Pascal’s Wager inverted: all atheists go to heaven?
Pascal’s Wager is one of the most famous arguments for belief in God. Pascal’s Wager does not offer proof or evidence for God. Instead, it is an exercise in probability that suggests belief in God is a better than non-belief.
But it turns out there are many flaws with Pascal’s Wager. There is a good summary of these in the Wikipedia entry for Pascal’s Wager.
This article intends to highlight another flaw (as a teaching device). It appears Pascal specified the game too broadly. When logically extended, Pascal’s Wager can be flipped on its head. The exact reason that theism has an infinite payout would also imply that many beliefs–including atheism–have an infinite payout.
First, let’s recap Pascal’s Wager and then go into the new interpretation which depends on analyzing beliefs and the infinite payout.
(The point here is not to support any particular belief, but rather to explore and teach game theory with the fun example of Pascal’s Wager. If this sort of discussion offends you, consider other reading).
Pascal’s Wager
There are many ways to state Pascal’s Wager. The argument usually goes like this:
“If you believe, and God exists, then you will be rewarded with an infinite payout. If you disbelieve, and God exists, then you lose out on the reward. If God does not exist, then either way nothing happens.”
The wager can be translated into the following decision table:
| God exists | God does not exist | |
| Belief | +∞ (gain all) | nothing |
| Disbelief | nothing | nothing |
The idea here is to consider the consequences of possible decisions. One can either choose to believe or not believe. The believer has everything to gain and nothing to lose. Therefore, if one has to wager, it would be better to act as though God existed and believe.
It is not so obvious why this logic is wrong. But it is, and here is one of the flaws that turns the wager on its head.
–The flaw: beliefs are probabilities, not absolutes–
Pascal’s Wager creates a false dichotomy between believing and not believing. In fact, beliefs are rarely absolutes. It is telling that beliefs, as defined in game theory, are probability distributions over states of the world or other players’ strategies.
It might seem strange to think about beliefs as probabilities, but it is something that most of us already do. When we say things like “I believe it will rain,” or “I believe he is innocent,” we are not speaking about absolutes. We are summarizing probability distributions over various states of the world. And as a rule of thumb, when we are uncertain, we rarely assign a zero probability to an event. We keep our beliefs open for updating.
The same logic can be extended to belief in God. An atheist who says, “I don’t believe in God,” is usually not saying God has a zero probability. He is summarizing his probability distribution over about various theories of the world, and he happens to think the probability for God is low. But the probability for God is not zero-typically an atheist does not summarily eliminate options but rather puts them at low probability.
Now consider Pascal’s Wager again. Realistic beliefs are not absolutes between the choices of believe or not believe. They are probability distributions, like “99 percent believe and 1 percent not” or “1 percent believe and 99 percent not.” I would argue that even faith is not 100 percent because theists typically question their faith from time to time.
And here is the kicker. The infinite payout of God rewarding believers will yield an infinite payout to almost all beliefs. That is, any non-zero belief in God will have an infinite payout. Therefore, an atheist who give a small but non-zero chance to God’s existence would also receive an infinite payout. This is an absurd conclusion, to be sure, but it is a consequence of the gambit Pascal set up.
This is one illustration of how an infinite payout is problematic. Although most theists would think a stronger belief in God should be rewarded, Pascal’s Wager does not specify this contingency. In the end, this assumption makes the wager lose much of its power.
In conclusion
Pascal’s Wager is an interesting idea, but this article illustrates some of its flaws. The two main problems are its assumption of absolute beliefs and its assignment of an infinite payout. These flaws ultimately turn the wager on its head to yield the opposite conclusion. This is yet another reason to reject Pascal’s Wager.
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23 Responses to “Pascal’s Wager inverted: all atheists go to heaven?”
I’ve had frequent dealings with Pascals wager and I can say that I’ve never thought of this angle. It’s very interesting!
I do have one nit pick. You say that:
“An atheist who says, “I don’t believe in God,” is usually not saying God has a zero probability. He is summarizing his probability distribution over about various theories of the world, and he happens to think the probability for God is low.”
With regards to god as a general concept, I agree. But allowing for the possibility of the concept as a general does not forbid one from eliminating specific possibilities.
For example, I probably would not go as far to say, with 100% certainty, that a given person has absolutely no money. This is a general statement. But it is certainly within the realm of probability that I could make a specific statement, such as: they do not have a real one dollar bill with the serial number 123456789. Especially if I have the real one in my possession.
That is, I can leave open a general possibility of them having money, but rule out specific examples of that.
By Scott on Aug 11, 2009
The standard wager has the states being “Act as if God exists” and “Act as if God doesn’t exist”. This formulation is not open to your charge.
