (MMS), envy-based relaxations, Pareto efficiency, and competitive equilibrium, and extends the model to unequal entitlements, non-additive structured valuations, and scheduling-constrained allocation (Garg et al., 2024, Hsu, 2024, Bogomolnaia et al., 2017, Jansen et al., 2024).

1. Formal setting and variants

A canonical mixed-manna instance is a tuple $(N,M,V,w)$, where $N=[n]$ is the agent set, $M=[m]$ is the item set, $V=(v_i)_{i\in N}$ is a valuation profile, and $w=(w_i)_{i\in N}$ is an entitlement vector when unequal claims are modeled. In the additive case,

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

Items may therefore be strictly positive, strictly negative, or zero for a given agent (Garg et al., 2024).

In the weighted mixed-manna model with unequal entitlements, an item $j$ is called a good if $v_i(j)>0$ for some agent $i$, neutral if $(N,M,V,w)$0 for some $(N,M,V,w)$1 and $(N,M,V,w)$2 for all agents, and a chore if $(N,M,V,w)$3 for all agents. A good is pure if $(N,M,V,w)$4 for every agent, but mixed manna allows non-pure goods that are good for some agents and bad for others (Garg et al., 2024). A closely related classification writes the item set as the union of mixed items, goods, and bads: $(N,M,V,w)$5

$(N,M,V,w)$6

which makes the sign heterogeneity explicit (Aleksandrov, 2020).

The divisible analogue is an exchange or Fisher-type economy in which each item may carry a positive, negative, or zero equilibrium price. Under additively separable piecewise linear concave utilities, mixed manna includes goods that everyone likes, bads that everyone dislikes, and items that some agents like while others dislike (Chaudhury et al., 2020). A further generalization embeds mixed manna into scheduling: items become jobs, agents become machines, each job has a processing time and a deadline, and an allocation is feasible only if each machine’s assigned jobs admit an earliest-deadline-first schedule satisfying all deadlines. Setting all processing times and deadlines to zero recovers classical fair division of indivisible mixed goods and chores (Jansen et al., 2024).

2. Fairness notions used in mixed manna

The central benchmark in the MMS literature is the maximin share. For an agent $(N,M,V,w)$7,

$(N,M,V,w)$8

where $(N,M,V,w)$9 is the set of $N=[n]$0-partitions of the item set. An allocation $N=[n]$1 is an MMS allocation if $N=[n]$2 for every $N=[n]$3 (Hsu, 2024). In mixed manna, $N=[n]$4 may be positive or negative, and an $N=[n]$5-MMS guarantee is therefore sign-sensitive: $N=[n]$6 where $N=[n]$7 denotes the maximin share of agent $N=[n]$8 (Kulkarni et al., 2020).

Envy-based notions are more varied because both removing a good from another bundle and removing a chore from one’s own bundle can mitigate envy. With unequal entitlements, weighted envy-freeness compares utility per entitlement, $N=[n]$9, and weighted envy-freeness up to one item (WEF1) allows either removing one good from the envied bundle or one chore from the envying agent’s own bundle. The same paper introduces weighted envy-freeness up to one transfer (WEF1T), which permits a single-item transfer operation as the corrective move and admits a polynomial-time algorithm (Garg et al., 2024).

A separate line develops “proximate” envy relaxations. Introspective envy-free up to one item (IEF1) requires that each agent can eliminate envy toward all other agents by adding one item to, or removing one item from, that agent’s own bundle. Formally, an allocation $M=[m]$0 is IEF1 if for every agent $M=[m]$1 there exists $M=[m]$2 with $M=[m]$3 such that

$M=[m]$4

where $M=[m]$5 denotes symmetric difference. In chores-only instances, IEF1 coincides with EF1 for chores (Barman et al., 23 Sep 2025).

Another global relaxation is envy-freeness up to $M=[m]$6 reallocations (EFR-$M=[m]$7). An allocation $M=[m]$8 is EFR-$M=[m]$9 if there exists a set $V=(v_i)_{i\in N}$0 of at most $V=(v_i)_{i\in N}$1 items such that, for each agent $V=(v_i)_{i\in N}$2, reassigning only items from $V=(v_i)_{i\in N}$3 produces an allocation $V=(v_i)_{i\in N}$4 that is envy-free for $V=(v_i)_{i\in N}$5. This notion is tailored to mixed manna because it controls the distance from exact envy-freeness by a common reallocation budget rather than by pairwise single-item deletions (Barman et al., 5 Jul 2025).

