Despite the mathematical proof defending the logic of expected value, in the real world, when hearts and minds do battle, the heart — one’s fears and hopes — often prevails. Our gut instinct knows that focusing on the average result may work in the long run, but as individuals we are more concerned about the specific case: How much can we lose? What’s the likelihood of a bad result occurring?
Plausibility Theory replaces the Bayesian expected-value calculation with a risk threshold that is more comfortable for most people. Although developed only in the last five years, it shows great promise as a way to drive rigorous decision analysis while focusing on the real priority of most decision makers: downside risk. This new theory still examines the range of possible outcomes but focuses on the probability of hitting a threshold point — such as a net loss — relative to an acceptable risk.
For example, using Plausibility Theory to analyze the coin-tossing bet would yield different conclusions about the appropriateness of the one-time bet versus the portfolio of 100 bets. A conservative decision maker might set as a risk threshold no more than a 1 percent chance of losing money. Using the calculus of Plausibility Theory, the gamble on a single coin toss — which presents a 50 percent chance of losing $50,000 — would be rejected. But the gamble of flipping the coin 100 times would be acceptable because the probability of a loss would be well under the risk threshold.
The use of a risk threshold also resolves another conundrum associated with Bayesian statistics: the problem of unknowable risk. Most business decisions involve a mix of knowable and unknowable risks. Knowable risks involve predictable odds. For example, Capital One Financial Corporation in Richmond, Va., amasses data on millions of customers, which allows the company to predict precisely the probability that a customer with a certain demographic profile will default on his or her credit card debt. Uncertainty over whether a particular customer will default remains, but the odds of default are understood well enough that the company can set interest rates high enough to profit. With enough data, such decisions are like the roulette wheel at a casino. Any one customer may win or lose, but “the house” will definitely come out ahead in the long run.
In contrast, unknowable risks cannot be defined with predictable odds. When Capital One first experimented with an auto loan business, it had no historical data to predict the behavior of this new type of customer. Bayesian decision analysis defines a probability for such unknowable risks by inference from the choices made by the decision maker. This approach, however, can also lead to nonintuitive results.
Consider another hypothetical gamble (a bit more complicated than a coin toss, but necessary to illustrate the point, so please bear with us). It’s based on randomly drawing a ball from an urn containing three balls. You have been assured the urn contains one red ball. All you know about the other two balls is that they are either blue or yellow: The urn could contain one red plus two blue balls; or it could have one red plus two yellow balls; or it could contain one red, one blue, and one yellow ball. The knowledge that there is one red ball provides an example of “knowable risk.” The uncertain mix of blue and yellow balls represents “unknowable risk.”
You are given the option to receive a payout of $1,500 based upon the color of one ball drawn randomly from the urn. You can pick red or blue — not yellow — as your winning color. A strict Bayesian view treats the two choices as equal, given the lack of information about the blue balls. But, since it is possible that the urn contains no blue balls, most people will choose red, for it offers the known probability of one chance in three of winning.