Models of Imprecise Cognition

Cognitive Imprecision and Stake-Dependent Risk Attitudes

NBER Working Paper no. 30417, September 2022. Revised October 2023.
(with Mel Win Khaw and Ziang Li)
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Abstract:   In an experiment that elicits subjects’ willingness to pay (WTP) for the outcome of a lottery, we confirm the fourfold pattern of risk attitudes described by Kahneman and Tversky. In addition, we document a systematic effect of stake sizes on the magnitude and sign of the relative risk premium, holding fixed both the probability that a lottery pays off and the sign of its payoff (gain vs. loss). We further show that in our data, there is a log-linear relationship between the monetary payoff of the lottery and WTP, conditional on the probability of the payoff and its sign. We account quantitatively for this relationship, and the way in which it varies with both the probability and sign of the lottery payoff, in a model in which all departures from risk-neutral bidding are attributed to an optimal adaptation of bidding behavior to the presence of cognitive noise. Moreover, the cognitive noise required by our hypothesis is consistent with patterns of bias and variability in judgments about numerical magnitudes and probabilities that have been observed in other contexts.

Individual Risk Attitudes Arise From Noise in Neurocognitive Magnitude Representations

Published in Nature Human Behaviour, 2023.
(with Miguel Barretto-Garcia, Gilles de Hollander, Marcus Grueschow, Rafael Polania, and Christian C. Ruff)
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Abstract:  We provide neural and behavioural evidence that risk aversion need not reflect diminishing marginal utility, but can also arise from a purely perceptual bias: the noisy logarithmic coding of numerical magnitudes can lead individuals to underestimate the size of larger monetary payoffs, leading to apparent risk aversion even when subjective valuation increases linearly with the estimated amount. A formal model of this process predicts that risk aversion should systematically increase when individuals represent numerical magnitudes more noisily. We confirm this prediction by measuring both the mental and neural acuity of magnitude representations during a purely perceptual task and relating these measures to individual risk attitudes during separate financial decisions. Computational model fitting suggests that subjects base both types of choices on similar mental magnitude representations, with correlated precision across the separate perceptual and risky choices. Increased stimulus noise due to the presentation format of risky outcomes leads to increased risk aversion, just as predicted by the model. The precision of the underlying neural magnitude representations was estimated with a numerical population receptive field model fitted to the fMRI data of the perceptual task. Subjects with more precise magnitude representations in parietal cortex showed both less variable behaviour and less apparent risk aversion in independent financial choices. Our results highlight that individual patterns of economic behaviour may, at least partially, be determined by capacity limitations in perceptual processing rather than by processes that assign subjective values to monetary rewards.

Imprecise Probabilistic Inference from Sequential Data

Posted on PsyArXiv, June 2023.
(with Arthur Prat-Carrabin)

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Abstract:  Although the Bayesian paradigm is an important benchmark in studies of human inference, the extent to which it provides a useful framework to account for human behavior
remains debated. We document systematic departures from the predictions of Bayesian inference, even on average, in the estimates by experimental subjects of the probability of a binary event following observations of successive realizations of the event. In particular we find under-reaction of subjects’ probability estimates to the evidence (‘conservatism’) after only a few observations, and at the same time over-reaction after a longer sequence of observations. This is not explained by an incorrect prior, nor by many common models of Bayesian inference. We uncover the autocorrelation in estimates, which suggests that subjects carry imprecise representations of the decision situations, with noise in beliefs propagating over successive trials. But even taking into account this internal imprecision, we find that the subjects’ updates are inconsistent with the rules of Bayesian inference. We show how subjects instead considerably economize
on the attention that they pay to the information relevant to decision, and on the degree of control that they exert over their precise response, while giving responses fairly adapted to the experimental task. A “noisy counting” model of probability estimation reproduces the several patterns that we exhibit in subjects’ behavior. In sum, human subjects in our task perform reasonably well while greatly minimizing the amount of information that they pay attention to. Our results emphasize that investigating this economy of attention is crucial for understanding human decisions.