The secular version of Pascal’s wager is often very seductive, and I wrote a short piece on it here:
http://www.bizop.ca/blog2/books/the-road-from-f.html
By michael webster on Aug 11, 2009
Pascal’s wager holds up just fine. Your critique is flawed. Michael nails it, but I’ll explain a little further.
The flaw in your argument is that you assume that the payoff rule is an “expected utility” rule. Your assumption is that if you think there is a 10% chance there is a God, then you’ll receive 0.10*(infinity)=infinity. This is incorrect, not only for religious belief but for earthly gambles as well.
Say you are playing roulette by wagering $100, and your distribution of belief is 1/40th for each number. After the wheel spins and the ball stops, you either win about $4000, or lose the entire $100. You don’t win 1/40th*$100. It’s an all or nothing proposition, just as Pascal’s wager is an all or nothing proposition.
By Brian on Aug 11, 2009
The gambling analogy does not accurately reflect how Pascal’s wager is being interpreted here.
To fix the analogy, you would be placing a bet on *all possible outcomes*. Each bet would be proportional to how likely you think that number is going to win. If you have equal belief for all numbers, then you would place equal bets.
The payout here is infinity, and having a non-zero belief for all possible options (God/not-God) guarantees a win, and also guarantees to an infinite payout.
Your analogy has the undue restriction of placing only a single bet, which would be the equivilent of ruling out – completely – a possibility. While that option is implied here, it is out of the main scope of this exercise.
There are atheists who do rule out all god-concepts completely. In this case, only they are the ones truly making the wager here. (In which case the wager fails for numerous other reasons).
With regards to the interpretation of Pascal’s wager (belief vs. action), I will concede that the standard intepretation seems to indicate that it is “living as god exists” rather than simply “believing god exists”.
This is not particularly significant as Pascal’s wager is often used for the Christian God, who specifical eschews actions in favor of belief (faith). Thus, for the Christian, living as God exists entails nothing more than believing God exists (Jesus was his son, died for sins, etc) so the dichotomy is erased.
By Scott on Aug 11, 2009
@Scott
Pascal thought that a non believer could become a believer by taking the first step: act as if god existed, faith would follow.
As for Pascal’s wager being limited to the Christian God, there is nothing in the argument that picks out any type of deity, the argument is insufficient.
By michael webster on Aug 11, 2009
I only noted what Pascal’s wager is “often” used for. I did not say that it is limited to that. The wager itself does not specify a certain god, correct, however Pascal created the wager specifically with regards to Christian apologetics so it seems odd to use it in a manner contrary to this.
The notion of other gods (with their inherent exclusivity) is another reason the wager fails, though that is outside the scope of this game theory article.
Taking Pascal’s intentions in mind, his wager pertains to the Christian god and all that is required to “act as of God existed” is to simply believe. Thus erasing the dichotomy.
By Scott on Aug 11, 2009
Scott writes: “Taking Pascal’s intentions in mind, his wager pertains to the Christian god and all that is required to “act as of God existed” is to simply believe. Thus erasing the dichotomy.”
This would be a good time for you to actually read what Pascal wrote about this very issue.
By michael webster on Aug 11, 2009
There is more to meaning than semantics. You must also consider context.
Pascal specifically constructed his wager as part of a collection of works specifically designed to support belief in Christianity. The notion that he intentionally created an argument that could be used equally well against non-Christian Gods is absurd.
Now, I should caveat that I fully believe that Pascal’s wager lends itself to generalization in this manner, using it for any specific god (Allah, Zeus, Thor) or simply “gods” as a general concept. In fact, this is one of the reasons why it fails so miserably.
But, I was clear when resolving this dichotomy between “belief” and “action” that I was talking under the context in which Pascal presented his wager: in support of Christianity – the Christian God.
If you still have an issue, it cannot, reasonably, be with whether or not Pascal was talking about the Christian God, but rather whether or not the Christian God requires more than just belief.
Everything I’ve read on the matter can be summed up as: acts are meaningless, faith is everything. That is, the only act that is required, is belief thus there is no difference, in this context between “act” and “belief”; they are one in the same.
By Scott on Aug 11, 2009
The extension of the wager to other gods or religions does not mean that it “fails miserably.” It’s simply part of the wager. If there are other religions that postulate eternal salvation, then the wager applies to them just as well. It’s just that there is only one right answer at the end of the day.
Basically, the wager says you’re no worse off believing in Jesus or Zeus as believing in materialism, or humanism, or whatever you please. Now it’s just a matter of who is right.
By Brian on Aug 11, 2009
The entire point of Pascal’s wager is to present belief in god as a dominant strategy and, thus, the only rational strategy to take.
Other presentations of the wager list the outcome of “God exists” & “Don’t believe” as eternal punishment – equivilent to negative infinity – which only highlights the dominant strategy of “Believe”.
In either case (“God exists” vs. “God doesn’t exist”), the strategy of “Believe” is equal to or better than “Don’t Believe”
This article attacks the wager from the angle of what it means to “believe”.
