Analysis of a class of decentralized dynamical systems: rapid convergence and efficiency of dynamical quantized auctions

In this paper, we study a class of progressive second price (PSP) auctions introduced by Lazar & Semret (1999, Design and analysis of the progressive second price auction for network bandwidth sharing. Technical Report 487-98-21. Columbia University Center for Telecommunications Research.) subject to various quantized pricing assumptions. The general PSP mechanism is employed here for the allocation of a divisible resource among arbitrary populations of agents in terms of two specific algorithms which are called, respectively, the aggressive–defensive qunatized progressive second price (ADQ-PSP) algorithm and the unique limit quantized progressive second price (UQ-PSP) algorithm, each of which derives from an associated set of quantized strategies. First, for the ADQ-PSP auction algorithm applied to agent populations with randomly and possibly widely distributed demand functions, it is shown 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. Second, for the UQ-PSP auction algorithm (developed as a modification of the ADQ-PSP algorithm) applied to general agent populations, the corresponding dynamical systems are such that (i) the limit price of all system trajectories is independent of the initial data and (ii) modulo the quantization level, the limiting resource allocation is efficient (i.e. the corresponding social welfare function, or summed individual valuation functions, is optimal).