Maximin share (MMS) is a criterion of fair item allocation. Given a set of items with different values, the 1-out-of-n maximin-share is the maximum value that can be gained by partitioning the items into n parts and taking the part with the minimum value. An allocation of items among n agents with different valuations is called MMS-fair if each agent gets a bundle that is at least as good as his/her 1-out-of-n maximin-share. MMS fairness was invented by Eric Budish as a relaxation of the criterion of proportionality - each agent gets a bundle that is at least as good as the equal split (1/n of every resource). Proportionality can be guaranteed when the items are divisible, but not when they are indivisible, even if all agents have identical valuations. In contrast, MMS fairness can always be guaranteed to identical agents, so it is a natural alternative to proportionality even when the agents are different.
Identical items. Suppose first that m identical items have to be allocated fairly among n people. Ideally, each person should receive m/n items, but this may be impossible if m is not divisible by n, as the items are indivisible. A natural second-best fairness criterion is to round m/n down to the nearest integer, and give each person at least floor(m/n) items. Receiving less than floor(m/n) items is "too unfair" - it is an unfairness not justified by the indivisibility of the items.
Different items. Suppose now that the items are different, and each item has a different value. For example, suppose n=3 and m=5 and the items' values are 1, 3, 5, 6, 9, adding up to 24. If the items were divisible, we would give each person a value of 24/3 = 8 (or, if they were divisible only to integer values as in the preceding paragraph, at least floor(24/3) = 8), but this is not possible. The largest value that can be guaranteed to all three agents is 7, by the partition {1,6},{3,5},{9}. Informally, 7 is the total value divided by n "rounded down to the nearest item".
The set {1,6} attaining this maximin value is called the "1-out-of-3 maximin-share" - it is the best subset of items that can be constructed by partitioning the original set into 3 parts and taking the least valuable part. Therefore, in this example, an allocation is MMS-fair iff it gives each agent a value of at least 7.
Different valuations. Suppose now that each agent assigns a different value to each item, for example:
Now, each agent has a different MMS:
Here, an allocation is MMS-fair if it gives Alice a value of at least 7, George a value of at least 8, and Dina a value of at least 3. For example, the in which George gets the first two items {1,7}, Alice gets the next two items {5,6}, and Dina gets the last item {17} is MMS-fair.
Interpretation. The 1-out-of-n MMS of an agent can be interpreted as the maximal utility that an agent can hope to get from an allocation if all the other agents have the same preferences, when he always receives the worst share. It is the minimal utility that an agent could feel entitled to, based on the following argument: if all the other agents have the same preferences as me, there is at least one allocation that gives me this utility, and makes every other agent (weakly) better off; hence there is no reason to give me less.
An alternative interpretation is: the most preferred bundle the agent could guarantee as divider in divide and choose against adversarial opponents: the agent proposes her best allocation and leaves all the other ones to choose one share before taking the remaining one.[1]
MMS-fairness can also be described as the result of the following negotiation process. A certain allocation is suggested. Each agent can object to it by suggesting an alternative partition of the items. However, in doing so he must let all other agents chose their share before he does. Hence, an agent would object to an allocation only if he can suggest a partition in which all bundles are better than his current bundle. An allocation is MMS-fair iff no agent objects to it, i.e., for every agent, in every partition there exists a bundle which is weakly worse than his current share.
When all n agents have identical valuations, an MMS-fair allocation always exists by definition.
Even when only n-1 agents have identical valuations, an MMS-fair allocation still exists and can be found by divide and choose: the n-1 identical agents partition the items into n bundles each of which is at least as good as the MMS; the n-th agent chooses a bundle with a highest value; and the identical agents take the remaining n-1 bundles.
In particular, with two agents, an MMS-fair allocation always exists.
Bouveret and Lemaître[1] performed extensive randomized simulations for more than 2 agents, and found that MMS-fair allocations exist in every trial. They proved that MMS allocations exist in the following cases:
Procaccia and Wang[2] and Kurokawa[3] constructed an instance with n=3 agents and m=12 items, in which no allocation guarantees to each agent the 1-out-of-3 MMS. In their instance, there are 12 objects, indexed by [math]\displaystyle{ i\in[3] }[/math] and [math]\displaystyle{ j\in[4] }[/math]. Each agent [math]\displaystyle{ k }[/math] values each object [math]\displaystyle{ (i,j) }[/math] by:
[math]\displaystyle{ v_k(i,j) = 1,000,000 + 1,000\cdot T_{i,j} + E_{i,j}^{(k)} }[/math]
where [math]\displaystyle{ T, E^{(1)}, E^{(2)}, E^{(3)} }[/math] are particular 3-by-4 matrices with values smaller than 100. They prove that every agent can partition the objects into 3 subsets of 4 objects each, such that the sum of values in each subset is 4,055,000, which is therefore the MMS of all agents. They prove that every MMS allocation must give exactly 4 particular objects to every agent, but such an allocation does not exist. Thus, every allocation gives at least one agent a value of at most 4,054,999.
