The paradox of The Bottle Imp

"The Bottle Imp" (1893) is a great horror story by Robert Louis Stevenson. (You might want to read it before continuing here, but it's not much of a spoiler if you do.)

Futility Closet: An Idler's Miscellany of Compendious Amusements, by Greg Ross, describes the paradox of the bottle:

In Robert Louis Stevenson's story “The Bottle Imp,“ the titular imp will grant its owner (almost) any wish, but if the owner dies with the bottle then he burns in hell. He may sell the bottle, but he must charge less than he paid for it, and the new buyer must understand these conditions.

Now, no one would buy such a bottle for one cent, as he could not then sell it again. (The imp can’t make you immortal, or support prices smaller than one cent, or alter the conditions.) And if 1 cent us too low a price, then so is 2 cents, for the same reason. And so on, apparently forever. It would be irrational to buy the bottle for any price.

But intuitively most people would consider $1,000 a reasonable price to pay for the use of a wish-granting genie. Who's right?

Side note: I learned about "The Bottle Imp" after someone pointed out that a short story, "Miniature Bottle", which I wrote for the anthology Significant Objects: 100 Extraordinary Stories About Ordinary Things, reminded them of Stevenson's story. Read the rest

The absent-minded driver’s paradox

Here's a good paradox from our friends at Futility Closet:

A driver is sitting in a pub planning his trip home. In order to get there he must take the highway and get off at the second exit. Unfortunately, the two exits look the same. If he mistakenly takes the first exit he’ll have to drive on a very hazardous road, and if he misses both exits then he’ll reach the end of the highway and have to spend the night at a hotel. Assign the payoff values shown above: 4 for getting home, 1 for reaching the hotel, and 0 for taking the first exit.

The man knows that he’s very absent-minded — when he reaches an intersection, he can’t tell whether it’s the first or the second intersection, and he can’t remember how many exits he’s passed. So he decides to make a plan now, in the pub, and follow it on the way home. This amounts to choosing between two policies: Exit when you reach an intersection, or continue. The exiting policy will lead him to the hazardous road, with a payoff of 0, and continuing will lead him to the hotel, with a payoff of 1, so he chooses the second policy.

This seems optimal. But then, on the road, he finds himself approaching an intersection and reflects: This is either the first or the second intersection, each with probability 1/2. If he were to exit now, the expected payoff would be

That’s twice the payoff of going straight!

Read the rest

20 amusing paradoxes and dilemmas to ponder

Cracked has compiled a collection of 20 fun-to-ponder paradoxes, including the famous Paradox of the Court. From Wikipedia:

The Paradox of the Court, also known as the counterdilemma of Euathlus, is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his infrastructure after he wins his first court case. After instruction, Euathlus decided to not enter the profession of law, and Protagoras decided to sue Euathlus for the amount owed.

Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case.

Euathlus, however, claimed that if he won, then by the court's decision he would not have to pay Protagoras. If, on the other hand, Protagoras won, then Euathlus would still not have won a case and would therefore not be obliged to pay.

The question is: which of the two men is in the right?, also known as the counterdilemma of Euathlus, is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his infrastructure after he wins his first court case. After instruction, Euathlus decided to not enter the profession of law, and Protagoras decided to sue Euathlus for the amount owed.

Read the rest

Unanimous decisions are less reliable

From the YouTube description of this TedEd video:

Imagine a police lineup where ten witnesses are asked to identify a bank robber they glimpsed fleeing the scene. If six of them pick the same person, there’s a good chance that’s the culprit. And if all ten do, you might think the case is rock solid. But sometimes, the closer you start to get to total agreement, the less reliable the result becomes. Derek Abbott explains the paradox of unanimity.

Read the rest

The Missing Dollar puzzle from Martin Gardner's Aha! Gotcha book series

Martin Gardner wrote Aha! Gotcha: Paradoxes to Puzzle and Delight and Aha! Insight in the early 1980s and I love them both. Both books have excellent brain teasers with charming illustrations. They are both out of print, which is criminal, but Amazon has used copies for $(removed) (plus $(removed) s&h).

Read the rest

Four sets of identical twins pull an epic NYC subway car time-machine prank

Improv Everywhere (previously) keeps on bringing the hits, but seriously, this one takes the cake. Read the rest

Saturday morning mind-benders: "Newcomb's Problem" and "Parfit's Hitchhiker" dilemma

In this video Julie Galef, host of the Rationally Speaking podcast (about philosophy, rationality, science) presents one of my favorite paradoxes - Newcomb's Problem (and the related and "Parfit's Hitchhiker" dilemma).

Before Carla and I started the bOING bOING zine, I published another zine in the mid-1980s called Toilet Devil (Koko the talking ape calls people and her pet kitties "dirty toilet devils" when she is mad at them). In the first issue I drew a comic about "Newcomb's Problem." I might scan it one day and post it.

In 2006, I posted about Newcomb's Problem:

Franz Kiekeben does a nice job of describing Newcomb's Paradox, which I've enjoyed contemplating, on and off, for many years.

A highly superior being from another part of the galaxy presents you with two boxes, one open and one closed. In the open box there is a thousand-dollar bill. In the closed box there is either one million dollars or there is nothing. You are to choose between taking both boxes or taking the closed box only. But there's a catch.

The being claims that he is able to predict what any human being will decide to do. If he predicted you would take only the closed box, then he placed a million dollars in it. But if he predicted you would take both boxes, he left the closed box empty. Furthermore, he has run this experiment with 999 people before, and has been right every time.

What do you do?

On the one hand, the evidence is fairly obvious that if you choose to take only the closed box you will get one million dollars, whereas if you take both boxes you get only a measly thousand.

Read the rest

Blackmail paradoxes

From Futility Closet:

We covered one paradox regarding blackmail in 2010: If it’s legal for me to reveal your secret, and it’s legal for me to ask you for money, why is it illegal for me to demand payment to keep your secret? In the words of Northwestern University law professor James Lindgren, “Why do two rights make a wrong?”

Here’s a second paradox: If you had initiated the same transaction — if you had offered to pay me for my silence, and I’d agreed — then we’d have the same outcome, but this time it’s legal. “It is considered paradoxical that the sale of secrecy is legal if it takes the form of a bribe, yet is illegal where the sale of secrecy takes the form of blackmail,” writes Loyola University economist Walter Block. “Why should the legality of a sale of secrecy depend entirely upon who initiates the transaction? Why is bribery legal but blackmail not?”

Read the rest