Fair item allocation

Fair item assignment is a kind of a fair division problem in which the items to divide are indivisible. The items have to be divided among several partners who value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios:

• Several heirs want to divide the inherited property, which contains e.g. a house, a car, a piano and several paintings.
• Several lecturers want to divide the courses given in their faculty. Each lecturer can teach one or more whole courses.

The indivisibility of the items implies that a fair division may not be possible. As an extreme example, if there is only a single item (e.g. a house), it must be given to a single partner, but this is not fair to the other partners. This is in contrast to the fair cake-cutting problem, where the dividend is divisible and a fair division always exists. In some cases, the indivisibility problem can be mitigated by introducing monetary payments or time-based rotation, or by discarding some of the items. But such solutions are not always available.

An item assignment problem has several ingredients:

1. The partners have to express their preferences for the different item-bundles.
2. The group should decide on a fairness criterion.
3. Based on the preferences and the fairness criterion, a fair assignment algorithm should be executed to calculate a fair division.

These ingredients are explained in detail below.

Preferences

Combinatorial preferences

A naive way to determine the preferences is asking each partner to supply a numeric value for each possible bundle. For example, if the items to divide are a car and a bicycle, a partner may value the car as 800, the bicycle as 200, and the bundle {car,bicycle} as 900 (see Utility functions on indivisible goods for more examples). There are two problems with this approach:

1. It may be difficult for a person to calculate exact numeric values to the bundles.
2. The number of possible bundles can be huge: if there are $m$ items then there are $2^{m}$ possible bundles. For example, if there are 16 items then each partner will have to present his preferences using 65536 numbers.

The first problem motivates the use of ordinal utility rather than cardinal utility. In the ordinal model, each partner should only express a ranking over the $2^{m}$ different bundles, i.e, say which bundle is the best, which is the second-best, and so on. This may be easier than calculating exact numbers, but it is still difficult if the number of items is large.

The second problem is often handled by working with individual items rather than bundles:

• In the cardinal approach, each partner should report a numeric valuation for each item;
• In the ordinal approach, each partner should report a ranking over the items, i.e, say which item is the best, which is the second-best, etc.

Under suitable assumptions, it is possible to lift the preferences on items to preferences on bundles. Then, the agents report their valuations/rankings on individual items, and the algorithm calculates for them their valuations/rankings on bundles.

Additive preferences

The most common such assumption is additivity - each partner has an additive utility function. Then:

• In the cardinal approach, it is easy to calculate the value of each bundle by summing up the values of its items.
• In the ordinal approach, additivity allows us to infer some rankings between bundles. For example, if a person prefers w to x to y to z, then he necessarily prefers {w,x} to {w,y} or to {x,y}, and {w,y} to {x}. This inference is only partial, e.g, we cannot know whether the agent prefers {w} to {x,y} or even {w,z} to {x,y}. Note that not all rankings can be inferred in this way.

The additivity implies that each partner can always choose a "preferable item" from the set of items on the table, and this choice is independent of the other items that the partner may have. This property is used by some fair assignment algorithms that will be described next.

The additivity assumption makes sense when the items are unrelated, i.e, they are independent goods. It does not hold when there are substitute goods or complementary goods.

Compact preference representation languages

Compact preference representation languages have been developed as a compromise between the full expressiveness of combinatorial preferences to the simplicity of additive preferences. They provide a succinct representation to some natural classes of utility functions that are more general than additive utilities (but not as general as combinatorial utilities). Some examples are:

• 2-additive preferences: each partner reports a value for each bundle of size at most 2. The value of a bundle is calculated by summing the values for the individual items in the bundle and adding the values of pairs in the bundle. Typically, when there are substitute items, the values of pairs will be negative, and when there are complementary items, the values of pairs will be positive. This idea can be generalized to k-additive preferences for every positive integer k.
• Graphical models: for each partner, there is a graph that represents the dependencies between different items. In the cardinal approach, a common tool is the GAI net (Generalized Additive Independence). In the ordinal approach, a common tool is the CP net (Conditional Preferences) and its extensions: TCP net, UCP net, CP theory, CI net (Conditional Importance) and SCI net (a simplification of CI net).
• Logic based languages: each partner describes some bundles using a first order logic formula, and may assign a value for each formula. For example, a partner may say: "For (x or (y and z)), my value is 5". This means that the agent has a value of 5 for any of the bundles: x, xy, xz, yz, xyz.
• Bidding languages: many languages for representing combinatorial preferences have been studied in the context of combinatorial auctions. Some of these languages can be adapted to the item assignment setting.

