I study the stability and doctor-optimality of doctors' proposing cumulative offer process in the many-to-one matching with contracts. First, I explore some conventional hospital-by-hospital conditions on each hospital's choice function, and show that unilateral substitutability is equivalent to observable substitutability across doctors combined with cumulative offer achievability, each of which is a necessary condition for cumulative offer process to be doctor-optimally stable in a sense that if a hospital does not satisfy the condition, then we could construct some choice functions for other hospitals such that cumulative offer process is not doctor-optimally stable for some doctors' preference profile. Then, I focus on the joint properties of the choice functions for the entire group of hospitals and introduce two joint conditions---independence of proposing order and group cumulative offer achievability---and show that when these conditions are satisfied, cumulative offer process is always doctor-optimally stable. And it is by far the weakest sufficient condition. Moreover, these two conditions are necessary in a sense that if not, then there exists a doctors' preference profile and a proposing order such that cumulative offer process is not doctor-optimally stable. At last, I also introduce doctor's preference monotonicity and show that when cumulative offer process is doctor-optimally stable, this condition guarantees its strategy-proofness.

This dissertation consists of two essays on microeconomics.

The first chapter explores leadership within hierarchical organizations.

For each hierarchy, I consider a dynamic signaling game

in which each player observes only the actions of his direct

superiors before choosing his action.

At the top of the hierarchy are the leaders, who learn the state from nature.

The hierarchy controls the flow of information and the timing of the game,

and determines the equilibrium output and welfare.

I show that the welfare-optimal hierarchy is the chain, because

it maximizes the incentive of players to ``lead by example'' for their

subordinates. The chain remains optimal even in the presence of verifiable

or unverifiable costly information acquisition by the leaders.

Lastly, I characterize optimal hierarchies when the number of layers or

the number of leaders is limited. Applications to fund-raising are also discussed.

The second chapter studies the optimal way to select projects or agents in environments where

information arrives in well defined rounds. Examples include academic

environments where review periods are set by policy, aptitude tests such as

those given by software developers to programmers applying for jobs, venture

capital protocols where the rounds of funding may be stopped before the

project is complete, and FDA testing, where drugs can be dropped at well

defined junctures. Sequential rounds of elimination reduce the cost of

selection, but also reduce the average quality of surviving projects. I

characterize the nature of the optimal screening process with and without

"memory." The second chapter is based on joint work with Suzanne Scotchmer.

This paper considers a general equilibrium model in which the distinction between un-certainty and risk is formalized by assuming agents have incomplete preferences over state-contingent consumption bundles, as in Bewley (1986). Without completeness, individual decision making depends on a set of probability distributions over the state space. A bundle is preferred to another if and only if it has larger expected utility for all probabilities in this set. When preferences are complete this set is a singleton, and the model reduces to standard expected utility. In this setting, we characterize Pareto optima and equilibria, and show that the presence of uncertainty generates robust indeterminacies in equilibrium prices and allocations for any specification of initial endowments. We derive comparative statics results linking the degree of uncertainty with changes in equilibria. Despite the presence of robust indeterminacies, we show that equilibrium prices and allocations vary continuously with underlying fundamentals. Equilibria in a standard risk economy are thus robust to adding small degrees of uncertainty. Finally, we give conditions under which some assets are not traded due to uncertainty aversion.

This dissertation studies models of dynamic choices under uncertainty with endogenous information acquisition. In particular we are interested in exploring the interactions between ambiguity attitudes and the incentive to collect new information.

The first chapter explores the link between intrinsic preferences for information and ambiguity attitudes in settings with subjective uncertainty. We enrich the standard dynamic choice model in two dimensions. First, we introduce a novel choice domain that allows

preferences to be indexed by the intermediate information, modeled as partitions of the underlying state space. Second, conditional on a given information partition, we allow preferences over state-contingent outcomes to depart from expected utility axioms. In particular

we accommodate ambiguity sensitive preferences. We show that aversion to partial information is equivalent to a property of static preferences called Event Complementarity. We show that Event Complementarity and aversion to partial information are closely related to

ambiguity attitudes. In familiar classes of ambiguity preferences, we identify conditions that characterize aversion to partial information.

The second chapter extends the basic model to allow for choices from non-singleton menus after partial information is revealed, and studies the value of information under ambiguity. We show that the value of information is not monotonic under ambiguity. Intrinsic aversion

to partial information in the basic model is equivalent to a preference for perfect information in the extended model. Moreover, the value of information is not monotone in the degree of ambiguity aversion.

The third chapter studies the impact of ambiguity in a classic information acquisition model-the K-armed bandit problem. We consider a particular family of ambiguity averse preferences, the multiple-priors model [Marinacci, 2002]. A previous paper [Li, 2012] shows that major classic characterizations of optimal strategies in the K-armed bandit problems

extend to incorporate ambiguity in the multiple-priors model. Here we explore new implications of ambiguity on the optimal incentive to experiment. First, increasing ambiguity in the unknown arm reduces the incentive to experiment, while increasing risk in the unknown arm typically increases the incentive to experiment. This suggests that ambiguity can offer an explanation for the widely observed under-experimentation in novel technology and consumer products. Second, optimal experimentation in the multiple-priors bandit problem generally cannot be reduced to that in a classic bandit problem with an equivalent

single prior. In particular, the lower envelope of the classic single-prior Gittins-Jones index for every prior lying in the multiple-priors set can be strictly higher than the generalized multiple-prior Gittins-Jones index. In one-dimensional parametric family, we identify monotonicity conditions under which this discrepancy disappears so an equivalent single prior

exists.