Cardinal versus ordinal criteria in choice under risk with disconnected utility ranges

This paper provides a formal justification for the existence of subjective random components intrinsic to the outcome evaluation process of decision makers and explicitly assumed in the stochastic choice literature. We introduce the concepts of admissible error function and generalized certainty equivalent, which allow us to analyze two different criteria, a cardinal and an ordinal one, when defining suitable approximations to expected utility values. Contrary to the standard literature requirements for irrational preferences, adjustment errors arise in a natural way within our setting, their existence following directly from the disconnectedness of the range of the utility functions. Conditions for the existence of minimal errors are also studied. Our results imply that neither the cardinal nor the ordinal criterion do necessarily provide the same evaluation for two or more different prospects with the same expected utility value. As a consequence, a rational decision maker may define two different generalized certainty equivalents when presented with the same prospect in two different occasions.