# A mean-field games formulation of network-based auction dynamics

**Peter Caines**, Mc Gill University, Canada

**Abstract:**

A decentralized quantity allocation problem over networks has been studied by Jia and Caines [2011] where two-level network based auction dynamics are formulated so as to achieve efficient resource allocations (in the sense of maximization of social welfare). In such a network based auction model, each vertex in the higher level network is regarded as a supplier for a uniquely associated lower level network, and each lower level network consists of a set of agents which represent buyers. Each lower level network with its associated supplier is assumed to constitute a local auction. The adjustment of the quantities provided to any supplier is facilitated via a cooperative dynamical system which exchanges quantities among its neighbors in the higher level network based upon the corresponding local auctions’ limit prices. This paper considers such a network based auction system with incomplete information and stochastic disturbances. First, in the lower level networks, each buyer does not know the complete bidding profile at each iteration, but each such agent is assumed to have a statistical distribution on the demand functions of the population of buyers. Assuming that all buyers apply so-called Mean Field (MF) Game strategies, efficient allocation is achieved immediately if a correct distribution of demand functions is known by each buyer. Second, in the higher level network, suppliers are associated with stochastic dynamics with inputs (local auctions’ limit prices) and outputs determining their quantity exchange rate. Each supplier only exchanges quantities among a time-varying random subset of the overall population. We analyze this stochastic dynamic game with the localized feedback MF control framework of Nourian, et.al. [2010]. We show that the set of the MF control laws have an epsilon-Nash equilibrium property where epsilon goes to 0 as the supplier population size goes to infinity, moreover, aweighted average consensus on quantities is reached asymptotically such that local limit prices are equal in all lower level networks, i.e., an efficient quantity allocation is achieved.