## Tuesday, June 25, 2013

### 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.

1. People are irrational; me included. Money is a part of life equation, accompanied by family, job content and amount, hobby, ego, etc. I think the goal should be, mathematically speaking, to find a cumulative max for all life factors. I'd call it life balance.

1. @AnatoliN: Seeking life balance is important, but I don't think we do it well. The trouble with money is that it affects everything else. If you overspend by \$50,000 on your car and \$200,000 on your house, there is a good chance that you'll have to stick with a job you hate because it pays well, and your hobbies and family relationships will suffer as well.

1. @Returns Reaper: Thanks for pointing out the error. It is fixed now.

3. Using Bernoulli's theory of decision making, I believe it would only be rational to take the chance to make \$300 or lose \$200 on the flip of a coin if your net worth is above a certain level (somewhere between \$800 and \$900 I believe).

1. @Blitzer68: The threshold is a net worth of \$600. However, any capable of even the most modest work can expect to have \$600 sometime in their future.

2. Interesting if you added some zeros though. I.e. would it only be rational to take the chance to make \$300,000 or lose \$200,000 on the flip of a coin if your net worth is above \$600,000? Seems pretty scary to me.

3. @Blitzer68: Yep. We're wired to find such things scary. According to Kahneman's research, even people with a net worth of \$600,000 tend to turn down a chance to flip for +\$30 or -\$20.

4. Suppose someone with a net worth of \$600,000 borrowed \$600,000 (interest free say for simplicity) and invested the whole \$1.2 million in the stock market expecting to either earn 25% (300,000) or lose 10% (120,000) yearly. This is based on a historical 7% expected yearly return for the stock market with an 18% standard deviation.

Using Bernoulli's theory of decision making, this would be a rational bet I believe. Seems kinda scary though.

1. @Blitzer68: If the stock market actually worked that way it might well be a good bet (although the compounded yearly return is only 6.1%). However, as Mandelbrot showed (in his book The Misbehavior of Markets), the distribution of stock market returns has fat tails. This means that big swings are far more likely to happen than a normal (Gaussian) distribution allows. Your hypothetical investor faces the possibility of complete ruin. For example, if he had been unfortunate enough to have begun his leveraged investing in the S&P 500 on 2007 Oct. 12, he would have lost 57% of his investment (including dividends) by 2009 Mar. 9. If the bank had demanded repayment on that day, he'd have been totally wiped out and left with some debt.

5. wrt the flip for +300 or -200, it would make most sense for a series of say 10 flips.

1. @Anonymous: Interestingly, Kahneman gives a detailed explanation of why people would accept 10 such bets, but not one. This is irrational, but consistent with how our brains are wired.

2. How is it irrational? 1 flip has a very high element of chance. 10 flips much less so. It seems rational to prefer a chance-eliminating option over the opposite.

3. @Anders: I would prefer 10 flips as well, but I'd still take just one flip if that's all I was allowed to have. To accept 10 flips but turn down one flip is consistent with how people act, but it is irrational.

6. Stumbled upon this page today as I am reading Kahneman and found myself pausing in the chapter about Bernoulli. Kahneman himself states that Bernoulli's theory on the diminishing marginal value of wealth explains risk aversion and that the model discusses an increase or decrease resulting in an outcome. That is, a change as opposed to a static.

Kahneman uses the example of Jack and Jill to debunk Bernoulli. Today, Jack and Jill both have 5 million dollars but yesterday Jack had just 1 and Jill had 9. "Are they equally happy? (Do they have the same utility?)", Kahneman asks?

Kahneman then states that "Bernoulli's theory assumes that the utility of their wealth is what makes people more or less happy."

But does it really? I haven't read Bernoulli, but I can't see anything in Kahneman's writing that validates that Bernoullis spoke of static utility. Didn't Bernoulli in fact speak of CHANGES in utility? Just like Kahneman does?

If that is the case, then obviously, both instinctively and according to Bernoulli's model, Jill and Jack are not equally happy today. Jack just went from 10 utility units to 70, Jill went from 96 to 70. So Jack is +60 happy today. Jill is -26 happy today.

Would love to hear your thoughts,

1. @Anders: I think Bernoulli and Kahneman are modeling completely different things. Bernoulli is attempting to model how people should act to serve their themselves best. He isn't trying to measure short-term happiness. I suspect Bernoulli would say that Jack and Jill should be equally happy with \$5 million if they are thinking rationally. And perhaps they will come to be equally happy in time as the memory of their sudden changes in wealth fades.

Kahneman is an expert in how people actually make decisions. I would trust his judgement in how Jack and Jill would react to their wealth changes (both immediately and over time). But I found nothing in Bernoulli's paper indicating he was trying to model how people actually act. He seemed to be trying to create a model of rational behaviour.

On the specific question of whether Bernoulli's model is static; it is. He has utility as proportional to the logarithm of wealth. Kahneman says this is a poor model of people's short-term happiness and also a poor model of their behaviour. But I see no evidence Bernoulli was trying to model these things.