Coherence,[1] also called uniformity[2]: Thm.8.3 or consistency, is a criterion for evaluating rules for fair division.
[2] The coherence requirement was first studied in the context of apportionment.
For example, these can be federal states or political parties.
For example, ti can be the fraction of votes won by party i.
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
the set of allocation vectors returned by
is called coherent if the following holds for every allocation vector
and any subset S of agents:[3]: Sec.4 In apportionment problems, the resource to allocate is discrete, for example, the seats in a parliament.
Therefore, each agent must receive an integer allocation.
One of the most intuitive rules for apportionment of seats in a parliament is the largest remainder method (LRM).
(the total number of seats to allocate).
Then each agent should get his normalized entitlement (often called quota) rounded down.
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.
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.
Suppose that the house size is 2, and there are two states A, B with quotas 0.4, 1.35.
This change should not affect the existing states A and B.
After recomputation of apportionment affected the number of seats because of other states: New York lost a seat, while Maine gained one.
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).
When coherency is combined with other natural requirements, it characterizes a structured class of apportionment methods.
[3]: Sec.1 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).
In bankruptcy problems, the resource to allocate is continuous, for example, the amount of money left by a debtor.
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.
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:[1]: Sec.6 Suppose the personal scores of some four patients A, B, C, D are 16, 21, 20, 23.
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.
For example, the bonus can be computed by the number of months in line, rather than by the fraction of patients.