An Ascending Auction with Multi-Dimensional Signals

Abstract

This paper examines a single-unit ascending auction where agents have interdependent values and observe multi-dimensional signals. The challenge is to characterize how the multi-dimensional signals observed by an agent are aggregated onto that agent’s one-dimensional bid. The challenge is solved by projecting an agent’s private signals onto a one-dimensional equilibrium statistic; the equilibrium bidding strategies are constructed as if each agent observed only his own equilibrium statistic. An agent’s equilibrium statistic aggregates this agent’s private signals while taking into account the additional information deduced from the other agents’ bids. The focus is on a symmetric model in which each agent observes two private signals, but the solution method extends to any (possibly asymmetric) Gaussian information structure. The equilibrium characterization reveals important aspects of bidding strategies that arise only when agents observe multidimensional signals. In contrast to one-dimensional environments, an ascending auction may have multiple symmetric equilibria that yield different social surpluses — the result of a strategic complementarity when signals are aggregated onto an agent’s bid. The aggregation of signals is also affected by the disclosure of a public signal; the disclosure may thereby jointly decrease the revenue and increase the social surplus. It follows that in multi- dimensional environments the linkage principle fails and public signals have an important effect on the social surplus.