Member-only story
Superrationality: How Decision Theory Resolves Any Dilemma
Chapter II: An unfortunate gamble
This is a follow-up to Chapter I.
In the television game show Deal or No Deal, the only contestant is faced with 26 cases with wildly varying monetary amounts. The contestant knows which amounts are present, but doesn’t know which case contains which amount.
In a series of rounds, the contestant has to remove boxes from the game, and at some point she gets to take home the amount of the last remaining case. That is, unless she takes a deal, which she is offered at different points during the game.
Such a deal works as follows: the contestant has to stop the game, and in return gets a certain amount of money (which of course depends on the monetary amounts left in the cases).
One particularly interesting game had a contestant play away all cases except two: one with $1, and one with $1,000,000. He was then offered a deal of $416,000. He said “No deal” — in other words, he took a 50/50 chance of winning $1 or winning $1,000,000 over a sure win of $416,000. After that, he opened the wrong case and won $1.
Was he being rational? Some people argued yes: the contestant went with the choice that had the highest expected monetary value. After all, a 50% probability of winning $1,000,000 and an equal probability of winning $1 is an expected value of 0.50 * $1,000,000 + 0.50 * $1 = $500,000 + $0.50 = $500,000.50. That’s more than the offered deal of $416,000!
But while the math here is correct, the argument isn’t: it assumes that the value the same amount of money has to the player doesn’t decrease as the player obtains more money. You have to ask yourself: is $1,000,000 more than twice as awesome to win as $416,000?
For me personally, it isn’t. I’d love to win $416,000, and I’d love to win $1,000,000 even more — but the difference between winning nothing and winning $416,000 is a lot bigger than the difference between winning $416,000 and winning $1,000,000.
And this valuation of money is subjective: a very poor person would probably value getting $100 way more than a billionaire would.
