Nonlinear Probability Weighting Research Articles

Toward a computational understanding of how reminiscing about positive autobiographical memories influences decision-making under risk.

Recent computational psychiatric research has dissected decision-making under risk into different underlying cognitive computational constructs and identified disease-specific changes in these constructs. Studies are underway to investigate what kind of behavioral or psychological interventions can restore these cognitive, computational constructs. In our previous study, we showed that reminiscing about positive autobiographical memories reduced risk aversion and affected probability weighting in the opposite direction from that observed in psychiatric disorders. However, in that study, we compared positive versus neutral memory retrieval by using a within-subjects crossover posttest design. Therefore, the change of decision-making from baseline is unclear. Furthermore, we used a hypothetical decision-making task and did not include monetary incentives. We attempt to address these limitations and investigated how reminiscing about positive autobiographical memories influences decision-making under risk using a between-subjects pretest posttest comparison design with performance-contingent monetary incentives. In thirty-eight healthy, young adults, we found that reminiscing about positive memories reinforced the commonly observed inverted S-shaped nonlinear probability weighting (f = 0.345, medium to large in effect size). In contrast, reminiscing about positive memories did not affect risk aversion in general. Given that the change in probability weighting after reminiscing about positive memories is in the opposite direction from that observed in psychiatric disorders, our results indicate that positive autobiographical memory retrieval might be a useful behavioral intervention strategy for amending the altered decision-making under risk in psychiatric diseases.

Second-best probability weighting

Non-linear probability weighting is an integral part of descriptive theories of choice under risk such as prospect theory. But why do these objective errors in information processing exist? Should we try to help individuals overcome their mistake of overweighting small and underweighting large probabilities? In this paper, we argue that probability weighting can be seen as a compensation for preexisting biases in evaluating payoffs. In particular, inverse S-shaped probability weighting is a flipside of S-shaped payoff valuation. Probability distortions may thus have survived as a second-best solution to a fitness maximization problem, and it can be counter-productive to correct them while keeping the value function unchanged.

Nonlinear probability weighting can reflect attentional biases in sequential sampling.

Nonlinear probability weighting allows cumulative prospect theory (CPT) to account for key phenomena in decision making under risk (e.g., certainty effect, fourfold pattern of risk attitudes). It describes the impact of risky outcomes on preferences in terms of a rank-dependent nonlinear transformation of their objective probabilities. The attentional Drift Diffusion Model (aDDM) formalizes the finding that attentional biases toward an option can shape preferences within a sequential sampling process. Here we link these two influential frameworks. We used the aDDM to simulate choices between two options while systematically varying the strength of attentional biases to either option. The resulting choices were modeled with CPT. Changes in preference due to attentional biases in the aDDM were reflected in highly systematic signatures in the parameters of CPT's weighting function (curvature, elevation). In a re-analysis of a large set of previously published data, we demonstrate that attentional biases are also empirically linked to patterns in probability weighting as suggested by the simulations. Our analyses also revealed a previously overlooked link between patterns in probability weighting and response times. These findings highlight that distortions in probability weighting can arise from simple option-specific attentional biases in information search, and suggest an alternative to common interpretations of weighting-function parameters in terms of probability sensitivity and optimism. They also point to novel, attention-based explanations for empirical phenomena associated with characteristic shapes of CPT's probability-weighting function (e.g., certainty effect, description-experience gap). The results advance the integration of two prominent computational frameworks for decision making. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Robustness of Rank Independence in Risky Choice

Bernheim and Sprenger (2020) devise and implement a novel test of rank-dependent probability weighting both in general and as formulated in cumulative prospect theory. They reject both hypotheses decisively. Cumulative prospect theory cannot simultaneously account for the rank independence of “equalizing reductions” for three-outcome lotteries, which it construes as indicating linear probability weighting, and the relationship between equalizing reductions and probabilities, which it interprets as indicating highly nonlinear probability weighting. In the current paper, we explore the robustness of the first finding, rank independence of equalizing reductions (and hence of decision weights), with respect to alternative experimental procedures.

