Pascal’s wager

Game theory is a concept in applied mathematics. Like any branch of mathematics, it throws around big sounding terms such as pay-off matrix and Nash equilibrium, but at its heart game theory is really very simple.

Game theory is about winning.

In any game where the rules are clearly defined, game theory can be used to try and find the best strategy. Take a classical game of chicken for example, where two drivers accelerate towards a head-on collision, the one who swerves being the chicken. A traditional pay-off matrix for chicken looks like this:

bob swerves bob goes straight
fred swerves tie bob wins
fred goes straight fred wins crash

The matrix is fairly easy to understand; if both players swerve, the game is a tie. Neither player is the chicken and neither player dies. If both players stay the course, they both die – the worst possible outcome. If one player swerves, only their pride will be hurt, while the other gains the honour of winning.

So what is the best way of winning? If you don’t swerve you could die – so you’d better swerve. The other guy will be thinking that as well though, so if you don’t swerve they probably will. But wait – what if he is thinking the same thing! Then you will both die!

Analysing the game in this way, trying to work out the right decisions to achieve the best outcome – most people call this strategy. Mathematicians call it game theory.

Blaise Pascal was a French mathematician who was also very religious. He used game theory’s sibling, decision theory, to try and tell the world that belief in God is the right decision. This is now known as Pascal’s Wager.

The wager states that belief in God is the correct decision because the pay-off is infinite, while the punishment is infinitely painful. The pay-off matrix as proposed by Pascal would look like this:

no god god exists
atheism oblivion hell
christianity oblivion heaven

It would seem Pascal is right. Time to throw away all my years of scepticism – praise be to God the almighty! I see the light! But wait, something tells me the matrix isn’t quite complete. That’s the problem with game theory (and decision theory): it can appear very authoritative until you realise that you haven’t accounted for all the variables.

Take my initial example of chicken. The analysis of the game looks complete, it looks like I have thought of every possible scenario. But what happens if both players swerve in the same direction. There’s a chance that if they both chicken out, they could actually swerve into each other and be killed. Taking this into consideration, the pay-off matrix now looks like this:

bob swerves left bob swerves right bob goes straight
fred swerves left tie crash bob wins
fred swerves right crash tie bob wins
fred goes straight fred wins fred wins crash

In the initial matrix, there was a 25% probability of winning and a 25% probability of crashing. In the new matrix there is only a 22% probability of winning and a 33% probability of crashing. Clearly it could lead to disaster if you base your decisions on the original matrix.

Pascal’s wager is equally flawed. For starters it doesn’t take other religions into account. Let’s add Islam:

no god god exists allah exists
atheism oblivion hell hell
christianity oblivion heaven hell
islam oblivion hell heaven

Or what about a God / Allah that doesn’t punish wrong choices in belief, and only judges based on how good you have been throughout your life?

no god god exists allah exists
good atheist oblivion heaven heaven
good christian oblivion heaven heaven
good muslim oblivion heaven heaven
bad atheist oblivion hell hell
bad christian oblivion hell hell
bad muslim oblivion hell hell

And what if God / Allah exists, but actively punishes belief? What if God / Allah wants us to think for ourselves?

no god god exists allah exists
atheism oblivion heaven heaven
christianity oblivion hell hell
islam oblivion hell hell

As you can see, decision theory isn’t very useful if you don’t know all the rules behind the game, and we know nothing of the rules behind the afterlife, or even whether it exists. This makes the wager an unconvincing argument.

Sorry Pascal, but I’m sticking with atheism.


18 thoughts on “Pascal’s wager

  1. Shouldn’t those “oblivion”s on the good atheist/christian/muslim matrix be changed to “Fulfilling life”s?

    And if a sentient deity wanted us to think for ourselves, why introduce false religion to most humans in the first place? Why not just let us all go to it? And wouldn’t “thinking for one’s self” simply then come under the banner of religion and/or theologically-based morality structures anyway (“Why are you thinking for yourself?” “Because The Deity wishes it so”) and defeat the purpose of being an atheist?

    That is, if the purpose of being an atheist is to deny external forces (consequences, lessons, etc) on one’s own moral behaviour …

    I’ll get back to my young adult novel now…

  2. Shouldn’t those “oblivion”s on the good atheist/christian/muslim matrix be changed to “Fulfilling life”s?

    The matrix shows the outcome for you in the afterlife, in terms of what you are able to consciously perceive. Besides, are you suggesting that an evil existence can’t be fulfilling? I disagree. Muhahahah.

    And if a sentient deity wanted us to think for ourselves, why introduce false religion to most humans in the first place?

    The deity wouldn’t need to introduce false religion – I’m sure we’d come up with that all by ourselves. Besides, I’m not seriously advocating the idea of a deity who punishes belief, I am merely using it as a tool to demolish Pascal’s wager. My point is that even if a god does exist, we can never be sure what he/she/it/they really want from us. Believers will tell you that they are sure, but they can never prove it.

    This doesn’t mean that you can’t (or shouldn’t) believe in god. It does mean that you can’t use mathematics to suggest that belief in god is the right choice, which is what Pascal tried to do.

  3. Pingback: Carnival of the Godless: The Book of ‘Pod Edition | Mind on Fire.

  4. Pingback: Bay of Fundie » Blog Archive » Carnival of the Godless is Here! Wanna Bet?

  5. I like your post. There is one more major reason for rejecting Pascal’s wager: The matrix is not only incomplete, it is basically fallacious. There could be a God but no afterlife, and there could be aferlife but no God. There is no logical reason to link the two, the only reason is that religion links them.

