Here's a three-question quiz to determine how rational you are. This will work best if you stop and answer each question before going on to the next.
Imagine that each of your three fabulously wealthy cousins offers you a choice of two Christmas gifts. In each case, choose the one you'd prefer.
1. Cousin Snip offers you a choice of:
A. $1 million in cash.
B. A lottery ticket. The ticket gives you a 10-percent chance of winning $5 million, an 89-percent chance of winning $1 million, and a 1-percent chance of winning nothing at all.
2. Cousin Snap offers you a choice of:
A. A lottery ticket that gives you an 11-percent chance of winning $1 million.
B. A lottery ticket that gives you a 10-percent chance of winning $5 million.
3. Cousin Snurr offers you a choice of:
A. $1 million in cash.
B. A lottery ticket that gives you a 10/11 chance of winning $5 million.
Now that you've made your choices, you can read on to discover whether you're a rational creature. "Rational" does not mean "risk-neutral." A risk-neutral person is one who is indifferent when given a choice between 50 cents and a 50-50 chance of $1. A risk-neutral person would choose B in all three cases. In Snip's offer, 10 percent of $5 million ($500,000) plus 89 percent of $1 million ($890,000) equals $1.39 million, which trumps $1 million. In Snap's offer, 10 percent of $5 million ($500,000) trumps 11 percent of $1 million ($110,000). In Snurr's offer, 10/11 of $5 million is $4.55 million, which trumps $1 million.
But it's equally rational to avoid risk or to seek it out. The insurance and gambling industries are based on these proclivities. Even so, rationality does imply some logical consistency in your choices about risk. It would be embarrassing if a lot of Slate readers failed this test, so I'm going to make it easy by adopting a very broad definition of rationality. As long as you satisfy two simple criteria, I'm willing to call you rational.
Here's my first criterion: If you prefer A to B, then you should prefer a chance of winning A to an (equally large) chance of winning B. And here's the test to see whether you've met that criterion: Your answers to Questions 2 and 3 should be the same. That's because Snap's choice A is an 11-percent shot at a million bucks, and Snurr's choice A is a million bucks. Therefore Snap's A is an 11-percent shot at Snurr's A. Meanwhile, Snap's choice B amounts to an 11-percent shot at Snurr's choice B. (Do the math: A 10-percent chance of winning $5 million is the same as an 11-percent chance of winning a 10/11 chance of winning $5 million. 0.11 x 10/11 = 0.10) So, if you prefer Snurr's A to Snurr's B, you should prefer Snap's A to Snap's B.
Here's my second criterion of rationality: If you're choosing between two lotteries with identical chances to win, then your preference should be unaffected if I throw in a consolation prize that you get if you lose in either case. You pass that test if your answers to Questions 1 and 3 are the same. This is why:
Snip's choice A is $1 million. Another way to say $1 million--weird, but bear with me--is "an 11-percent chance to win $1 million, with a consolation prize of $1 million for losing." Snurr's choice A is also $1 million. So Snip's A is an 11-percent shot at Snurr's A with a $1-million consolation prize.
Snip's choice B is a 10-percent chance of winning $5 million plus a 1-percent chance of winning nothing plus an 89-percent chance of winning $1 million. The first two items, taken together, amount to an 11-percent chance of a 10/11 chance of winning $5 million. The third item means you get $1 million if that 11-percent chance doesn't come through. Snurr's choice B is a 10/11 chance to win $5 million. Snip's choice B is therefore an 11-percent shot at Snurr's choice B with a $1-million consolation prize.
So if you prefer Snip's A to Snip's B, you should prefer Snurr's A to Snurr's B--if you're rational.
To sum up, if you are even minimally rational, your answers to Questions 1, 2, and 3 should all be the same. But they probably aren't. According to survey data collected by Nobel laureate Maurice Allais--and duplicated by several subsequent researchers--most people answer A to Question 1 and B to Question 2. There is no way to reconcile that combination of answers with the most rudimentary theory of rationality, no matter how you answer Question 3. In other words, people prefer the cash over the lottery ticket--to an extent that rational risk aversion can't explain.
Economists have variously viewed the "Allais Paradox" as a warning, a trifle, an opportunity, and a challenge. If you're looking to explain all human behavior on the basis of a few simple axioms, it's a warning. If you don't believe that casual answers to abstract survey questions constitute an important part of human behavior, it's a trifle. If the survey responses mean that people are less rational than they ought to be, it's an opportunity for economists to teach better decision-making skills. If you conclude that there's a critical element missing from our theory of rationality, it's a challenge to identify that element.
O ne missing element is regret. When you choose a lottery instead of a sure thing (as in Question 1), you risk not just losing the lottery but also feeling regretful about your recklessness. But when you choose between two lotteries (as in Question 2), you can always reconcile yourself to a loss by thinking, "Well, I'd probably have lost no matter what I chose." Maybe that's why most people go for the sure thing in Question 1 but are willing to go for the slightly riskier of the two bets in Question 2. (Click for an experiment that could test this hypothesis.)
Here's another thought-experiment that indicates the importance of avoiding regret. Suppose you belong to a company of 10 soldiers, of whom one must be chosen for the distasteful task of executing a prisoner. Which of the following do you prefer? A) One soldier is selected at random to shoot the prisoner? Or B) all 10 soldiers fire at once, without knowing which one of the 10 has been issued live ammunition? Either way, you'd have a 10-percent chance of being the executioner, so simple theories of rationality suggest that you should be indifferent when asked to choose between the two options. Yet most people prefer B), because in case B) you never know whether you've been unlucky.
The analogy between the soldiers and the Allais survey respondents is imperfect; the soldiers who choose method B) are trying to avoid regret over bad luck, while the survey respondents are, perhaps, trying to avoid regret over bad decisions. But in either case, ignoring the human impulse toward regret-avoidance might give a social scientist cause for regret.
Note to readers: A week after this article was posted, Landsburg clarified the "simple test." Read his addendum in "E-Mail to the Editors."