They generalized this instance and showed that for every n ≥ 3 there is such an instance with 3n+4 items.
Following the non-existence result for MMS allocations, Procaccia and Wang introduced the multiplicative approximation to MMS: an allocation is r-fraction MMS, for some fraction r in [0,1], if each agent's value is at least a fraction r of the value of his/her MMS.
They presented an algorithm that always finds an rn-fraction MMS, where
[math]\displaystyle{ r_n := \frac{2\cdot \text{oddfloor}(n)}{3\cdot \text{oddfloor}(n) -1} = \begin{cases} \frac{2n}{3n-1} & n ~ \text{ odd} \\ \frac{2n-2}{3n-4} & n ~ \text{ even} \end{cases} }[/math]
where oddfloor(n) is the largest odd integer smaller or equal to n. In particular, r3 = r4 = 3/4, it decreases when n increases, and it always larger than 2/3. Their algorithm runs in time polynomial in m, when n is constant.
Amanatidis, Markakis, Nikzad and Saberi[4] proved that, in randomly-generated instances, MMS-fair allocations exist with high probability. They presented several improved algorithms:
Aziz, Rauchecker, Schryen and Walsh[5] presented:
Barman and Krishnamurthy[6] presented:
Ghodsi, Hajiaghayi, Seddighin, Seddighin and Yami[7] presented:
Garg, McGlaughlin and Taki[8] presented a simple algorithm for 2/3-fraction MMS whose analysis is also simple.
Garg and Taki[9] presented a simple algorithm for 3/4-fraction MMS, which does not need to know the MMS value, and thus runs in strongly polynomial time. They also prove the existence of [math]\displaystyle{ (\frac{3}{4} + \frac{1}{12 n}) }[/math]-fraction MMS.
To date, it is not known what is the largest r such that an r-fraction MMS allocation always exists. It can be any number between 3/4 and slightly less than 1.
Huang and Lu[10] prove that a 11/9-fraction MMS allocation for chores always exists.
Kulkarni, Mehta and Taki[11] study MMS allocation with mixed valuations, i.e., when there are both goods and chores. They prove that:
Amanatidis, Birmpas and Markakis[12] presented truthful mechanisms for approximate MMS-fair allocations (see also Strategic fair division):
Budish[13] introduced a different approximation to the 1-of-n MMS—the 1-of-([math]\displaystyle{ n+1 }[/math]) MMS (each agent receives at least as much as he could get by partitioning into n+1 bundles and getting the worst one). He showed that the Approximate Competitive Equilibrium from Equal Incomes always guarantees the 1-of-([math]\displaystyle{ n+1 }[/math]) MMS, However, this allocation may have excess supply, and - more importantly - excess demand: the sum of the bundles allocated to all agents might be slightly larger than the set of all items. Such an error is reasonable when allocating course seats among students, since a small excess supply can be corrected by adding a small number of seats. But the classic fair division problem assumes that items may not be added.
Without excess supply and demand, the following approximations are known:
To date, it is not known what is the smallest N such that a 1-out-of-N MMS allocation always exists. It can be any number between n+1 and 3n/2. The smallest open case is n=4.
The ordinal MMS condition can also be applied to asymmetric agents (agents with different entitlements).[16]
Various normalizations can be applied to the original problem without changing the solution. Below, O is the set of all objects.
If, for each agent i, all valuations are scaled by a factor ki (which can be different for different agents), then the MMS for each agent is scaled by the same factor; therefore, every MMS allocation in the original instance is an MMS allocation in the scaled instance. It is common to scale the valuations such that the MMS of evey agent is exactly 1. After this scaling, the MMS approximation problems can be stated as:
The above scaling requires to compute the MMS of each agent, which is an NP-hard problem (multiway number partitioning). An alternative scaling, that can be done faster, is:[8]
If we remove one object o from O. Then for each agent, the 1-out-of-(n-1) MMS w.r.t. the remaining set O \ o is at least his 1-out-of-n MMS w.r.t. the original set O. This is because, in the original MMS partition, n-1 parts remain intact.[1] Now, suppose we aim to give each agent a value of r. If some object o1 is worth at least r to at least one agent, then we can give o1 to one such agent arbitrarily, and proceed to allocate the remaining objects to the remaining agents. Therefore, we can assume w.l.o.g. that:
This normalization works even with the fast scaling, and with arbitrary monotone valuations (even non-additive).[7]
Denote an object, that is valued by at most s by all agents, as an "s-small object". Suppose that all objects are s-small. Take an empty bag and fill it with object after object, until the bag is worth at least r to at least one agent. Then, give the bag to one such agent arbitrarily. Since all objects are s-small, the remaining agents value the bag at most r+s; if this value is sufficiently small, then the remaining value is sufficiently large so that we can proceed recursively.[8] In particular, bag-filling gives as the following solutions:
These bag-filling algorithms work even with the fast scaling, so they run in polynomial time - they do not need to know the exact MMS value.[8] In fact, both algorithms can be stated without mentioning the MMS at all:
Modified bag filling: The condition that all objects are s-small can be relaxed as follows.[8] Take some s<r. Denote an object that is not s-small (i.e., valued at least s by at least one agent) as an "s-large object". Suppose at most n objects are s-large. Take one s-large object o1, put it in a bag, and fill it with s-small objects until one agent indicates that it is worth for him at least r. There must be at least one such agent, since some agent i values o1 at some x>s. For this agent, there are at most n-1 remaining s-large objects. By the previous normalization, these objects are still r-small, so their total value for i is at most r(n-1), so the value of remaining s-small objects is at least n-r(n-1)-x = r(n-1)+r-x ≥ r-x.
