Mathematics Awareness Week 1996

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Mathematics, Decision Making, and Risk Perception

by Sam C. Saunders

Mathematics, Decision Making, and Risk Perception

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.

Sam C. Saunders, Professor of Applied Mathematics/Statistics
Washington State University; Pullman, WA 99164-3113

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.