A Mean Field Games formulation of network based auction dynamics

A decentralized quantity allocation problem over networks has been studied in (Jia and Caines, 2013) where two-level network based auction dynamics are formulated so as to achieve efficient resource allocations (in the sense of maximization of social welfare). The adjustment of the quantities provided to any supplier in the higher-level network is facilitated via a cooperative dynamical system which exchanges quantities among the supplier´s neighbors based upon the corresponding lower-level buyer auctions´ limit prices. This paper considers such a network based auction system with incomplete information and stochastic disturbances. First, in the lower-level networks, assuming that all buyers apply a so-called Mean Field Games (MFG) strategy, efficient allocation is achieved rapidly when the distribution of the demand functions is known to each buyer. Second, in the higher-level network, suppliers are associated with a stochastic dynamical system with inputs (i.e., the local auctions´ limit prices) and outputs determining their quantity exchange rate. We analyze this stochastic dynamical game within the MFG control framework. We show that the set of MFG control laws has an ∈-Nash equilibrium property where ∈ goes to zero as the supplier population size goes to infinity; moreover, a weighted average consensus on quantities is reached asymptotically with all local limit prices being equal in the lower-level networks, i.e., efficient quantity allocation is achieved asymptotically. A potential application of this work is to small cell communication networks.