Sunday, September 03, 2006

Pascal's Wager

This is an old argument thought up by the French physicist, mathematcian and philosopher Blaise Pascal. I'd be interested to see people's thoughts on it.

It's not meant to be an argument that says Christianity is true - Pascal helped invent probability theory in maths; it's meant to be an argument about which way is the better bet, Christianity or atheism.

The Argument

The argument basically goes like this...

Lets assume there are only two possibilities for belief - Christianity and Atheism and that one and only one of them is correct. For the sake of argument, lets say that the probability of the Christians being correct is p, so the probability of the Atheists being correct is 1-p.Each belief has a group of people who follow it - the Christians and the Atheists.

If Atheism is correct, then both groups will end up exactly the same - pushing up the daisies. If Christianity is correct, then the Christians will end up very much better off than the Atheists. Lets call the amount the Christians will be better off than the Atheists in this scenario H.

So the Christians stand chance p of gaining H - their expected gain is pH.

The Atheists, by contrast, have no expected gain. Whether they are right or not, they do not end up better off than the Christians. So, said Pascal, if you're betting, it's much better to bet on being a Christian than an Atheist.


The usual criticisms I hear of Pascal's Wager go something along these lines:

1. Probability theory doesn't work on this.

2. Pascal only gets the results he does because he has massively oversimplified the situation. There are far more possibilities than just Christianity and Atheism. And surely leading a good life has to count for something, even if I don't believe in God.

3. It doesn't work, because Atheists get to make the most of their life now, whereas Christians are busy worrying about the next one.

My Analysis

I don't understand 1. Why shouldn't probability theory work on this?

2. really doesn't work either. Yes, you could introduce other ideas - Islam, Buddhism, etc. But it wouldn't change the fact that Christians end up better off than Atheists on average unless you start introducing silly religions that treat Atheists better than Christians in the afterlife, and give them a higher probability than Christianity. I don't see any of those religions around, and I don't think I've ever heard any arguments for their validity... Of course, you might decide that Islam (for example) comes out as an even better bet, but that doesn't change the fact that Christianity comes out better than Atheism

The third objection is better. Being a Christian does cost stuff in this life (but also, I think, makes this life a lot more livable). Lets say that being a Christian in this life costs an extra amount c compared to being an Atheist.

In that case, Christians would gain pH and lose c, so their expected gain would be pH - c. Atheists would still have zero gain.

So in order for Atheism to be a better bet than Christianity, c > pH. In other words, the cost of being a Christian would have to be bigger than the probability of Christianity being true multiplied by the gain Christians get if they are right in the afterlife.

In other words, to be an Atheist rather than a Christian, you are either very poor at betting or sure of at least one of the following options:

  • that it would cost an immense amount to be a Christian, so great that even a decent chance of an eternity of paradise would not make up for it
  • the chance of Christianity being true is vanishingly small (to which I would say that if so many otherwise sensible people believe it, it's not going to be that unlikely)
  • the promised reward in Christianity is not that great.

So what do people think? Which is it?


Daniel Hill said...

I've been discussing this a lot with a non-Christian friend. One has to remember that we have to discuss actions rather than doctrines or beliefs. This is because doctrines don't have expected utilities, and beliefs aren't under our (direct) control. Pascal considers actions such as going to church, which seems reasonable enough. The problem is that if the potential gain is infinite, as the Christian asserts, then, since every action has a non-zero (epistemic) probability of delivering this infinite gain, every action has the same expected utility (infinite). (Incidentally, it's the infinity that leads some (but not me) to say that probability theory doesn't apply here.) So there is no reason to choose going to church over going to the mosque or going to the pub.

John said...

Ah yes - the infinity problem.

I deal with it by just calling it H, then letting H tend to infinity. Problems of multiple infinities can then be allowed to vanish by renormalisation.

Anonymous said...

For the second time I typed out a reply to one of your posts, only to lose it via a 'page expiry' warning! I shall attempt to type it out from memory.

Congratualtions, Custard on an excellent post. Had me thinking for a time, and I finally understand where the term pH comes from ;)

As to your analysis of the three main criticisms of Pascal's Wager:

point 1 - I don't know enough probability theory to make a claim as to whether or not it applies here, so I shall take your word that it does

point 2 - it depends on what you define as a religion. Given a sufficiently loose definition, then the number of possible religions are infinite, meaning that all choices are infinite long shots, in betting terms!

point 3 - to me, this is irrelevent without having criticism 2 debunked to my satisfaction, and you know me well enough to know that I will not be satisfied with such an imprecise comment as 'silly religions' as part of your answer!


John said...