What the issue of multiple gods does is reveal that there isn’t actually a dominant strategy.
Instead of viewing the games as Atheism vs. Theism, view it as Jehovah vs. Allah.
In this case the possible outcomes are “Jehovah Exists” & “Allah Exists” and the possible strategies are “Believe Jehovah Exists” and “Believe Allah exists”
(Note: I understand that Jehovah and Allah are different interpretations of one Judeo-Christian god, However, at most only one of these interpretations can be correct; it is not an issue to discuss them as though they are different entities).
In this case, “Believe Jehovah Exists” yields infinite gain only if “Jehovah Exists” while “Believe Allah exists” yields infinite gain only if “Allah Exists”.
Neither of these strategies has the advantage of being dominant. Thus the entire point of Pascals wager evaporates. This issue becomes even more clear when you include ALL known god-concepts.
Now, one could say that, in regards to theism vs. atheism, theistic strategies still outweigh atheistic ones (since only theistic strategies allow for the possibility of infinite gain) but even this fails for a number of reasons:
1) You still have no way of choosing *which* theistic belief. Given the number and interpretations of god, the actual probabilities heavily imply that you will be picking the wrong one. Since failure is pretty much guaranteed, you might as well remain true to your own beliefs.
2) Pascal’s wager also operates under the premise that god is going to be fooled by (or won’t care) that you are simply believing to hedge your bets, rather than as a genuine display of faith. It is certainly quite possible that some specific god (perhaps one that has yet to be described accurately by humans) would rather reward things other than irrational belief (I say “irrational” because Pascal’s wager was invented specifically to address situations where rationality failed to yield a clear answer). Perhaps a god would rather you be true to reason and be an atheist, rather than abandon reason to become a theist.
In almost all of these instances (included in this post and in this article) the underlying problem of Pascal’s wager is that he oversimplified the problem. Thus the conclusion he derived is erroneous.
By Scott on Aug 11, 2009
Scott writes:
“If you still have an issue, it cannot, reasonably, be with whether or not Pascal was talking about the Christian God, but rather whether or not the Christian God requires more than just belief.
Everything I’ve read on the matter can be summed up as: acts are meaningless, faith is everything. That is, the only act that is required, is belief thus there is no difference, in this context between “act” and “belief”; they are one in the same.”
Scott, I’d suggest taking a look at James Chapter 2. There it shows what that the Christian belief is that “faith without deeds is useless”. So, I don’t think that “act” and “belief” are on in the same.
Actually, there are plenty of references as to how the Christian God wants people to act. If a person acted as if the Christian God existed, they would (or perhaps, should) not merely maintain that their faith alone is enough.
Having said that, I do agree with the issues of so many deities to choose from. And also, supposing that a God is omniscient, he/she/it would know that one is moved by this “hedging of the bets” and such a motivation would do little to impress him/her/it, I suspect.
By Mike on Aug 11, 2009
RE: Act vs. Belief
I don’t doubt that there are plenty of interpretations to support either side of the issue here but I think I’ll refrain from continuing to hijack Mr. Talwalkar’s blog for three reasons:
1. There is no conclusive, objective “correct” interpretation. (The argument would proceed ad infinitum/nauseum)
2. It is outside the scope of the game theory issues this article was designed to tackle.
3. In the same manner that an atheist does not necessarily completely exclude the possibility of the existence of a god, neither does an atheist necessarily act in a manner that is completely, 100% non-Christian-like.
Since the point here was to show that any non-zero belief in favor of God yields the full outcome, the same would apply to any non-zero compliance to typical “Christian” acts (Do unto others… for example). Thus the difference between act and belief yields no difference on the outcome of the application of game theory to Pascal’s wager.
By Scott on Aug 11, 2009
>> the point here was to show that any non-zero belief in favor of God yields the full outcome (…) the difference between act and belief yields no difference on the outcome
Considering that any ordinary life would consist of both good and bad deeds, aren’t you implicitly concluding, then, that belief systems don’t make any difference in the outcome so long as you help someone once?
The wager, however, starts from the premise that we cannot reason about God or God’s motives. Stealing a bike and asking for forgiveness technically would work in theory, but if it turned out that God is omniscient and vengeful, then chances are pretty good that you’d burn in hell.
By Leo Horie on Aug 13, 2009
“[A]aren’t you implicitly concluding, then, that belief systems don’t make any difference in the outcome so long as you help someone once?”
To generalize, it doesn’t make any difference so long as you have not excluded it completely.
Take the belief system at had (Christianity). It explicitly states that we humans are incapable of being the purely good beings required to achieve the reward. This was why it was necessary to set up an alternative. It just happens to be that this alternative requires only belief. Yes, it encourages certain acts, but only requires belief.
