Inference from Auction Prices

TL;DR

This paper presents an algorithm for inferring agents' private values in dominant-strategy mechanisms from observable outcomes and prices, with efficient solutions for certain social choice functions.

Contribution

It introduces a method to invert payment identities for value inference in mechanisms, extending existing results to new notions of concavity.

Findings

Unique invertibility for single-unit proportional weights functions

Efficient inversion algorithm for specific social choice functions

Extension of Nash equilibrium uniqueness to Gale-Nikaido concavity

Abstract

Econometric inference allows an analyst to back out the values of agents in a mechanism from the rules of the mechanism and bids of the agents. This paper gives an algorithm to solve the problem of inferring the values of agents in a dominant-strategy mechanism from the social choice function implemented by the mechanism and the per-unit prices paid by the agents (the agent bids are not observed). For single-dimensional agents, this inference problem is a multi-dimensional inversion of the payment identity and is feasible only if the payment identity is uniquely invertible. The inversion is unique for single-unit proportional weights social choice functions (common, for example, in bandwidth allocation); and its inverse can be found efficiently. This inversion is not unique for social choice functions that exhibit complementarities. Of independent interest, we extend a result of Rosen (1965), that the Nash equilbria of "concave games" are unique and pure, to an alternative notion of concavity based on Gale and Nikaido (1965).