Fortune’s Formula by William Poundstone

Author:William Poundstone
Language: eng
Format: mobi, epub
Publisher: Hill and Wang
Published: 2010-06-01T05:00:00+00:00

Logarithmic Utility

This means that the terms in the infinite series need to be adjusted downward to account for the diminishing returns of large winnings. Though the series is still infinite, it becomes one of those well-mannered infinite series that converges. You can add up ½ + ¼ + 1/8 + 1/16…and never quite reach 1, no matter that the series is endless. When Bernoulli’s series of expectations is adjusted this way, it too converges to a finite and modest sum.

Economic thinkers were infatuated with logarithmic utility for the next couple of centuries. British economist William Stanley Jevons (1835–1882) maintained that logarithmic utility applied to consumer goods as well as wealth: “As the quantity of any commodity, for instance, plain food, which a man has to consume, increases, so the utility or benefit derived from the last portion used decreases in degree.” You might say this explains how all-you-can-eat restaurants stay in business. In 1954 Leonard Savage called the logarithmic curve a “prototype for Everyman’s utility function”—a reasonable approximation to how most people value money, most of the time, over the range of dollar values they normally encounter.

Not everyone agreed. By Savage’s time, logarithmic utility had taken on a fusty, old-fashioned cast. One blow to the concept was the realization that logarithmic utility is not an entirely satisfying resolution to the St. Petersburg paradox. In the 1930s, Vienna mathematician Karl Menger pointed out that it is easy to come back with revised versions of the St. Petersburg wager where Bernoulli’s solution fails. All you have to do is to sweeten the payoffs. Instead of offering 1, 2, 4, 8 ducats on successive throws, offer something like 2, 4, 16, 256 ducats…You can arrange to have the prizes escalate so fast that the expected utility is again infinite.

Menger’s most devilish counterexample was to have the wager’s prizes not in dollars or ducats but utiles. A utile is a hypothetical unit of utility. You would win 1, 2, 4, 8…utiles, depending on how many tosses it takes. The value of the wager, now in expected utility, is infinite. A rational person would supposedly give up anything he’s got to play this game—which is still absurd because he’s likely to win the utile equivalent of chump change.

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