The first chapter is based on a paper with Jin Li in fair division. It was recently discovered that on the domain of Leontief preferences, Hurwicz (1972)'s classic impossibility result does not hold; that is, one can find efficient, strategy-proof and individually rational rules to divide resources among agents. Here we consider the problem of dividing l divisible goods among n agents with the generalized Leontief preferences. We propose and characterize the class of generalized egalitarian rules which satisfy efficiency, group strategy-proofness, anonymity, resource monotonicity, population monotonicity, envy-freeness and consistency. On the Leontief domain, our rules generalize the egalitarian-equivalent rules with reference bundles. We also extend our rules to agent-specific and endowment-specific egalitarian rules. The former is a larger class of rules satisfying all the previous properties except anonymity and envy-freeness. The latter is a class of efficient, group strategy-proof, anonymous and individually rational rules when the resources are assumed to be privately owned.
The second and third chapters are based on two working papers of mine in decision making under uncertainty. In the second chapter, I study the wealth effect under uncertainty --- how the wealth level impacts a decision maker's degree of uncertainty aversion. I axiomatize a class of preferences displaying decreasing absolute uncertainty aversion, which allows a decision maker to be more willing to take uncertainty-bearing behavior when he becomes wealthier. Three equivalent preference representations are obtained. The first is a variation on the constraint criterion of Hansen and Sargent (2001). The other two respectively generalize Gilboa and Schmeidler (1989)'s maxmin criterion and Maccheroni, Marinacci and Rustichini (2006)'s variational representation. This class, when restricted to preferences exhibiting constant absolute uncertainty aversion, is exactly Maccheroni, Marinacci and Rustichini (2006)'s ariational preferences. Thus, the results further enable us to establish relationships among the representations for several important classes within variational preferences.
The three representations provide different decision rules to rationalize the same class of preferences. The three decision rules correspond to three ways which are proposed in the literature to identify a decision maker's perception about uncertainty and his attitude toward uncertainty. However, I give examples to show that these identifications conflict with each other. It means that there is much freedom in eliciting two unobservable and subjective factors, one's perception about and attitude toward uncertainty, from only his choice behavior. This exactly motivates the work in Chapter 3.
In the third chapter, I introduce confidence orders in addition to preference orders. Axioms are imposed on both orders to reveal a decision maker's perception about uncertainty and to characterize the following decision rule. A decision maker evaluates an act based on his aspiration and his confidence in this aspiration. Each act corresponds to a trade-off line between the two criteria: The more he aspires, the less his confidence in achieving the aspiration level. The decision maker ranks an act by the optimal combination of aspiration and confidence on its trade-off line according to an aggregating preference of his over the two-criterion plane. The aggregating preference indicates his uncertainty attitude, while his perception about uncertainty is summarized by a generalized second-order belief over the prior space, and this belief is revealed by his confidence order
