Risks and Rewards

Oct. 2, 2005 — -- Money, money, money. Everyone wants more, but, alas, the second million won't do for you what the first one did, nor will you be as willing to take the same risks to get it.

It was Daniel Bernoulli, the 18th century Swiss mathematician, who wrote that people's enjoyment of any increase in wealth (or regret at any decrease) is "inversely proportionate to the quantity of goods previously possessed." The more dollars you have, the less you appreciate getting one more and the less you fear losing one.

What's important is the "utility" to you of the dollars you receive, and their utility drops off, often logarithmically, as you receive more of them. Gaining or losing $1 million means much more to most people than it does to Warren Buffett or Bill Gates. People consider not the dollar amount at stake in any investment or game, but the utility of the dollar amount for them.

Note that the declining average utility of money provides part of the rationale for progressive taxation and higher tax rates on greater wealth.

A less weighty illustration than progressive taxation is provided by a recent British study of the show "Who Wants To Be a Millionaire." It confirms that contestants behave as considerations of utility would suggest. Once they've reached a high rung on the winnings ladder, they more often quit while ahead rather than risk falling to a much lower level.

On the show and in general, people tend to be risk averse and usually choose the sure thing.

Likewise, despite the equal expected values of the following alternatives, almost everyone offered the choice between i.) $100 million or ii.) a 1 percent chance at $10 billion will choose the sure $100 million. (Note that since 1 percent of $10 billion is $100 million, their expected values are equal.)

The notion of utility also resolves the famous St. Petersburg paradox. The paradox usually takes the form of a game requiring that you flip a coin repeatedly until a tail first appears. If a tail appears on the first flip, you win $2. If the first tail appears on the second flip, you win $4. If the first tail appears on the third flip, you win $8, and, in general, if the first tail appears on the Nth flip, you win 2^N dollars. How much would you be willing to pay to play this game?

In a sense you should be willing to pay any amount to play this game. To see why, recall that the probability of a sequence of independent events such as coin flips is obtained by multiplying the probabilities of each of the events. Thus the probability of getting the first tail, T, on the first flip is 1/2; of getting a head and then the first tail on the second flip, HT, is (1/2)^2 or 1/4; of getting the first tail on the third flip, HHT, is (1/2)^3 or 1/8, and so on.

Multiplying the probabilities for the various outcomes of the St. Petersburg game by the size of these outcomes and adding these products gives us the expected value of the game: ($2 x 1/2) + ($4 x 1/4) + ($8 x 1/8) + ($16 x 1/16) + … (2^N x [1/2]^N) + … Each of these products equals $1, there are infinitely many of them, and so their sum is infinite, and this is why it can be argued that you should be willing to pay any price to play this game. No matter how much you bet each time you play, you'll still come out way ahead on average.