Beliefs and Dynamic Consistency,

In this chapter, we adopt the decision theoretic approach to the representation and updating of beliefs. The standard bayesian approach, as developed by Savage (1954), provides the conditions under which a decision maker whose preferences satisfy a certain number of “rationality” axioms (among which the famous “sure thing principle”) must have probabilistic beliefs. Thus, under these axioms, any decision maker is able (or acts as if he were able) to come up with a probability distribution over the states of nature (i.e., the sources of uncertainty) and, furthermore, acts as if maximizing an expected utility with respect to this prior. This construction underlies most of economic theory. However, as early as 1921, Keynes (1921) and Knight (1921) were already concerned that probabilistic beliefs might miss an important aspect, namely the confidence that agents have in their own beliefs. Doubts about the fact that probabilistic beliefs could represent any type of uncertainty were also raised by Shackle (1952). These (theoretical) concerns were backed by experimental evidence showing that subjects can have problems in dealing with probabilistic information in a way predicted by the theory (Allais (1953)), and more, drastically, had difficulties to come up with a probabilistic prior when no such information is a priori at hand (Ellsberg (1961)). In more recent years, more general axiomatic models of decision under uncertainty have emerged, in which agents could well have non-additive beliefs (Schmeidler (1989)) or multiple prior beliefs (Gilboa and Schmeidler (1989)). These models are precisely meant to deal with “Ellsberg paradox” while retaining enough structure so as to have some normative appeal on top of their better descriptive ability. This paradox has been central in the development of decision theory under uncertainty and we hence briefly recall it here. Consider an urn in which there are 90 balls. The decision maker is told that there are 30 red balls and that the ∗CNRS–EUREQua, [email protected] †CNRS–EUREQua, [email protected]