Players' beliefs in extensive form games
The epistemic program in game theory uses formal models of interactive reasoning to provide foundations for various game-theoretic solution concepts. Much of this work is based around the (static) Aumann structure model of interactive epistemology, but more recently dynamic models of interactive reasoning have been developed, most notably by Stalnaker (Economics and Philosophy 1996) and Battigalli and Siniscalchi (Journal of Economic Theory 1999), and used to analyze rational play in extensive form games. But while the properties of Aumann structures are well un- derstood, without a formal language in which belief and belief revision statements can be expressed, it is unclear exactly what are the properties of these dynamic models. In chapter 1, "Dynamic In- teractive Epistemology", we investigate this question by defining such a language. A semantics and syntax are presented, with soundness and completeness theorems linking the two. Chapter 2, "Algorithmic Characterization of Ratioalizability in Extensive Form Games", uses the framework of chapter 1 to construct a dynamic epistemic model for extensive form games, which generates a hierarchy of beliefs for each player over her opponents' strategies and beliefs, and tells us how those beliefs will be revised as the game proceeds. We use the model to analyze the implications of the assumption that the players possess common (true) belief in rationality, thus extending the concept of rationalizability to extensive form games. Chapter 3, "The Equivalence of Bayes and Causal Rationality in Games", takes as its starting point a seminal paper of Aumann (Econometrica 1987), which showed how the choices of rational players could be analyzed in a unified state space framework. His innovation was to include the choices of the players in the description of the states, thus abolishing Savage's distinction between acts and consequences. But this simplification comes at a price: Aumann's notion of Bayes ratio nality does not allow players to evaluate what would happen were they to deviate from their actual choices. We show how the addition of a causal structure to the framework enables us to analyze such counterfactual statements, and use it to introduce a notion of causal rationality. Under a plausible causal independence condition, the two notions are shown to be equivalent. If we are prepared to accept this condition we can dispense with the causal apparatus and retain Aumann's original framework. In chapter 4, "The Deception of the Greeks", it is argued that the standard model of an extensive form game rules out an important phenomenon in situations of strategic interaction: deception. Using examples from the world of ancient Greece and from modern-day Wall Street, we show how the model can be generalized to incorporate this phenomenon. Deception takes place when the action observed by a player is different from the action actually taken. The standard model does allow imperfect information (modeled by non-singleton information sets), but not deception: the actual action taken is never ruled out. Our extension of extensive form games relaxes the assumption that the information sets partition the set of nodes, so that the set of nodes considered possible after a certain action is taken might not include the actual node. We discuss the implications of this relaxation, and show that in certain games deception is inconsistent with common knowledge of rationality even along the backward induction path.