Information-Constrained Coordination of Economic Behavior

April 2023
(with Guy Aridor and Rava Azeredo da Silveira)
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Abstract:   We analyze a coordination game with information-constrained players. The players’ actions are based on a noisy compressed representation of the game’s payoffs in a particular case, where the compressed representation is a latent state learned by a variational autoencoder
(VAE). The VAE is trained over a finite sample of past experiences, and is optimized to trade off the average payoff obtained over a distribution of possible games against the reconstruction error involved in predicting game payoffs on the basis of the latent state. Our use of the average payoff in the training objective requires us to generalize the basic VAE framework of Kingma and Welling (2014); we also compare our proposal to rationally inattentive decision making as modeled by Sims (2003) and Yang (2015). We apply our model to the coordination game in the experiment of Frydman and Nunnari (2022), and show that it offers an explanation for two salient features of the experimental evidence: both the gradual adjustment of the players’ action probabilities with changes in the game payoffs, and the dependence of the degree of stochasticity of players’ choices on the range of game payoffs encountered on different trials. Our approach also provides an account of the way in which play should gradually adjust to a change in the distribution of game payoffs that are encountered, offering an explanation for the history-dependent play documented by Arifovic et al. (2013).

Modeling Imprecision in Perception, Valuation and Choice

Published in  Annual Review of Economics, volume 12, 2020
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Abstract: Traditional decision theory assumes that people respond to the exact features of the options available to them, but observed behavior seems much less precise. This review considers ways of introducing imprecision into models of economic decision making, and stresses the usefulness of analogies with the way that imprecise perceptual judgments are modeled in psychophysics — the branch of experimental psychology concerned with the quantitative relationship between objective features of an observer’s environment and elicited reports about their subjective appearance. It reviews key ideas from psychophysics, provides examples of the kinds of data that motivate them, and proposes lessons for economic modeling. Applications include stochastic choice, choice under risk, decoy effects in marketing, global game models of strategic interaction, and delayed adjustment of prices in response to monetary disturbances.

Cognitive Imprecision and Small-Stakes Risk Aversion

Published, Review of Economic Studies, 2021
(with Mel Win Khaw and Ziang Li)
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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.

Efficient Coding of Numbers Explains Decision Bias and Noise

Published, Nature Human Behaviour, 2022
(with Arthur Prat-Carrabin)
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Abstract: Human subjects differentially weight different stimuli in averaging tasks. This has been interpreted as reflecting biased stimulus encoding, but an alternative hypothesis is that stimuli are encoded with noise, then optimally decoded. Moreover, with efficient coding, the amount of noise should vary across stimulus space, and depend on the statistics of stimuli. We investigate these predictions through a task in which participants are asked to compare the averages of two series of numbers, each sampled from a prior distribution that differs across blocks of trials. We show that subjects encode numbers with both a bias and a noise that depend on the number. Infrequently occurring numbers are encoded with more noise. A model combining efficient coding and Bayesian decoding best captures subjects’ behaviour. Our results suggest that Wei and Stocker’s “law of human perception”, which relates the bias and variability of sensory estimates, also applies to number cognition.

Efficient Sampling and Noisy Decisions

Published, eLife, September 2020
(with Joseph A. Heng and Rafael Polania)
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Abstract:  The precision of human decisions is limited by both processing noise and basing decisions on finite information. But what determines the degree of such imprecision? Here we develop an efficient coding framework for higher-level cognitive processes, in which information is represented by a finite number of discrete samples. We characterize the sampling process that maximizes perceptual accuracy or fitness under the often-adopted assumption that full adaptation to an environmental distribution is possible, and show how the optimal process differs when detailed information about the current contextual distribution is costly. We tested this theory on a numerosity discrimination task, and found that humans efficiently adapt to contextual distributions, but in the way predicted by the model in which people must economize on environmental information. Thus, understanding decision behavior requires that we account for biological restrictions on information coding, challenging the often-adopted assumption of precise prior knowledge in higher-level decision systems.

