Economics - General

# Shapley Value

What is the Shapley Value?  The Shapley value determines the relative importance of different individuals within a cooperative game.  For instance, who is more important: the owner or the workers?  Without an owner supplying capital, there would be business to start.  Without workers to produce goods, there would be no output to sell.  Thus, the more specific question the Shapley value answers is what share of the total surplus is each person entitled to.

Wikipedia gives the answer to this question:

Consider a simplified description of a business. We have an owner o, who does not work but provides the crucial capital, meaning that without him no gains can be obtained. Then we have k workers w1,…,wk, each of whom contributes an amount p to the total profit. So N = {ow1,…,wk} andv(S) = 0 if o is not a member of S and v(S) = mp if S contains the owner and m workers. Computing the Shapley value for this coalition game leads to a value of mp/2 for the owner and p/2 for each worker.

Here is an example of how to calculate the Shapley value for the glove game.  The glove game has two people with a right-handed glove and one person with a left-handed glove.  Only coalitions that include both a right and left-handed glove are successful.  Unsurprisingly, the person with the single left-handed glove collects a larger share of the surplus than the two individuals with the right-handed glove.

The Shapley value was introduced by Lloyd Shapely in 1953.  A recent article in Health Economics discusses using the Shapley value to decompose redistributive, vertical and horizontal effects of health care finance by factor components.

Here is a more technical discussion of the Shapley value.

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