Strategic decision making involves deliberating about the consequences of every available course of action. This in turn, requires hypothesizing about the beliefs of the other players and their possible reactions to observed play. Unless individuals have previous experience with similar games or players, they will have to resort to their own assessment and reasoning in order to justify their strategies.
Decisions that arise out of introspection are full of counterfactual quandaries. Had I chosen an action that the other players did not expect, would they have rejected the hypothesis that I was rational? Or, would they have instead believed that I played in this unexpected way intentionally? Game theorists used to think that common knowledge of rationality was enough guidance to decide about the truth of these counterfactual statements. In the late 80s it became clear that the game theoretic framework lacked a model to understand this type of deliberation and that without such a model, the whole agenda of refining the predictions and prescriptions of game theory would become seriously jeopardized. Only by modeling the way in which players make inferences, namely by introducing a theory of how to analyze counterfactuals, it is possible to evaluate these deliberations. As the philosopher Robert Stalnaker clearly argued, this type of reasoning is of vital importance because players cannot choose an action or decide to comply with any suggested equilibrium if they cannot deliberate about such counterfactuals.
This book examines the work of two philosophers who provided theories to evaluate counterfactual reasoning, namely, David Lewis and Jonathan Bennett, to assess their impact upon the foundations of backward induction and sequential equilibrium. First, we test the consistency of the results implied by these approaches and analyze the consequences of having players form beliefs in the manner prescribed by our interpretation of these theories of counterfactuals. According to our interpretation of Lewis's approach and in our version of the centipede game, common knowledge of rationality leads to the backward induction outcome. According to our interpretation of Bennett's approach, backward induction and common knowledge may turn out to be inconsistent. Second, we explore the consequences that alternative ways of drawing inferences from deviations have upon refinements of sequential equilibrium in signaling games. These refinements assume that off-the-equilibrium path signals could be intentional. We show that this feature may lead to an inconsistency whenever the game cannot encompass common knowledge of rationality with fully intentional off-the-equilibrium path behavior.