Why We Distort Probability

Summary: People distort probability to compensate for perceptual and cognitive limitations.

Source: NYU

The chances of a commercial airliner crashing are vanishingly small — and yet many people are uncomfortable flying. Vaccination for many common childhood diseases entail almost no risk — but parents still worry. Human perception of probabilities — especially very small and very large probabilities — can be markedly distorted and these distortions can lead to potentially disastrous decisions.

But why we distort probability is unclear. While the question has been previously studied, there is no consensus on its causes.

A team of scientists from New York University and Peking University, using experimental research, has now concluded that our cognitive limitations lead to probability distortions and to subsequent errors in decision-making. The researchers have developed a model of human cognitive limitations and tested its predictions experimentally, as reported in the latest issue of the journal Proceedings of the National Academy of Sciences.

The team, which included New York University’s Laurence Maloney as well as the University of Peking University’s Hang Zhang, a professor, and Xiangjuan Ren, a post-doctoral fellow, initiated the analysis by examining the nature of distortions as a potential clue for explaining this phenomenon.

“Probability distortion limits human performance in many tasks, and we conjectured that the observed changes in probability distortion with task was a kind of partial compensation for human limitations,” explains Maloney. “A marathon runner with a sprained ankle will not run as well as she might have with ankle intact, but the awkward, limping gait we observe could in fact be an optimal compensation for injury.”

The key step in the model is the recoding of probabilities that depends on the range of probabilities in a task.

“Much like a variable magnification microscope, the brain can represent a wide range of probabilities, but not very accurately, or a narrow range at high precision,” explains Maloney. “If, for example, a task involves reasoning about the probability of various causes of death, for example, then the probabilities are all very small (thankfully) and small differences are important. We can set the microscope to give us high resolution over a limited window of very small probabilities. In another task we might accept less precision in return for the ability to represent a much wider range of probabilities.”

This shows question marks
The key step in the model is the recoding of probabilities that depends on the range of probabilities in a task. Image is in the public domain.

Zhang, Ren, and Maloney set out to test this model in two experiments, one in which subjects made typical economic decisions under risk (e.g. choosing between a 50:50 chance of $200 and the certainty of $70) and one involving judgements of relative frequency (the relative frequency of black and white dots appearing on a computer screen). The two experiments together tapped into the basic ways we use probability and frequency in everyday life. The researchers found that their model predicted human performance far better than any previous model.

They discovered that — like the marathon runner — people’s limitations were costly but, subject to those limitations, we do as well as we possibly can.

Zhang and Ren are part of Peking University’s School of Psychological and Cognitive Sciences; Maloney is a professor in NYU’s Department of Psychology and Center for Neural Science.

Funding: This research was supported by the National Eye Institute of the National Institutes of Health (EY019889).

About this neurology research article

American Heart Association
James Devitt – NYU
Image Source:
The image is in the public domain.

Original Research: Open access
“The bounded rationality of probability distortion” by Hang Zhang, Xiangjuan Ren, and Laurence T. Maloney. PNAS.


The bounded rationality of probability distortion

In decision making under risk (DMR) participants’ choices are based on probability values systematically different from those that are objectively correct. Similar systematic distortions are found in tasks involving relative frequency judgments (JRF). These distortions limit performance in a wide variety of tasks and an evident question is, Why do we systematically fail in our use of probability and relative frequency information? We propose a bounded log-odds model (BLO) of probability and relative frequency distortion based on three assumptions: 1) log-odds: probability and relative frequency are mapped to an internal log-odds scale, 2) boundedness: the range of representations of probability and relative frequency are bounded and the bounds change dynamically with task, and 3) variance compensation: the mapping compensates in part for uncertainty in probability and relative frequency values. We compared human performance in both DMR and JRF tasks to the predictions of the BLO model as well as 11 alternative models, each missing one or more of the underlying BLO assumptions (factorial model comparison). The BLO model and its assumptions proved to be superior to any of the alternatives. In a separate analysis, we found that BLO accounts for individual participants’ data better than any previous model in the DMR literature. We also found that, subject to the boundedness limitation, participants’ choice of distortion approximately maximized the mutual information between objective task-relevant values and internal values, a form of bounded rationality.

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