If you are then offered another gamble from an identical urn, your choices can easily appear, to a Bayesian, even more irrational. Suppose you are offered $750 if either one of a pair of selected colors — blue/yellow or red/yellow — is drawn. In this scenario, the choices are again between a known and unknown risk. Although you don’t know the mix of blue and yellow, you do know that only one ball is red. So the first option of selecting the pair of blue and yellow as your winning colors produces the “known” probability, a two-thirds chance of winning. The second option, choosing red and yellow, returns us to the “unknown,” because we don’t know how many yellow balls are in the urn: There could be zero, one, or two, and each scenario would produce a very different probability of winning.
So, most people choose the first option because it offers a precisely quantifiable, known probability of two-thirds versus an unknown probability.
Bayesians are troubled by this behavior. If you chose red in the first gamble, it suggests you believe that it is more likely that the urn contains two yellow balls than two blue balls. But, if this is true, you should then prefer the combination of red and yellow in the second bet. From a Bayesian point of view, you are behaving inconsistently based on the contradictory probabilities implicit in your decisions.
Plausibility Theory finds no fault with these intuitive choices. We are rationally choosing knowable risks over unknowable risks because they allow us to examine our decision against a risk threshold.
Back to Business
Analogous gambles occur regularly in companies whenever unknowable risks with no historical precedents drive the profitability of a business strategy. Go back to the Webvan story. According to a February 2001 report in the Wall Street Journal, a venture capitalist told Webvan’s founder, retailing entrepreneur Louis Borders, “Louis, I think this is going to be a billion-dollar company.” Mr. Borders replied, “Naw, it’s going to be $10 billion. Or zero.”
In a sense, the colloquy is like the urn example, for it underscores the limited value of treating all decisions with a common metric of “expected value” presumably equally relevant to any “rational player.” Mr. Borders and the venture capitalist each had different risk thresholds that made them willing to take the bet on the unknown. A successful and wealthy entrepreneur, Mr. Borders implicitly understood that he was making a one-off gamble on an unknowable risk with potentially extreme outcomes. The venture capitalist’s willingness to wager was based on his ownership of a portfolio of risky businesses, wherein a few winners more than justify the majority of startups that bomb. Unfortunately, the thousands of individual investors caught up in the hype surrounding Webvan’s public offering clearly had far lower risk thresholds than either Mr. Borders or the venture capitalist, but most failed to appreciate the unknowable nature of the risks in Webvan. A more explicit recognition of their individual risk thresholds could have saved many naive investors from squandering their nest eggs.
Establishing a risk threshold helps to define downside risks. The financial-services industry, for instance, has begun to embrace a rigorous analysis of downside risk rather than a simple examination of expected value. Using historical data, regulators can assess the amount of money that a bank stands to lose with a probability of some threshold percentage over a specified period of time. The Basel Committee on Banking Supervision recently set forth detailed guide-lines for the calculation of a risk-threshold limit called “value-at-risk,” to determine a bank’s required minimum capital holdings (www.bis.org). These capital adequacy rules are proposed for the Basel II Accord.