| This research project studies the role of the principle of diversification (Convexity) in Economics, particularly, for situations of choice under Risk and Ambiguity.In the first chapter, we study convex preferences over lotteries and over menus of lotteries. We consider a set of consequences C and we characterize complete, transitive, and convex binary relations over lotteries on the set C. We prove that convex preferences correspond to a decision criterion in which the Decision Maker reveals pessimism and a lack of confidence in the evaluation of consequences or his future tastes. We show in a context of choice over menus of lotteries how convex preferences translate into Maxmin Expected Utility on a Subjective State Space. Finally, we show how convex preferences can be interpreted as a cautious criterion of completion.In the second chapter, we study uncertainty averse preferences, that is, complete and transitive preferences that are convex and monotone in an Anscombe and Aumann setting. We establish a representation result, which is at the same time general and rich in structure. Many objective functions commonly used in applications are special cases of this representation. Our purpose in this chapter is to put some order in this class of models by providing a common representation that, through its properties, allows to unify and classify them.In the last chapter, we introduce a notion of complete monotone quasiconcave duality and we show that it holds for important classes of quasiconcave functions. This chapter develops the mathematical theory on which the previous two chapters rest.The results presented in the second and third chapter are joint work with F. Maccheroni, M. Marinacci, and L. Montrucchio. |
