Pair-wise Maximin Share (PMMS) Fairness

Updated 2 July 2025

Pair-wise Maximin Share (PMMS) fairness is a refined fair division criterion that guarantees each agent receives a bundle at least as valuable as the worst split in every two-agent partitioning of combined goods.
It bridges traditional maximin share (MMS) and envy-freeness notions by offering a stronger local fairness guarantee through pairwise comparisons.
Practical algorithms, such as modified envy-cycle elimination and round-robin strategies, achieve approximate PMMS bounds of 2/3 to 4/7 under additive valuations.

Pair-wise Maximin Share (PMMS) fairness is a central concept in the algorithmic paper of fair division, particularly for indivisible goods. PMMS refines the classic maximin share (MMS) criterion by evaluating fairness at the level of every agent pair, rather than with respect to the entire population. PMMS and its approximate relaxations have received significant attention because they offer a more nuanced local fairness guarantee, bridging the gap between share-based notions and relaxations of envy-freeness.

1. Theoretical Foundations of PMMS Fairness

The pair-wise maximin share (PMMS) of an agent $i$ with respect to another agent $j$ is defined as follows. Consider the goods assigned to both $i$ and $j$ in a given allocation, i.e., $A_i \cup A_j$ . Agent $i$ considers herself as the partitioning agent: she partitions this set into two bundles and the other agent selects their preferred bundle first, leaving $i$ with the bundle of least value. The best guarantee $i$ can achieve is: $\mu^2_i(A_i \cup A_j) = \max_{(B_1, B_2) \in \Pi_2(A_i \cup A_j)} \min\{ v_i(B_1), v_i(B_2) \}$ An allocation is called PMMS if for all $i$ and $j$ : $v_i(A_i) \ge \mu^2_i(A_i \cup A_j)$ Approximate PMMS fairness is defined by requiring $v_i(A_i) \geq \alpha \cdot \mu^2_i(A_i \cup A_j)$ for some $0 < \alpha < 1$ .

PMMS strictly strengthens MMS: every PMMS allocation is MMS, but not vice versa (1806.03114). PMMS is also strictly stronger than EFX (envy-freeness up to any good), but, perhaps surprisingly, does not provide better worst-case MMS approximation than EFX (1806.03114).

2. Algorithmic and Approximation Results

Existence and Computation

Additive Valuations: For additive agents, a $2/3$-PMMS allocation exists and can be computed efficiently via a modification of envy-cycle elimination and agent ordering strategies (1909.07650). The general round-robin algorithm guarantees a $1/2$-PMMS allocation in polynomial time (1806.03114).
EFX and PMMS: For all $n \geq 4$ , both EFX and PMMS allocations guarantee each agent a $4/7$-approximation to her MMS; for $n = 3$ , the bound is $2/3$ (1806.03114). These bounds are tight up to small tolerances.
Identical Ordinal Preferences: With all agents sharing the same preference order over goods, one can efficiently compute a $2/3$-PMMS allocation (1806.03114).

Impossibility and Lower Bounds

It is demonstrated that an exact MMS allocation does not imply any constant-factor PMMS guarantee, and vice versa (1806.03114).
For PMMS, there exist instances where no allocation achieves better than a $4/7$-approximation for MMS, and approximate PMMS can provide no guarantee of even approximate EFX (1806.03114).

3. Practical Algorithms

Key approaches for approximating PMMS include:

Envy-cycle elimination and round-robin algorithms: These are adapted to ensure that, after each agent takes their turn acquiring an item, cycles of envy among agents are identified and eliminated by reassigning bundles accordingly (1909.07650). This approach enforces both EF1 and strong PMMS (up to $2/3$-approximate) in practice.
Modified envy graphs: Fine-tuning the envy graph—by only preserving edges representing strong envy (e.g., where an agent values the other's bundle significantly more)—can further improve PMMS approximation up to $0.717$ in theory (1909.07650).
Fractional and partial guarantee algorithms: There are algorithms which, rather than insisting on PMMS for all agents, ensure that most agents (for example, at least $2/3$ of the population) receive their MMS, and thus PMMS fairness is satisfied within a large subpopulation (2105.09383).

4. PMMS in Broader Fairness Landscapes

PMMS interacts in non-trivial ways with other fairness notations:

Relation to MMS and EFX: PMMS is strictly stronger than both, but does not ensure strictly better MMS approximations. Both PMMS and EFX guarantee the same worst-case MMS bound ($4/7$ for $n \geq 4$ ) (1806.03114).
Groupwise MMS (GMMS): GMMS strengthens PMMS by enforcing that, for every coalition of agents, each agent gets at least her MMS with respect to that group. GMMS allocations (when they exist) are also PMMS, but not conversely (1711.07621).
Chores and Submodular Costs: For chores or submodular utilities, PMMS and MMS do not imply each other, and the price of fairness can become unbounded as the number of agents increases (2101.07435). Efficient algorithms under such general valuations are either unknown or only provide very weak guarantees.

5. Illustrative Example and Tightness Constructions

An instructive case (adapted from (1806.03114)):

Agent	a	b	c	d	e
1	3	1	1	1	4
2	4	3	3	1	4
3	3	2	1	3	4

For $n=3$ , a PMMS allocation corresponds to each agent receiving their $2$-maximin share with respect to any other agent. In certain allocations (carefully constructed with "large" and "small" bundles), only the minimal theoretical guarantee (e.g., $2/3$ of MMS) can be achieved, highlighting the necessity and tightness of approximation bounds.

6. Open Questions and Future Directions

Active research questions include:

Is there a polynomial-time algorithm achieving a PMMS guarantee better than $2/3$ for all agents and general additive valuations?
How can PMMS guarantees be meaningfully extended to settings with submodular or other non-additive utilities, or under combinatorial or connectivity constraints?
What is the precise boundary of compatibility between PMMS, Pareto efficiency, and other pragmatic considerations in fair division applications (chore division, resource scheduling, etc.)?

Empirical analysis suggests that in many practical random instances, PMMS (and even stronger groupwise fairness) is almost always attainable by simple algorithms (1711.07621, 2105.09383). However, theoretical impossibility results persist for worst-case constructions, indicating that new algorithmic paradigms may be necessary to achieve robust and strong PMMS guarantees across all domains.

7. Summary Table: PMMS Approximation for Additive Valuations

Algorithmic Setting	PMMS Guarantee	Existence	Efficiency
Additive, general	$2/3$ (tight)	Always	Polynomial-time
Additive, identical ordering	$2/3$	Always	Polynomial-time
Round-robin, general	$1/2$	Always	Polynomial-time
EFX/EF1 via envy elimination	$0.667 \rightarrow 0.717$	Always	Poly (with tuning)
Submodular	$O(0.2)$ (inspired)	Always	Poly (weak)
Chore division	$3/2$ (EF1 implies)	Always for 2	Poly for EF1
Groupwise MMS (GMMS)	$\approx 0.5$	Always	Poly

PMMS fairness thus constitutes a robust, pairwise, and local-fairness guarantee with strong algorithmic implications and clear connections to broader fairness criteria. While efficient and strong PMMS approximations exist for additive valuations, key challenges—especially for richer utility models, chores, and more intricate combinatorial constraints—remain the subject of active investigation.