I haven't written much on Pascal's Wager, but I had this small thought. Perhaps someday I'll flesh this out a bit more.

## Introduction

In the standard Pascal’s Wager, the calculation goes something like this. You have two actions:

1. Believe in God (B)
2. Not believe in God (~B)

Then you have certain rewards and punishments, given the existence of God.

1. If there is a God, and you believe, the reward is huge: $$U(B|G)\gg 1$$
2. If there is a God, and you dont believe, the penalty is huge: $$U(\sim B|G)\ll -1$$
3. If there is no God, and you dont believe, no big deal: $$U(\sim B|~G)\approx 0$$
4. If there is no God, and you do believe, perhaps you’ve wasted your time a bit: $$U(B|~G) \approx -{\rm small}$$

So you should believe! Essentially it is an expected value problem, but what if that is not the proper calculation?

## Where Expected Value Breaks Down

Here is an example from E. T. Jaynes (see his book here - the chapter on Decision Theory):

Suppose that you are offered the following opportunity. You can bet any amount you want on the basis that, with probability ($$1 − 10^{−6}$$), you will lose your money; but with probability $$10^{−6}$$, you will win 1000001 times the amount you had wagered. Again, the criterion of maximizing expected profit says that you should bet all the money you have. Common sense rejects this solution even more forcefully.

Quoting Laplace we have:

[I]t is apparent that one franc has much greater value for him who possesses only a hundred than for a millionaire. We ought then to distinguish in the hoped-for benefit its absolute from its relative value.

The solution is to not use the expected value of the actual money, $$M$$, but the log of that, $$\log(M)$$.

## Back to the Wager

So, in the case of Pascal's Wager we have a similar thing. If there is no God, then the life we have is the only life we will have. Gambling that away on the basis of poor evidence is not rational. This life has more value to the non-believer, so the possibility of a great afterlife isn’t enough to offset the more likely wasting of the only life we know to actually have. It is for this reason that atheists find meaning in life, and further, they tend to find notions of an afterlife cheap and distracting.