We present some limiting properties of a network resource allocation mechanism known as the Progressive Second Price (PSP) auction. This mechanism aims at efficiently allocate network resources, such as bandwidth or buffer capacity, in an environment characterized by competing users; the PSP auction seeks to solve or at least to ameliorate congestion in a network demanding a low signalling burden between the auctioneer and the users, and solving the allocation problem of an (theoretically) infinitely divisible resource. The allocation rule is inspired in the second price (Vickrey) auction. Our analysis of the PSP auction explores its limiting properties, namely, how the allocation changes in the presence of a polarized set of users. A polarized set of users is a mixture of users of two types: high valuation, low demand users and low valuation, high demand users. Mechanisms such as auctions are becoming increasingly popular to handle the resource allocation problem in networks facing congestion, such as the access to Internet-based services.