## Research

**Decision making under uncertainty:**

** Three representations of preferences with decreasing absolute uncertainty aversion**

*(Job market paper)*

This paper axiomatizes 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. In our first main result, we obtain three equivalent representations. The first is a variation on the constraint criterion of Hansen and Sargent (2001). The other two 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 variational preferences. In our second main result, we establish relationships among the representations for several important classes within variational preferences.

** Aspiration and confidence under uncertainty**

This paper develops a model of uncertainty in which 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. To reveal the decision maker’s perception about uncertainty, this paper introduces confidence orders in addition to preference orders; the confidence orders compare the decision maker’s confidence in all aspiration levels of all acts. Axioms are imposed on both confidence and preference orders, which yields a capacity over all priors to represent the confidence order, and the above decision rule to represent the preference order. The aggregating preference over the aspiration and confidence criteria plane is endogenously determined.

**Fair allocation:**

**Egalitarian division under Leontief preferences **

*(with Jin Li, Economic Theory, forthcoming)*

It was recently discovered that on the domain of Leontief preferences, Hurwicz (1972)’s classic impossibility result does not hold; that is, we can find efficient, strategy-proof and individually rational rules to allocate resources. In this paper, we consider the problem of fairly 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.

**Fair division with probabilistic demand **

*(in progress)*

This paper considers the problem of dividing a limited resource among agents with probabilistic demands. A key feature of such problems is the possibility that resources are wasted when the realized demand is less than the allocated amount. Two different families of rules are characterized which respect both fairness and non-wastefulness.

In the first family, each rule regards a random demand as a corresponding deterministic demand according to a parental preference over all the demands. These rules extend all of the established rules for deterministic demands to situations where agents have probabilistic demands. The second family of rules explicitly takes into account the agents’ expected waste of the resource in an allocation. Each division rule in this family is associated with a function that specifies how the resource allocated to each agent grows with the increase of the total resource. Under reasonable axioms, this function favors the agent whose expected waste increases more slowly in the resource that he receives.