Criterion for evaluating rules for fair division
Coherence, also called uniformity or consistency, is a criterion for evaluating rules for fair division. Coherence requires that the outcome of a fairness rule is fair not only for the overall problem, but also for each sub-problem. Every part of a fair division should be fair.
The coherence requirement was first studied in the context of apportionment. In this context, failure to satisfy coherence is called the new states paradox: when a new state enters the union, and the house size is enlarged to accommodate the number of seats allocated to this new state, some other unrelated states are affected. Coherence is also relevant to other fair division problems, such as bankruptcy problems.
Definition.
There is a "resource" to allocate, denoted by formula_1. For example, it can be an integer representing the number of seats in a "h"ouse of representatives. The resource should be allocated between some formula_2 "agents". For example, these can be federal states or political parties. The agents have different "entitlements", denoted by a vector formula_3. For example, "ti" can be the fraction of votes won by party "i". An "allocation" is a vector formula_4 with formula_5. An "allocation rule" is a rule that, for any formula_1 and entitlement vector formula_3, returns an allocation vector formula_4.
An allocation rule is called coherent (or uniform) if, for every subset "S" of agents, if the rule is activated on the subset of the resource formula_9, and on the entitlement vector formula_10, then the result is the allocation vector formula_11. That is: when the rule is activated on a subset of the agents, with the subset of resources they received, the result for them is the same.
Handling ties.
In general, an allocation rule may return more than one allocation (in case of a tie). In this case, the definition should be updated. Denote the allocation rule by formula_12, and Denote by formula_13 the set of allocation vectors returned by formula_12 on the resource formula_1 and entitlement vector formula_16. The rule formula_12 is called coherent if the following holds for every allocation vector formula_18 and any subset "S" of agents:
Coherence in apportionment.
In apportionment problems, the resource to allocate is "discrete", for example, the seats in a parliament. Therefore, each agent must receive an integer allocation.
Non-coherent methods: the new state paradox.
One of the most intuitive rules for apportionment of seats in a parliament is the largest remainder method (LRM). This method dictates that the entitlement vector should be normalized such that the sum of entitlements equals formula_1 (the total number of seats to allocate). Then each agent should get his normalized entitlement (often called "quota") rounded down. If there are remaining seats, they should be allocated to the agents with the largest remainder – the largest fraction of the entitlement. Surprisingly, this rule is "not" coherent. As a simple example, suppose formula_24 and the normalized entitlements of Alice, Bob and Chana are 0.4, 1.35, 3.25 respectively. Then the unique allocation returned by LRM is 1, 1, 3 (the initial allocation is 0, 1, 3, and the extra seat goes to Alice, since her remainder 0.4 is largest). Now, suppose that we activate the same rule on Alice and Bob alone, with their combined allocation of 2. The normalized entitlements are now 0.4/1.75 × 2 ≈ 0.45 and 1.35/1.75 × 2 ≈ 1.54. Therefore, the unique allocation returned by LRM is 0, 2 rather than 1, 1. This means that in the grand solution 1, 1, 3, the internal division between Alice and Bob does "not" follow the principle of largest remainders – it is not coherent.
Another way to look at this non-coherence is as follows. Suppose that the house size is 2, and there are two states A, B with quotas 0.4, 1.35. Then the unique allocation given by LRM is 0, 2. Now, a new state C joins the union, with quota 3.25. It is allocated 3 seats, and the house size is increased to 5 to accommodate these new seats. This change should not affect the existing states A and B. In fact, with the LRM, the existing states "are" affected: state A gains a seat, while state B loses a seat. This is called the new state paradox.
The new state paradox was actually observed in 1907, when Oklahoma became a state. It was given a fair share 5 of seats, and the total number of seats increased by that number, from 386 to 391 members. After recomputation of apportionment affected the number of seats because of other states: New York lost a seat, while Maine gained one.
Coherent methods.
Every divisor method is coherent. This follows directly from their description as picking sequences: at each iteration, the next agent to pick an item is the one with the highest ratio (entitlement / divisor). Therefore, the relative priority ordering between agents is the same even if we consider a subset of the agents.
Properties of coherent methods.
When coherency is combined with other natural requirements, it characterizes a structured class of apportionment methods. Such characterizations were proved by various authors. All results assume that the rules are homogeneous (i.e. it depends only on the percentage of votes for each party, not on the total number of votes).
Coherence in bankruptcy problems.
In bankruptcy problems, the resource to allocate is "continuous", for example, the amount of money left by a debtor. Each agent can get any fraction of the resource. However, the sum of entitlements is usually larger than the total remaining resource.
The most intuitive rule for solving such problems is the "proportional rule", in which each agent gets a part of the resource proportional to his entitlement. This rule is definitely coherent. However, it is not the only coherent rule: the Talmudic rule of the contested garment can be extended to a coherent division rule.
Coherence in organ allocation.
In most countries, the number of patients waiting for an organ transplantation is much larger than the number of available organs. Therefore, most countries choose who to allocate an organ to by some priority-ordering. Surprisingly, some priority orderings used in practice are not coherent. For example, one rule used by UNOS in the past was as follows:
Suppose the personal scores of some four patients A, B, C, D are 16, 21, 20, 23. Suppose their waiting times are A ; B ; C ; D. Accordingly, their bonuses are 10, 7.5, 5, 2.5. So their sums are 26, 28.5, 25, 25.5, and the priority order is B ; A ; D ; C. Now, after B receives an organ, the personal scores of A, C, D remain the same, but the bonuses change to 10, 6.67, 3.33, so the sums are 26, 26.67, 26.33, and the priority order is C ; D ; A. This inverts the order between the three agents.
In order to have a coherent priority ordering, the priority should be determined only by personal traits. For example, the bonus can be computed by the number of months in line, rather than by the fraction of patients.
References.
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