  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).
