Risk Aversion as a Perceptual Bias∗

The theory of expected utility maximization (EUM) proposed by Bernoulli explains risk aversion as a consequence of diminishing marginal utility of wealth. However, observed choices between risky lotteries are difficult to reconcile with EUM: for example, in the laboratory, subjects’ responses on individual trials involve a random element, and cannot be predicted purely from the terms offered; and subjects often appear to be too risk averse with regard to small gambles (while still accepting sufficiently favorable large gambles) to be consistent with any utility-of-wealth function. We propose a unified explanation for both anomalies, similar to the explanation given for related phenomena in the case of perceptual judgments: they result from judgments based on imprecise (and noisy) mental representation of the decision situation. In this model, risk aversion is predicted without any need for a nonlinear utility-of-wealth function, and instead results from a sort of perceptual bias — but one that represents an optimal Bayesian decision, given the limitations of the mental representation of the situation. We propose a specific quantitative model of the mental representation of a simple lottery choice problem, based on other evidence regarding numerical cognition, and test its ability to explain the choice frequencies that we observe in a laboratory experiment. ∗An earlier version of this work, under the title “Cognitive Limitations and the Perception of Risk,” was presented as the 2015 AFA Lecture at the annual meeting of the American Finance Association. We thank Colin Camerer, Tom Cunningham, Xavier Gabaix, Arkady Konovalov, Ifat Levy, Rosemarie Nagel, Charlie Plott, Rafael Polania, Antonio Rangel, Christian Ruff, Hrvoje Stojic, Shyam Sunder, and Ryan Webb for helpful comments, and the National Science Foundation for research support. One of the most commonplace observations about economic life is that people often appear to be risk averse: they are unwilling to accept fair bets, and indeed pass up opportunities that would offer them a higher expected monetary reward for the sake of reduced uncertainty about the outcome. The standard explanation for such behavior, dating back to Bernoulli (1954 [originally 1738]), proposes that people do not choose so as to maximize their expected wealth, but instead to maximize expected utility, where utility is hypothesized to be a strictly concave (rather than linear) function of wealth. The consequences of this theory for choice under risk are now a staple element of undergraduate pedagogy, and the cornerstone of the modern theory of finance. The theory of expected utility maximization (EUM), however, fails to account for a number of robust features of observed behavior, clearly documented by laboratory studies of choices between small monetary gambles. For one, EUM implies that choice should be a deterministic function of the monetary payoffs offered and their associated probabilities. In the laboratory, instead, choices appear to be random, in the sense that the same subject will not always make the same choice when offered the same set of simple gambles on different occasions (Hey and Orme, 1994; Hey, 1995, 2001). This was evident (though little remarked upon) already in Mosteller and Nogee (1951), one of the earliest experimental studies of the empirical support for EUM. Figure 1 (reproduced from their paper) plots the responses of one of their subjects to a series of questions of a particular type. In each case, the subject was offered a choice of the form: are you willing to pay five cents for a gamble that will pay an amount x with probability 1/2, and zero with probability 1/2? The figure shows the fraction of trials on which the subject accepted the gamble, in the case of each of several different values of x. The authors used this curve to infer a value of x for which the subjects would be indifferent between accepting and rejecting the gamble, and then proposed to use this value of x to identify a point on the subject’s utility function. The fact that the indifference point is at a value of x greater than 10 cents (the case of a fair bet) is taken by the authors to indicate a concave utility function. But in fact, no utility function is consistent with the data shown in the figure; for EUM implies that the probability of acceptance should be zero for all values of x below the indifference point, and one for all values above it. Instead one observes probabilistic choice for a range of values of x, with the probability of acceptance increasing monotonically with x. This randomness is often de-emphasized in discussions of the experimental evidence for particular types of preferences over risky gambles, by simply focusing on modal or median responses. Studies that model the randomness in individual responses (e.g., Loomes and Sugden, 1995; Ballinger and Wilcox, 1997; Holt and Laury, 2002; Loomes, 2005; Wilcox, 2008) typically treat the randomness as something that can be specified independently of a “core” deterministic model of preference over lotteries (such as EUM), which is supposed to explain subjects’ See Friedman et al. (2014) for an overview of the literature.