Sandwich games

The extension of set functions (or capacities) in a concave fashion, namely a concavification, is an important issue in decision theory and combinatorics. It turns out that some set-functions cannot be properly extended if the domain is restricted to be bounded. This paper examines the structure of those capacities that can be extended over a bounded domain in a concave manner. We present a property termed the sandwich property that is necessary and sufficient for a capacity to be concavifiable over a bounded domain. We show that when a capacity is interpreted as the product of any sub group of workers per a unit of time, the sandwich property provides a linkage between optimality of time allocations and efficiency.