Bernoulli’s Model of Risky Decisions

In 1738, Daniel Bernoulli devised a simple model of risk aversion (English translation here). Nobel Prize winner and author of Thinking Fast and Slow Daniel Kahneman criticizes Bernoulli’s theory extensively, describing it as “Bernoulli’s Error.” I disagree with Kahneman. I think Kahneman misunderstands Bernoulli’s claims.

Bernoulli’s theory of decision-making is best described with some examples. He claims that doubling your net worth is as positive as dividing it by 2 is negative. So, if your net worth is $200,000, receiving another $200,000 is as positive as losing $100,000 is negative.

Bernoulli applies the same type of rule to smaller changes as well. Going from $200,000 to $250,000 is multiplying by 1.25. Going from $200,000 to $160,000 is dividing by 1.25. So, winning $50,000 is as good for you as losing $40,000 is bad for you. (For the more mathematically-inclined, the utility of your net worth is proportional to its logarithm.)

Kahneman’s extensive research has shown that people don’t think this way. They tend to be more risk-averse than Bernoulli’s model indicates. When trying to avoid losses, people tend to be more risk-seeking than Bernoulli’s model. Across several pages in Thinking Fast and Slow that Kahneman devotes to criticizing Bernoulli’s theory, the argument seems to boil down to the fact that people don’t make decisions consistent with Bernoulli’s model.

I agree with this. However, after reading Bernoulli’s paper, I see no evidence that Bernoulli was trying to model human behaviour. He was trying to model rational behaviour. In section 7, Bernoulli looks at how much people “should be willing to venture,” not how much they are willing to venture. In section 14, he says that anyone who accepts a certain type of gamble “acts irrationally.” In section 15, he says that offering certain types of insurance is “foolish” and “unwise.” It seems clear that Bernoulli was not trying to model actual decision-making; he was modeling rational decision-making.

Some may think that we should make decisions about gambles based on simple mathematical expectation. So, if you’re offered a chance to either win $10,000 or lose $9999.99 on the flip of a coin, it is rational to accept. This is incorrect. We can see this if we take it to the extreme. Imagine you’re given a chance to gamble for everything you own: double (plus a dollar) or nothing. This is a bad bet. The misery you’d face in your future if you lost far exceeds to benefit you’d get if you won. A certain amount of risk-aversion is perfectly rational, and Bernoulli sought a rule to decide which risks are rational to accept.

It’s quite true that Bernoulli’s model has its challenges. It will fail in some narrow circumstances such as desperately needing money for a life-saving operation. Another challenge is deciding what counts in your net worth. How do you model future income (human capital)? How do bankruptcy laws factor in?

Despite these challenges, I’ve found Bernoulli’s theory to be an excellent model of rational behaviour. I’ve learned from Kahneman’s research that Prospect Theory is an excellent model of the way people actually make decisions. Faced with a chance to make $300 or lose $200 on the flip of a coin, Prospect Theory explains why people turn down this gamble, and Bernoulli explains why it is rational to accept the gamble. Any tension between the two theories is easily explained by the fact that people are sometimes irrational.