  6. This doesn’t mean that you can’t (or shouldn’t) believe in god. It does mean that you can’t use mathematics to suggest that belief in god is the right choice, which is what Pascal tried to do.

    you don’t really need pascal to come to that conclusion.
    interesting article nevertheless

  7. You fucking rock dude. That was a cool article. You might consider getting away from the default wordpress theme though, but only because I’m getting fucking SICK OF SEEING IT EVERYWHERE ON THE INTERNET NOW!! Sorry 4 the rant.. Anyway, I’d love to see the same kind of article about Nash’s Equilibrium, if you feel like writing that one too. That one montage in a beautiful mind just didn’t do it for me.

  8. Firstly, i’d like to praise you for the amazing statistics-theory, especially pointing out the unceartancy of the calculation. The problem with Pascal’s Wager, is that it reminds me of the statistic calculations of whether God exists or not. Rubbish in, rubbish out.
    And there is one parameter you did not account for: There is not an equal statistic possibility that God exists. Imagine doing a matrix as above but with ten billion columns saying “no god” and one saying “god exists” Furthermore, choosing christianity when there is no god should display all the disadvantages christianity brings, such as intercepting science and wasting time.

  9. There is yet another consideration omitted by Pascal. He reasoned that (a) everyone’s life-span is finite and (b) the after-life is eternal. Thus the ratio of time spent in religious devotion to its payoff (eternal life) was a finite number divided by infinity, giving a result very close to zero. In other words, the pious get something for nothing; a slam-dunk for the deists.

    Unfortunately, Pascal’s beliefs blinded him to the other side of that argument. Viz., if there is no after-life, then the ratio of a lifetime of devotion to its payoff (nothing) is a finite number divided by zero. This time the result approaches infinity so, in the mathematics of this scenario, devotion is an infinite waste of time.

  10. Actually, Pascal argues that the payoff for a lifetime of devotion if there is no God and afterlife is that you have lived a fulfilling life, which is what franzy was getting at early on when suggesting that you replace “oblivion” with “fulfilling life”. Taking that tack, the ration of a lifetime of devotion to a lifetime of fulfillment is 1, which is still pretty good, even if it is sandwiched between the two infinities (which Pascal talks more about).

    Actually, why don’t you pick up a copy of the Pensees and read the whole thing, including the famous Wager, so that you can have a better view of what you’re arguing against. No offense intended, but it doesn’t sound like you read even the whole Wager, much less the greater work.


  11. Simply put: the biggest error in Pascal’s reasoning is that he was ‘begging the question.’ The result of his reasoning is belief in Christianity, but his entire argument requires the belief in the Christian thesis of redemption and damnation.

    However, I disagree with your last chart. If there was a God, and he wanted us to think for ourselves, why is this automatically aligned with ‘atheism’? Here, I believe you are presupposing something: that atheism is identical to ‘thinking for yourself’ and that religiosity is not. However, are we to say that St. Augustine or St. Thomas Aquinas never thought for themselves? Or the works of Soren Kierkegaard, who supposes this very criterion for Christianity, that it is ultimately about the singular person in an individual relation to the absolute as opposed to the blind following of group belief (which also falls on the side of atheism)?

    Finally, Pascal’s argument is no longer held even by theologians and Christian apologetics. The idea that one’s faith should derive from an exchange or in “game theory” has been not only disputed by atheists but by believers as well (see the above notes on Kierkegaard). Further, the work of Levinas argues that God (or divinity) is so radically different from us, there is no way we could understand or commune with it. Hence, recognizing divinity as an economic exchange (belief exchanged for redemption) is quite impossible with the ‘proper’ understanding of divinity.

    Arguing against Pascal and claiming a victory for atheism is akin to ID proponents refuting Lamarkism or saltationism and claiming they have refuted the theory of evolution.

  12. Dear god, I never “claimed a victory for atheism”. I only claimed a victory against Pascal’s wager, as I mentioned in a previous comment:

    This doesn’t mean that you can’t (or shouldn’t) believe in god. It does mean that you can’t use mathematics to suggest that belief in god is the right choice, which is what Pascal tried to do.

    As for the last table, the statement “What if God / Allah wants us to think for ourselves?” was a bit rash on my part. The first bit, “what if God / Allah exists, but actively punishes belief?” better represents that last table.

    LT, no I haven’t read the whole thing, just the wikipedia article. If you have any criticisms based on the whole thing, be my guest, but as “god himself” has said, even the well informed theists have dropped the wager as a useful argument. If only all theists did the same, then this post would be redundant.

  13. Nice Article… I just stumbled upon your site. I’ll check out some of your other work when I get some time.

    You’ve missed the biggest flaw in the premise of Pascal’s Wager. Pascal’s Wager assumes that belief in the Deity out of fear of the consequences suffered by non-believers is good enough to get you into Heaven. Since I don’t believe in Heaven or God for any reason, I can’t say if this is true or not. But in Biblical terms, this assumption is complete fallacious.

  14. Mike J makes a great point. I might just add that there are several little pieces in the whole christian game that need to be taken into account.
    Those being the words of Jesus…( supposedly thats what they are going by right)….ok then, well he even stated that not all followers would receive the prize of after life….no????? Shall i remind you then…..”Why do you call me lord lord and do not do what i say….Get away from me i never knew you”…ring a bell?
    Nuff said….so would a mere belief as opposed to a devout faith cut the mustard.
    A further thing to consider ….are you guaranteed a ticket to ride?
    You must employ some rules and funnily enough thats means about 95% of christians are in for a rude shock. Pfffft it doesn’t really work then does it. Anyway they say you can’t take it with you therefore i’m not going!

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