Nonlinear Probability Weighting in Depression and Anxiety: Insights From Healthy Young Adults.

Both depressive and anxiety disorders have been associated with excessive risk avoidant behaviors, which are considered an important contributor to the maintenance and recurrence of these disorders. However, given the high comorbidity between the two disorders, their independent association with risk preference remains unclear. Furthermore, due to the involvement of multiple cognitive computational factors in the decision-making tasks employed so far, the precise underlying mechanisms of risk preference are unknown. In the present study, we set out to investigate the common versus unique cognitive computational mechanisms of risk preference in depression and anxiety using a reward-based decision-making task and computational modeling based on economic theories. Specifically, in model-based analysis, we decomposed risk preference into utility sensitivity (a power function) and probability weighting (the one-parameter Prelec weighting function). Multiple linear regression incorporating depression (BDI-II) and anxiety (STAI state anxiety) simultaneously indicated that only depression was associated with one such risk preference parameter, probability weighting. As the symptoms of depression increased, subjects’ tendency to overweight small probabilities and underweight large probabilities decreased. Neither depression nor anxiety was associated with utility sensitivity. These associations remained even after controlling covariates or excluding anxiety-relevant items from the depression scale. To our knowledge, this is the first study to assess risk preference due to a concave utility function and nonlinear probability weighting separately for depression and anxiety using computational modeling. Our results provide a mechanistic account of risk avoidance and may improve our understanding of decision-making deficits in depression and anxiety.

Maybe favors: How to get more good deeds done.

While previous research has revealed several reasons why humans generally do good deeds, we explore a simple nudge that might get more of them done: the "maybe favor." We first show conceptually that, compared to a conventional favor, humans are more willing to grant a favor to a stranger on which they might eventually not have to make good. Furthermore, we conducted a series of fully incentivized experiments (total N = 3,475) where participants could make actual donations to charity. Introducing a "maybe" into our donation proposals by randomly revoking some donations not only led to significant increases in donation rates but also increased the total amount of donations. That is, due to biased perceptions of costs and benefits combined with nonlinear probability weighting, the donations we revoked due to the "maybe" were overcompensated by an increased overall willingness-to-donate. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Supplemental Material for Nonlinear Probability Weighting Can Reflect Attentional Biases in Sequential Sampling

Supplemental Material for Nonlinear Probability Weighting Can Reflect Attentional Biases in Sequential Sampling

Give Me a Challenge or Give Me a Raise

I study the effect of task difficulty on workers’ effort. I find that task difficulty has an inverse-U effect on effort and that this effect is quantitatively large, especially when compared to the effect of conditional monetary rewards. Difficulty acts as a mediator of monetary rewards: conditional rewards are most effective at the intermediate or high levels of difficulty. The inverse-U pattern of effort response to difficulty is inconsistent with many popular models in the literature, including the Expected Utility models with the additively separable cost of effort. I propose an alternative mechanism for the observed behavior based on non-linear probability weighting. I structurally estimate the proposed model and find that it successfully captures the behavioral patterns observed in the data. I discuss the implications of my findings for the design of optimal incentive schemes for workers and for the models of effort provision.

Give me a challenge or give me a raise

I study the effect of task difficulty on workers’ effort. I find that task difficulty has an inverse-U effect on effort and that this effect is quantitatively large, especially when compared to the effect of conditional monetary rewards. Difficulty acts as a mediator of monetary rewards: conditional rewards are most effective at the intermediate or high levels of difficulty. The inverse-U pattern of effort response to difficulty is inconsistent with many popular models in the literature, including the Expected Utility models with the additively separable cost of effort. I propose an alternative mechanism for the observed behavior based on non-linear probability weighting. I structurally estimate the proposed model and find that it successfully captures the behavioral patterns observed in the data. I discuss the implications of my findings for the design of optimal incentive schemes for workers and for the models of effort provision.

