On the Rapid Probabilistic Convergence of a Class of Progressive Second Price Auctions ⋆

A quantized progressive second price auction mechanism called ADQ-PSP is presented which improves upon the performance of the so-called Q-PSP mechanism presented in (Qu, Jia & Caines 2007, Jia, Qu & Caines 2008). The Q-PSP mechanism was developed for the fast convergence properties that can be proven for it subject to the assumption that all agents (i.e. buyers) share similar demand (i.e. marginal valuation) functions. For the ADQ-PSP mechanism applied to agent populations with randomly distributed demand functions, it is shown here that the states (i.e. bid prices and quantities) of the corresponding dynamical systems rapidly converge with high probability to a quantized (Nash) equilibrium with a common price for all agents. This property holds for ensembles containing populations which have significantly different demand functions and, moreover, for elastic supply. Furthermore, the convergence rates are independent of the number of quantization levels, and of the number of sellers and their strategies in a double auction. Finally, the ǫ-efficiency of the quantized equilibria is established and numerical examples are given.