Imagine that one day, a big bloke with wings taps you on the shoulder. It’s OK, he says, Brian sent me. To offer you this potential wager, on behalf of God, who has more or less given up on the human race except as a subject for philosophy conundrums.
In the envelope in my left hand, he says, I have a number, called X. At some point in the recent past, X was drawn by God from a uniform distribution over the real numbers from 0 to 1 inclusive. You can have a look at it if you like.
In my right hand, he says, I have a mobile telephone which will allow me to receive a message from God with another number, Y, which will also be drawn by God from a uniform distribution on the line 0 to 1 inclusive.
The wager is this; if you accept the wager, and X and Y are equal, then every human being currently alive on the planet earth will be horribly tortured for the next ninety million trillion years and then killed. If you accept the wager but X and Y are not equal, then a small, relatively undeserving child somewhere, will be given a lollipop.
So, do you take the wager or not?
“Go on”, says the angel. “Look at it as a problem of utility maximisation. Just look at the utility associated with each possible outcome, multiplied by the probability of that outcome.
“In fact, it’s quite easy. Define p as the probability that Y equals X. In the favourable outcome (which occurs with probability 1-p), a child gets a lollipop, which increases the sum of utilities by a small amount. In the unfavourable outcome (occuring with probability p), the sum utilities is reduced by a massive amount. So what is p?
“Well, what’s the probability that a continuous random variable Y will be equal to a particular value X?
“Therefore, if you look at the calculation, accepting the bet gives a zero chance of a very horrible outcome, and therefore a (1-0) certain small increment to utility, so you should take it”
“Hurray”, you say, aware that angels have in the past not be so forthcoming in explaining the mathematics. But you are still nagged by doubts; wouldn’t the laugh be rather terribly on you if Y turned out to equal X after all?