an instance is ordered if all agents have the same ordinal ranking on the objects, i.e, the objects can be numbered o1, ..., om such that, for each agent i, vi(o1) ≥ ... ≥ vi(om). Intuitively, ordered instances are hardest, since the conflict between agents is largest. Indeed, the negative instance of [2] is ordered - the order of the objects is determined by the matrix [math]\displaystyle{ T }[/math], which is the same for all agents. This can also be proved formally. Suppose we have an algorithm that finds, for every ordered instance, an r-fraction MMS allocation. Now, we are given a general item-allocation instance P. We solve it in the following way.[1][6]
So when looking for r-fraction MMS allocations, we can assume w.l.o.g. that:
Suppose we find two objects o1 and o2, that one agent i values at least r, while the other agents value at most 1. Then these two objects can be allocated to i. For the other agents, the 1-out-of-(n-1) MMS w.r.t. the remaining set is at least his 1-out-of-n MMS w.r.t. the original set O. This is because, in the original MMS partition, at least n-2 parts remain intact, while the two parts that are not intact can be combined to form a single part with a value of at least 1. This normalization works only with additive valuations.[7]:Lem.3.2
Moreover, suppose that the instance is ordered, and suppose we remove from O the two objects on, on+1 (i.e., the n-th and (n+1)-th highest-valued items). Then for each agent, the 1-out-of-(n-1) MMS w.r.t. the remaining set is at least his 1-out-of-n MMS w.r.t. the original set O. This is because, by the pigeonhole principle, at least one MMS part of each agent must contain two or more objects from the set {o1, ..., on+1}. These items can be used to replace the objects given away, which results in n-1 parts with a value of at least 1. This means that, if the objects on, on+1 have a value of at least the MMS for some agent i, we can give them to i and proceed to allocate the remaining objects to the remaining agents. Therefore, we can assume w.l.o.g. that:
This normalization works even with the fast scaling. Combining it with modified bag-filling leads to the following simple algorithm for 2/3-fraction MMS.[8]
The guarantee of this algorithm can be stated even without mentioning MMS:
Several basic algorithms related to the MMS are:
An allocation is called pairwise-maximin-share-fair (PMMS-fair) if, for every two agents i and j, agent i receives at least his 1-out-of-2 maximin-share restricted to the items received by i and j. It is not known whether a PMMS allocation always exists, but a 0.618-approximation always exists.[20]
An allocation is called groupwise-maximin-share-fair (GMMS-fair) if, for every subgroup of agents of size k, each member of the subgroup receives his/her 1-out-of-k maximin-share restricted to the items received by this subgroup.[21]
With additive valuations, the various fairness notions are related as follows:
GMMS allocations are guaranteed to exist when the valuations of the agents are either binary or identical. With general additive valuations, 1/2-GMMS allocations exist and can be found in polynomial time.[21]
An allocation is called envy-free-except-c-items (EFc) for an agent i if, for every other agent j, there exists a set of at most c items that, if removed from j's bundle, then i does not envy the remainder. An EF0 allocation is simply called envy-free. EF1 allocations can be found, for example, by round-robin item allocation or by the envy-graph procedure.
An allocation is called proportional-except-c-items (PROP*c) for an agent i if there exists a set of at most c items outside i's bundle that, if removed from the set of all items, then i values his bundle at least 1/n of the remainder. A PROP*0 allocation is simply called proportional.
EF0 implies PROP*0, and EF1 implies PROP*(n-1). Moreover, for any integer c 0, EFc implies PROP*((n-1)c).[23]:Lem.2.3 The opposite implication is true when n=2, but not when n>2.
The following maximin-share approximations are implied by PROP*(n-1), hence also by EF1:[23]:Lem.2.7
The above implications are illustrated below:
Envy-free | → | Proportional | → | Maximin-share |
---|---|---|---|---|
↓ | ↓ | ↓ | ||
→ | 1/n-fraction MMS | |||
EF1 | → | PROP*(n-1) | ↔ | MMS for binary valuation |
↓ | ↓ | → | 1-out-of-(2n-1) MMS | |
↓ | ||||
EFc | → | PROP*((n-1)c) | → | 1-out-of-((n-1)c+n) MMS |
An allocation is called envy-free-except-any-item (EFx) for an agent i if, for every other agent j, for any single item removed from j's bundle, i does not envy the remainder. EFx is strictly stronger than EF1. It implies the following MMS approximations:[22]:Prop.3.3-3.4
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