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.
- 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.)