A decision frame theory for uncertainty with applications to regret and choice acclimatization

Abstract

An axiomatization of expected utility under uncertainty is extended in several steps to characterize more complicated decision models. Central to each step is a bijective mapping that, applied to the set of prospects, changes the framing of the decision problem. Static models of subjective expected utility and reference dependence result from stretch mappings and translations (i.e., shifts), respectively. Dynamic models, such as regret and aspiration learning, involve groups of models each of which applies a translation by a different reference prospect (aspiration). The resulting regret model accommodates preference cycles in a higher-dimensional decision space without violating transitivity. An equilibrium of such a dynamic model is characterized as the prospect that is a maximal element in reference to itself. Under aspiration learning, asymptotic stability thus ensures that the aspiration (eventually) matches the equilibrium prospect. The concepts of regret and aspiration learning are combined to a model of choice acclimatization for the purpose of equilibrium selection. The valuation function of this equilibrium selection model is then further specified to accommodate cumulative regret that accrues during the acclimatization process.