I agree with the second paragraph as well, and would think that – if there is an all powerful being out there – he is unlikely to be fooled by this crude ruse.
By Scott on Aug 13, 2009
The problem here is that you got you table wrong!
God exists God does not exist
Belief +∞ (gain all) nothing
Disbelief does not = nothing nothing
Disbelief=Lose everything for eternity!
By Jon Rackley on Sep 25, 2009
In my opinion, there are many problems with Pascal’s wager. Perhaps the worst flaw: the wager is used as a bullying tactic. It reminds me of an old mafia saying: you can pay us a if you want, but if you do not, there is a chance we might not “protect” you.
I appreciate Scott for defending a secular and rational view.
Michael Webster always brings up interesting points, so I will have to take a closer, historical look at Pascal before I make additional comments.
By Presh Talwalkar on Sep 26, 2009
Wagers
1. Christian God Exists
2. Another God(s) Exists
3. Christian God Exists but is co-opted by religious leaders to control you
4. Another God Exists but is co-opted by religious leaders to control you
5. Neither God exists
Costs
1.Hell for angering Christian god
2.Hell for angering other god
3.Heaven for pleasing Christian god
3.Heaven for pleasing other god
4.Giving up rational thought
5.Putting yourself under control of others
6.Nothing
And that even over simplifies all possible gods, for example:
There is a god, it just doesn’t concern itself with human affairs
There are multiple gods
There are multiple universes all with there own god/gods
We are all gods in training
We are all different facets of the same god
We are in a computer simulations
There is no god now, but the universe will eventually create one through evolution
There was a god, but he is dead now
There is a god and anti god fighting it out and if the antigod wins you’re screwed for backing the wrong one, etc, etc.
By tvscifi on Sep 29, 2009
Brian said: “The extension of the wager to other gods or religions does not mean that it “fails miserably.” It’s simply part of the wager. If there are other religions that postulate eternal salvation, then the wager applies to them just as well. It’s just that there is only one right answer at the end of the day.
Basically, the wager says you’re no worse off believing in Jesus or Zeus as believing in materialism, or humanism, or whatever you please. Now it’s just a matter of who is right.”
No, the wager definitely does not apply to all other possible gods. The wager fails miserable because it works under the presumption that non-belief would either elicit punishment or a loss of reward.
We need only postulate the existence of a god that happens to reward disbelief and punish faith to make an inverted Pascal’s Wager, in which the dominant tactic is to disbelieve regardless of whether the disbelifophile god exists. Granted, it is a capricious presumption. However, I honestly fail to see how it is less so than Pascal’s original.
By SketchSepahi on Sep 30, 2009
This entire discussion exemplifies the fundamental issue with “beliefs”. Anyone can believe anything – regardless of it being rational or irrational. Is there a God? Who knows? Is there no God? Who knows? Many gods? Same answer. One thing can be said with certainty – game theory exists and is applicable to a wide array of real-world problems. And while God or gods may or may not exist, it seems many of the real-world problems have been caused at the very least in their Name or names.
If you have any interest in understanding how the Christian church has over time come to view scientific thinking I would suggest the course “Science and Religion” from the Teaching Company.
If you are interested, it is here: http://www.teach12.com/ttcx/CourseDescLong2.aspx?cid=4691
BTW – I have no relationship with this organization, I simply have found their materials useful and insightful.
As it happens, much of traditional theological thought and scientific thought are actually aligned. It has only been in relatively recent times there has occurred such a divisive perspective between the two.
Thanks for an interesting article and perspective.
By Gary on Oct 4, 2009
It’s obvious that only atheists go to heaven. God is quite able to make himself non-existant. As a result of this only people whom don’t believe in God are actually speaking the truth ; all those who do believe in God are limiting God’s power. This is just to show how Pascal’s wager turned on it’s head might be thought of in practice. Also you might try backing up your argument with a quote from the bible ‘In the last days there will be many christians whom when I asked them what they did say we preached your word and did miracles in your name. Evildoers.’
By Paul on Oct 11, 2009
It is very interesting indeed. I would go a bit further though, because there are more flaws in Pascal’s game theoretic model (of course, it is a bit pretentious to give this label to this theory). Another critique would be that it does not take into account the fact that the preference rankings of different individuals may skew his results. Moreover, the payoffs can only be finite due to the law of diminishing returns. There certainly is a point where the cost function meets the benefit function in this situation, thus rendering Pascal’s theory rather artificial.
By Paul on Oct 31, 2009
Hello,
why use the name Pascal when, as Jon pointed out,
it is *not* Pascal’s wager we are talking about.
Let’s call it – say – Mr. Banana wager, just to clarify matter.
Cheers,
C
By Carlo Paggi on Nov 7, 2009
pascals wager is fine, but you could apply it to all gods, ra, thor, allah, odin, the list goes on, so which god should you believe in
By derek on Jan 6, 2010