Multiple Beliefs, Dominance and Dynamic Consistency

This paper investigates multiperiod decisions under multiple beliefs. We explore the dynamic consistency of both complete and incomplete orderings. We focus on a dominance concept that supports decision-making under multiple characterizations of uncertainty by ruling out strategies that are dominated across a set of beliefs. We uncover a distinction between two types of dynamic inconsistency, which we label fallacious and fallible inconsistency. Fallacious inconsistency occurs when an a priori optimal strategy is suboptimal in the second period, thus requiring the decision-maker to depart from the original strategy. Fallible inconsistency occurs when an a priori suboptimal second-period action ceases being suboptimal from the perspective of the second-period preferences. We introduce corresponding definitions of dynamic consistency and show that the two types of consistency are equivalent for complete orderings, but differ for incomplete orderings. Subjective expected utility is dynamically consistent and non-expected-utility decision rules, such as minmax, are not. We show that the dominance relation over beliefs falls between these two: It is immune to the more severe fallacious inconsistency, but not to the less problematic fallible inconsistency. We illustrate the method and concepts using a numerical example addressing a focal, real-world problem of risk and ambiguity regarding climate change.