Against the Gods: The Remarkable Story of Risk, by Peter L. Bernstein Review date: November 23, 1996 Reviewed by: Kevin Drum Overall grade: A-

Now here's a subject you don't hear about very often: risk in human life, how we measure it, how we control it, and how we react to it. The first 100 pages of Against the Gods are devoted to the first faint stirrings of the mathematics of probability (take note that although not much math is required to understand this book, it definitely helps if you're not intimidated by it), after which the pace picks up and it is mostly devoted to human reaction to risk, rather than the mathematical measurement of it.

An interesting paradox in risk assessment comes courtesy of Daniel Bernoulli, who posed the following problem in 1738. Known as the Petersburg Paradox, it proposes a game between Peter and Paul in which Peter tosses a coin continuously until it comes up heads. If it comes up heads on the first toss, Paul wins one ducat and the game is over. If it takes two tosses, Paul wins two ducats. If it takes three tosses, he wins four ducats, and so forth. The question is, how much should Paul be willing to pay to play this game with Peter?

The paradox is this: the standard calculation of expected value indicates that the value of playing the game is infinite. The calculation looks like this:

(1/2 * 1) + (1/4 * 2) + (1/8 * 4) …. = 1/2 + 1/2 + 1/2…. = infinity

Nevertheless, no ordinary person would be willing to pay more than a few ducats to play this game. Bernoulli's explanation is that our assessment of the value of the winnings decreases as the probability of the winnings gets smaller. Mathematically, the probability of the game going to, say, twenty tosses, is 1/1,048,576, and the payoff is 524,288 ducats. However, although the expected value is still 1/2, the probability is so small that we effectively evaluate it as zero. In our minds, the expected value of the game is not the infinite series above, but a series that stops after 10 or 20 terms.

(As an experiment, I wrote a program to play this game a few million times and tell me the average payout. It came out to around 10 ducats. I imagine you have to play an infinite number of games to get the infinite payoff, which certainly justifies our instinctive reluctance to value the game very highly if we only get to play once.)

This eventually leads to a discussion of Prospect Theory, which examines the way people rank risks. An interesting result of Prospect Theory is that people aren't generally risk averse, they are loss averse. For example, give a group of people the choice of $1000 or a 50% chance of winning $2000, and most will choose the sure thing. However, give them a choice of losing $1000 immediately vs. a 50% chance of losing $2000, and most will take the risk. Interesting, no? Apparently, this basic pattern of being unwilling to accept a sure loss has been verified over and over in a variety of circumstances (losing money, losing lives, losing employment, etc.) and certainly says something basic about human nature.

Prospect Theory is still relatively young, but the results so far (and Bernstein's summary of them) are both interesting and thought provoking. Against the Gods is highly recommended if you have an interest in this kind of thing.

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