We consider the question, “Under what conditions is the extremum of a random function over a random set itself a random object?” The answer is relevant to problems in both game theory and econometrics, as we illustrate with examples. Our purpose here is to bring the powerful tools of the theory of analytic sets as developed by Dellacherie and Meyer (1978) to the wider attention of the economics profession and to distill Dellacherie and Meyer's work in such a way as to provide some readily accessible theoretical results that will permit relatively easy treatment of economically or econometrically relevant applications.

We consider a principal-agent contracting problem under incomplete information where some of the agent's actions are imperfectly observable. Contracts take the form of reward schedules based on the noisy observation of the agent's action. We first review situations where the principal can reach the same utility as in the absence of noise. Then we focus on the use of linear reward schedules, which allow universal implementation, i.e. implementation of a given mechanism for any unbiased noise of observation, and on quadratic reward schedules, which only require the knowledge of the variance of the noise. We exhibit sufficient conditions under which linear reward schedules implement a given mechanism. Finally, we characterize necessary conditions for a mechanism to be implementable under noisy observation by a linear schedule, and by quadratic schedules. We give the geometric intuition behind all results.

This paper examines the problem of non-existence of a single-wage equilibrium in a simple search model with asymmetric information. A pure-strategy, symmetric Nash equilibrium fails to exist because adverse selection arising from steady-state considerations causes a non-concavity in the payoff function.