## Wednesday, January 22, 2014

### Choosing a zero probability of an infinite good over a certainty of a finite good

Sometimes it is rational to choose a zero probability of an infinite good over a certainty of a finite good.

Suppose there are uncountably many benevolent people, each of whom is assigned a number in (0,1), the interval from zero to one non-inclusive.  A random number Y is chosen in (0,1) with a continuous distribution (say, a uniform one, or a cut-off Gaussian).  The people aren't informed of its value, but they know the setup of the story.
Person number x is now given this choice:
• wager: if Y=x, then everyone gets \$1; else, nothing happens.
• don't wager: the person with number x/2 gets \$1.
Then:
• If everybody wagers, then everybody gets \$1.
• If nobody wagers, then all and only the people with numbers in (0,1/2) get \$1.
So surely at least some, and probably all, should wager.  But if you wager, you're choosing a zero probability of an infinite good (since the probability that your number matches Y is zero) over the certainty of a finite good.  (The goods are to others, but since you're benevolent, that doesn't matter.)

Sebastian said...

"else nothing happens."
Not really true.
Rather:
"else, if all the other benevolents wager - as they should - everybody gets a dollar."

Alexander R Pruss said...

Nothing else happens as a result of your choice.

Sebastian said...

"Some, if not all, should wager."

Is there some optimal fraction of players that don't wager? Even if they can't communicate among themselves to coordinate their behavior, they could then choose to wager/not wager according to a random generator showing "do wager" that fraction of times.