Maximin share

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. Now, rounding down to the nearest integer may not be the right solution. For example, suppose n=3 and m=5 and the items' values are 1, 3, 5, 6, 9. The sum of values is 24, and it is divisible by 3, so ideally we would like to give each person a value of at least 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:

• Alice values them at 1,3,5,6,9;
• George values them at 1,7,2,6,8;
• Dina values them at 1,1,1,4,17.

Now, each agent has a different MMS:

• Alice's MMS is still {1,6} as above;
• George's MMS is {1,7} or {2,6} or {8}, by the partition {1,7},{2,6},{8} (all these bundles are equivalent for him);
• Dina's MMS is {1,1,1}, by the partition {1,1,1},{4},{17}.

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.[2]

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[2] 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:

• Binary valuations - each agent either likes an item (values it at 1) or dislikes it (values it at 0).
• Identical multisets - agents may value the items differently, but the multisets of the agents' values are the same.
• Few items - m ≤ n+3.

Procaccia and Wang[3] and Kurokawa[4] constructed an example with n=3 agents and m=12 items, in which no allocation guarantees to each agent the 1-out-of-3 MMS. Moreover, they showed that for every n ≥ 3 there is such an example 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 r_n-fraction MMS, where ${\displaystyle r_{n}:={\frac {2\cdot {\text{oddfloor}}(n)}{3\cdot {\text{oddfloor}}(n)-1}}}$, 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[5] proved that, in randomly-generated instances, MMS-fair allocations exist with high probability. They presented several improved algorithms:

• A simple and fast 1/2-fraction MMS algorithm;
• A 2/3-fraction MMS algorithm that runs in polynomial time in both m and n;
• A 7/8-fraction MMS algorithm for 3 agents;
• An MMS-fair algorithm for the case of ternary valuations - each value is 0 or 1 or 2.

Barman and Krishnamurthy[6] presented:

• A simple and efficient algorithm for 2/3-fraction MMS with additive valuations - based on the envy-cycles procedure.
• A simple algorithm for 1/10-fraction MMS for the more challenging case of submodular valuations - based on round-robin item allocation.

Ghodsi, Hajiaghayi, Seddighin, Seddighin and Yami[7] presented:

• For additive valuations: a polynomial-time algorithm for 3/4-fraction MMS.
• For n=4 additive agents: an algorithm for 4/5-fraction MMS.
• For submodular valuations: a polynomial-time algorithm for 1/3-fraction MMS, and an upper bound of 3/4-fraction.
• For XOS valuations: a polynomial-time algorithm for 1/8-fraction MMS, an existence proof for 1/5-fraction, and an upper bound of 1/2-fraction.
• For subadditive valuations: an existence proof for log(m)/10-fraction MMS, and an upper bound of 1/2-fraction.

Garg, McGlaughlin and Taki[8] presented a simple algorithm for 2/3-fraction MMS whose analysis is also simple.

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.

Amanatidis, Birmpas and Markakis[9] presented truthful mechanisms for approximate MMS-fair allocations (see also Strategic fair division):

• For n agents: an 1/O(m)-fraction MMS.
• For 2 agents: a 1/2-fraction MMS, and a proof that no truthful mechanism can attain more than 1/2.

Xin and Pinyan[10] study MMS-fair allocation of chores (items with negative values) and show that a 9/11-fraction MMS allocation always exists.

Budish[1] introduced a different approximation to the 1-of-n MMS—the 1-of-(${\displaystyle n+1}$) 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-(${\displaystyle n+1}$) 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.

For any number of agents with additive valuations, any envy-free-except-1 (EF1) allocation gives each agent at least the 1-out-of-(2n-1) MMS.[11] EF1 allocations can be found, for example, by round-robin item allocation or by the envy-cycles procedure.

Moreover, a 1-out-of-(2n-2) MMS allocation can be found using envy-free matching.[12]

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 2n-2. The smallest open case is n=4.

The ordinal MMS condition can also be applied to asymmetric agents (agents with different entitlements).[13]

Several basic algorithms related to the MMS are:

• Calculating the MMS of a given agent. It is NP-hard even for agents with additive valuations: it can be reduced from the partition problem. Woeginger[14] developed a PTAS for it.
• Deciding whether a given allocation is 1-of-${\displaystyle n}$ MMS is co-NP complete for agents with additive valuations.
• Deciding whether a given instance admits any MMS allocation is in ${\displaystyle NP^{NP}}$, i.e., it can be solved in nondeterministic-polynomial time using an oracle to an NP problem (the oracle is needed to calculate the MMS of an agent). However, the exact computational complexity of this problem is still unknown: it may be level 2 or 1 or even 0 of the polynomial hierarchy.[2][15]

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. [16]

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.[17]

With additive valuations, the various fairness notions are related as follows:

• Envy-freeness implies GMMS-fairness;[17]
• GMMS-fairness implies MMS-fairness (by taking the subgroup of size n) and PMMS-fairness (by taking subgroups of size 2);
• PMMS-fairness implies 1/2-MMS-fairness;[16]
• PMMS-fairness implies EFX, which implies EF1.
• MMS-fairness and PMMS-fairness do not imply each other.

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.[17]

• Envy-free item allocation