Optimally Imprecise Memory and Biased Forecasts

Presented at the conference “Expectations in Macroeconomic and Financial Models,” Becker-Friedman Institute, June 2020

Revised November 2020
(with Rava Azeredo da Silveira and Yeji Sung)
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Abstract: We propose a model of optimal decision making subject to a memory constraint. The constraint is a limit on the complexity of memory measured using Shannon’s mutual information, as in models of rational inattention; but our theory differs from that of Sims (2003) in not assuming costless memory of past cognitive states. We show that the model implies that both forecasts and actions will exhibit idiosyncratic random variation; that average beliefs will also differ from rational-expectations beliefs, with a bias that fluctuates forever with a variance that does not fall to zero even in the long run; and that more recent news will be given disproportionate weight in forecasts. We solve the model under a variety of assumptions about the degree of persistence of the variable to be forecasted and the horizon over which it must be forecasted, and examine how the nature of forecast biases depends on these parameters. The model provides a simple explanation for a number of features of reported expectations in laboratory and field settings, notably the evidence of over-reaction in elicited forecasts documented by Afrouzi et al. (2020) and Bordalo et al. (2020).

Outlier Blindness: A Neurobiological Foundation for Neglect of Financial Risk

Published, Journal of Financial Economics, 2022
(with Elise Payzan-LeNestour)
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Abstract:  How do people record information about the outcomes they observe in their environment? Building on a well-established neuroscientific framework, we propose a model in which people are hampered in their perception of outcomes that they expect to seldom encounter. We provide experimental evidence for such ‘outlier blindness’ and discuss how it provides a microfoundation for neglected tail risk by investors in financial markets.

Individual Differences in the Perception of Probability

Published, PLoS Computational Biology, 2021
(with Mel Win Khaw and Luminita Stevens)
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Abstract:  In recent studies of humans estimating non-stationary probabilities, estimates appear to be unbiased on average, across the full range of probability values to be estimated. This finding is surprising given that experiments measuring probability estimation in other contexts have often identified conservatism: individuals tend to overestimate low probability events and underestimate high probability events. In other contexts, repulsive biases have also been documented, with individuals producing judgments that tend toward extreme values instead. Using extensive data from a probability estimation task that produces unbiased performance on average, we find substantial biases at the individual level; we document the coexistence of both conservative and repulsive biases in the same experimental context. Individual biases persist despite extensive experience with the task, and are also correlated with other behavioral differences, such as individual variation in response speed and adjustment rates. We conclude that the rich computational demands of our task give rise to a variety of behavioral patterns, and that the apparent unbiasedness of the pooled data is an artifact of the aggregation of heterogeneous biases.

Adaptive Efficient Coding: A Variational Autoencoder Approach

May 2020
(with Guy Aridor and Francesco Grechi)
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Abstract:    We study a model of neural coding with the structure of a variational auto-encoder. The model posits that the encoding of individual stimulus values is optimally adjusted for a finite training sample of stimuli retained in memory. We demonstrate that this model can rationalize existing experimental evidence on both perceptual discrimination thresholds and neural tuning curve widths in multiple sensory domains. Finally, since our model implies that encoding is optimized for a sample from the environment, it also provides predictions about the adaptation of neural coding as the environmental frequency distribution changes.

Multiple Conceptions of Resource-Rationality

Published, Brain and Behavioral Sciences, March 2020
(with Wei Ji Ma)
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Abstract:   A commentary on Falk Lieder and Thomas L. Griffiths, “Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources,” Brain and Behavioral Sciences, 2019.  Resource rationality holds great promise as a unifying principle across theories in neuroscience, cognitive science, and economics. The target article clearly lays out this potential for unification. However, resource-rational models are more diverse and less easily unified than might appear from the target article. Here, we explore some of that diversity.