Fairness criteria

Individual guarantee criteria

An individual guarantee criterion is a criterion that should hold for each individual partner, as long as the partner truthfully reports his preferences. Five such criteria are presented below. They are ordered from the weakest to the strongest (assuming the valuations are additive):

1. Max-min fair-share (MFS): The max-min-fair-share (also called: max-min-share-guarantee) of an agent is the most preferred bundle he could guarantee himself as divider in divide and choose against adversarial opponents. An allocation is called MFS-fair if every agent receives a bundle that he weakly prefers over his MFS. The MFS 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 can be considered as the minimal amount of utility that an agent could feel to be 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. It is also the maximum utility that an agent can get for sure in the allocation game “I cut, I choose last”: the agent proposes her best allocation and leaves all the other ones choose one share before taking the remaining one. MFS-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 MFS-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.

2. Proportional fair-share (PFS): The proportional-fair-share of an agent is 1/n of his utility from the entire set of items. An allocation is called proportional if every agent receives a bundle worth at least his proportional-fair-share.

3. Min-max fair-share (mFS): The min-max-fair-share of an agent is the minimal utility that she can hope to get from an allocation if all the other agents have the same preferences as her, when she always receives the best share. It is also the minimal utility that an agent can get for sure in the allocation game “Someone cuts, I choose first”. An allocation is mFS-fair if all agents receive a bundle that they weakly prefer over their mFS. mFS-fairness can be described as the result of the following negotiation process. A certain allocation is suggested. Each agent can object to it by demanding that a different allocation be made by another agent, letting him choose first. Hence, an agent would object to an allocation only if in all partitions, there is a bundle that he strongly prefers over his current bundle. An allocation is mFS-fair iff no agent objects to it, i.e., for every agent there exists a partition in which all bundles are weakly worse than his current share.

For every agent with subadditive utility, the mFS is worth at least $1/n$. Hence, every mFS-fair allocation is proportional. For every agent with superadditive utility, the MFS is worth at most $1/n$. Hence, every proportional allocation is MFS-fair. Both inclusions are strict, even when every agent has additive utility. This is illustrated in the following example:

There are 3 agents and 3 items:

• Alice values the items as 2,2,2. For her, MFS=PFS=mFS=2.
• Bob values the items as 3,2,1. For him, MFS=1, PFS=2 and mFS=3.
• Carl values the items as 3,2,1. For him, MFS=1, PFS=2 and mFS=3.

The possible allocations are as follows:

• Every allocation which gives an item to each agent is MFS-fair.
• Every allocation which gives the first and second items to Bob and Carl and the third item to Alice is proportional.
• No allocation is mFS-fair.

The above implications do not hold when the agents' valuations are not sub/superadditive.

4. Envy-freeness (EF): every agent weakly prefers his own bundle to any other bundle. Every envy-free allocation of all items is mFS-fair; this follows directly from the ordinal definitions and does not depend on additivity. If the valuations are additive, then an EF allocation is also proportional and MFS-fair. Otherwise, an EF allocation may be not proportional and even not MFS. In general, an EF allocation of all items is not guaranteed to exist (the simplest case is when there is a single item). But, if the agents have additive preferences that are drawn from probability distributions satisfying some independence criteria, and the number of items is sufficiently large relative to the number of agents, then an EF allocation exists with high probability. Particularly:

• If the number of items is sufficiently large: $m=\Omega (n\log n)$, then w.h.p. an EF allocation exists (the probability goes to 1 as m goes to infinity).
• If the number of items is not sufficiently large, i.e, $m=n+o(n)$, then w.h.p. an EF allocation does not exist.

