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size to the base
b
, but the relative frequency of occurrence of that event decreases to
base
a
. The index for the scaling is typically expressed in terms of the parameters
a
and
b
; in fact it is in terms of the parameter
ln
b
as found earlier.
One measure of play is, as we said, the expected outcome of a wager. If the
j
th wager
μ
=
ln
a
/
w
j
the expected winnings are determined by summing over all the wagers
is
∞
W
=
1
w
j
p
j
,
j
=
which for the St. Petersburg game diverges. To circumvent this infinity Daniel Bernoulli
argued that not everyone should wager in the same way since the subjective value or
utility associated with a unit of currency depends on one's total wealth. Consequently,
he proposed estimating the “winnings” by using the average utility of a wager
∞
U
w
j
p
j
,
U
=
j
=
1
not the wager itself. It is the average utility that one ought to maximize, not the expected
winnings. The mathematical wrangling over the resolution of the St. Petersburg paradox
lasted another two centuries, but that need not concern us here, and we move on.
2.2.2
Temporal scaling
Scaling relates the variations of time series across multiple time scales and has been
found to hold empirically for a number of complex phenomena, including many of
physiologic origin. The experimentalist sees the patterns within fluctuating time series
as correlations produced by interactions that mitigate the randomness. Scientists with a
bias towards theory recognize time series as solutions to equations of motion, sometimes
with the web of interest coupled to the environment and other times with the web having
nonlinear dynamics with chaotic solutions. All these perspectives, as well as those of
the computer modeler, need to be considered in order to understand complex webs,
particularly since there is not a unique way to generate the fractal properties discussed
in the previous section. As a matter of fact, most biological phenomena are described
by the “pathological” behavior just discussed, and it is common to use life models as
paradigms of complex webs.
One way physicists understand the source of scaling relations in experimental data is
through the use of renormalization groups that tie together the values of certain func-
tions at different time scales. An unknown function
Z
(
t
)
that satisfies a relation of the
form [
74
]
Z
(
bt
)
=
aZ
(
t
)
(2.33)
is said to scale. However, this equation expresses a dynamical relation, when
t
is the
time, in the same sense as a differential equation relates a dynamical variable with its
derivatives and consequently the scaling relation has a solution. It is worth emphasizing
here that, although (
2.33
) is analogous to an equation of motion, it may describe a