Neighborhood-Based Information Costs

Published, American Economic Review, 2021
(with Benjamin Hébert)
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Technical Appendix

Abstract:   We propose a new measure of the cost of information structures in rational inattention problems, the “neighborhood-based” cost functions, given that many applications involve states with a topological structure. These cost functions summarize the results of a sequential information sampling problem, and also capture a notion of perceptual distance. This second property allows neighborhood-based cost functions, unlike mutual information, to make accurate predictions about behavior in perceptual experiments. We compare the implications of our neighborhood-based cost functions with those of a mutual-information cost function in a series of applications: security design, global games, modeling perceptual judgments, and linear-quadratic-Gaussian problems.

Rational Inattention when Decisions Take Time

Published, Journal of Economic Theory, 2023
(with Benjamin Hébert)
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Technical Appendix

Abstract:  Decisions take time, and the time taken to reach a decision is likely to be informative about the cost of more precise judgments. We formalize this insight in the context of a dynamic rational inattention (RI) model. Under standard conditions on the flow cost of information in our discrete-time model, we obtain a tractable model in the continuous-time limit. We next provide conditions under which the resulting belief dynamics resemble either diffusion processes or processes with large jumps. We then demonstrate that the state-contingent choice probabilities predicted by our model are identical to those predicted by a static RI model, providing a micro-foundation for such models. In the diffusion case, our model provides a normative foundation for a variant of the drift-diffusion models studied in mathematical psychology.

Noisy Memory and Over-Reaction to News

Published AEA Papers and Proceedings, 2019
(With Rava Azeredo da Silveira)
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Abstract:    We propose a model of optimal decision making subject to a memory constraint. The constraint is a limit on the complexity of memory measured using Shannon’s mutual information, as in models of rational inattention. We show that the model implies that both forecasts and actions will exhibit idiosyncratic random variation; that beliefs will fluctuate forever around the rational-expectations (perfect-memory) beliefs with a variance that does not fall to zero; and that more recent news will be given disproportionate weight. The model provides a simple explanation for a number of features of expectations in laboratory and field settings.

Efficient Coding of Subjective Value

Published Nature Neuroscience, 2019
(With Rafa Polania and Christian C. Ruff)
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Abstract:  Preference-based decisions are essential for survival, for instance, when deciding what we should (not) eat. Despite their importance, preference-based decisions are surprisingly variable and can appear irrational in ways that have defied mechanistic explanations. Here we propose that subjective valuation results from an inference process that accounts for the structure of values in the environment and that maximizes information in value representations in line with demands imposed by limited coding resources. A model of this inference process explains the variability in both subjective value reports and preference-based choices, and predicts a new preference illusion that we validate with empirical data. Interestingly, the same model explains the level of confidence associated with these reports. Our results imply that preference-based decisions reflect information-maximizing transmission and statistically optimal decoding of subjective values by a limited-capacity system. These findings provide a unified account of how humans perceive and valuate the environment to optimally guide behavior.

Diverse Motives for Human Curiosity

Published Nature Human Behaviour, 2019
(With Kenji Kobayashi, Silvio Ravaioli, Adrien Baranes, and Jacqueline Gottlieb )
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Abstract:  Curiosity—our desire to know—is a fundamental drive in human behaviour, but its mechanisms are poorly understood. A classical question concerns the curiosity motives. What drives individuals to become curious about some but not other sources of information? Here we show that curiosity about probabilistic events depends on multiple aspects of the distribution of these events. Participants (n = 257) performed a task in which they could demand advance information about only one of two randomly selected monetary prizes that contributed to their income. Individuals differed markedly in the extent to which they requested information as a function of the ex ante uncertainty or ex ante value of an individual prize. This heterogeneity was not captured by theoretical models describing curiosity as a desire to learn about the total rewards of a situation. Instead, it could be explained by an extended model that allowed for attribute-specific anticipatory utility—the savouring of individual components of the eventual reward—and postulates that this utility increased nonlinearly with the certainty of receiving the reward. Parameter values fitting individual choices were consistent for information about gains or losses, suggesting that attribute-specific anticipatory utility captures fundamental heterogeneity in the determinants of curiosity.

