Bernoulli’s Errors
Kahneman criticizes evaluating choices solely through final wealth rather than gains and losses from a reference point. The supplied endnotes also explain Bernoulli’s logarithmic utility idea and why diminishing utility can make an infinite-expected-value gamble unattractive.
THE IDEA
25: BERNOULLI’S ERRORS
This excerpt contains the chapter’s endnotes, not its main discussion.
1 Mathematical Psychology: Clyde H. Coombs, Robyn M. Dawes, and Amos Tversky, Mathematical Psychology: An Elementary Introduction (Englewood Cliffs, NJ: Prentice-Hall, 1970).
2 for the rich and for the poor: This rule applies approximately to many dimensions of sensation and perception. It is known as Weber’s law, after the German physiologist Ernst Heinrich Weber, who discovered it. Fechner drew on Weber’s law to derive the logarithmic psychophysical function.
3 $10 million from $100 million: Bernoulli’s intuition was correct, and economists still use the log of income or wealth in many contexts. For example, when Angus Deaton plotted the average life satisfaction of residents of many countries against the GDP of these countries, he used the logarithm of GDP as a measure of income. The relationship, it turns out, is extremely close: Residents of high-GDP countries are much more satisfied with the quality of their lives than are residents of poor countries, and a doubling of income yields approximately the same increment of satisfaction in rich and poor countries alike.
4 “St. Petersburg paradox”: Nicholas Bernoulli, a cousin of Daniel Bernoulli, asked a question that can be paraphrased as follows: “You are invited to a game in which you toss a coin repeatedly. You receive $2 if it shows heads, and the prize doubles with every successive toss that shows heads. The game ends when the coin first shows tails. How much would you pay for an opportunity to play that game?” People do not think the gamble is worth more than a few dollars, although its expected value is infinite—because the prize keeps growing, the expected value is $1 for each toss, to infinity. However, the utility of the prizes grows much more slowly, which explains why the gamble is not attractive.
5 “history of one’s wealth”: Other factors contributed to the longevity of Bernoulli’s theory. One is that it is natural to formulate choices between gambles in terms of gains, or mixed gains and losses. Not many people thought about choices in which all options are bad, although we were by no means the first to observe risk seeking. Another fact that favors Bernoulli’s theory is that thinking in terms of final states of wealth and ignoring the past is often a very reasonable thing to do. Economists were traditionally concerned with rational choices, and Bernoulli’s model suited their goal.
When to use it
Evaluating the attractiveness of a lottery or gamble based on potential gains and losses relative to current wealth.
Understanding why a person might prefer a certain gain over a gamble with higher expected value (risk aversion) or why someone might take a risky gamble to avoid a certain loss (risk seeking).
Analyzing public policy decisions, such as whether to invest in a project with uncertain outcomes, by considering how people evaluate outcomes relative to the status quo.
FROM THEORY TO EVERYDAY LIFE
In practice
St. Petersburg Paradox
In a game where you toss a coin repeatedly, the prize starts at $2 and doubles each time heads appears, stopping at first tails. Although the expected value is infinite, people are only willing to pay a few dollars to play, illustrating that utility grows more slowly than monetary payoff.
Diminishing Sensitivity to Wealth
Bernoulli's theory predicts that the same absolute gain (e.g., $10 million) yields less additional satisfaction for a wealthy person than for a poor person. This aligns with the observation that a doubling of income gives roughly the same happiness boost in rich and poor countries, supporting a logarithmic utility function.
A MOMENT TO REFLECT
Put it to work.
Think of a recent financial decision you made. Did you evaluate it in terms of final wealth or in terms of gains and losses from a reference point? How might that have affected your decision?
Consider the St. Petersburg paradox. Why do you think people are unwilling to pay more than a few dollars to play a game with infinite expected value? What does this reveal about the psychology of risk?
How might evaluating a change from your current wealth differ from evaluating the final wealth alone? What would serve as your reference point?
Often confused with
References
- Mathematical Psychology: An Elementary Introduction — Clyde H. Coombs, Robyn M. Dawes, and Amos Tversky (1970)
CONTINUE THE CONVERSATION
Source texts
- Thinking, Fast and Slow (Daniel Kahneman)