Thinking, Fast and Slow

25: Bernoulli’s Errors
Mathematical Psychology: Clyde H. Coombs, Robyn M. Dawes, and Amos
Tve r s k y, 
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

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“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.