Maximin Share Criterion: Fairness in Indivisible Goods

• Maximin share (MMS) is a fairness standard ensuring each agent receives at least the value of the worst bundle in an optimal partition.
• It applies to both goods and chores, adapting additive valuations and approximation strategies when exact fairness is unattainable.
• MMS frameworks extend to constrained and structured settings, using techniques like dynamic reductions and ordinal relaxations to overcome feasibility and incentive challenges.

Maximin share (MMS) is a share-based fairness notion for the allocation of indivisible items. In the standard goods model with additive valuations, the maximin share of agent $i$ is the value she can guarantee for herself by partitioning the goods into $n$ bundles and then receiving the least valuable bundle according to her own valuation. An allocation is MMS if every agent receives at least her own MMS value. MMS occupies a central position in fair division because envy-freeness can always be satisfied in the divisible setting but often cannot be satisfied in the indivisible setting, and MMS is one of the principal relaxations developed for that gap (Hsu, 2022).

1. Formalization and core interpretation

For a set $M$ of indivisible goods and additive valuation functions $v_i:2^M\to \mathbb{R}_{\ge 0}$, the maximin share of agent $i$ is

$$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$

where $\Pi_n(M)$ is the set of all partitions of $M$ into $n$ bundles. An allocation $(A_1,\ldots,A_n)$ is an MMS allocation if

$n$0

The interpretation is the standard divide-and-choose guarantee under adversarial choice: agent $n$1 chooses the partition, the environment chooses the worst bundle for her, and the maximin optimization selects the best such worst-case guarantee. This same logic appears in constrained, graphical, and ordinal variants of the criterion, where the partition space is restricted or the bundle comparison relation is generalized (Hsu, 2022).

For indivisible chores, the direction is reversed. If $n$2 is the additive non-negative workload or cost assigned by agent $n$3 to subset $n$4, then the maximin share is

$n$5

and an allocation is MMS if

$n$6

Thus, for goods the agent maximizes the minimum bundle value, whereas for chores the agent minimizes the maximum bundle cost (Huang et al., 2019).

Approximate versions are standard. For goods, an allocation is $n$7-MMS if every agent receives at least $n$8. For chores, an allocation is $n$9-MMS if every agent receives cost at most $M$0. These approximations are central because exact MMS allocations need not exist in many settings (Heidari et al., 12 Oct 2025).

2. Exact existence, thresholds, and non-existence

Exact MMS allocations are not guaranteed in full generality. In the additive goods setting, MMS allocations need not exist when $M$1, and the literature has therefore focused both on exact existence in special regimes and on universal approximation guarantees (Akrami et al., 2023).

A sharp existence threshold is known when the number of goods is close to the number of agents. Previous work showed that MMS allocations are guaranteed to exist for all instances with $M$2 players and $M$3 goods if $M$4. This was extended to $M$5, and the same guarantee fails for $M$6. Equivalently, a maximin share allocation is guaranteed to exist whenever $M$7, and there exists an instance with $M$8 agents and $M$9 goods for which no MMS allocation exists (Hsu, 2022).

Connectivity constraints produce a different exact/non-exact boundary. For indivisible goods arranged on a cycle, MMS allocations are not always guaranteed: there is an explicit counterexample for three agents on a 9-cycle, while MMS allocations always exist for two agents, for three agents and at most 8 goods, and whenever the number of goods is less than $v_i:2^M\to \mathbb{R}_{\ge 0}$0 (Lonc et al., 2019). By contrast, in graphical cake cutting, when the underlying graph is a forest, an allocation satisfying maximin share fairness always exists, and this remains true even when positive separation constraints are imposed (Elkind et al., 2021).

Special valuation classes can also restore exact existence. Under cost utilities, an MMS allocation always exists for three agents, and under laminar set approvals MMS allocations are guaranteed for any number of agents (Botan et al., 2024). These results indicate that exact MMS is highly sensitive to combinatorial structure: small $v_i:2^M\to \mathbb{R}_{\ge 0}$1, acyclic topology, and restricted valuation classes can all move the problem from impossibility to universal existence.

3. Approximation guarantees and algorithmic development

Because exact existence fails in general, the main algorithmic program studies the largest universal constant $v_i:2^M\to \mathbb{R}_{\ge 0}$2 for which $v_i:2^M\to \mathbb{R}_{\ge 0}$3-MMS allocations always exist. In additive valuations, the long-standing sequence of guarantees moved from $v_i:2^M\to \mathbb{R}_{\ge 0}$4 to $v_i:2^M\to \mathbb{R}_{\ge 0}$5, then to $v_i:2^M\to \mathbb{R}_{\ge 0}$6, and later to $v_i:2^M\to \mathbb{R}_{\ge 0}$7. The $v_i:2^M\to \mathbb{R}_{\ge 0}$8 guarantee was the first to break the $v_i:2^M\to \mathbb{R}_{\ge 0}$9 barrier, using new reduction rules and refined bag-filling analysis; the subsequent $i$0 guarantee uses dynamic reduction rules, deferred matching, prioritization by agent color, pre-filled bags, and calibration functions in the analysis (Akrami et al., 2023, Heidari et al., 12 Oct 2025).

