DEFINITION of Shapley Value

In game theory, the Shapley value is a solution concept of fairly distributing both gains and costs to several actors working in coalition. The Shapley value applies primarily in situations when the contributions of each actor are unequal. The Shapley value ensures each actor gains as much or more as they would have from acting independently. This is important because otherwise there is no incentive for actors to collaborate.


A famous example of the Shapley value in practice is the airport problem. In the problem, an airport needs to be built in order to accommodate a range of aircraft which require different lengths of runway. The question is how to distribute the costs of the airport to all actors in an equitable manner. The solution is simply to spread the marginal cost of each required length of runway among all the actors needing a runway of at least that length. In the end, actors requiring a shorter runway pay less, and those needing a longer runway pay more. However, none of the actors pay as much as they would have if they had chosen not to cooperate.

Essentially, the Shapley value is the average expected marginal contribution of one player after all possible combinations have been considered. While not perfect, this has proven a fair approach to allocating value. In theory, a player can be a product sold in a store, an item on a restaurant menu, a party injured in an auto accident, or a group of investors in a lottery ticket fund. The Shapley value has utility in economic models, product line distribution, procurement measures for embassies and industry, market mix models, and calculations for tort damages. Strategists are continuously discovering new methods to use the solution concepts behind the Shapley value.