# Detectability, Duality, and Surplus Extraction

## TL;DR

This paper examines surplus extraction in complex environments, providing two proofs—geometric and duality-based—that extend understanding of when full surplus extraction is possible or impossible.

## Contribution

It offers two alternative proofs of McAfee and Reny's main theorem, one geometric for finite states and one duality-based for infinite states, broadening the theoretical framework.

## Key findings

- Full extraction is impossible without additional conditions.
- Geometric proof applies to finite state spaces.
- Duality proof extends to infinite state spaces.

## Abstract

We study surplus extraction in the general environment of McAfee and Reny (1992), and provide two alternative proofs of their main theorem. The first is an analogue of the classic argument of Cremer and McLean (1985, 1988), using geometric features of the set of agents' beliefs to construct a menu of contracts extracting the desired surplus. This argument, which requires a finite state space, also leads to a counterexample showing that full extraction is not possible without further significant conditions on agents' beliefs or surplus, even if the designer offers an infinite menu of contracts. The second argument uses duality and applies for an infinite state space, thus yielding the general result of McAfee and Reny (1992). Both arguments suggest methods for studying surplus extraction in settings beyond the standard model, in which the designer or agents might have objectives other than risk neutral expected value maximization.