Mixed manna has also generated further variants, including jealousy-freeness up to one item and stronger “by parts” refinements such as EFX$V=(v_i)_{i\in N}$6, EFX$V=(v_i)_{i\in N}$7, and EF1$V=(v_i)_{i\in N}$8, designed to treat goods, bads, mixed items, and dummies explicitly in additive settings (Aleksandrov, 2020, Aleksandrov et al., 2020).

3. Existence, non-existence, and the present frontier

Exact MMS allocations do not behave in mixed manna as they do in goods-only models. Nonetheless, there are positive existence theorems. For additive valuations under same-order preferences, an MMS allocation exists whenever $V=(v_i)_{i\in N}$9 and at least one of the following holds: $w=(w_i)_{i\in N}$0; every agent is a chores agent; or the instance contains a non-negative agent, meaning an agent whose MMS guarantee is non-negative. These are stated as the first non-trivial exact MMS existence results for mixed manna (Hsu, 2024).

The general existence picture is sharply negative for approximation by a fixed factor. For every $w=(w_i)_{i\in N}$1, there are mixed-manna instances with no $w=(w_i)_{i\in N}$2-MMS allocation at all. This rules out the goods-style strategy of fixing a universal approximation factor and always targeting that benchmark (Kulkarni et al., 2020). Envy-based notions show a similar split. Under unequal entitlements, WEF1 is not known to exist in full generality; the literature identifies failures of ordinal methods and failures of two-phase “goods then chores” or “chores then goods” constructions, and introduces WEF1T precisely because WEF1 remained elusive in general mixed-manna settings (Garg et al., 2024).

More recent work establishes weaker but systematic fairness-efficiency combinations. One result proves that every mixed-manna instance with additive valuations admits an allocation that is both Pareto optimal and IEF1. Because IEF1 coincides with EF1 for chores, this extends the chores-only PO+EF1 result to the mixed setting through a different fairness notion (Barman et al., 23 Sep 2025). Another result proves that every mixed-manna instance admits an allocation that is both EFR-$w=(w_i)_{i\in N}$3 and Pareto optimal, and that the personalized envy-free allocations induced by the EFR-$w=(w_i)_{i\in N}$4 witness are themselves Pareto optimal. The same work shows that the $w=(w_i)_{i\in N}$5 bound is tight for chores and that $w=(w_i)_{i\in N}$6 is tight for goods (Barman et al., 5 Jul 2025).

The coexistence of stronger fairness and efficiency remains incomplete. The existence of Pareto-optimal EF1 allocations for general indivisible mixed manna is still presented as an open problem (Barman et al., 23 Sep 2025). On the negative side, jealousy-freeness up to one item may fail to exist even in small mixed instances, and deciding whether a JF1 allocation exists is NP-hard (Aleksandrov, 2020).

4. Competitive division and market formulations

For divisible mixed manna, competitive equilibrium is the dominant formal framework. In homothetic, concave, continuous economies, a competitive division $w=(w_i)_{i\in N}$7 consists of an allocation $w=(w_i)_{i\in N}$8, prices $w=(w_i)_{i\in N}$9, and a common budget $v_i(S)=\sum_{j\in S} v_i(j).$0. The theory partitions problems into positive, negative, and null cases according to the position of the zero utility profile relative to the feasible utility set. If the zero profile is Pareto dominated, the competitive utility profile is unique and maximizes the product of utilities; if the zero profile is unfeasible, the competitive utility profiles are the critical points of the product of disutilities on the efficiency frontier; and in null problems the competitive utility profile is the zero vector (Bogomolnaia et al., 2017).

Under additively separable piecewise linear concave utilities, mixed manna admits a direct computational treatment. A simplex-like algorithm based on Lemke’s complementary pivoting scheme computes competitive allocations in the exchange setting, strictly generalizing the linear-utility case. The same framework yields a constructive existence proof under strong connectivity, establishes PPAD membership, gives rational-valued solutions, and proves the odd-number-of-solutions property conjectured for mixed-manna equilibria (Chaudhury et al., 2020).

