Analysis of Decentralized Quantized Auctions on Cooperative Networks

This Technical Note considers a decentralized quantity allocation problem over networks. By employing the so-called UQ-PSP auction algorithm, distributed auctions on a two-level network are developed so as to achieve efficient resource allocations (in the sense of maximization of social welfare). In the formulation in this Technical Note, each vertex in the connected 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 UQ-PSP auction, generically denoted by Al. The adjustment of the quantities supplied to any Al is via a consensus-based dynamical system which exchanges quantities depending upon the limit prices of the recursive bidding processes of the local auctions in the neighborhood of Al in the higher-level network. Such a consensus auction system is formulated as a discrete-time weighted-average consensus problem with an associated family of time-varying and asymmetric Perron matrices. Subject to continuous-valued pricing, the corresponding dynamical system converges to a global limit price which is independent of the initial data and corresponds to an efficient quantity allocation. Exponential convergence is established by using the passivity property of UQ-PSP auctions, and using the stochastic, indecomposable, and aperiodic (SIA) properties of the family of Perron matrices.