PARIS OR THE BAHAMAS? (The Inability to Visualize Probabilistic States)

PARIS OR THE BAHAMAS?

You have two options for your next brief vacation in March. The first is to fly to Paris; the second is to go to the Caribbean. You expressed indifference between the two options; your spouse will tip the decision one way or the other. Two distinct and separate images come to you when you think of the possibilities. In the first one, you see yourself standing at the Musée d’Orsay in front of some Pissaro painting depicting a cloudy sky—the gray Parisian wintry sky. You are carrying an umbrella under your arm. In the second image, you are lying on a towel with a stack of books by your favorite authors next to you (Tom Clancy and Ammianus Marcellinus), and an obsequious waiter serving you a banana daiquiri. You know that the two states are mutually exclusive (you can only be in one place at one time), but exhaustive (there is a 100% probability that you will be in one of them). They are equiprobable, with, in your opinion, 50% probability assigned to each.

You derive great pleasure thinking about your vacation; it motivates you and makes your daily commute more bearable. A probability-weighted description assigns 50% to one destination and 50% to the other—what is mathematically a linear combination of the states. That description is not a literal holiday with your feet in Caribbean waters and your head in Parisian rain. The images are concrete; the uncertainty between them is abstract. Now try to picture an 85%/15% combination. The difficulty illustrates the gap between imagining an outcome and representing a distribution.

Consider a bet you make with a colleague for the amount of $1,000, which, in your opinion, is exactly fair. Tomorrow night you will have zero or $2,000 in your pocket, each with a 50% probability. In purely mathematical terms, the fair value of a bet is the linear combination of the states, here called the mathematical expectation, i.e., the probabilities of each payoff multiplied by the dollar values at stake (50% multiplied by 0 and 50% multiplied by $2,000 =$1,000). Can you imagine (that is visualize, not compute mathematically) the value being $1,000? It is easier to picture either 0 or $2,000 than a value that will not itself be paid. Left to our own devices, we are likely to bet in an irrational way, as one of the states would dominate the picture—the fear of ending with nothing or the excitement of an extra $1,000. Expected utility and the ability to bear a loss also matter when deciding whether to accept the bet.