States, Models and Unitary Equivalence I: Representation Theorems and Analogical Reasoning

I show that virtually any model of decision making under uncertainty is associated to a certain structure. This contains three fundamental ingredients: (1) The domain of the acts; (2) Another set, which is called the set of models for the decision maker; and (3) The decision maker’s information about the set of models (an algebra of subsets of the set of models). A consequence of this finding is that that the decision maker’s choices can be viewed as the outcome of a two-stage process. First, the set of acts is mapped into a system of hypothetical bets on the set of models. Then, the latter are ranked by the decision maker. I show that this procedure can be thought of as describing a general form of analogical reasoning. I also observe that the appearance of two different sets implies that the decision maker is uncertain about two different objects and that he may receive information about any of them. In particular, information about the set of models affects the decision maker’s ranking of the available alternatives. In the sequel to this paper, I show that certain natural information structures lead to an inherent inability of assigning probabilities on the domain of the acts. In a formal sense, their properties describe the idea of Knightian Uncertainty. ∗I benefited from comments and suggestions from Larry Epstein, Paolo Ghirardato, Tzachi Gilboa, Fabio Maccheroni, Massimo Marinacci, Rich McLean, Ket Richter, Aldo Rustichini, Marco Scarsini, Paolo Siconolfi, Max Stinchcombe, William Thomson, Stan Wellisz and Bill Zame. I also wish to thank seminar participants at University of Iowa, RUD 2003, CalTech, University of Minnesota, University of Texas-Austin, Columbia University, University of Rochester, Rutgers University, University of Turin, Ohio State University, Columbia University Workshop on Multiple Priors. †Department of Economics, Columbia University, 1019 IAB, 420 W 118th street, NY, NY 10027. Email: [email protected].