My dissertation devotes to the understanding of people's interactions under uncertainty.. It contains four essays on Microeconomics with Incomplete Information. 1 Chapter 1 focuses on the existence of rational bubbles in an Allen-Morris-Postlewaite (1993) setting, and finds positive and negative results for bubbles in an asset market featuring rational expectations equilibrium. An expected bubble is said to exist if it is mutual knowledge that the price of the asset is higher than the expected

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... ividend. Similarly we call it a strong bubble if everyone knows that the price is higher than the maximum possible dividend. Substituting common knowledge for mutual knowledge, I develop the new concepts of a common expected bubble and a common strong bubble. In a simple finite horizon model with asymmetric information and short sales constraints, I show that the following results hold for any finite number of agents. First, under the implicit assumption of perfect memory, common strong bubbles never exist in any rational expectations equilibrium. Second, it is possible to have one that is both a strong bubble and a common expected bubble in a rational expectations equilibrium. Based on these results, this paper, as well as Conlon (2004) and many others, provides a partial answer to the question: What properties do rational bubbles have in a rational expectations equilibrium? In Chapter 2, I study the relationship between information improvement and welfare outcomes in a finite-player finite-state model with incomplete information. In a context 1 Chapter 3 is based on joint work with John Conlon (University of Mississippi). ii of strategic interactions, it is possible that people may prefer to be ignorant rather than knowledgeable. Three simple examples are studied carefully in order to provide economic insight for this observation: if players were allowed to (not forced to) forget at no cost, they might have incentives to do so in equilibrium, and their expected payoff could actually be improved. In a general setting where players simultaneously choose whether to forget or not before the state of the world is realized, I show that players' actions would reveal additional information and that their preferences must be negatively correlated, for forgetfulness to be part of a possible equilibrium strategy. This finding indicates that in a world of incomplete information, people may not be made better off by obtaining more information, and they may even have incentive to be forgetful. These results will have important applications in policy design. Many economic models of rational bubbles are not very robust to perturbations. The existence of bubbles in these models requires strong conditions to be satisfied. In Chapter 3, we first study the bubble examples in the first Chapter and show that those bubbles are robust to both strongly symmetric perturbations in beliefs and very symmetric perturbations in dividends, but not robust to general perturbations. Then we construct a new three-period two-agent robust bubble example where small variations in parameters do not eliminate the bubble equilibria. The idea is that assuming continuum of states can lead to a robust bubble equilibrium where each bad type of the seller pools with some good type of the seller. This provides a new answer to the question: How robust can rational bubbles be in a finite horizon model? iii Morris, Shin and Postlewaite (1993) show an upper bound of asset prices in Rational Expectations Equilibrium. Chapter 4 is a note that strengthens their result by providing a tighter upper bound and hence offers a better answer to the question: How large can a bubble be in equilibrium? iv Acknowledgements After six years of doctoral studies in the Department of Economics, at Washington University in St. Louis, I have finally produced my Ph.D. thesis. This achievement would not have been possible without the supports from many people and institutes.