Unawareness and Partitional Information Structures

We claim ̄rst that simple uncertainty is not an adequate model of a subject's ignorance, because a major component of it is the inability to give a complete description of the states of the world, and we provide a formal model of unawareness. In Modica and Rustichini (1994) we showed a di±culty in the project, namely that without weakening of the inference rules of the logic one would face the unpleasant alternative between full awareness and full unawareness. In this paper we study a logical system where non full awareness is possible, and prove that a satisfactory solution to the problem can be found by introducing limited reasoning ability of the subject. A determination theorem for this system is proved, and the appearance of partitional informational structures with unawareness is analysed. 1. Uncertainty and Ignorance Decision theory under uncertainty models the behavior of a subject who has to take an action, but does not know which of a list of possible states of the world is the true state. In the theory as presently structured, however, the ignorance of the subject is limited to this lack of knowledge: the description of the world in his mind is in fact correct and exaustive. It is perhaps obvious, but still important to note that this is by no means necessarily true; rather it is an assumption on the rationality of the decision maker. The assumption is that he may be uncertain about the true state, but he has no ignorance about the state space. The purpose of this paper is to model situations where such ignorance is possible. We provide a model where some of the facts that determine which state of nature occurs are not present to the subject's mind, and this lack of awareness makes him incapable of giving a complete description of the `objective' state space. The idea which we formalize here is that the individual's `subjective' description of the world is constructed on the basis of those events of which the subject has awareness. Our formal de ̄nition of awareness, ̄rst introduced in Modica and Rustichini (1994), is based on the idea that there are three possible states of a subject's knowledge regarding the truth/falsity of a proposition p (for example, `it rains'). First, he may be certain of the truth value (true or false) of p; second, he may be consciously uncertain about it, in the sense of not knowing and knowing of not knowing. These are the two possibilities that are usually considered in Decision Theory. The third possibility we want to consider is the following: he may not know P , not know he does not know it, not know he does not know he does not know it, and so on ad in ̄nitum. The last possibility represents the situation of Date: this version October 1997. We thank Eddie Dekel, Itzhak Gilboa, Bart Lipman, Philippe Mongin, Herakles Polemarchakis, Roy Radner, David Schmeidler, Dov Samet, for long discussions on this topic. We thank an associate editor and two referees for a great number of suggestions and comments that helped to improve the presentation and the substance of the paper. We also thank Robert Stalnaker for the insightful discussion on the issue of unawareness, knowledge and belief.