The decision paradoxes motivating Prospect Theory: The prevalence of the paradoxes increases with numerical ability

AbstractProspect Theory (PT: Kahneman & Tversky, 1979) of risky decision making is based on psychological phenomena (paradoxes) that motivate assumptions about how people react to gains and losses, and how they weight outcomes with probabilities. Recent studies suggest that people’s numeracy affect their decision making. We therefore conducted a large-scale conceptual replication of the seminal study by Kahneman and Tversky (1979), where we targeted participants with larger variability in numeracy. Because people low in numeracy may be more dependent on anchors in the form of other judgments we also manipulated design type (within-subject design, vs. single-stimuli design, where participants assess only one problem). The results from about 1,800 participants showed that design type had no effect on the modal choices. The rate of replication of the paradoxes in Kahneman and Tversky was poor andpositivelyrelated to the participants’ numeracy. The Probabilistic Insurance Effect was observed at all levels of numeracy. The Reflection Effects were not fully replicated at any numeracy level. The Certainty and Isolation Effects explained by nonlinear probability weighting were replicated only at high numeracy. No participant exhibited all 9 paradoxes and more than 50% of the participants exhibited at most three of the 9 paradoxes. The choices by the participants with low numeracy were consistent with a shift towards a cautionary non-compensatory strategy of minimizing the risk of receiving the worst possible outcome. We discuss the implications for the psychological assumptions of PT.

Game-Theoretic Protection Against Networked SIS Epidemics by Human Decision-Makers

Abstract We study networks of human decision-makers who independently decide how to protect themselves against Susceptible-Infected-Susceptible (SIS) epidemics. Motivated by studies in behavioral economics showing that humans perceive probabilities in a nonlinear fashion, we examine the impacts of such misperceptions on the equilibrium protection strategies. In our setting, nodes choose their curing rates to minimize the infection probability under the degree-based mean-field approximation of the SIS epidemic plus the cost of their selected curing rate. We establish the existence of a degree based equilibrium under both true and nonlinear perceptions of infection probabilities (under suitable assumptions). When the per-unit cost of curing rate is sufficiently high, we show that true expectation minimizers choose the curing rate to be zero at the equilibrium, while curing rate is nonzero under nonlinear probability weighting.

From Aggregate Betting Data to Individual Risk Preferences

As a textbook model of contingent markets, horse races are an attractive environment to study the attitudes towards risk of bettors. We innovate on the literature by explicitly considering heterogeneous bettors and allowing for very general risk preferences, including non-expected utility. We build on a standard single-crossing condition on preferences to derive testable implications; and we show how parimutuel data allow us to uniquely identify the distribution of preferences among the population of bettors. We then estimate the model on data from US races. Within the expected utility class, the most usual specifications (CARA and CRRA) fit the data very badly. Our results show evidence for both heterogeneity and nonlinear probability weighting.

Nonlinear decision weights or moment-based preferences? A model competition involving described and experienced skewness

The predictive power of cumulative prospect theory and expected utility theory is typically compared using decisions from description, where lotteries’ outcome values and probabilities are explicitly stated. In decisions from experience, individuals sample (in the sampling paradigm without cost) from the return distributions to learn outcome values and their relative frequencies; here cumulative prospect theory and expected utility theory require the calculation of probabilities from experience. Individuals, however, may be more attuned to the experienced moments of outcome distributions, rather than the probabilities. We therefore test the mean–variance–skewness model, and retrieve the proportion of expected utility theory (over income), cumulative prospect theory, and mean–variance–skewness populations using a latent-class hierarchical Bayesian model across six large datasets. For simple lotteries (with 1–2 outcomes), we find a mixture of cumulative prospect theory and mean–variance–skewness populations in decisions from both description and experience. For more complex lotteries (with 2–3 outcomes), all participants are classified as cumulative prospect theory types in decisions from description, but as mean–variance–skewness types in decisions from experience. This suggests that in decisions from experience with more complex return distributions, preferences for skewness are more predictive than nonlinear probability weighting.

