A Dynamic Game under Ambiguity: Repeated Bargaining with Interactive Learning1

Conventional Bayesian games of incomplete information are limited in their ability to represent complete ignorance of an uninformed player about an opponent’s private information. Using an illustrative example of repeated bargaining with interactive learning, we analyze a dynamic game of incomplete information that incorporates a multiple-prior belief system. We consider a game in which a principal sequentially compensates an agent for his effort on a novel experiment – a Poisson process with unknown hazard rate. The agent has knowledge to form a single prior over the hazard rate, but the principal has complete ignorance, 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 infers the agent’s prior from the observed history of the game via maximum likelihood updating. The new equilibrium concept embodies a novel model of learning under ambiguity in the context of a dynamic game. The unique (Markov) equilibrium outcome determines a unique bargaining solution. The underlying Markov Perfect Objectivist Equilibria are all belief-free, in sharp contrast to Markov Perfect Bayesian Equilibria, which hinge on subjective pretense of knowledge and predict a 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)