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Convexity and closure in optimal allocations determined by decomposable measures. (English) Zbl 1426.91141

Summary: A general optimal allocation problem is considered, where the decision-maker controls the distribution of acting agents, by choosing a probability measure on the space of agents. The notion of a decomposable family of probability measures is introduced, in the spirit of a decomposable family of functions. It provides a sufficient condition for the convexity of the feasible set, and the concavity of the value function. Together with additional conditions, closure properties also follow. The notion of a decomposable family of measures covers, both the case of set-valued integrals and the case of convexity in the space of probability measures.

MSC:

91B32 Resource and cost allocation (including fair division, apportionment, etc.)
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
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