Cognitive Imprecision and Small-Stakes Risk Aversion

Abstract

Observed choices between risky lotteries are difficult to reconcile with expected utility maximization, both because subjects appear to be too risk averse with regard to small gambles for this to be explained by diminishing marginal utility of wealth, as stressed by Rabin (2000), and because subjects’ responses involve a random element. 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 representations of the decision situation. In this model, risk aversion results from a sort of perceptual bias — but one that represents an optimal decision rule, given the limitations of the mental representation of the situation. We propose a quantitative model of the noisy mental representation of simple lotteries, based on other evidence regarding numerical cognition, and test its ability to explain the choice frequencies that we observe in a laboratory experiment.

Individual differences in the perception of probability

“…In Table 2 , the columns titled BIC calibration and BIC validation respectively report the average BIC values obtained from the calibration datasets and the remaining validation partitions. We follow Khaw, Li, and Woodford [ 29 ] in reporting a composite Bayes factor K = K 1 ⋅ K 2 , taking into account observations of both calibration and validation datasets. For any two models, and …”

Normative principles for decision-making in natural environments

“…One important intuition is that some phenomena may be rational for an agent whose cognitive systems have finite precision, because they have limited numbers of neurons (Dasgupta & Gershman 2021;Franconeri et al 2013) or noisy inference (Drugowitsch et al 2016). For example, the concave mapping of monetary value to utility that is a ubiquitous property of econometric functions is adaptive if lower-valued goods are more frequently encountered than higher-valued goods (as seems plausible), even in the absence of any assumptions about risk preference (Khaw et al 2018). If valuation is conceived as a sampling process, and preferences are dynamically constructed by retrieving memories of past experiences to a current prospect, then utility and probability functions will be naturally tailored to match previously encountered distributions of value and probability.…”

“…In combination with Bayesian decoding, a lognormal encoding scheme will ensure that utility functions exhibit the concavity that is characteristic of diminishing marginal utility, and that value estimates are contracted towards the prior mean (Khaw et al 2018). Of course the natural statistics of number, value, or quantity may be hard to measure, and so we must make some assumptions.…”

Individual Differences in the Perception of Probability

“…In Table 2, the columns titled BIC calibration and BIC validation respectively report the average BIC values obtained from the calibration datasets and the remaining validation partitions. We follow Khaw, Li, and Woodford [29] in reporting a composite Bayes factor K = K 1 � K 2 , taking into account observations of both calibration and validation datasets. For any two models, M 1 and…”