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Pascal Wager
What is Pascal's Wager

What is Pascal's wager? It is a probabilistic formulation of expected values on how a reasonable person should behave with regards to religion. The question is tries to answer is: Based on the fact that we cannot know whether there is a God or not, what are the consequences of being religious or being atheist.

It is based on the probability theory which states that in order to choose under uncertainty, you can compute expected values by multiplying outcomes with their probabilities. Let us take a simple example:

I propose you to roll a dice. If a 6 comes out, I give you 10 Francs, otherwise you give me 5 Francs. Should you accept the wager?
The probability to roll a 6 is 1/6. The outcome is 10 Francs, therefore 10/6.
The proability not to roll a 6 is 5/6. The outcome is -5 Francs (loss), therefore -25/6.
If I add 10/6 and -25/6 the total is -15/6, i.e. you will be loosing statistically 2.5 Francs. You should refuse the wager.
This can be displayed in a value table, and the rational decision is the highest expected value:

ScenarioRoll a 6
prob = 1/6
Not Roll a 6
prob = 5/6
Expected value
Accept Wager1/6*105/6*-5-2.5
Refuse Wager1/6*05/6*00

Returning to Pascal's wager, the "dice" is the existence of God, and the money is the possibility for an eternal afterlife of bliss. Pascal proposes the following table, based on the fact that an eternal afterlife of bliss is infinitely more valuable than anything finite in our temporal lives:

ScenarioGod exists
prob = p
God does not exist
prob = 1-p
Expected value
For any p > zero
Be religiousInf*pa*(1-p)Infinite
Be atheistb*pb*(1-p)b

Whatever values you personally choose to give for a and b (quality of temporal life as a religious or atheist person) and whatever value you give for p, as long as it is non-zero, the expected value is higher for the religious life.

The consequence is that we should all choose to be religious!

The beauty of this demonstration is that it inverts the "burden of proof" - normally, an atheist will say to a religious person: show me evidence of your God, without evidence why should I believe? But with Pascal's wager the answer is: unless you are absolutely sure that no God exists, you should believe... the atheists must now prove that p=0 or appear irrational in not becoming religious.

In the following, we will discuss several weaknesses of the wager. We will begin with specific critics and zoom out to more general issues.

The use of infinity

The use of Infinity is necessary in order to "reverse the burden of proof" as we have seen above. If the reward of eternal life is not infinite, but merely very large (let's call it L), then it may be reasonable not to believe - indeed the expected values become:

Religious: L*p + a*(1-p)
Atheist: b

So if b > a+(L-a)*p, then the best option is atheism.

Let us put some values for a and b. We can arbitrarily set a base value to zero and fix all the other values in relation. Let us put a=0.
What is a acceptable value for b? The major difference is that a religious life must bend to many external rules: for example you have to go to church every sunday at 10, even if you had a party the previous evening and feel like staying in bed. In absence of a God, we can surely agree that therefore the extra freedom to do as you see fit means that b>a.
We can again put an arbitrary value - say 1 - to b, as long as we accept that L>>b to take the eternity of afterlife into account. The exact numerical values have no impact on the reflections, only the signs are important. If you think (remember - in absence of God) that b is smaller than a then you should stop reading this, as this means you want to be religious regardless of the existence or not of God and therefore Pascal's wager does not apply to you.

This leaves us with the following expected values:
Religious: L*p
Atheist: 1
So if p<1/L it is reasonable to be atheist.

If L is infinite, then 1/L = 0, so p cannot be smaller than 1/L. But if L is finite, then the wager is not helping, because it reverts to the question of setting a probability for God's existence. An atheist will say that the probability is very small (otherwise he would not be atheist) and therefore will not be convinced by the argument. A religious person will put a higher probability but do not need the argument to be convinced...

Therefore it is necessary (and resonable for eternal bliss) to set L as infinite.

The problem is that Infinity does not behave like a normal number when making maths with it - and this is what we do in computing expected values. In particular, it follows the following rules:
x * Inf = Inf if x > zero
x * Inf = -Inf if x < zero
Inf + x = Inf if x is not -Infinity
Inf - x = Inf if x is not Infinity
Note: the above rule are the usually accepted definition of infinity in the extended real numbers. There are some other possible definitions of infinity, such as the the hyperreal numbers, but this does not change the conclusions if you translate "very small" into hyperreal-infinitesimal in the discussion above.

So let add some possible behavior. I can become religious now, or I could throw a dice and if a 6 comes out, then (and only then) will I become religious. Let us investigate this new scenario and its expected value:
Religious: Inf
Atheist: 1
Dice-decides: I have 1/6 probability of having the religious reward and 5/6 for the atheist, therefore 1/6*Inf + 5/6*1 = Inf + 5/6 = Inf.

Because the expected values are the same, it is exactly as rational to decide on a dice-roll than to become religious!.

Of course, this is just ridiculous. But let me reformulate: as long as I consider that I have a non-zero probability to change my mind before dying, it is as reasonable to keep living as an atheist (stay in bed on Sunday mornings) than to live religiously (and stand up on sunday mornings to follow the rules). As the former is more confortable, is it not reasonable to remain atheist?

To conclude this chapter, we have seen an intrinsic mathematical weakness in the mathematical approach of Pascal, and because of this weakness, Pascal's wager cannot be used to prove the irrationality of atheism.

Hidden Assumptions

Pascal's wager has several hidden assumptions about God and his behavior. In particular, the key assumption is that God will reward believers with eternal bliss, and will not similarly reward unbelievers - which is equivalent to say that He will "punish" them by not "saving" them.

