Bottle imp paradox

Imagine you buy a magic lamp. After buying it you summon its genie within, and once summoned the genie grants you one and only one wish. And it's a nice genie, too, not a jinn that interprets your wish overly literally and transforms your pleasant wish into something terrible. No, this is a beneficent genie who will give you one thing you want, and you'll be better off for it. But there's a catch: once the genie has granted you your wish, you must sell the lamp, and you must sell the lamp for twice as much money as you paid for it. If you fail to sell the lamp in this way—say, within a week—then not only will your wish be reversed (whatever that means), but the most terrible of all fates will befall you—for example, you'll be tortured for the rest of your life. Also, the buyer to whom you sell the lamp becomes subject to the same stipulation, that the buyer must find another buyer who buys the lamp for double the previous sum—that's quadruple what you paid for it—or else the first buyer suffers that terrible fate. And so it goes to infinity, with each buyer required to find an ever richer buyer.

This is known as the bottle imp paradox. What makes it a paradox is that the lamp is theoretically worthless. That's because no matter how little you pay for the lamp—say, $1—after a few dozen transactions the lamp will cost more than the richest man in the world has. Thus, the last buyer—call him Mr. Z—will suffer a terrible fate. But Mr. Z would know his fate before buying the lamp (having done the math and checked it against the latest Forbes list of the world's wealthiest people to know he'd be the last possible buyer), and so Mr. Z wouldn't buy the lamp. That instead means the lamp's previous owner—call him Mr. Y—would be stuck with the lamp and the terrible fate. But Mr. Y too would have foreseen this, for Mr. Y would've gone through the same logic as the hypothetical Mr. Z did to know there would be no Mr. Z to buy the lamp, so Mr. Y too wouldn't have bought the lamp. But by the same logic, the previous owner, Mr. X, wouldn't have bought the lamp either, knowing there would be no Mr. Y. Nor would Mr. W, Mr. V, or anyone else have bought the lamp—not even Mr. A, who could have bought the lamp for a measly dollar.

And I'll point out, for you creative logic-lawyer-types, that there are no loopholes around the lamp's rules. For example, Mr. A can't wish himself a quadrillion dollars and then use some of that money to buy the lamp from Mr. Z, thus allowing for a possible infinite cycle of buyers. That's because the lamp can't be owned by the same person twice. Or whatever—any loophole can be closed with an explicit rule we imagine. The point isn't beating the game but dealing with the paradox.

That said, the bottle imp paradox demonstrates a rational counter to the greater fool theory. While the paradox says that the lamp is worthless because all potential buyers will have accurately valuated the lamp and will thus refuse to buy it, the greater fool theory says you should go ahead and buy the lamp for a $1 because you're sure to find a fool who'll buy it for $2. But that fool isn't quite so foolish because there'll be another fool who'll buy the lamp for $4, and so on. The greater fool theory works out well—until you get to the fool who can't find another fool, at which point the bubble pops and someone is in a lot of pain for the rest of their life.

For a long time, I've thought the stock of any company that forever refuses to pay out a dividend should be worthless. I concluded this using the same logic as used in the bottle imp paradox: that if a stock never pays anything back to the shareholders, then its only value lies in shareholders finding other fools who're willing to buy the stock in hope of finding yet other fools. I'm no fool—or so I like to think—so I figured companies shouldn't refuse to pay dividends on principle. Rather, they should fail to pay dividends for a good reason, such as their being broke.

Specifically, I thought Warren Buffett was foolish for being anti-dividend. How can his company, Berkshire Hathaway, be of any value if it doesn't eventually pay back investors? Without an eventual payback, its value must be entirely the result of a bubble of greater fools. But it turns out I was wrong: Berkshire Hathaway is indeed paying back investors. Recently, Buffett announced that Berkshire is running out of places to invest its mountain of cash, and so the company is returning some of that money to investors, who presumably will have better things to do with it. But Berkshire isn't paying its shareholders with cash dividends. Rather, it's paying back exclusively with stock buybacks. The buybacks work by reducing the number of outstanding shares, which raises the remaining shares' value. The reason for doing a buyback rather than a cash dividend has to do with incurring less tax liability, but in effect a buyback is the same a dividend.

Here's an article that explains these silent dividends.