On the distributed mean-variance paradigm

In this paper we study the distributed mean-variance paradigm with linear state dynamics of mean-field type in discrete time and several control inputs. The goal is to reduce the variance and the mean of the state in a fully distributed manner. We formulate and explicit solve the problem using recent development of mean-field-type games. We show that there is unique best response strategy to the mean of the state and provide a simple sufficient condition of existence and uniqueness of mean-field equilibrium. We also provide a closed-form expression of the global optimum as a state-and-mean-field feedback strategy.