Essays in game theory
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Abstract
This dissertation studies the nature and the role of strategic uncertainty, which is uncertainty about the strategy choice of the other players. This uncertainty is typically represented by the player's probabilistic belief about the other players. In these models, analysts rely on the expected utility hypothesis to predict players' behavior. However, the expected utility model is often criticized since it excludes much interesting behavior. In the first chapter of the dissertation, I analyze a very general class of games and study a notion of rationalizability, showing it captures rationality and common knowledge of rationality in this framework. The framework is general enough to accommodate many non-expected utility models. I show that any rationalizable action profile can be made uniquely so by perturbing higher order uncertainty slightly. This result implies that the set of the rationalizable action profiles is generically a singleton. When there are multiple rationalizable action profiles, any refinement is not stable because there is another model with slightly different higher order beliefs where another action profile is uniquely rationalizable. This result also suggests that relaxing the expected utility axioms such as independence does not change behavior of the rational agents very much in strategic settings. In particular, if an action profile is uniquely rationalizable in the expected utility model, this action profile is still uniquely rationalizable even if the agents' risk attitude or ambiguity attitude changes slightly.
In the second chapter, I propose a new framework to study the role of strategic uncertainty in the information design problem while maintaining the expected utility hypothesis about the players' behavior. In the information design problem, a designer chooses an information structure, which is a probability distribution over signals she discloses to the player about the state. The designer's goal is to affect the players' decisions and implement her preferred outcome. After receiving information, the players update their belief about the other players. I characterize what is implementable when the designer takes into account potential misunderstandings of information and misspecification of beliefs of the players. In particular, I require that an outcome close to the desired one is implementable in every nearby information structure. I show that under a richness condition, an outcome is robustly implementable if and only if it is strictly dominant for all realizations of the signal. As a result, the designer's ability to affect the outcome is severely restricted and she cannot gain by disclosing information in many interesting cases. Another main result suggests that the designer has to send a more informative signal to implement a particular outcome compared to the standard case.