Coherence (fairness)

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. There is a resource to allocate, denoted by . For example, it can be an integer representing the number of seats in a house of representatives. The resource should be allocated between some agents. For example, these can be federal states or political parties. The agents have different entitlements, denoted by a vector . For example, ti can be the fraction of votes won by party i. An allocation is a vector with . An allocation rule is a rule that, for any and entitlement vector , returns an allocation vector . 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 , and on the entitlement vector , then the result is the allocation vector . 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. 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 , and Denote by the set of allocation vectors returned by on the resource and entitlement vector . The rule is called coherent if the following holds for every allocation vector and any subset S of agents: That is, every part of every possible solution to the grand problem, is a possible solution to the sub-problem.