Thinking, Fast and Slow

25: Bernoulli’s Errors
Mathematical Psychology: Clyde H. Coombs, Robyn M. Dawes, and Amos Tversky,
Mathematical Psychology: An Elementary Introduction
(Englewood Cliffs, NJ: Prentice-
Hall, 1970).
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.
$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.
“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.
“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.