Wrong Solutions to Math Problems Are Not Due to Cognitive Biases

Michael Harris
7 min readJun 13, 2021

According to Wikipedia, cognitive biases are systematic patterns of deviation from norm and/or rationality in judgment. They are often studied in psychology and behavioral economics.

The Resolve Asset Management Riffs last Friday, June 11, 2021, had a interesting discussion with Annie Duke on Biases and Optimal Decision Making in Markets and Life.

About 18.56 minutes into the conversation Annie Duke mentions the Bat-and-Ball Problem included in Cognitive Reflection Tests (CRT).

A bat and ball together cost $1.10. The bat costs $1.00 more than the ball. How much does the ball cost?

A frequent wrong answer of $0.10, instead of $0.05, has been argued as evidence of bias towards intuitive answers rather than higher-order rational considerations by committing to attribute substitution, which is a process that substitutes a more complicated problem by a simpler problem[1]. The simpler problem is the following:

A bat and ball together cost $1.10. The bat costs $1.00. How much does the ball cost?

I argue the psychologists are wrong and some of the people with alleged bias are right. In fact I argue that psychologists try to trick people with problems in order to create artificial images to attack.

This is the justification: Have you ever been to a store where there was no value for the ball and the price tag stated that the total price is $1.10 but the bat costs $1.00 more than the ball?

Customer: How much is the ball?
Store sales: You have to figure that out. The bat and ball together cost $1.10. The bat costs $1.00 more than the ball.
Customer: OK, I will call the father of a friend of my uncle’s ex wife who is a mathematician.
Store sales: No, answer now, because the Psychology department of the local university is doing a study and I am collecting data for them.
Customer: The results of the study will be biased by the confusion you create to people. Usually there are price tags on everything.
Store sales: They want to prove people are biased in their decision making.
Customer: Their conclusions will be biased by the confusion they cause to people.

It is more complicated than this. Do you expect most people to run y = 2x+a in their heads for this problem? People are not HP calculators. Human brain works with matching patterns, not with solving equations. Of course, some people are trained in math and will come up with the right answer of $0.05 but that does not mean those who said $0.10 are under some fancy attribute substitution bias.

I fact, those who err are considering the situation normal and interpret “The bat costs $1.00 more than the ball” as “The bat costs $1.00.” Because this is normal. This is how things are priced in stores. So people who answer $0.10 may be thinking of natural language “slippage” in stating the price rather than being the victims of cognitive bias(es).

Natural language does not map to mathematics well, not even categorical logic. An example is this proposition (from an elective course I took in analytical philosophy in college instead of taking that tennis course with all the exciting people):

p: The King of America is bald

How does one negate p?

“The King of America is not bald” does not reflect the fact that there is no American king. Neither, “It’s not the case that the King of America is bald” does the negation job.

In fact, philosophers dealing with p have not been able to conclude whether the sentence has any meaning at all despite Russell’s suggestion of using definite descriptions. In a similar way, I argue that the following problem

A bat and ball together cost $1.10. The bat costs $1.00 more than the ball. How much does the ball cost?

makes no sense to ask in a world dominated by well-defined price tags in stores, unless you are looking to intentionally confuse people. It is a meaningless problem unless posed to a math class. In order to uncover true biases, psychologists may have to look for questions, or problems, where there is no prior massive use of an easy alternative.

In essence then, I am claiming some of these behavioral psychologists when they ask questions in the sphere of the ball and bat problem, they are conflating natural language uncertainty and imprecision with math, through a problem that in everyday life has a straight forward expected answer. They are creating an artificial image to attack that is as biased as the bias they think they are discovering.

Let us go to the next problem mentioned in the Resolve Asset Management Riff with Annie Duke. It is as follows:

  1. A woman comes into a jewelry store and buys a necklace.
  2. But she doesn't have cash so she gives to store owner a $100 check.
  3. Store owner happens not to have any change so she goes to the store next door and she gives the $100 check and that person gives back $100 in change.
  4. Store owner comes back and she gives back the woman $22 because the price of the necklace was $78. The cost for the necklace was $39.
  5. Later on the store owner finds out the check was counterfeit and she has to give the person $100 back, which she does.
  6. The question is how much money did she lose?

