INTRODUCTION

In this chapter, I shall develop a theory of equilibrium in normal-form games based on a formalization of counterfactuals. This theory has its starting point in the decision-theoretic framework of Jeffrey (1965), but develops this framework in ways that diverge from the spirit of Jeffrey's own theory. In order to motivate the theory, and to explain why it diverges from Jeffrey's, let us start by describing the main features of Jeffrey's framework for decisions.

In abolishing Savage's (1954) distinctions between “acts,” “states of the world,” and “consequences,” Jeffrey (1965) was able to construct a unified framework for decisions in which the consequences of an individual's action and the action itself is as much a part of the description of the world as any other feature of the world. To choose an act in Jeffrey's framework is to make a certain proposition true. Thus, when Ω is the state space consisting of all states of the world ω, an act can be seen as a subset of Ω. To choose an act ak is to ensure that the true state of the world is an element of ak.

Let {a1, a2, …, am} be the set of acts for the decision maker. This set partitions Ω, reflecting the condition that one (and only one) act is chosen. It is assumed that the decision maker has a probability distribution p over Ω and that u(ω), the desirability of the state ω, is thus the value of a random variable u at ω.