Decision making is a part of the human condition and skill at it,
in part, determines success or failure. How should persons
proceed, when they are forced to make vital decisions? There is a
branch of mathematics called "Decision Theory" which has as its
subject the process of thinking carefully about how to formulate
such problems in mathematical terms and then solve them.
First one must posit: (i) the appropriate set of possible
decisions; (ii) the nature and the likelihood of every possible
response to each decision whether due to nature or to the actions
of an adversary (in mathematics this is called "stochastic;" (ii)
the result (pay-off) for each possible contingency. Then an
optimal solution to the problem of making the best choice is
sought. First one must decide upon a philosophy or strategy, which
is deemed suitable for the circumstances, which can determine among
any set of outcomes which one is best. Then a method of
optimization (i.e., calculating the best strategic choice)
must be developed.
But cannot good decisions be made without this mathematical
folderol? Certainly but analyzing Nelson's battle plan at
Trafalgar showed that he intuitively choose what we know to be the
optimal strategy. We know that Lee certainly made different
decisions on battle fields of the Civil War than did McClellan.
And the same was true of Patton and Montgomery in World War II.
Both Lee and Patton followed bold strategies of maximizing one's
probability of success while McClellan and Montgomery closely
followed a conservative strategy of minimizing their probability of
losing. This is called minimizing maximum regret.
Rather than discussing either of these criteria, let's briefly
examine the optimization principle: Choose the action which
minimizes one's risk where the risk of any decision (or action) is
the expected value of its anticipated loss (or gain).
Nothing would seem to demarcate the human species from lower life
forms more clearly than our powers of reason concerning good
decisions in our self interest or preservation as is so clearly
shown by the story of frog behavior!
It is reported that if a frog is immersed in hot water it will
immediately scramble out but if it is placed in cool water
that is gradually heated the frog will remain complacently in
the water until it is cooked.
But there is a subtle distinction between anticipated loss and
perceived loss. One cannot anticipate what one does not perceive.
For example, it is commonly known that smoking one pack of
cigarettes per day over a period of time will reduce ones life
expectancy. But this fact does not dissuade the myriad of smokers
who say that smoking is addictive (which it is). But to non-
smokers the guarantee of early death would seem overwhelmingly
repugnant. Let us assume the approximately correct but convenient
figures: by smoking one pack of cigarettes per day for 40 years,
starting at age 20, one will, on average, die at age 60 rather than
at 70 and moreover die of either heart disease or lung cancer. But
this means smoking one pack of cigarettes per day for one year will
reduce one's life expectancy by one-quarter year. So, at about
three minutes per cigarette, one spends one hour smoking per day
which reduces ones life expectancy by six hours! However this is
only on the average and everyone knows a Mr. R. Crabtree who
started smoking at age 16 and died at 90. Such an example of a
deviation from the statistical expectation convinces those persons,
who have little knowledge of the behavior of stochastic phenomena,
only that there is no certainty and they, being lucky, will escape
too. However Mr. Crabtree might have lived to age 100 had he not
smoked. Moreover since this loss comes at the end of life ( of
course any fatality comes at the end of life) the
perceived risk to a young person is imperceptibly small. This self
deceit, when combined with the addictive pleasure of smoking, means
that millions of persons will die years before their time.
The ways in which we make decisions are intriguing. Let's suppose,
on the other hand, that cigarettes were not harmful at all except
for one indistinguishable cigarette in one package, among 18,250
packages, contains an undetectable explosive ingredient which when
ignited will blow off the head or hand of the person smoking or
holding it. If that were the case the perceived risk to public
health and safety would be so great as to cause the immediate
cessation of cigarette manufacture and sale, not by law but by the
virtual elimination of smoking! If 30 million packages of such
cigarettes were sold each day one could expect 1,600 deaths or
disfigurements daily. The daily death-accident toll would rival
that of automobiles! Yet the total expected loss of life or health
to smokers using dynamite-loaded (but otherwise harmless)
cigarettes over forty years would not be as great as with ordinary
filtered tobacco! If a person age 20 is selected at random and
(s)he smokes one pack of ordinary cigarettes per day the
expectation of life, with death due only to that cause, is
approximately forty years but if that same person were to
smoke only explosive cigarettes, assuming that this real ``coffin
nail" is totally indistinguishable and is distributed at random
among the packs, the expectation of life is fifty years!
This great difference in perceived risk is due to sudden
unsuspected consequences rather than the gradual increase in
breathlessness and breathing discomfort; consequently in such
circumstances human decision making seems more like that of frogs.
If one could smoke for a year without effect and then have the
cumulative damage suddenly imposed upon ones breathing and physical
well being, virtually no one would continue smoking after
experiencing the first saltus.