The current upper barrier in the additive setting remains close to 1. For some instances, no allocation can guarantee a factor better than $i$1 of maximin share value to all agents. This places the known constructive lower bounds and non-existence upper bounds within the same asymptotic regime but still leaves a substantial gap for finite $i$2 (Heidari et al., 12 Oct 2025).

For chores, the approximation landscape is different because the objective is minimization. A polynomial-time $i$3-approximation MMS allocation exists for arbitrary instances, improving on the previous best $i$4, and the same work gives a polynomial-time $i$5-approximation using lower-bound estimation of MMS values. The analysis is explicitly connected to First Fit Decreasing and to makespan minimization in job scheduling (Huang et al., 2019).

Another approximation axis relaxes fairness across the population rather than the value ratio. There exist allocations that guarantee MMS for $i$6 of agents, and for up to nine agents this bound can be achieved in polynomial time. A key implication is the existence of allocations that guarantee $i$7, improving the previously known guarantee of $i$8 (Hosseini et al., 2021). This suggests that population-based and partition-based relaxations can provide stronger universal guarantees than standard multiplicative approximation.

4. Ordinal, groupwise, and entitlement-sensitive generalizations

A major generalization replaces the standard $i$9-out-of-$$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$0 guarantee by $$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$1-out-of-$$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$2 maximin share. Given a finite set $$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$3 and an ordering $$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$4 over subsets, the $$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$5-out-of-$$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$6 maximin-share is

$$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$7

This notion induces a dominance relation on parameter pairs: $$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$8 dominates $$\operatorname{MMS}_i := \max_{(P_1,\dots,P_n) \in \Pi_n(M)} \min_{j \in [n]} v_i(P_j),$$9 if, for every finite set $\Pi_n(M)$0 with any subset ordering $\Pi_n(M)$1, $\Pi_n(M)$2. The relation admits a complete characterization: if $\Pi_n(M)$3 with $\Pi_n(M)$4 and $\Pi_n(M)$5, then $\Pi_n(M)$6 dominates $\Pi_n(M)$7 if and only if

$\Pi_n(M)$8

This characterization yields an algorithm for finding all non-dominated pairs relevant to ordinal maximin-share fairness (Segal-Halevi, 2019).

The ordinal approach has its own universal guarantees. For additive goods, $\Pi_n(M)$9-out-of-$M$0 MMS allocations always exist for every integer $M$1, and there is a polynomial-time algorithm for $M$2-out-of-$M$3 MMS when $M$4 (Hosseini et al., 2021). The motivation is robustness: ordinal MMS approximations depend only on bundle rankings, whereas multiplicative MMS guarantees are sensitive to small perturbations in cardinal valuations (Hosseini et al., 2021).

MMS has also been strengthened in an ex-post direction. Groupwise maximin share guarantee (GMMS) requires the maximin guarantee not only with respect to the grand bundle, but also among all subgroups of agents. Formally, for every $M$5 and every subgroup $M$6, one requires

$M$7

GMMS strictly strengthens MMS and PMMS, implies approximate envy-freeness, and in additive valuations a $M$8-approximate GMMS allocation always exists and can be computed in polynomial time (Barman et al., 2017). The same paper emphasizes a common criticism of MMS: MMS is not sufficient to rule out unsatisfactory allocations, and MMS does not imply EF1 (Barman et al., 2017).

Unequal entitlements expose another limitation. For arbitrary entitlements, previous attempts to extend MMS have shortcomings, and the AnyPrice share (APS) was introduced as an alternative benchmark. Even in the equal-entitlement case, APS is new and satisfies $M$9, where the inequality is sometimes strict. For additive valuations and arbitrary entitlements, there is a polynomial-time algorithm that gives every agent at least a $n$0-fraction of her APS (Babaioff et al., 2021). A plausible implication is that MMS remains central for equal claims, but entitlement-sensitive settings may require a different share benchmark.

