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Playing It Smart: Are you paid by how often you win, or by how much?21 July 2008
In "Fooled by Randomness" (Random House), Nassim Nicholas Taleb tells of a high-level meeting at a Wall Street brokerage. The honchos asked whether he thought the stock market would go up or down the following week. "Up," he replied. And, when pressed, predicted a strong probability it would do so about 70 percent. Someone in attendance pointed out, though, that Taleb had been selling rather than buying. This was essentially betting on the 30 percent likelihood that the market would go down. So his forecast seemed to contradict his action.
He disagreed. Rather, he explained, stock prices were highly favored to rise, but the gains were apt to be small. Conversely, in the admittedly lesser chance of prices falling, he speculated that losses would be massive.
Put some numbers on the projections for the amounts by which the market might rise or fall. Suppose that prices seemed poised to climb on the average of $0.20 but if they tumbled the drop could easily be $2.00. Then, instead of juggling probabilities and prices separately, combine the two into a single term. A suitable index is the statistically expected result, or "expectation." This equals probability times amount. For the two scenarios hypothesized, it's an expected gain of 70 percent times $0.20 or $0.14 and an expected loss of 30 percent times $2.00 or $0.60.
Were probability the be-all and end-all, buying on a good chance of prices rising would make sense. But combining both factors and working with expectations, the opposite would hold. And, Taleb points out, stock traders (and casino aficionados alike, there isn't as much difference as the moguls on Wall Street want to believe) are paid or penalized in dollars, not in probabilities. Anyone who proceeds solely on the basis of frequency, he asserts, is inept at acting under uncertainty and coping with risk.
Slot machine sessions in casinos illustrate the contrast between probability alone and expectation. Solid citizens rarely think about individual pulls or intermediate returns, but of venturing their stakes on shots at substantial jackpots. Here, normally, chances of what a responsible gambler would consider a modest setback are quite high and those of a huge payday very low. Even without the erosive effect of the edge, someone coming with $100 and hell-bent on leaving with $10,000 has only 1 percent chance of success and the complimentary 99 percent of failure.
Would you take such a risk? Many bettors do. This, principally because they consider $100 a reasonable fee for the opportunity to try and the entertainment value of the gamble, and $10,000 a worthwhile target despite the dim prospects for attainment.
But, again ignoring edge, you'd have a 75 percent chance of leaving the casino with $10,000 versus 25 percent of going bust if you start with $7,500. Intuitively, this fat a wad lets you weather some severe cold spells and still be a contender for the biggie. Not many folks would try, though. The probabilities may seem favorable, but the size of the possible loss is daunting.
In the first slot instance, expectation associated with victory is 1 percent of the $9,900 profit or plus $99. That with defeat is 99 percent of $100 or minus $99. In the second example, profit expectation is 75 percent of the $2,500 earnings or plus $1,875 while loss expectation is 25 percent of $7,500 or minus $1,875.
Either way, plus or minus $99 or $1,875, net session expectation is zero a wash owing to the simplification of no house advantage. However, the situations aren't equivalent. In one, the negative component of expectation is $99, which a typical grownup casino visitor can presumably manage. In the other, it's $1,875, which most recreational gamblers would have trouble justifying.
Taleb summarizes by noting "very few people take home a check linked to how often they are right or wrong. What they get is a profit or loss." The punter's poet, Sumner A Ingmark, was as canny when he composed this catchily cogent couplet:
'Though chances are low, if a failure costs mightily,
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