Are there other unperceived risks that may affect humanity? At the
present time various persons might list:
- The risk of higher rates of skin cancer and eye injury in
the human and animal populations due to the increased
ultra-violet radiation penetrating to the earth's surface from
the CFC's reduction of the ozone layer.
- The risk of the eventual contamination of water near nuclear
waste repositories. After all the last ice-age was a "period
of intermittent glaciation starting about 500,000 years ago
and ending about 20,000 years ago." How, if it was
intermittent, do we know it ended then? No nuclear waste
repository presently contemplated would withstand a glacier.
- The risk of global warming, which is commonly understood to
mean that the temperature everywhere on earth will uniformly
increase by the some small amount. There are mathematical
models which predict that global warming, since the earth
radiates energy into space (which cannot be heated), will mean
principally more violent local weather extremes such as the
ones we are presently experiencing.
- The risk of a virus developing, which is as easily spread
and as lethal as was the bubonic plague in the middle ages,
and is immune to all known antiviral medicines. If the
infection should have an incubation period of a few days,
during which it might be inadvertently spread world wide by
air travel, it could be a devastating pandemic within a few
weeks.
- The risk of the loss of potable water sources. There are
numerous cities which obtain water from aquifers which are
gradually becoming polluted by, at least one of, increasing
amounts of: (i) fecal bacteria from the untreated septic tanks
that are in common use in adjacent communities; (ii)
pesticides that are being used by farmers on their fields
which are upflow; (iii) toxic wastes that are now percolating
into the aquifer from previous years of ground-dumping of
now-illegal wastes at various nearby sites nearby the aquifer.
What is the perceived risk? What is the expected loss? For
example, at present only a few cases of illness due to water
contamination are reported that can be shown conclusively not to
have come from alternate sources. What effort should be made in a
democracy, where more than 50 percent of the voters must be
persuaded before tax supported governmental action can be taken, to
explain risks where the expected loss necessarily must be born by
future generations.
In the choice of optimal investment schemes, where the profit or
loss has only financial consequence, the methods for optimal
decision making have been whetted by mathematical
economists/investment councilors, who rely on the power of the
computing machine to calculate (not divine) the risk of such
arcane instruments as futures and derivatives. But in situations
where the loss is not so easily quantified we humans find ourselves
not doing much better than frogs and we too may suffer the fate of
putting up with environmental inconvenience until we too are
"cooked."
Maybe Aristophane's comedic play, Frogs, with its croaking chorus,
serves as a more apt simile than we realized, not only for the
ancient, but also for the modern predicament of mankind.
How can Statistics Help?
Choose the action which maximizes one's probability of success.
Consider whether a person should play one of the state pick-six
lotteries. An oft heard saying is "The best chance you have to
become a multi-millionaire is to win the state lottery and you
can't win if you don't bet." All that is true and the more tickets
you've purchased, the higher the probability of winning. So to
maximize one's probability of success one should invest as much as
possible in the lottery: borrow on the mortgage, sell the car, cash
retirement investments? But this strategy ignores the expected
return for each ticket purchased, which is always negative! The
more you "invest" the more you are expected to lose. Always? Yes
always! If you could buy all possible combinations in, say, the
New York lottery (which is 54 pick 6) you would have invested
$12,913,000 which means that the present value of the prize, which
is to be paid over 20 years, would have to exceed what was spent,
i.e., the prize should exceed c. $26,000,000. Moreover one must
take into account the probability of having to split the prize with
other winners which means that the value of the prize must be
higher yet. But the higher the prize the more participants there
are, and the higher the likelihood of a further split. In fact,
the more one spends the more one is expected to lose (otherwise the
states would not be making money.)
Let us now consider the sanitation of cities during previous
centuries. There was a time when city residents daily threw the
contents of their chamber pots from an upstairs window into the
street below. (This caused pedestrians to call out the genteel
French phrase "garde l'eau" - hold the water - as they passed.)
The small gain in a residant's personal convenience outweighed the
inconvenience of the odor. But not until epidemics occurred and
reoccurred, with high morbidity concentrated along the open sewers,
was the city government persuaded that underground sewers were
necessary. This persuasive statistical evidence of the correlation
(a modern word) between incidence of disease and propinquity to
fecal matter, came long before the germ theory of disease was even
proposed, let alone accepted. The cause was dimly understood since
the origin of many diseases was thought to be "bad air." The
associated diseases were called malaria.
But long before science could explain the mechanism of infection
and death in bubonic plague, perceptive people (who had the means,
e.g., Isaac Newton during the 17th century plague in England) left
the cities to spend time in the countryside until the plague had
abated. They took such action because of the perceived correlation
between death and disease and the density of people. In fact
persons who collected in churches to pray for the intervention of
heaven were not usually successful. Here success, i.e., not
becoming infected and dying of the plague, is everything. Clearly,
whatever the cost of maximizing the probability of one's success,
that strategy should be followed. So there are instances other
than war in which this strategy is advisable.