Market ideas also reappear in indivisible mixed manna with unequal entitlements. A mixed-manna Fisher market uses positive prices for goods, negative prices for chores, and zero prices for neutral items, together with best-bang-per-buck conditions adapted to sign heterogeneity. Market equilibria in that framework are fractionally Pareto-optimal, and for two agents a local-search algorithm on integral equilibria yields a WEF1 allocation that is also fractionally Pareto-optimal (Garg et al., 2024).

5. Complexity and algorithmic methods

The algorithmic complexity of mixed manna is highly sensitive to structure. In the generalized scheduling formulation of MMS, computing exact or optimal approximate MMS allocations is hard in general, but fixed-parameter tractable algorithms are available under three minimal parameterizations: by the number of jobs $v_i(S)=\sum_{j\in S} v_i(j).$1; by $v_i(S)=\sum_{j\in S} v_i(j).$2 for grouped instances; and by $v_i(S)=\sum_{j\in S} v_i(j).$3 for deadline-structured instances. The same framework computes optimal multiplicative MMS allocations, optimal additive shortfall, and welfare-optimal MMS, and the $v_i(S)=\sum_{j\in S} v_i(j).$4 dependence for the $v_i(S)=\sum_{j\in S} v_i(j).$5-parameterization is essentially optimal under ETH (Jansen et al., 2024).

For indivisible mixed manna with additive utilities, one tractability route is approximation under strong structure. When the number of agents is constant and, for every agent, the absolute value of the total utility is at least a constant fraction of the total absolute value of all goods or all chores, there is a PTAS that, given $v_i(S)=\sum_{j\in S} v_i(j).$6, finds an $v_i(S)=\sum_{j\in S} v_i(j).$7-MMS and $v_i(S)=\sum_{j\in S} v_i(j).$8-PO allocation for the highest possible $v_i(S)=\sum_{j\in S} v_i(j).$9. If either of these two conditions is dropped, finding an $j$0-MMS allocation for any $j$1 becomes NP-hard, even when an exact MMS allocation exists (Kulkarni et al., 2020).

Envy-based relaxations admit more direct algorithms. EFR-$j$2 allocations can be computed in polynomial time for general mixed manna, and EFR-$j$3+PO can be computed efficiently when $j$4 is fixed (Barman et al., 5 Jul 2025). WEF1T under unequal entitlements also admits a polynomial-time algorithm obtained by computing WEF1 allocations separately on goods and chores and then combining them (Garg et al., 2024). By contrast, deciding the existence of JF1 or JF1+PO allocations is NP-hard in settings already contained in mixed manna (Aleksandrov, 2020).

6. Structured valuation classes, applications, and terminological disambiguation

Mixed manna has also been studied beyond additivity. Under order-neutral submodular valuations with marginal values in $j$5, a polynomial-time algorithm computes leximin allocations. These leximin allocations are Lorenz dominating and approximately proportional; under the further restriction to additive valuations, the same leximin allocations are approximately envy-free and guarantee each agent their maximin share. The same paper also proves that computing leximin allocations becomes NP-hard when $j$6 is rational (Cousins et al., 2023).

Application domains in the literature are explicitly mixed in both sign and feasibility. Unequal-entitlement models motivate settings with assets and liabilities, enjoyable and undesirable tasks, and inheritance containing both property and obligations (Garg et al., 2024). The scheduling generalization models rides, shifts, or tasks with agent-dependent positive or negative utilities, together with deadlines and processing times, and therefore treats fairness and capacity constraints in a unified way (Jansen et al., 2024).

The term should be distinguished from the unrelated statistical-physics usage of “Manna,” where it refers to the Manna sandpile or Manna universality class rather than to a resource endowment. In that literature, “Manna” denotes absorbing-state and self-organized-critical systems, including exact mappings from conserved directed percolation to quenched Edwards–Wilkinson interface models and analyses of dynamic hyperuniformity and density relaxation in conserved Manna sandpiles (Doussal et al., 2014, Mukherjee et al., 2022, Tapader et al., 2020).