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 beyond his control. Following Luce and Raiffa (Games and decisions: introduction and critical survey. Wiley, New York, 1957), we adopt a natural representation of such a situation which takes as primitives a set of conceivable actions A, a set of states S and a consequence function $$\rho {:}\,A\times S\rightarrow C$$ ρ : A × S → C . Each action induces a map from states to consequences, a Savage act, and each mixed action induces a map from states to probability distributions over consequences, an Anscombe–Aumann act. Under an axiom of consequentialism, preferences over pure or mixed actions yield corresponding preferences over the induced acts. This observation allows us to relate the Luce–Raiffa description of a decision problem to the most common framework of modern decision theory which directly takes as primitive a preference relation over the set of all Anscombe–Aumann acts. The key advantage of the latter framework is the possibility of applying powerful convex analysis techniques as in the seminal work of Schmeidler (Econometrica 57:571–587, 1989) and the vast literature that followed. 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 many applications and experiments. We argue that our framework is more expressive as it allows us to be both 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.