Three essays on subjective ambiguity

The Subjective Expected Utility (SEU) Theory axiomatized by Savage is widely referred to as providing a subjective theory of probability because a probability measure 'p' is derived from preference. However, the domain of 'p' is assumed to be given exogenously, thus having an objective status in the theory. Exogeneity of the domain is not a limitation of a 'subjective theory' if it is believed that decision makers assign probabilities to all events that are relevant to the context being modeled. But choice behavior such as exhibited in the Ellsberg paradox and related evidence have demonstrated that many decision makers do not assign probabilities to all events. Decision makers may differ not only in the probabilities assigned to given events, but also in the identity of the events to which they assign probabilities. The main purpose of this thesis is to derive a fully subjective theory of probability in the sense that both the values and domain of the probability are derived from preference. The 'key' definition of the thesis--'subjectively unambiguous events'--is given in chapter 1. An implication is that the set <math> <f> <sc>A</sc></f> </math> of unambiguous events is in general not an algebra due to the fact the intersection of two unambiguous events is not necessarily unambiguous. However, <math> <f> <sc>A</sc></f> </math> is a [lambda]-system, a class of sets closed with respect to complements and disjoint unions. Thus one step towards in providing a fully subjective theory of probability is to characterize those likelihood relations defined on a [lambda]-system that can be represented numerically by an additive probability measure. Such a characterization is provided in chapter 3. After imposing some axioms on preference, we derive both a probability measure on <math> <f> <sc>A</sc></f> </math> and a representation for preference on the corresponding set of 'unambiguous' acts, namely acts that are measurable with respect to <math> <f> <sc>A</sc></f> </math>. The noted representation is that preference is 'based on probabilities' in the sense of probabilistic sophistication as in Machina and Schmeidler (MS). The major novelty relative to MS is that while their axioms deliver probabilistic sophistication on the domain of all acts, ours deliver probabilistic sophistication only on the domain of subjectively unambiguous acts, leaving the nature of the preference ranking of ambiguous acts unrestricted. A well-known alternative to the Savage model is Choquet Expected Utility (CEU) Theory axiomatized by Schmeidler (1989) and Gilboa (1987). Schmeidler uses the Anscombe-Aumann set-up to deliver the CEU model, but this approach presumes the existence of objective lotteries. Gilboa avoids this drawback by using the Savage set-up, but his axioms are hard to interpret. Moreover, it is not clear from these models 'why' and 'how' aversion to ambiguity leads a decision maker to use a nonadditive probability measure to represent her beliefs and a CEU model to represent her preferences over acts. Adopting the Savage set up, in chapter 2 we provide a simple axiomatization of the Choquet Expected Utility model where the capacity is an inner measure. Two attractive features of the model are its specificity and the transparency of its axioms. The key axiom states that the decision-maker uses unambiguous acts to approximate ambiguous ones. The notion of 'ambiguity' is subjective and derived from preference.