#### The Imp in the Bottle

A quite different illustration of our short-sightedness comes courtesy of Robert Louis Stevenson's "The Imp in the Bottle." The story tells of a genie in a bottle able and willing to satisfy your every romantic whim and financial desire. You're offered the opportunity to buy this bottle and its amazing denizen at a price of your choice. There is a serious limitation, however.

When you've finished with the bottle, you have to sell it to someone else at a price strictly less than what you paid for it. If you don't sell it to someone for a lower price, you will lose everything and will suffer excruciating and unrelenting torment. What would you pay for such a bottle?

Certainly you wouldn't pay 1 cent because then you wouldn't be able to sell it for a lower price. You wouldn't pay 2 cents for it either since no one would buy it from you for 1 cent since everyone knows that it must be sold for a price less than the price at which it is bought. The same reasoning shows that you wouldn't pay 3 cents for it since the person to whom you would have to sell it for 2 cents would object to buying it at that price since he wouldn't be able to sell it for 1 cent. Likewise for prices of 4 cents, 5 cents, 6 cents, and so on.

We can use mathematical induction to formalize this argument, which proves conclusively that you wouldn't buy the genie in the bottle for any amount of money. Yet you would almost certainly buy it for 1,000 dollars. I know I would. At what point does the argument against buying the bottle cease to be compelling?

The question is more than academic since in countless situations people prepare exclusively for near-term outcomes and don't look very far ahead. They myopically discount the future at an absurdly steep rate.

On the other hand, in certain quite different circumstances this tendency can also be seen as a rationally exuberant "seize the day" mentality that has much to recommend it. Live large before the music stops ... but be a little skeptical.

John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.

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