Allocating the gains from resource pooling with the Shapley Value

To make their cost structure more efficient, firms often pool their critical resources: small divisions of a large firm may negotiate a joint contract to benefit from volume discounts; or firms may outsource their call centres to an independent provider who is able to increase utilization by reducing variability since demand is now pooled. Since pooling demand reduces total joint costs, an immediate question is how the realized savings should be shared. We model the problem as a cooperative game and use the resulting allocation schemes to distribute the savings. One popular scheme is the Shapley Value, which always exists and, we show, represents each player's incremental value to the pool. When the pooled savings depend on the sum of each player's demand, we label the game coalition symmetric and propose, for those games, an algorithm that makes pseudo-polynomial the computation of the Shapley Value.