Set-valued capacities: multi-agenda decision making

We study the problem in which a set of agents are required to produce across several different projects (or more generally, agendas), and we consider environments in which resources are constrained and investing (say, time or effort) in one agenda reduces the ability to invest in other agendas. To this end, we introduce a class of capacities we refer to as set-valued: the value of each coalition is a subset of a vector space. For a particular coalition, each vector in its value is associated with a different distribution of the resources invested across the different agendas. In this context, the Choquet and the concave integrals are defined, characterized and shown to be identical if and only if the underlying set-valued capacity is supermodular. We apply the tools developed and introduce a new decision theory.