Coping With Ignorance: Unforeseen Contingencies and Non-Additive Uncertainty

Abstract

In real-life decision problems, decision makers are never provided with the necessary background structure: the set of states of the world, the outcome space, the set of actions. They have to devise all these by themselves. I model the (static) choice problem of a decision maker (DM) who is aware that her perception of the state space is too coarse, as for instance when there might be unforeseen contingencies. After making assumptions on the way the DM perceives the decision problem, I present a set of axioms on her preferences which imply that they can represented by a (generalized) expectation with respect to a non-additive measure, called a belief function. As it turns out, the very natural axioms presented have strong implications on the way the DM copes with the type of ignorance described above. I show how some decision rules that have been studied in the literature can be obtained as a special case of the model presented here (though they have to be interpreted differently). I then show that this formulation of the problem can yield very natural results on the comparative statics of beliefs as the DM's understanding of the decision problem becomes deeper, for instance when unforeseen events become foreseen. I present the implications of these results for a simple asset pricing model. Finally I argue that if we are willing to make assumptions on the faithfulness of the DM's perception with respect to reality, then the DM described here will, in the limit as her perception becomes finer and finer, resemble a Savage DM.