Annie Duke claims the answer is $61. This is the correct answer based on cost of the necklace. The $100 transaction is a wash. The store owner took $100 and returned back $100. Furthermore, the store owner paid $39 for the necklace and also gave $22 to the crook. That’s a $61 loss based on cost. But based on value of necklace the correct answer is $100.

This is because if this was a high demand item, probably it would have been sold quickly and the owner lost not just the cost but also the profit.

If fact, in the case of retail stores, there is case law that deals with retail and wholesale value lost because of stole goods [2]

“If the goods were stolen from a retail merchant, the value is its retail value; while if stolen from a wholesale merchant the value is its wholesale value. See United States v. Robinson, 687 F.2d 359 (11th Cir. 1982). “

Therefore the correct answers are $61 or $100, depending on how “loss” is defined. But this is not the main issue either.

Many answer this wrong and there seems to be a problem. According to a poll I ran in Twitter, close to 55% answered it wrong. Some had even wrong answers and were not able to vote because they were not listed. So I estimate about 60% answered this wrong. Is this due to a cognitive bias?

Are you going to ask 100 people to solve an equation even if they don’t know math? If 60% answer wrong, will you conclude this is due to a cognitive bias, or lack of math skills, in the case of the above problem?

Instead of focusing on the inability of people to model simple problems and solve them mathematically, we are trying to discover cognitive biases to justify them. In my opinion, the reason for the wrong answers are not cognitive biases but lack of education in modeling and solving real world problems. There is a significant gap in these two views and significantly different approaches resulting from them.

The solution in my opinion is more practical math education with modeling and solutions. Teach people how to model simple reality and get solutions. The human brain is very powerful pattern matching engine that will probably never be challenged by any artificial machine. But machines are very good in doing calculations people cannot do. As use of mobile devices increases, the capability of people to solve basic problem will decrease since they are constantly relying on these devices for answers.

Is the failure to come up for an answer in a second to how much 766,758 times 907,847 yields, a cognitive bias? But a person with no math education will be able to recognize a fly sitting on the face of a camel while a machine with fancy machine learning algorithm may fail.

The educational establishment, at least part of it, should maybe stop treating people as biased for maybe winning tenure credit. Human beings survived hundreds of thousands of years with all their biases and before even discovering math, computers and establishing universities. Actually they may have survived because of the biases.

Are we treating humans as transhumans actually with all these cognitive tests involving hidden math problems? This can only result in degradation of human quality in favor of machines. Fine, if that is the objective. For me, it’s not fine.

In many problems posed that allegedly prove there are cognitive biases, the biases are frequently with those that pose them against human qualities.

Below is an example from a bet with positive expectation people refuse to take and this is attributed to cognitive bias.

People are skeptical of “too good to be true” bets because they know money is not free, at least for the 99.99% of the population. This is not a bias, it is reality. Below is another example of a bet people won’t take and behavioral economist attribute to bias and my response.

I like to defend common sense and the average person who is under constant attack by part of an academic elite. I argue that problems exist but most are not due to cognitive biases but due to low quality education or complete lack thereof. After four years of undergrad education and six years in grad school, I think a legitimate task of a scientist is to defend common sense above all.

Biases exist; sometimes they are useful and sometimes are harmful. But human history shows over the longer-term the expectation from biases is positive. Is there room for improvement? Yes, there is huge room, but first we have to raise the level of education and teach people how to model simple problems with math. It’s hard but this is the profitable route, not trying to prove people suffer from cognitive biases using questions that actually prove people lack basic math skills.

You can find me in Twitter if you have any questions or suggestions.

[1] The Bat-and-Ball Problem: Stronger evidence in support of a conscious error process




Michael Harris

Ex-fixed income and ex-hedge fund quant, blogger, book author, and developer of DLPAL machine learning software. No investment advice. priceactionlab.com/Blog/