5. Competitive equilibrium from Equal Incomes (CEEI): This criterion is based on the following argument: the allocation process should be considered as a search for an equilibrium between the supply (the set of objects, each one having a public price) and the demand (the agents’ desires, each agent having the same budget for buying the objects). A competitive equilibrium is reached when the supply matches the demand. The fairness argument is straightforward: prices and budgets are the same for everyone. CEEI implies EF regardless of additivity. When the agents' preferences are additive and strict (each bundle has a different value), CEEI implies Pareto efficiency.

Several recently-suggested fairness criteria are:

6. Envy-freeness-except-1 (EF1): For each two agents A and B, if we remove from the bundle of B the item most valuable for A, then A does not envy B (in other words, the "envy level" of A in B is at most the value of a single item). Under additivity, an EF1 allocation always exists.

7. Envy-freeness-except-cheapest (EFx): For each two agents A and B, if we remove from the bundle of B the item least valuable for A, then A does not envy B. EFx is strictly stronger than EF1. It is not known whether EFx allocations always exist.

Global optimization criteria

A global optimization criterion evaluates a division based on a given social welfare function:

• The egalitarian social welfare is minimum utility of a single agent. An item assignment is called egalitarian-optimal if it attains the maximum possible egalitarian welfare, i.e, it maximizes the utility of the poorest agent. Since there can be several different allocations maximizing the smallest utility, egalitarian optimality is often refined to leximin-optimality: from the subset of allocations maximizing the smallest utility, it selects those allocations that maximize the second-smallest utility, then the third-smallest utility, and so on.
• The Nash social welfare is the product of the utilities of the agents. An assignment called Nash-optimal or Maximum-Nash-Welfare if it maximizes the product of utilities. Nash-optimal allocations have some nice fairness properties.

An advantage of global optimization criteria over individual criteria is that welfare-maximizing allocations are Pareto efficient.

Assignment algorithms in the cardinal model

Max-min-share fairness

The problem of calculating the MFS of an agent is NP-complete: it can be reduced from the partition problem.

MFS allocations exist in almost all cases, but not always. There are very rare cases in which they do not exist.

The problem of deciding whether an MFS allocation exists is in $NP^{NP}$, i.e., it can be solved in nonderetministic-polynomial time using an oracle to an NP problem (the oracle is needed to calculate the max-min-share of an agent). However, the exact computational complexity of this problem is still unknown.

Allocations that guarantee each partner 2/3 of the above value always exist. This division procedure have been implemented in the spliddit website.

Proportionality

The problem of deciding whether a proportional allocation exists is NP-complete: it can be reduced from the partition problem.

Min-max-share fairness

The problem of calculating the mFS of an agent is coNP-complete.

The problem of deciding whether an MFS allocation exists is in $NP^{NP}$, but its exact computational complexity is still unknown.

Envy-freeness (without money)

The problem of deciding whether an envy-free allocation exists is NP-complete.

The problem of deciding whether an envy-free and Pareto-efficient allocation exists is ΣP2-complete.

Envy-freeness (with money)

Demange, Gale and Sotomayor showed a natural ascending auction that achieves an envy-free allocation using monetary payments for unit demand bidders (where each bidder is interested in at most one item).

Fair by Design is a general framework for optimization problems with envy-freeness guarantee that naturally extends fair item assignments using monetary payments.

Egalitarian-optimal allocations

With general cardinal valuations, finding egalitarian-optimal allocations, or even approximately-optimal allocations, is NP-hard. This can be proved by reduction from the partition problem. When the valuations are more restricted, better approximations are possible:

• With additive utilities, for every integer $k$, a $(1-1/k)$ fraction of the agents receive utility at least $OPT/k$. This result is obtained from rounding a suitable linear programming relaxation of the problem, and is the best possible result for this linear program.
• With two classes of goods, an $O({\sqrt {n}})$ approximation exists.
• With submodular utility functions, an $(m-n+1)$ approximation exists.

and present some stronger hardness results.

The special case of optimizing egalitarian welfare with additive utilities is called "the Santa Claus problem". The problem is still NP-hard and cannot be approximated within a factor > 1/2, but there is an $O(n)$ approximation and a more complicated $O({\sqrt {n}}\cdot \log ^{3}{n})$ approximation.

Nash-optimal allocations

and prove hardness of calculating utilitarian-optimal and Nash-optimal allocations.

present an approximation procedure for Nash-optimal allocations.