Temporal Discounting and Search Habits: Evidence for a Task-Dependent Relationship

Published in Frontiers in Psychology: Decision Neuroscience, 2018
(With Mel Win Khaw and Ziang Li)
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Abstract:  Recent experiments suggest that search direction causally affects the discounted valuation of delayed payoffs. Comparisons between options can increase individuals’ patience toward future payoff options, while searching within options instead promotes impatient choices. We further test the robustness and specificity of this relationship using a novel choice task. Here individuals choose between pairs of delayed payoffs instead of single delayed outcomes. We observe a relationship between search styles and temporal discounting that are the opposite of those previously reported. Integrators—those who tend to compare attributes within alternatives—discount and choose more slowly than comparators—those who are more likely to compare between alternatives. This finding supports and augments the view that individuals’ search strategy is predictive of subsequent discount rates. In particular, the direction of this relationship is further modifiable based on the spatial layout and varying information within an individual’s decision-making environment.

Discrete Adjustment to a Changing Environment: Experimental Evidence

Published Journal of Monetary Economics, 2017
(With Mel Win Khaw and Luminita Stevens)
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Abstract: We conduct a laboratory experiment to shed light on the cognitive limitations that may affect the way decision makers respond to changes in their economic environment. The subjects solve a tracking problem: they estimate the probability of drawing a green ring out of a box with green and red rings, which changes stochastically. The subjects draw rings from the box and indicate their draw-by-draw estimate. Our subjects depart from the optimal Bayesian benchmark in systematic ways, but these deviations are not simply the result of some boundedly rational, but deterministic rule. Rather, there is a random element in the subjects’ response to any given history of evidence. Moreover, subjects adjust their forecast in discrete jumps rather than after each new ring draw, even though there are no explicit adjustment costs. They adjust by both large and small amounts, contrary to the predictions of a simple Ss model of optimal adjustment subject to a fixed cost. Finally, subjects prefer to report “round number” probabilities, even though that requires exerting additional effort. We develop a model of inattentive adjustment and compare its quantitative fit with alternative models of stochastic discrete adjustment.

Optimal Evidence Accumulation and Stochastic Choice

August 2016
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Abstract: Additional details (and some minor corrections) of the analysis underlying the results reported in “Stochastic Choice: An Optimizing Neuroeconomic Model” (below).

Stochastic Choice: An Optimizing Neuroeconomic Model

January 2014
Published in AEA Papers and Proceedings, May 2014
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Abstract: A model is proposed in which stochastic choice results from noise in cognitive processing rather than random variation in preferences. The mental process used to make a choice is nonetheless optimal, subject to a constraint on available information-processing capacity that is partially motivated by neurophysiological evidence. The optimal information-constrained model is found to offer a better fit to experimental data on choice frequencies and reaction times than either a purely mechanical process model of choice (the drift-diffusion model) or an optimizing model with fewer constraints on feasible choice processes (the rational inattention model).

Inattentive Valuation and Reference-Dependent Choice

May 2012
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Abstract: In rational choice theory, individuals are assumed always to choose the option that will provide them maximum utility. But actual choices must be based on subjective perceptions of the attributes of the available options, and the accuracy of these perceptions will always be limited by the information-processing capacity of one’s nervous system. I propose a theory of valuation errors under the hypothesis that perceptions are as accurate as possible on average, given the statistical properties of the environment to which they are adapted, subject to a limit on processing capacity. The theory is similar to the “rational inattention” hypothesis of Sims (1998, 2003, 2011), but modified for closer conformity with psychophysical and neurobiological evidence regarding visual perception. It can explain a variety of aspects of observed choice behavior, including the intrinsic stochasticity of choice; focusing effects; decoy effects in consumer choice; reference-dependent valuations; and the co-existence of apparent risk-aversion with respect to gains with apparent risk-seeking with respect to losses. The theory provides optimizing foundations for some aspects of the prospect theory of Kahneman and Tversky (1979).

Prospect Theory as Efficient Perceptual Distortion

January 2012
Published in AEA Papers and Proceedings, May 2012
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