520 18. PROBABILITY AND STATISTICSand as a result, the total utility of personal wealth is a logarithmic function of
total wealth. One consequence of this assumption is a law of diminishing returns:
The additional satisfaction from additional wealth decreases as wealth increases.
Bernoulli used this idea to explain why any rational person would refuse to play
the game. Obviously, the expected gain in utility from each of these wins, being
proportional to the logarithm of the money gained, has a finite total, and so one
should be willing to pay only an amount of money that has an equal utility to
the gambler. A different explanation can be found in Problem 18.4 below. This
explanation seems to have been given first by the mathematician John Venn (1834-
1923) of Caius^9 College, Cambridge in 1866.
The utility y, which Bernoulli called the emolumentum (gain), is an important
tool in economic analysis, since it provides a dynamic model of economic behavior:
Buyers exchange money for goods or services of higher personal utility; sellers
exchange goods and services for money of higher personal utility. If money, goods,
and services did not have different utility for different people, no market could exist
at all.^10 That idea is valid independently of the actual formula for utility given
by Bernoulli, although, as far as measurements of pyschological phenomena can be
made, Bernoulli's assumption was extremely good. The physiologist Ernst Heinrich
Weber (1795-1878) asked blindfolded subjects to hold weights, which he gradually
increased, and to say when they noticed an increase in the weight. He found
that the threshold for a noticeable difference was indeed inversely proportional to
the weight. That is, if S is the perceived weight and W the actual weight, then
dS = kdW/W, where dW is the smallest increment that can be noticed and dS
the corresponding perceived increment. Thus he found exactly the law assumed by
Bernoulli for perceived increases in wealth.^11 Utility is of vital importance to the
insurance industry, which makes its profit by having a large enough stake to play
"games" that resemble the Petersburg paradox.
Mathematically, there was an important concept missing from the explanation
of the Petersburg paradox. Granted that one should expect the "expected" value
of a quantity depending on chance, how confidently should one expect it? The
question of dispersion or variance of a random quantity lies beneath the surface
here and needed to be brought out. It turns out that when the expected value is
infinite, or even when the variance is infinite, no rational projections can be made.
However, since we live in a world of finite duration and finite resources, each "game"
will be played only a finite number of times. It follows that every actual game has
a finite expectation and variance and is subject to rational analysis using them.1.7. Laplace. Although Laplace is known primarily as an astronomer, he devel-
oped a great deal of theoretical physics. (The differential equation satisfied by
harmonic functions is named after him.) He also understood the importance of
probabilistic methods for processing the results of measurements. In his Theorie
analytique des probabilites, he proved that the distribution of the average of ran-
dom observational errors that are uniformly distributed in an interval symmetric
about zero tends to the normal distribution as the number of observations increases.
(^9) Pronounced "Keys."
(^10) One feels the lack of this concept very strongly in the writing on economics by Aristotle and
his followers, especially in their condemnation of the practice of lending money at interest.
(^11) Weber's result was publicized by Gustave Theodor Fechner (1801-1887) and is now known as
the Weber-Fechner law.