Dynamics of multidimensional Simple Clock Auctions

TL;DR

This paper analyzes the dynamics of Simple Clock Auctions in spectrum markets, establishing a continuous-time model with unique solutions and piecewise linear value functions, supported by analysis of a real-world spectrum auction.

Contribution

It introduces a continuous-time differential inclusion model for Simple Clock Auctions and proves the existence and uniqueness of solutions, linking discrete and continuous auction dynamics.

Findings

Unique solution in the sense of Filippov for the continuous model

The value function is piecewise linear, possibly discontinuous

Model validated through analysis of the 2017 Australian spectrum auction

Abstract

Simple Clock Auctions (SCA) are a mechanism commonly used in spectrum auctions to sell lots of frequency bandwidths. We study such an auction with one player having access to perfect information against straightforward bidders. When the opponents' valuations satisfy the ordinary substitutes condition, we show that it is optimal to bid on a fixed lot overtime. In this setting, we consider a continuous-time version of the SCA auction in which the prices follow a differential inclusion with a piecewise-constant dynamics. We show that there exists a unique solution in the sense of Filippov. This guarantees that the continuous-time model coincides with the limit of the discrete-time auction when price increments tend to zero. Moreover, we show that the value function of this limit auction is piecewise linear (though possibly discontinuous). Finally, we illustrate these results by analyzing a simplified version of the multiband Australian spectrum auction of 2017.