Noisy preferences in risky choice: A cautionary note.

We examine the effects of multiple sources of noise in risky decision making. Noise in the parameters that characterize an individual’s preferences can combine with noise in the response process to distort observed choice proportions. Thus, underlying preferences that conform to expected value maximization can appear to show systematic risk aversion or risk seeking. Similarly, core preferences that are consistent with expected utility theory, when perturbed by such noise, can appear to display nonlinear probability weighting. For this reason, modal choices cannot be used simplistically to infer underlying preferences. Quantitative model fits that do not allow for both sorts of noise can lead to wrong conclusions.

Contests with group size uncertainty: Experimental evidence

In many contest situations, the number of participants is not observable at the time of investment. We design a laboratory experiment to study individual behavior in Tullock (lottery) contests with group size uncertainty. There is a fixed pool of n potential players, each with independent probability q∈(0,1] of participating. We independently manipulate each of the parameters and test the implied comparative statics predictions. Our results provide considerable support for the theory, both in terms of comparative statics and point predictions. Most surprisingly, we find no evidence of overbidding in treatments where there is a nontrivial probability that group size is one. This stands in stark contrast to the robust overbidding observed in experimental contests with deterministic group size. We propose a one-parameter model that incorporates nonlinear probability weighting and a modified version of joy of winning, which we call Constant Winning Aspirations (CWA), and show that it neatly organizes all of our results.

Nonlinear Decision Weights or Skewness Preference? A Model Competition

An extensive literature exists comparing the predictive power of Cumulative Prospect Theory (CPT) and Expected Utility Theory (EUT) in decisions from description. CPT accommodates the empirically observed deviations from EUT primarily by assuming a nonlinear probability weighting function and loss aversion. An alternative explanation is that decision makers have preferences for higher-order moments, beyond the mean and variance of lotteries — we extensively test the Mean-Variance-Skewness (MVS) preference model. We estimate the proportion of EUT, CPT and the MVS populations using a latent-class hierarchical Bayesian model for six different datasets, including both decisions from description and experience. In decisions from description, subjects are explicitly presented with the probabilities and outcome values of the lotteries. In decisions from experience subjects must sample (without cost) from the lottery distributions to learn both the probabilities and outcome values. In the latter, CPT and EUT require the explicit calculation of probabilities from experience; however, the MVS model only requires that decision-makers hold estimates of the three moments of the outcome distributions. In decisions from experience and description with simple lotteries (one safe option and one risky lottery with only two outcomes), we find significant subject heterogeneity — some subjects are classified as CPT and others as MVS. In the remaining four datasets with multiple (two to three) outcomes per lottery, a single class is sufficient to characterize subjects. Specifically, in two datasets of decisions from experience all subjects are classified as MVS, thus implying that preferences for skewness are more predictive than nonlinear probability weighting. In two datasets with decisions from description all subjects are classified as CPT. We find no evidence of an EUT population in any of the datasets. We conclude that the scope of CPT is primarily limited to decisions from description, while the scope of MVS to decisions from experience. Furthermore, out of the four datasets with a CPT population, only in one dataset did we observe the standard inverse-S shaped probability weighting function; this was the only dataset with simple lotteries (as defined above) and no multi-outcome lotteries.