This assumption is inline with christian teachings. Or is it? At least Calvinists assume that our acts cannot save us - we are foreordained to be saved at the discretion of God. Being religious is a consequence of being selected to be saved. Alternatively, we could deem that the way we act (e.g. generously, nice) is as important (or more) than being religious and going to church. Or that God is all-benevolent and will save everybody that wants it anyway. Or that God will give one more chance to believe in Him after death when evidence will be available.

But anyway the purpose is to convince atheists, and atheists will definetly not accept such assumptions just because they would be based on the Bible - that they don't believe to be written by God (which God?) anyway.

Let me take another set of (reasonable) assumptions and see the consequences.

God created us as rational creatures with free-will (another hidden assumption here: we are rational because we can act upon rational, calculated, expected values, and have the free-will to decide of our behavior). Would it not be a reasonable assumption that God expects us to behave like he created us? That he expects of us that we use our free-will rationally? And that therefore we shoud believe in Him only if we have sufficient evidence of His existence? And that therefore he will reward the people that acted honestly and followed their reason?

Let us summarize this in a new version of the wager:

ScenarioGod existsGod does not existExpected value
Case 1: I have enough evidence for God's existence
Be religiousInf0*Infinite
Be atheist1**1*1
Case 2: I do not have enough evidence for God's existence
Be religious0***0***0
Be atheistInf1Infinite

Notes: Based on 1 for the value of unconstrained life (atheist) and zero for religious life (external rules); see above for rationale. *) Obviously mistaken, the evidence I considered (case 1: I have enough evidence) was not valid. **) Stubborn, refuses evidence. ***) actually not rational as believing without evidence.

We can read from this table that it is rational to be religious if you think you have enough evidence that God exists - which is almost equivalent to say "if you believe in God". And it is rational to be atheist if you have no evidence - which is almost equivalent to say "if you do not believe in God".
If you are unsure about your evidence? Obviously it is then not sufficient therefore you should not believe. Only people sure that they have sufficient evidence should indeed believe in God.
I sure hope that all church-goers are sure of their evidence otherwise they risk missing the jackpot!

Therefore with this other set of assumption - we suppose that if God gifted us with reason, He wants us to be reasonable - it is perfectly rational to stay atheist in absence of sufficient evidence of God's existence. The fact that one cannot prove that the probability of my tables being wrong is exactly zero make them exactly equivalent to Pascal's original table. Indeed even if the probability of my table being right is very small, the infinite rewards make them weight exactly the same.

I could produce many other tables, by introducing several incompatible religions or by changing my assumptions about God and obtain almost any result. I could make a table where atheists always win - and convince believers that they are wrong... I will not because the fact that different tables can be drawn based on different assumptions about God and lead to different results is only showing that this is not a valid base of decision. So any table I create will just be as weak as Pascal's initial table and only valid within my (subjective) assumptions. The fact that Pascal's wager is supposed to convince atheists that by definition will not share the "Faith" in a given God as defined by a tradition or a holy book - and therefore will refuse the factuallity of the hidden assumptions - makes the wager unfit to convinve them.

I want to believe

The last chapter of this essay will focus on another assumption that we mentioned already: we are able to decide to believe.

Are we? Are we really able to believe without any evidence? Can we decide that starting tomorrow we will believe that Superman or Father Christmas exist? Or is belief actually a consequence of reaching (subjectively) sufficient evidence - i.e. independant from the will?
Acting based on a belief is then an act of will. Or searching for evidence in absence of belief is an act of will.

Even Pascal doubted this assumption, I let you savor a citation (english translation from Wikipedia) from the passage of Les Pensées just following the description of the wager:
"Follow the way [...] by acting as if they believed, bless yourself with holy water, have Masses said, and so on; by a simple and natural process this will make you believe, and will dull you - will quiet your proudly critical intellect...".

So Pascal actually means you should be rational enough to understand the wager, and henceforth try to dull your spirit to make you believe. Interesting. Does it work like this and you become dumb when visiting churches? From the priests I know (most very intelligent and sharp people) I doubt it... At this point the use of drugs may be safer than going to church to dull your critical intellect. Because as a non-believer, by the way, in which church are you going? As only one is probably true (is God schizophrenic?), that means the "rules" of the others may be untrue (incompatible with the true rules) and may not bring you closer, but further, to God...

I believe that I cannot choose to believe. Therefore the options in Pascal's wager are not to believe (be religious) or not to believe (be atheist) but act religious or act atheist. And the core question is now whether God will reward unbeliever acting as believers (lying!), i.e. whether acting against your beliefs is really bringing an advantage for Judgement Day...

In my personal opinion, we should not pretend to believe against our feelings if we have no Faith - in this case we should keep looking for evidence - and this quest may lead us sometimes into different churches among other potential sources of evidence. This is probably something that God will approve, if he exists, more than fake adoration.


We have highlighted several weaknesses of Pascal's wager. It remains of very important historical significance (first use of uncertainty and expected value in decision theory) and I recommend everybody to read more from the genius that Pascal was.

But it can not be used to show that unbelievers are irrational.

Of course, it does not bring any element to the debate whether God does exist or not (it is not its purpose), therefore the fact that the wager is not valid does not in anyway give a negative indication about the validity of religions.

Final word: this text is based on thoughts developed in very interesting discussions on the Forum Beyond the Fourth Wall - a forum dedicated to a comic and not to theology usually...


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