How the brain deals with its own limitations while handling numbers
Everyday, people make decisions based on numbers, from choosing the cheapest flight to judging whether the new restaurant around the corner is too expensive. Numbers offer accurate objective values—but that’s not always how people use them. In fact, people often make decisions based on their own distorted representations of numbers instead.
Mathematics tells us to use linear weightings of numbers, just like you learned in primary school: The distances between neighboring numbers are equal. For example, the difference between 3 and 4 is the same as the difference between 8 and 9.
However, people often use nonlinear weightings, distorting numbers in either of two ways: towards extremes or away from them. For example, a single $40 pizza on a menu may make you think a restaurant is quite expensive, even if most of the other pizzas cost less than $15. In nonlinear weighting, a single item on a list may have more influence on your decision than many other items combined.
When do people give different weights to different numbers—and is it really that irrational? We found that the way people weigh numbers reflects how the human brain deals with its own limitations.
Compression and anti-compression
There are two types of nonlinear weighting: compression and anti-compression. Compression indicates that extreme values are being underweighted, meaning that people are more sensitive to average numbers. Anti-compression indicates the opposite: Extreme values are being overweighted and therefore people are more sensitive to very small or very large values.
Our theory is that the type of distortion that occurs depends on the context. Compression is especially common and can be found in everyday tasks that do not even need to feel particularly challenging. For example, it’s easy to tell that your 2 kg backpack is lighter than your friend’s 4 kg backpack, because these are weights you encounter often—they’re part of your average daily experience. You’re therefore more sensitive to this difference of 2 kg than you would be if you had to distinguish a 40 kg suitcase from a 42 kg suitcase: You encounter these heavier weights less often, so you’re less sensitive to these more extreme values.
Anti-compression is more often found in tasks that are more mentally taxing, for example when an overwhelming amount of information has to be processed quickly. In this case, extreme values get overweighted.
Why do people distort numbers in the first place?
Humans are not machines. First, brains are noisy—like a dusty vinyl record, there are small inaccuracies in the brain’s neural processes. Because brains are delicate, they have to devise robust strategies that can withstand any number of hiccups. Second, brains are limited by basic physiological needs (e.g., whether enough glucose is available for the cells to do their work). Once a state of healthy functioning has been reached, the brain will be able to process information, but won’t go beyond its limits. Compression and anti-compression are ways to deal with these challenges.
Dealing with a noisy and limited brain
Using computational modeling, we simulated a human decision maker, with all its biological limitations, and found that compression and anti-compression can actually lead to decisions that are more, not less, accurate.
Compression generally counteracts the noise of the brain. It is similar to ignoring outliers in statistics. For example, a soccer scout will look at a player’s performance across the season rather than focus on a single game where the player scored 7 goals.
Sometimes noise isn’t the only problem. When a human brain reaches its limits, it turns to anti-compression, where extreme values are given more weight. If bombarded with an overwhelming amount of information, we have to focus on some things and ignore others. Imagine practicing tennis with an automated ball machine set on high speed. With balls coming in rapid fire, it’s better to focus on hitting some balls well than to chase after each one (and consequently miss most).
Quick: Judge this number stream!
To test our theory that the type of distortion depends on the context, we asked participants in a study to determine the average of a sequence of 8 numbers shown in less than 3 seconds. In an easier scenario, participants had to decide whether the average of the sequence was higher or lower than a given number. Computational modeling revealed that they used compression to deal with the task, focusing on average values.
In a more difficult scenario, the numbers were colored red or blue, and participants were asked which color had the higher average. Here they had to integrate both the number and color— much like in the pizza GIF above, where you had to consider both price and whether the pizza is vegetarian. In this scenario, participants used anti-compression, overweighting extreme values—and we think you did too1.
Next, we recorded participants' neural signals while they performed the same task in order to understand the neural processes behind compression and anti-compression.
For anti-compression, the results were clear: Neural data mirrored the results from the behavioral data. However, for compression, the results were less clear, and generally the pattern of neural activity indicated a more linear weighting. One possible explanation is that anti-compression might be people’s default response to extreme values, while compression happens on a higher analytic level that was not captured in our neural recording technique.
Sometimes it pays to ignore information
Overall, we found that seemingly irrational behavior can actually be a smart adaptation that allows the human brain to deal with noise and its own limited capacity. While the brain might not be as accurate as a computer, it has strategies for making good decisions about challenges people face in their day-to-day lives. That’s how people might focus on just part of a big picture but still do a good job.
We are not so irrational after all.
1By the way, the average cost of a vegetarian pizza in the GIF above is $6.
Appelhoff, A., Hertwig, R., & Spitzer, B. (2022). EEG-representational geometries and psychometric distortions in approximate numerical judgment. bioRxiv
Clarmann von Clarenau, V., Pachur, T., & Spitzer, B. (2022). Over- and underweighting of extreme values in decisions from sequential samples. PsyArXiv.
Spitzer, B., Waschke, L., & Summerfield, C. (2017). Selective overweighting of larger magnitudes during noisy numerical comparison. Nature Human Behaviour, 1(0145). doi.org/10.1038/s41562-017-0145