Essays on Robustness and Uncertainty in Game Theory

Ziegler, Gabriel

doi:https://doi.org/10.21985/n2-zept-eh75

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Essays on Robustness and Uncertainty in Game Theory

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In this dissertation I examine issues related to uncertainty and robustness in game theory. In Chapter 1 a strategic setting is analyzed where players face Knightian uncertainty about the strategic choices of their opponents. That is, in contrast to the usual Bayesian framework and in line with experimental evidence, players might not be able to form probabilistic belief about the uncertainty they face. Instead players entertain a set of beliefs about this uncertainty. In joint work with Peio Zuazo-Garin, I provide a general model that captures this situation and show that, under certain assumptions, well-known solution concepts, such as rationalizability, are still appropriate. Chapter 2 reinterprets the model with Knightian uncertainty by interpreting cautiousness as robustness to ambiguity. As before, each player’s strategic uncertainty is represented by a possibly non-singleton set of beliefs, but now a rational player also wants to make a choice that is robust to this ambiguity. Thus, a rational player chooses a strategy that is a best-reply to every belief in this set. Again in joint work with Peio Zuazo-Garin, I show that the interplay between these two features precludes the conflict between strategic reasoning and cautiousness and therefore solves the inclusion-exclusion problem raised by Samuelson (1992). Notably, my approach provides a simple foundation for the iterated elimination of weakly dominated strategies. In Chapter 3 I study an information provider who commits to provide information to multiple receivers seeking robustness towards the reasoning of these receivers. The robustness consideration arises naturally in a setting where information is provided bilaterally. Such a scenario precludes the possibility of commitment to a grand information structure. Consequently, in a strategic situation, each receiver needs to reason about what information other receivers get. Since the information provider does not know this reasoning process, a motivation for a robustness requirement arises: the provider seeks an information structure that performs well no matter how the receivers actually reason. In this chapter, I provide a general method to study how to optimally provide information under these constraints. The main result is a representation theorem, which makes the problem tractable by reformulating it as a linear program in belief space. Furthermore, I provide novel bounds on the correlation among receivers’ beliefs, which provide even more tractability in some special cases. I illustrate the main result by solving for the optimal provision of information in a stylized model of contract research organizations, which are an integral part of the pharmaceutical industry.

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