Consistency and Aggregation in Individual Choice Under Uncertainty

It is common in studies of individual choice behavior to report averages of the behavior under consideration. In the social sciences the mean is, indeed, often the quantity of interest, but at times focusing on the mean can be misleading. For example, it is well known in labor economics that failure to account for individual differences may lead to incorrect inference about the nature of hazard functions for unemployment duration. If all workers have constant hazard functions independent of duration, simple aggregation will nonetheless lead to the inference that the hazard function is state-dependent, with the hazard of leaving unemployment declining with duration of unemployment. Similarly, a recent study in psychology has shown that the “learning curve,” a monotonically increasing function of response to a stimuli, is better understood as an average representation of individual response functions that are, in fact, more step-function-like. As such, the learning curve as commonly understood is a misleading representation of the behavior of any one individual. These observations motivate us to consider the question of possible aggregation bias in the realm of choice under uncertainty. In particular, Cumulative Prospect Theory posits a weighting function through which probabilities are transformed into decision weights. An inverted S-shaped weighting function is commonly taken to be “the” appropriate weighting function, based on quite a number of experimental studies. This particular version of the weighting function implies, in simple two outcome lotteries, that an individual will tend to overweight small (near 0) probabilities and to underweight large (near 1) probabilities. A natural question to ask, suggested by both the hazard function and the learning curve examples, is whether this weighting function is not, similarly, an artifact of aggregation. Of course, no one believes that every individual’s behavior can be accounted for by a single weighting function. Studies have shown that there can be considerable variation in estimated weighting functions across individuals. But no one, to our knowledge, has systematically addresses the question of whether, in fact, one can meaningfully use a single weighting function, even as a rhetorical device, to accurately discuss individual choice behavior. If most individuals indeed do have an inverted S-shaped weighting function, then this representation of choice behavior is not misleading, provided it is clear that one is discussing the behavior of “most,” not all, individuals. We focus on the reliability of estimated weighting functions. We study the problem of determining the parameters of the cumulative prospect theory function. Using responses to paired sets of choice questions, it is possible to derive estimates for a two-parameter version of the Cumulative Prospect Theory choice function (using a power function for the value function and Prelec’s one parameter version of the weighting function). By analyzing multiple such pairs of choice questions, we are able to also investigate the consistency of these estimates. Our main finding is that there is, in general, considerable variation at the individual level in the choice parameters implied by the responses to the different pairs of choice questions. The modal choice pattern observed is one consistent with expected value maximization, and there is considerably less variation (again, at the individual level) in the parameters implied by those who appear to be maximizing expected value on one pair of choice questions than for those who never choose in this way. But these individuals account for only about one-fifth to one-sixth of subjects. For the rest of the subjects, it is rare that any two pairs of estimates are the same, and often the implied parameters