Expected Utility Theory with “Small Worlds”

We formulate a new theory of expected utility under uncertainty based on the notion of an event-lattice, which is a natural generalization of a Savage state space. The modelling of uncertainty is based on the idea that the decision maker for each group of related decisions to be taken creates a ”small world” which functions as a local state space. We introduce a set of preference axioms similar in spirit to the Savage axioms, and show that they lead to a generalization of the standard von Neumann-Savage theory of expected utility. The generalization allows for an intuitive distinction between risk and uncertainty. In each ”small world” risk is described by an additive probability measure; and these local risk measures all appear as restrictions of a common integrated additive expectation functional which is defined on the ”grand world”, thereby providing numerical expressions to the notion of uncertainty. We illustrate the use of the theory for the Ellsberg paradox and for some portfolio decisions which cannot be captured by the standard von Neumann-Savage theory.