Useful point about the infinite religions, AC - I'd forgotten that one.

I think there are two ways round it, depending on exactly what you mean.

If you mean that there are potentially infinite slight variations on Christianity, for example, then you can just treat the different views as grouped data with grouped probabilities in the same way that you would, for example, with a Maxwell-Boltzmann distribution. In that case, the argument is essentially unchanged.

If you mean that there are potentially infinite completely different religions, I'd argue that because religions claim to be about getting access to something that is beyond our experience, they have to be based on the idea that God (or whoever) has revealed something to us, and there are only a finite number of groups that claim that, and the argument still holds.

If God exists, hasn't revealed anything to us, and has a set of completely arbitrary and unrevealed criteria for admission to heaven, then we could count that as a seperate "religion", which would just add the same amount onto everyone's expected outcome as everyone would have an equal chance... It still doesn't change the argument siginificantly, unless you start saying such deities are more likely to be nice to atheists than Christians, which I don't see any basis for believing.

Anonymous said...

I would argue that there are an infinite number of 'potential' religions completely different from each other and variations on those currently acknowledged, though I will concede that there is merit in lumping sections of the latter into distinct groups. Again it comes back to what one is willing to accept as a religion.

I find mathematical analysis of metaphysical issues deeply unsatisfying, as I can never just accept the assumptions made, even when I agree with the outcome! That's not to devalue the worth of such study - it really does make you think, and I am always willing to hear that I am wrong


Daniel Hill said...

`If God exists, hasn't revealed anything to us, and has a set of completely arbitrary and unrevealed criteria for admission to heaven, then we could count that as a seperate "religion", which would just add the same amount onto everyone's expected outcome as everyone would have an equal chance.'
The point that I was making was that for every action there is a non-zero (epistemic) chance that it will get us into Heaven, and therefore the expected utility of every action will tend to infinity in just the same way as that in which Pascal argues the expected utility of church attendance tends to infinity. Therefore, Pascal's wager doesn't provide us with a reason for anything.

John said...

Sorry Daniel - I thought I'd dealt with that on my comment regarding (re)normalisation.

Could you explain why the method I suggested wouldn't work?

John said...

AC - If you can find a sensible way of applying mathematics to metaphysics which ends up with atheism coming out ahead of Christianity, I'd be very interested.

Daniel Hill said...

I don't know what 'renormalisation' means, I'm afraid, Custard, though Wikipedia says that it is a tool used in quantum theory. The method of letting H tend to infinity rather than merely be infinite won't work because it will equally tend to infinity for every action.

John said...

Conceptual steps: 1) Write value that will end up big as H.
2) Sum all expected values for all actions. Call this sum Z.
3) Divide each expected value by Z (or a fraction of Z).
4) Then let H tend to infinity and take the limiting value of (expected outcome)/Z for each expression. They'll all be finite.

Alternatively, just write it in terms of H. 0.5H > 0.3H for all H, even in lim H-> infinity.

It's all just standard mathematical tricks for dealing with big numbers.

Daniel Hill said...

But surely as H tends to infinity the difference between the expected utilities of the actions tends to zero?

Have a look at this article, which is the source of my ideas. It's written by Alan Hajek, who has studied statistics, maths, and philosophy:

John said...

Nope - the Hs will cancel out and you'll be left with constants.

Whoever wrote that stuff doesn't seem to be very good at coping with big numbers in maths. I'm pretty sure I did it in A-level further maths when we were thinking about limiting values and so on.

It also comes up in several areas of physics - most famously in renormalisation.

John said...

Of course, when the Hs cancel out, they'll still leave their multiplying factors intact.

They'll also reduce any terms not containing H - any views which don't include heaven as a possible benefit - as those terms will then have a 1/H multiplier, which will tend to zero.

So instead of having, say, inifinity:infinity:1, you might have 2:1:0.

Anonymous said...

Whilst browsing I came across this tongue-in-cheek analysis of Pascal's Wager. No doubt you will not consider it serious, Custard, but there is hope for me yet!

Quote: 'Therefore only intellectually committed but critical nontheists are genuinely good and will go to heaven.'


Anonymous said...

Oops - I lost part of the link. Apologies:


Anonymous said...

Argh! So I did post it correctly!
I shall try it again ... _carrier/heaven.html


Anonymous said...

Ho hum - add 'ard' to to the end of 'rich' and remove the space before the underscore.


Anonymous said...

Final one - next time I will preview my comment before I post!


John said...

Want me to clean these up AC? Feel free to post a "definitive version"...

John said...

And here's the link you wanted...

Yes, I don't take it seriously, but it's sad that they consider the suffering thing sufficient reason for not believing in God.