The ordinal model

Two partners, envy-free allocation

Suppose there are two partners, Alice and George, whose valuations are ordinal. I.e, each partner can order the items from the most preferred to the least preferred (without ties).

An item assignment is called envy-free for Alice if there is an injection f from George's items to Alice's items, such that for each item x received by George, Alice prefers f(x) to x. The property envy-free for George is defined analogously. An item assignment is called envy-free (EF) if it is envy-free for both partners. Note that in an EF assignment, Alice and George receive the same number of items.

The following division procedure can be used to find an EF allocation:

• Put all items on the table.
• While there are items on the table, do:
  • Ask the partners to choose their favorite item from all items on the table.
  • If the choices are different, then give each partner his/her favorite item and continue.
  • If the choices are identical, then move the chosen item to the Contested Pile. It will not be allocated.

This procedure, while very simple, is not very efficient, since many items are discarded to the contested pile. A slightly more complicated procedure finds an EF allocation which is also Pareto efficient (PE), i.e., there is no other EF allocation which is weakly better for both partners. The idea is that, before moving an item to the contested pile, the procedure tries to allocate it to one partner while compensating the other partner with another item. Only if this doesn't succeed, the item is sent to the contested pile.

For example, suppose there are 4 items (1, 2, 3, 4) and the preferences of the partners are:

• Alice: 1 > 2 > 3 > 4
• George: 2 > 3 > 4 > 1

The first procedure gives 1 to Alice and 2 to George, since these are their favorites and they are different. Then, both Alice and George choose 3 so it is discarded. Then, both choose 4 so it is also discarded. The final allocation is: Alice←{1}, George←{2}. It is EF but not PE.

The second procedure also starts by giving 1 to Alice and 2 to George. Then, instead of discarding item 3, it is given to Alice, and George is compensated with item 4. The final allocation is: Alice←{1,3}, George←{2,4}. It is EF and PE.

Both procedures are manipulable: a partner can gain by reporting incorrect preferences. However, such manipulation requires knowledge of the other partner's preferences, so it is difficult in practice.

Empirical evidence

Several experiments have been conducted with people, in order to find out what is the relative importance of several division criteria. In particular, what is the importance of fairness versus efficiency? Do people prefer divisions which are fair but inefficient, or efficient but unfair?

In one experiment, subjects were asked to answer questionnaires regarding the division of indivisible items between two people. The subjects were shown the subjective value that each (virtual) person attaches to each item. The predominant aspect considered was equity - satisfying each individual's preferences. The efficiency aspect was secondary. This effect was slightly more pronounced in economics students, and less pronounced in law students (who chose a Pareto-efficient allocation more frequently).

In another experiment, subjects were divided into pairs and asked to negotiate and decide how to divide a set of 4 items between them. Each combination of items had a pre-specified monetary value, which was different between the two subjects. Each subject knew both his own values and the partner's values. After the division, each subject could redeem the items for their monetary value.

The items could be divided in several ways: some divisions were equitable (e.g., giving each partner a value of 45), while other divisions were Pareto efficient (e.g., giving one partner 46 and another partner 75). The interesting question was whether people prefer the equitable or the efficient division. The results showed that people preferred the more efficient division, but only as long as it was not "too much" unfair. A difference of 2-3 units of value was considered sufficiently small for most subjects, so they preferred the efficient allocation. But a difference of 20-30 units (such as in the 45:45 vs. 46:75 example) was perceived as too large: 51% preferred the 45:45 division.

The effect were less pronounced when the subjects were only shown the rank of the item combinations for each of them, rather than the full monetary value.

This experiment also revealed a recurring process which was used during the negotiation. The subjects first find the most equitable division of the goods. They take this as a reference point, and try to find Pareto improvements. An improvement is implemented only if the inequality it causes is not too large. Hence the authors call this process CPIES: Conditioned Pareto Improvement from Equal Split.

See for another experiment focusing on envy-freeness.

See also

• Rental harmony - a fair division problem where indivisible items and a fixed total cost have to be divided simultaneously.
• Price of fairness - a general measure of the trade-off between fairness and efficiency, with some results about the item assignment setting.