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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: 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.:285 But such solutions are not always available. An item assignment problem has several ingredients: These ingredients are explained in detail below.
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:
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 [math]\displaystyle{ 2^m }[/math] 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:
Under suitable assumptions, it is possible to lift the preferences on items to preferences on bundles. ^{[1]}^{:44-48} Then, the agents report their valuations/rankings on individual items, and the algorithm calculates for them their valuations/rankings on bundles.
To make the item-assignment problem simpler, it is common to assume that all items are independent goods (so they are not substitute goods nor complementary goods). ^{[2]} Then:
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.^{[5]}^{:287–288}
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:^{[5]}^{:289–294}
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):^{[6]}
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.^{[7]} 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.^{[6]} 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.^{[6]} 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 [math]\displaystyle{ 1/n }[/math]. Hence, every mFS-fair allocation is proportional. For every agent with superadditive utility, the MFS is worth at most [math]\displaystyle{ 1/n }[/math]. 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:^{[6]}
The above implications do not hold when the agents' valuations are not sub/superadditive.^{[8]}
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.^{[8]} See envy-free item assignment for more details.
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.^{[6]}
Several recently-suggested fairness criteria are:^{[9]}
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.
A global optimization criterion evaluates a division based on a given social welfare function:
An advantage of global optimization criteria over individual criteria is that welfare-maximizing allocations are Pareto efficient.
The problem of calculating the MFS of an agent is NP-complete: it can be reduced from the partition problem.^{[6]}
MFS allocations exist in almost all cases, but not always. There are very rare cases in which they do not exist.^{[10]}
The problem of deciding whether an MFS allocation exists is in [math]\displaystyle{ NP^{NP} }[/math], 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.^{[6]}
Allocations that guarantee each partner 2/3 of the above value always exist.^{[10]} This division procedure have been implemented in the spliddit website.^{[11]}
1. Suppose the agents have cardinal utility functions on items. Then, the problem of deciding whether a proportional allocation exists is NP-complete: it can be reduced from the partition problem.^{[6]}
2. Suppose the agents have ordinal rankings on items, with or without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial time: it can be reduced to the problem of checking whether a bipartite graph admits a feasible b-matching (a matching when the edges have capacities). ^{[12]}
For two agents, a simpler algorithm exists.^{[13]}
3. Suppose the agents have ordinal rankings on items, without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial time. It is not known whether the same is rue when the agents are allowed to express indifferences.^{[12]}
The problem of calculating the mFS of an agent is coNP-complete.
The problem of deciding whether an mFS allocation exists is in [math]\displaystyle{ NP^{NP} }[/math], but its exact computational complexity is still unknown.^{[6]}
Envy-freeness becomes easier to attain when it is assumed that agents' valuations are quasilinear in money, and thus transferable across agents.
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).^{[14]}
Fair by Design is a general framework for optimization problems with envy-freeness guarantee that naturally extends fair item assignments using monetary payments.^{[15]}
Cavallo^{[16]} generalizes the traditional binary criteria of envy-freeness, proportionality, and efficiency (welfare) to measures of degree that range between 0 and 1. In the canonical fair division settings, under any allocatively-efficient mechanism the worst-case welfare rate is 0 and disproportionality rate is 1; in other words, the worst-case results are as bad as possible. This strongly motivates an average-case analysis. He looks for a mechanism that achieves high welfare, low envy, and low disproportionality in expectation across a spectrum of fair division settings. He shows that the VCG mechanism is not a satisfactory candidate, but the redistribution mechanism of ^{[17]} and ^{[18]} is.
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:^{[19]}
^{[20]} and ^{[21]} present some stronger hardness results.
The special case of optimizing egalitarian welfare with additive utilities is called "the Santa Claus problem". ^{[22]} The problem is still NP-hard and cannot be approximated within a factor > 1/2, but there is an [math]\displaystyle{ O(n) }[/math] approximation^{[23]} and a more complicated [math]\displaystyle{ O(\sqrt{n}\cdot \log^3{n}) }[/math] approximation.^{[24]} See also ^{[25]} for a branch-and-bound algorithm for two partners.
The Decreasing Demand procedure returns a egalitarian-optimal division in an ordinal sense: it maximizes the rank, in the linear-ranking of bundles, of the agent with the lowest rank. It works for any number of agents with any ordering of bundles.
^{[20]} and ^{[21]} prove hardness of calculating utilitarian-optimal and Nash-optimal allocations.
^{[26]} present an approximation procedure for Nash-optimal allocations.
A picking sequence is a simple protocol where the agents take turns in selecting items, based on some pre-specified sequence of turns. The goal is to design the picking-sequence in a way that maximizes the expected value of a social welfare function (e.g. egalitarian or utilitarian) under some probabilistic assumptions on the agents' valuations.
Most research about item assignment assumes that all agents have equal entitlements. But, in many cases there are agents with different entitlements. One such case is when dividing cabinet ministries among parties in the coalition. It is common to assume that each party should receive ministries according to the number of seats it has in the parliament. See ^{[27]} for a discussion of this problem and some solutions.
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,^{[28]} 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,^{[29]} 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 ^{[30]} for another experiment focusing on envy-freeness.