Backwards Induction, and Contradictory Assumptions
The Bottle Imp is a story by Robert Louis Stephenson. The titular bottle contains a magical imp that can grant wishes, but there is a catch (of course there is a catch): the owner of the bottle is doomed to go to Hell when they die. They cannot give away the bottle, they can only sell it for less than they paid for it. This creates a desperate situation where owners try to rid themselves of the bottle, but face an obvious end point as the bottle must be sold, it cannot be given away.
The use of Backwards Induction (“look ahead, reason back”) shows that no one should purchase the bottle in the first place. Why? First, assume for simplicity that the lowest denomination of currency is $1. Whoever buys the bottle for $1 would then be unable to sell it. They know this before they purchase it, so no one would purchase it for $1.
Now step back and think if anyone would purchase it for $2. They know that they will be unable to sell it, so they would not purchase it for $2.
Continue this way indefinitely: no one would purchase the bottle for $X, because they know that no one will buy it for $(X - 1). The end result is that no one would ever purchase the bottle, seeing where the series ends.
And yet, there is something tempting about purchasing it for, say, $1,000,000. It seems so very likely that you could find someone who would buy it for $999,999 or less, so maybe it’s worth taking a chance. The Bottle Imp seems to have a Push-Your-Luck element to it, one that is precluded by Backwards Induction.
Indeed, this story inspired Günter Cornett to design The Bottle Imp, a trick-taking card game currently published by Stronghold. This game wouldn’t be any fun, or wouldn’t even work, if no one ever took the bottle. But players do, because there’s a chance they’ll be able to sell it before the round ends. A very different game, QE (Quantitative Easing) by Gavin Birnbaum, has a similar dilemma. In QE, players bid money to win various prizes, but the catch is that (1) they can bid however much money they want, and (2) whoever bids the most overall straight-up loses the game. A similar analysis by Backwards Induction says that no one should want to win an auction, ever.
Is this a case of players simply being irrational? Or is there something wrong with Backwards Induction itself? There’s a strong case that it is in fact the latter.
One of the assumptions of Backwards Induction, indeed one of the assumptions of Game Theory itself, is that players are perfectly rational. Backwards Induction asks what those perfectly rational players would do in a situation where they have to sell a bottle imp for $2, say. But the conclusion of Backwards Induction says that rational players will never find themselves in such a situation.
It asks “what would a rational player do, here at this point that no rational player would ever get to?” This is contradictory. If the player truly is rational, they won’t ever be in that situation, so you cannot ask what they would do if they got there. And if the player is not rational, then they are not bound by whatever the analysis says.
The solution proposed by Backwards Induction for these sorts of sequential games (including this Pirate Puzzle) are tempting and alluring, but they may not be as air-tight as they first appear. Can this contradiction inherent to Backwards Induction ever be resolved? That is an open question in the Philosophy of Game Theory (sorry - no answer here!)