The bottle imp paradox

I can calculate the movement of the stars, but not the madness of men–Isaac Newton, after losing £20,000 in the South Sea Bubble in 1720

One of my first stock purchases was Lucent Technologies in the late 90s. It was an unusual buy. I only did casual research. And what little research I did was worrisome. But I was taken into the stock market craze. I was sure the stock would rise. I thought someone else would definitely buy it for more. In short, I was relying on the greater fool theory.

My decision did not turn out so well. Lucent unraveled just around the time of my purchase, and quickly few would touch the stock. I proved to be the greatest fool and was stuck with the bill. I was as confused as Isaac Newton after his disastrous investment.

To this day, few if any have been able to predict the madness of the market. The problem is knowing when the tide will shift and when the bubble will burst. This is a fiendishly complex problem, as the following mathematical puzzle will vividly illuminate.

The bottle imp

image credit: lel4nd

The Bottle Imp is an 1891 story by Robert Lewis Stevenson. The bottle imp paradox is adapted from the story’s narrative, and the setup is something like this:

One day you are greeted by an elderly gentleman. He is nice and very wealthy. He takes a liking to you, and he wants to help you. He offers to sell you a magic bottle. The bottle has a genie that will grant you any wish. If you buy the bottle, you too will find success and wealth.

But there is just one catch. To assure your success, you must also sell the bottle at a loss. That is, you must sell the bottle for a price lower than what you will pay him. If you do not do this, then you will be condemned to eternal damnation in Hell. What do you do?

Specifically, you are thinking about the following questions:

–Do you buy the bottle?

–What price do you pay?

–What is the lowest price one should buy the bottle for?

It turns out these questions are not so easy to answer.

The bottle imp paradox

You first consider the price. On the one hand, you do not want to pay too high a price. You worry about shelling out cash which you cannot recover until you sell. On the other hand, you do not want to pay too low a price, or else you risk not finding another buyer.

What price is sensible? Let’s start from the beginning and work our way up. Suppose you offer to pay only one cent. This turns out to be a very bad price. The reason is there is no lower denomination and hence it will be impossible to find another buyer. You will be stuck with the bottle. Buying the bottle at one cent is equivalent to buying your own eternal damnation. So clearly this is a bad price.

But what about two cents? At first, this seems okay. If you buy at two cents, then you could theoretically sell for one cent. The problem is that you will be hard pressed to find a buyer. The reason is the person you buys from you is buying at one cent. And as argued just above, it is stupid to buy the bottle at one cent. Therefore, no one would want to buy the bottle at two cents.

Indeed, this logic can be extended. No one should want to buy at three cents, or four cents, and so on. Inductively one can reason there is no “safe” price to buy the bottle. Thus, the bottle should never be bought because it will be hard or impossible to find a buyer.

But in practice, this conclusion feels wrong. You would expect a buyer at a high enough price. If you buy the bottle for $100, for example, you can likely find someone who will want to buy at $99.99. And they will fell safe, reasoning that they can find someone willing to pay $99.98, and so on.

The bottle imp paradox is that inductive reasoning and practical reasoning come to contradictory conclusions. Is the bottle never to be bought, or is there some high enough price range?

How can we resolve this paradox? I’ll present two reasonable resolutions.

Resolution 1: the sinner saves the day

The paradox could be readily resolved with the existence of an atheist buyer. There could be someone who buys the bottle without expecting to sell it. This may be someone who does not believe in a supernatural afterlife with damnation.

Or alternately, it could be a buyer who is a sinner that cannot be saved. Since his life is already destined for damnation, having the bottle does not add an additional penalty.

The latter situation is more or less the resolution offered in Stevenson’s story The Bottle Imp.

Resolution 2: foreign currencies

Another trick is that are currencies with money worth less than one penny. In Stevenson’s story, the protagonist travels to Tahiti in search of a coin worth one-fifth of an American penny.

Introducing foreign currencies also allows for the bottle to be sold indefinitely. The reason is that currencies fluctuate in values on the foreign exchange market. One could buy the bottle for a low price in one currency, and sell it when the currency appreciates. The bottle could be sold back and forth in accordance with the swings of the market.

Of course, now we are back to the situation of betting on the market and the madness of men, which leaves us in a situation that even Isaac Newton could not solve.