Using methods from machine learning to evaluate behavioral models of choice under risk and ambiguity

How can behavioral science incorporate tools from machine learning (ML)? We propose that ML models can be used as upper bounds for the “explainable” variance in a given data set and thus serve as upper bounds for the potential power of a theory. We demonstrate this method in the domain of uncertainty. We ask over 600 individuals to make a total of 6000 choices with randomized parameters and compare standard economic models to ML models. In the domain of risk, a version of expected utility that allows for non-linear probability weighting (as in cumulative prospect theory) and individual-level parameters performs as well out-of-sample as ML techniques. By contrast, in the domain of ambiguity, two of the most widely studied models (a linear version of maximin preferences and second order expected utility) fail to compete with the ML methods. We open the “black boxes” of the ML methods and show that under risk we “rediscover” expected utility with probability weighting. However, in the case of ambiguity the form of ambiguity aversion implied by our ML models suggests that there is gain from theoretical work on a portable model of ambiguity aversion. Our results highlight ways in which behavioral scientists can incorporate ML techniques in their daily practice to gain genuinely new insights.

Real options in the laboratory: An experimental study of sequential investment decisions

Many real-life risky decisions in finance and management are dynamic and decision policies can be adapted as uncertainty is reduced by the arrival of new information. In this type of situation, called a real options problem, a decision maker must choose how much of his finite resources to invest in a dynamic risky environment. In two laboratory experiments, we test a well-defined decision problem with the central characteristics of a real options framework and do so in such a way that it is amendable to formal modeling. We find that people choose differently than the expected value maximizing policy, consistent with risk aversion and non-linear probability weighting. We conclude that although real options analysis is useful as a normative valuation method, its recommendations are sometimes contrary to people’s innate tendencies when making risky choices and this counterintuitiveness should be considered when implementing real options analysis in training and practice.

Risk Preferences, Probability Weighting, and Strategy Tradeoffs in Wildfire Management.

Wildfires present a complex applied risk management environment, but relatively little attention has been paid to behavioral and cognitive responses to risk among public agency wildfire managers. This study investigates responses to risk, including probability weighting and risk aversion, in a wildfire management context using a survey-based experiment administered to federal wildfire managers. Respondents were presented with a multiattribute lottery-choice experiment where each lottery is defined by three outcome attributes: expenditures for fire suppression, damage to private property, and exposure of firefighters to the risk of aviation-related fatalities. Respondents choose one of two strategies, each of which includes "good" (low cost/low damage) and "bad" (high cost/high damage) outcomes that occur with varying probabilities. The choice task also incorporates an information framing experiment to test whether information about fatality risk to firefighters alters managers' responses to risk. Results suggest that managers exhibit risk aversion and nonlinear probability weighting, which can result in choices that do not minimize expected expenditures, property damage, or firefighter exposure. Information framing tends to result in choices that reduce the risk of aviation fatalities, but exacerbates nonlinear probability weighting.

Economic choices reveal probability distortion in macaque monkeys.

Economic choices are largely determined by two principal elements, reward value (utility) and probability. Although nonlinear utility functions have been acknowledged for centuries, nonlinear probability weighting (probability distortion) was only recently recognized as a ubiquitous aspect of real-world choice behavior. Even when outcome probabilities are known and acknowledged, human decision makers often overweight low probability outcomes and underweight high probability outcomes. Whereas recent studies measured utility functions and their corresponding neural correlates in monkeys, it is not known whether monkeys distort probability in a manner similar to humans. Therefore, we investigated economic choices in macaque monkeys for evidence of probability distortion. We trained two monkeys to predict reward from probabilistic gambles with constant outcome values (0.5 ml or nothing). The probability of winning was conveyed using explicit visual cues (sector stimuli). Choices between the gambles revealed that the monkeys used the explicit probability information to make meaningful decisions. Using these cues, we measured probability distortion from choices between the gambles and safe rewards. Parametric modeling of the choices revealed classic probability weighting functions with inverted-S shape. Therefore, the animals overweighted low probability rewards and underweighted high probability rewards. Empirical investigation of the behavior verified that the choices were best explained by a combination of nonlinear value and nonlinear probability distortion. Together, these results suggest that probability distortion may reflect evolutionarily preserved neuronal processing.