Mixed Extensions of Decision Problems under Uncertainty

In a decision problem under uncertainty, a decision maker considers a set of alternative actions whose consequences depend on uncertain factors outside his control. Following Luce and Raiffa (1957), we adopt a natural representation of such situation that takes as primitives a set of conceivable actions A, a set of states S and a consequence function from actions and states to consequences in C. With this, each action induces a map from states to consequences, or Savage act, and each mixed action induces a map from states to probability distributions over consequences, or Anscombe-Aumann act. Under a consequentialist axiom, preferences over pure or mixed actions yield corresponding preferences over the induced acts. The most common approach to the theory of choice under uncertainty takes instead as primitive a preference relation over the set of all Anscombe-Aumann acts (functions from states to distributions over consequences). This allows to apply powerful convex analysis techniques, as in the seminal work of Schmeidler (1989) and the vast descending literature. This paper shows that we can maintain the mathematical convenience of the Anscombe-Aumann framework within a description of decision problems which is closer to applications and experiments. We argue that our framework is more expressive, it allows to be explicit and parsimonious about the assumed richness of the set of conceivable actions, and to directly capture preference for randomization as an expression of uncertainty aversion.