5. Constraints, topology, and alternative valuation domains

A substantial body of work studies MMS under feasibility constraints. Under cardinality constraints, where items are partitioned into categories and each category has an upper bound on the number of items that may contribute to a bundle, a polynomial-time $n$1-approximate MMS allocation exists for goods in the general case, and a $n$2-approximate allocation exists for single-category instances. In the same model, exact MMS allocations always exist when the single-category threshold satisfies $n$3, while MMS allocations do not necessarily exist when $n$4. For chores, the corresponding guarantees are $n$5-approximate in the general case and $n$6-approximate for single-category instances (Hummel et al., 2021).

Lower quotas further enlarge the feasible-allocation model. With arbitrary lower and upper quotas on bundle sizes, a $n$7-MMS allocation of goods and a $n$8-MMS allocation of chores can be computed in polynomial time in the single-category case. In the multi-category case, the guarantees become $n$9-MMS for goods and $(A_1,\ldots,A_n)$0-MMS for chores (Kinoshita et al., 9 Feb 2026). These results extend earlier work on upper-only cardinality constraints.

Hereditary set systems provide a non-additive constrained model in which each agent values a bundle by the maximum total value of an independent subset. In this model, a $(A_1,\ldots,A_n)$1-approximate MMS allocation always exists, improving on an earlier $(A_1,\ldots,A_n)$2 guarantee; the existence proof is constructive but does not directly yield a polynomial-time algorithm. With valuation oracles, a $(A_1,\ldots,A_n)$3-approximate MMS allocation can be found in polynomial time, while $(A_1,\ldots,A_n)$4-approximate MMS allocations do not always exist for every $(A_1,\ldots,A_n)$5 (Hummel, 2024).

Topology changes the partition space even more sharply. For goods on cycles, exact MMS may fail, but every instance admits a polynomial-time allocation giving each agent at least $(A_1,\ldots,A_n)$6 of her MMS, with stronger guarantees of $(A_1,\ldots,A_n)$7 for at most three types and $(A_1,\ldots,A_n)$8 for three agents (Lonc et al., 2019). In graphical cake cutting, if the graph is a forest then maximin allocations exist both without separation and with positive separation constraints, whereas for general graphs one can guarantee ordinal relaxations using the feedback vertex set number (Elkind et al., 2021). Cost utilities form another structured domain: for three agents, MMS allocations always exist, and with laminar set approvals they exist for any number of agents (Botan et al., 2024).

6. Strategy, adjusted supply, and limited sharing

The MMS criterion also interacts strongly with incentive constraints. In truthful mechanism design without money, no truthful deterministic mechanism can guarantee strictly better than $(A_1,\ldots,A_n)$9-approximation in the cardinal model for $n$00 and $n$01, and the same impossibility holds in the ordinal model. In the public rankings model, stronger positive results are possible: for $n$02, a truthful exact MMS mechanism exists; for general $n$03, a truthful $n$04-approximation is achievable for two players; and for general $n$05, a picking-sequence mechanism guarantees $n$06-approximation (Amanatidis et al., 2016). These results make clear that truthfulness can substantially lower the best attainable MMS ratio.

Another line of work changes the resource model rather than the algorithmic objective. Exact MMS fairness can be achieved via limited duplication of goods or limited disposal of chores. Under monotone valuations for goods, there always exists an assignment such that every agent receives at least her maximin share and no single good is allocated to more than $n$07 agents; under additive valuations, there always exists an MMS assignment in which no single good is allocated to more than $n$08 agents and the total number of goods assigned is at most $n$09. For chores under monotone costs, there exists an MMS assignment in which at most $n$10 remain unassigned, and this bound is essentially tight (Barman et al., 6 Feb 2025).

Limited sharing yields a different relaxation. In cost-sensitive $n$11-sharing, each good may be allocated to up to $n$12 agents while incurring a sharing cost. Under the equal-share cost model, if $n$13, exact MMS is guaranteed when $n$14 is even, and for odd $n$15 one obtains an $n$16 guarantee. The Shared Bag-Filling Algorithm guarantees a $n$17-approximate MMS allocation, where $n$18 is the maximum cost of sharing a good, and it recovers exact MMS when $n$19. The same work introduces Sharing Maximin Share (SMMS), proves existence under identical utilities and for two agents, and gives a counterexample showing that universal existence of SMMS allocations is impossible (Salavcova et al., 24 Feb 2026).

Taken together, these developments show that the maximin share criterion is not a single theorem but a large research program. The core definition remains the divide-and-choose lower bound for indivisible allocation, yet the modern literature studies exact thresholds, multiplicative approximations, ordinal relaxations, groupwise strengthenings, entitlement-sensitive alternatives, feasibility constraints, topology, incentive compatibility, and resource-model relaxations. This suggests that MMS is best understood as a family of closely related fairness benchmarks whose behavior is governed by the geometry of feasible partitions and by the information and exclusivity assumptions imposed on the allocation problem.