This article introduces a new approach to modeling the expanding universe of decision makers in the wake of growing awareness, and invokes the axiomatic approach to model the evolution of decision makers' beliefs as awareness grows. The expanding universe is accompanied by extension of the set of acts, the preference relations over which are linked by a new axiom, invariant risk preferences, asserting that the ranking of lotteries is independent of the set of acts under consideration. The main results are representation theorems and rules for updating beliefs over expanding state spaces and events that have the flavor of "reverse Bayesianism." (JEL D81, D83)According to the Bayesian paradigm, as new discoveries are made and new information becomes available, the universe shrinks: with the arrival of new information, events replace the prior universal state space to become the posterior state space, or universe of discourse. This process of "destruction" reflects the impossibility, in the Bayesian framework, of expanding the state space and of updating the probabilities of null events, coupled with the fact that conditioning on new information renders null events that, a priori, were nonnull. Yet, experience and intuition alike contradict this view of the world. Becoming accustomed to possibilities that were once inconceivable is part of history and our own life experience. There is a sense, therefore, in which our universe expands as we become aware of new opportunities.In this paper we take a step toward modeling the process of growing awareness and expansion of the universe, or state space, in its wake. 1 To model the evolution of beliefs in response to growing awareness, we invoke the theory of choice under uncertainty; borrowing its language and structure while modifying it to fit our purpose. In particular, we allow for new consequences and feasible acts to be discovered and for new evidence to establish, in the mind of decision makers, new links between acts and consequences. The interpretation of the updating is somewhat different for the discovery of new feasible acts and consequences on the one hand 1 Dekel, Lipman, and Rustichini (1998) argue that standard state spaces preclude unawareness. A choice theoretic approach therefore needs a more general point of departure than Savage (1954) and Anscombe and Aumann (1963).

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This paper extends our earlier work on reverse Bayesianism by relaxing the assumption that decision makers abide by expected utility theory, assuming instead weaker axioms that merely imply that they are probabilistically sophisticated. We show that our main results, namely, (modified) representation theorems and corresponding rules for updating beliefs over expanding state spaces and null events that constitute "reverse Bayesianism," remain valid.

This paper studies an environment in which a decision maker choosing between acts may initially be unaware of certain consequences. We follow the approach of Karni and Vierø (2013) to modeling increasing awareness, which allows for the decision maker's state space to expand as she becomes aware of new possible consequences. We generalize the main result in Karni and Vierø (2013) by allowing the discovery of new consequences to nullify some states that were non-null before the discovery. We also provide alternative assumptions which strengthen the predictions of the belief updating model.