A Dynamic Game under Ambiguity: Contracting for Delegated Experimentation

Conventional (single-prior) Bayesian games of incomplete information are limited in their ability to capture the extent of informational asymmetry. In particular, they are not capable of representing complete ignorance of an uninformed player about an unknown parameter of the environment. Using a framework of contracting for delegated experimentation, we formulate and analyze a dynamic game of incomplete information that incorporates a multiple-prior belief system. Specifically, we consider a game with a principal contracting with an expert agent for his (observable) effort on a novel experiment – a Poisson process with unknown hazard rate. Although the expert agent has suffi cient knowledge to form a single prior over the hazard rate, the principal initially has complete ignorance and her ambiguous beliefs are represented by the set of all plausible prior distributions over the hazard rate. We propose a new equilibrium concept – Perfect Objectivist Equilibrium – in which the principal, who has ambiguity aversion, draws inference about the agent’s prior from the observed history of the game via maximum likelihood updating. The new equilibrium concept thus also embodies a novel model of learning under ambiguity in the context of a dynamic game. Although the game is rich in its contractual space and strategic interactions, the unique (Markov) equilibrium outcome is a remarkably simple pooling contract with appealing economic properties. In addition, the underlying Markov Perfect Objectivist Equilibria are all belief-free. These are in sharp contrast with the set of Markov Perfect Bayesian Equilibria, which not only hinge on subjective pretence of knowledge, but also predict multiple continuum of equilibrium outcomes. “Acknowledging what is known as known, what is not known as unknown, that is knowledge.” —Analects of Confucius “What has now appeared is that the mathematical concept of probability is inadequate to express our mental confidence or diffi dence in making such inferences, and that the mathematical quantity which appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability. To distinguish it from probability, I have used the term ‘likelihood’to designate this quantity.” —Sir R. A. Fisher (1925)