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.
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11 Responses to “The bottle imp paradox”
Buy it w/ gold by weight. Weight is a real number, and can be broken up in 1/2, then 1/3, 1/4, etc, down I guess to the atomic level, where you could trade it out for more common elements or materials.
By drobviousso on Jan 5, 2010
What if you take the bottle for free and give someone a penny to take it from you?
(Buy for 0, Sell for -1)
By J on Jan 5, 2010
This reminds me of the pop quiz paradox:
A teacher says that, at some point during the week, he will give a surprise quiz. One student reasons that this is impossible.
If the quiz is given on a Friday, then by Thursday we can reason that, since we haven’t had a quiz yet, it must be on Friday, and they won’t be surprised.
Since it can’t be on Friday, if they haven’t had a quiz by Wednesday, they will know it must be on Thursday, and won’t be surprised, so it can’t be on Thursday.
Likewise for all the rest of the days of the week, ergo there can be no surprise quiz.
Other possible solutions:
1. Caveat Emptor:
Don’t reveal the stipulation to whomever you are selling it do. Lacking this knowledge, they have no reason not to buy the bottle at a low price.
2. Genie = Patsy:
Buy the bottle at two cents, then use your last wish to wish that the Genie would buy the bottle from you at 1 cent.
@drobviousso:
At some point you would get to a component of matter that is not yet practically possible to split. While the issue of practicality seems like a temporary barrier, there are two possibilities:
1. Probing and splitting smaller and smaller chunks of matter requries increasing amounts of energy. At some point it would require more energy than is available in the entire universe.
2. There may be levels of matter that cannot be split any further.
@J:
There is only a finite amount of wealth in the world, which would seem to put a limit on this tactic.
I don’t see that there is any feasible way to keep the chain going on forever. Even if we were using electronic transactions, we are limited to the numbers a computer can process (be they small, or large). While we would think that, so long as we keep the change in price smaller than the increase in computing power, we would be able to maintain this indefinitely, there has to be an upper limit: a computer made out of the entire universe.
By Scott on Jan 5, 2010
This also reminds me of War Games. To out-wit the computer they forced it to think ahead far enough to realize (via inductive reasoning) that the only way to “win” was not to play.
By Scott on Jan 5, 2010
This is only a tangent, but there is a clever trick taking game for three based on this story.
http://www.boardgamegeek.com/boardgame/619/flaschenteufel
Basically the lower half of the deck is trump, with the highest trump taking the trick and the bottle, but reducing the range of trump to all cards lower than the card that took the bottle. Whoever has the bottle at the end of the hand loses points instead of gaining them.
What I find interesting about it is that novice players tend to try to get rid of the trump cards as quickly as possible and/or force the value of trump to be as low as possible as quickly as possible (so they don’t have any trump left in their hand), but this means the player with the highest cards gets most of the points.
With more experience, players learn how to manipulate the trump so they take some tricks with high trump but then lose the bottle before the end of the hand.
By Malachi on Jan 5, 2010
That reminds me of another paradox about a pile of sand:
No one would claim that a single grain of sand is a pile. Adding another grain of sand does not a pile make, either. But a pile of sand certianly exists so which is the grain of sand that does it?
Same with the price of the bottle. When you start at one penny and work your way up, nothing seems reasonable. But if you start with $50 (a pile of pennies) then working your way down seems reasonable.
By Eyal on Jan 6, 2010
One more thing, @Scott: The pop quiz paradox is easy to solve. The confusion is about surprise and predictability. When you flip a coin, you can’t predict how it will land but neither heads or tails would be a surprise. (If it turned into a banana, that would be a surprise.)
Same for the pop quiz. You can’t predict when it’ll happen but when it does, it won’t be a surprise.
By Eyal on Jan 6, 2010
Presh, I like this story and was not familiar with it.
The conditions of the game rule out J’s ingenious solution because you cannot give the bottle away, which I assume also covers paying someone to take the bottle.
I also like Scott’s solution, which could be generalized to having the genie find a person damned already but who wants to end his/her own life in wealth.
There is another obvious solution: make the payments in real and not nominal dollars. A penny today is worth more than a penny tomorrow, so the bottle can be resold.
What is interesting, to me, is that this problem may prove to be an antidote to network marketers who are constantly wrong about the span of their network of friends and believe that it is possible to sell an opportunity to sell an opportunity to sell x, when the
opportunity to sell x is a lousy deal. A striking example of the greater fool theory.
By michael webster on Jan 6, 2010
The bottle imp problem is a version of what philosophers call a sorites inference. Here is a nice review of the problems a sorites form can generate:
http://artsci.wustl.edu/~philos/people/sorensen/PAPERS/Precis%20of%20Vagueness%20and%20C.pdf
By michael webster on Jan 6, 2010
The Greater Fool Theory relies on the fact that you don’t know everyone in the world. If this was a game (not the card game) but a game that plays out just like the story, the difference is it’s a hand full of people that could buy the bottle (maybe 20) and you knew everyone, would people still play or will they think more rationally?
Of course in the story you don’t even have to wait for the madness of others; simply wish that a currency was lower (or the currency you paid for was higher). Though, I bet that would create madness.
By Gamer on Jan 6, 2010
There’s one key difference between the greater fool theory and the bottle imp paradox.
The prices in the bottle imp paradox goes down and have a clear lower limit, zero. The prices in the greater fool theory goes up and have no theoretical upper limit.
Actually, prices CAN keep going up indefinitely and it is called inflation. If stock prices goes up faster or slower than average inflation is another story.
By Mauricio on Jan 27, 2010