The Shapley Value, Average Productivity Differentials, and Coalition Size

The Shapley value is arguably the most well-known solution concept for cooperative, transferable utility games. In this Note we show, in contrast to its many marginal characterizations, that the Shapley value can also be viewed as a solution based on average productivity. Specifically, we show that the Shapley value can be axiomatized by means of symmetry, efficiency and a property we call coalition size neutrality. This property requires, roughly, that the payoff to each player depend only on that player’s overall relative average productivity and not on how that productivity is distributed over coalition size. In addition, we observe how a weakened version of coalition size neutrality may be used to characterize the vector space of all linear combinations of the Shapley value and the well-known equal division solution.