Authority distribution

The solution concept authority distribution was formulated by Lloyd Shapley and his student X. Hu in 2003 to measure the authority power of players in a well-contracted organization. The index generates the Shapley-Shubik power index and can be used in ranking, planning and organizational choice.

The organization contracts each individual by boss and approval relation with others. So each individual has its own authority structure, called command game. The Shapley-Shubik power index for these command games are collectively denoted by a power transit matrix Ρ.

The authority distribution π is defined as the solution to the counterbalance equation π=πΡ. The basic idea for the counterbalance equation is that a person's power comes from his critical roles in others' command game; on the other hand, his power could also be redistributed to those who sit in his command game as vital players.

For a simple legislative body, π is simply the Shapley-Shubik power index, based on a probabilistic argument (,).

Example 1. College ranking by applicants’ acceptance

Suppose that there are large numbers of college applicants to apply the colleges Each applicant files multiple applications. Each college then offers some of its applicants admissions and rejects all others. Now some applicants may get no offer from any college; the other then get one offer or multiple offers. An applicant with multiple offers will decide which college to go to and reject all other colleges which make offers to him. Of all applicants who apply to and receive offers from College i, we let P(i,j) be the proportion of those applicants who decide to go to college j. Such applicants of course apply to and receive offers from College j as well.

To rank the colleges by the acceptance rates of the applicants to whom offers were made, we can apply the authority distribution associated with the matrix P. The so-called “authority distribution” can be regarded as the measure of relative attractiveness of the colleges from the applicants’ point of view.

Example 2. Journal rankings by citations

Assume there are n journals in a scientific field. For any Journal i, each issue contains many papers, and each paper has its list of references or citations. A paper in journal j can be cited in another paper in Journal i as a reference. Of all papers cited by Journal i (repetition counted), we let P(i,j) be the proportion of those papers which are published on Journal j. So P measures the direct impact between any two journals and P(i, i) is the self-citation rate for Journal i . The authority distribution for π = πP would quantify the long-term influence of each journal in the group of journals and can be used to rank these journals.

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