Here is a relatively simple math problem:  A bat and a ball cost $1.10 combined. The bat costs $1 more than the ball. How much does the ball cost? (I will provide the answer below the fold.)

This problem is the basis of a large psychological literature on thinking systems in the human brain, discussed in Daniel Kahneman’s book: Thinking, Fast and Slow. The idea is that there are two parallel thinking systems in the brain, a fast intuitive system that provides quick answers which may or may not be strictly true, and a slow analytical system that will go through a problem systematically and check the results.

This basic scheme is fairly well established in the research literature, but there are many sub-questions. For example – what is the exact nature of the intuition for any particular problem? What is the interaction between the fast and slow system? What if multiple intuitions come into conflict by giving different answers to the same problem? Is it really accurate to portray these different thinking styles as distinct systems? Perhaps we should consider them subsystems, since they are ultimately part of the same singular mind. Do they function like subroutines in a computer program? How can we influence the operations or interaction of these subroutines with prompting?

A recent publication present multiple studies with many subjects addressing these subquestions. If you are interested in this question I suggest reading the original article in full. It is fairly accessible. But here is my overview.

The answer to the math problem I presented above is that the ball costs $0.05, and the bat costs $1.05. This fulfills both criteria, the total price is $1.10 and the bat costs $1 more than the ball. However, the vast majority of people do not give that answer. The intuitive answer to this problem is that the ball costs $0.10. Without further prompting, 76% of respondents give the intuitive wrong answer, while only 13% give the analytical correct answer. This result does depend on the population, with more “elite” subjects (not a value judgement, just the term used to refer to those with either higher education or likely familiarity with the subject matter) doing much better, but still largely wrong. But let’s get into some more detail.

Some studies also track how long it takes for each subject to answer. The intuitive wrong answer is typically given more quickly than the correct answer. This leads researchers to the hypothesis that generally people will mentally produce the intuitive answer, but then some subjects will check the answer, find that it is wrong, and revise the answer to the correct one. So perhaps the difference between those who get the question correct and those who don’t is that some people don’t check their intuitive answer, while others do. But now we can get into a host of revised study questions to see how it influences people’s behavior.

One subquestion regards the intuitive process. One hypothesis is that the intuition involves substitution – when confronted with a difficult problem we will substitute a simpler problem that we think will produce the same answer, or at least a similar answer. In this case the intuitive answer, $0.10, results from subtracting the difference in price ($1) from the total price ($1.10). Or, is the wrong answer the result of being thoughtless or careless? Perhaps some people simply can’t do math in their head well enough to solve the problem. The researchers explored this by asking subjects to later recall the problem. Of those who got the intuitive wrong answer, 23% remembered it as the simpler subtraction problem, suggesting they had substituted the simpler problem for the more difficult one. But still, most remembered the question correctly.

In another test they asked the subjects to give the price of the bat instead of the ball. The subtraction strategy would then result in the answer that the bat costs $0.10. The two responses were about the same. So perhaps there are three groups of subjects – those who simply subtract the numbers without thinking about it deeply, those who look to substitute a simpler problem, and those who check their answers analytically. But there is still more going on.

Another variable the researcher altered was the difference in the price between the bat and the ball. If you say the bat costs $0.90 more than the ball, the subtraction strategy results in a price for the ball of $020. This makes it more obvious that the bat does not, for this answer, cost $1 more than the ball. As this difference gets larger, fewer people give the intuitive wrong answer. This suggests that people are only OK with the wrong answer if it is close enough to the correct answer. I wrote recently about other research which showed that people will deem answers “correct” if they are close enough. We tend to take approximately correct as good enough to be correct. This is probably adaptive – small differences are likely not worth the mental energy to worry about. So if the intuitive answer seems close enough to the truth, we go with it. In fact, that may be the only check we mentally give – yeah, that sounds close enough to be true.

Could subjects be prompted into the analytical strategy? Yes, but not completely. The researchers tried all sorts of prompts – consider that the answer may be $0.05, or may not be $0.10, or whether it is either. In one iteration they even prompted – the answer is $0.05. All these prompts increased the percentage of people who gave the correct answer, but not by much. Only the one giving the correct answer had a big impact, but even then it was not 100%. Some people either didn’t trust the provided answer, or their intuitions were too strong to alter. This coincides with another measure in these experiments – how confident are people in their wrong intuitive answer? Most are either very or maximally confident, and it’s hard to push people off an answer in which they have so much confidence.

Putting this all together I think we are still left with the notion that there are multiple subsystems that make up our cognitive processes, and some of these subsystems are subconscious and intuitive, while others are more conscious and analytical. The ultimate analytical system may be what we call metacognition – thinking about how you are thinking. These analytical subsystems are also referred to as reflective, looking in at the thinking process to make sure it is valid. Using the analytical strategy, however, requires mental work and ability. So there is another principle in play – biology is lazy. By this I mean that biology favors efficiency, which conserves resources.

Mentally what this means is that if we can get to a 90% or so correct answer with very little effort, our adaptive laziness will often just go with that answer as good enough. It’s not worth the mental energy and the delay in action to make sure we have that last 10% correct. This system may be well adapted to survival situations, giving the statistically greatest chance of surviving. But it is not well adapted to a complex technological civilization, where being actually correct is often quite important. This is especially true when there are billions of dollars to be made exploiting your intuitions. This is where metacognition comes into place – we need to consciously substitute a more analytical strategy to replace our evolved intuition-dominated strategy, and have some sense of when it is worth the extra mental effort to do so.