A Defense of Pascal's Wager
Note: This piece is somewhat expatiating and philosophical. A more to-the-point discussion for those already familiar with Pascal's wager and Bayesian decision theory can be found here.
Summary. This piece responds to two common objections against Pascal's wager. Regarding the many-gods objection, I point out that in light of all the evidence we have about the world, not all religions are equally probable, so that the symmetry assumed by the objection fails to obtain. I mention two alternate approaches to infinite decision theory, one based on infinities that can be treated like ordinary real numbers and another based onabsolute infinities.The second objection--that the probability of, say, the Christian god is zero--is, in my opinion, an example of overconfidence.
When asked why they don't follow a particular religion, many people respond, Because I find the claims of that religion highly improbable.
As far as epistemic correctness is concerned, this is fine. But decision theory demands more than probabilities; it also includes consideration of the magnitude of utilities entailed by different scenarios.
The probability that you'll get into a car crash the next time you drive somewhere is small (around 0.0002, which I calculated by dividing 31 million car collisions per year in the US by a population of 300 million, assuming 2*365 car trips per year, assuming 3 passengers per collision, and assuming the reader of this article is twice as cautious as the average person). Yet, you still probably think it's a good idea to wear a seatbelt, which shows that your decision is not based only on the most likely outcome. Small probabilities deserve large weight in our decisions if the potential gains and losses are sufficiently great. Pascal's wager is a rather extreme version of this point: As far as heaven and hell are concerned, the consequences are infinite, so that even the smallest of (positive) probabilities of an outcome can dominate our choices.
The literature on Pascal's Wager is vast (see, e.g., the references in the entries of the Stanford Encyclopedia of Philosophy and Internet Encyclopedia of Philosophy). A relatively careful mathematical exposition of the wager can be found in Frank A. Chimenti's "Pascal's Wager: A Decision-Theoretic Approach" (JSTOR link). In what follows, I will address two common objections to Pascal's wager.
Objection 1: As Denis Diderot famously noted, "An Imam could reason just as well this way." Thus, the expected benefit of believing in Islam or perhaps other religions like Hinduism might also be infinite. If infinity equals infinity, why choose one religion over another? A particular case of this objection is that there might be an "anti-Christian God" or a "professor's God" who punishes believers and rewards skeptics.
I'll give two different possible responses. Each of them essentially boils down to the claim that, in my opinion, a Christian
-type god who punishes atheists is more probable than an anti-Christian
-type god who punishes those who convert to Christianity.
It's not always true that infinities are equal; there are a number of mathematical constructs that we can use to talk about different sizes of infinity--e.g., John Horton Conway's surreal numbers or Abraham Robinson's hyperreal numbers, the latter of which Jordan Howard Sobel discusses in the context of Pascal's wager in his Logic and Theism: Arguments for and Against Beliefs in God (Google Books entry). In the hyperreal number system, the standard infinimal, I, is identified with the sequence (1, 2, 3, 4, ...). For instance, we might imagine going to heaven, and feeling one unit of bliss on each day. We could record our accumulated amount of happiness, day by day, as the sequence (1, 2, 3, 4, ...). Since heaven lasts for eternity, our total happiness would be the infinite sequence, I. Similarly, if going to hell involved -5 happiness units per day, our total suffering would be represented by (-5, -10, -15, -20, ...) = -5 * I. It turns out that we can manipulate infinimals in the same ways as regular numbers, so we can use them in computing expected values.
Suppose we're deciding between Christianity and another potential religion.[1] For the sake of illustration, suppose we assign it a probability of, say, 0.0005 to Christianity and a probability of 0.0002 to the other religion we're considering. Assume the hells of both religions are equally bad (represented by -5I) and both heavens are equally good (represented by I). The gain due to believing in the correct religion is then 6I. The expected value of believing in Christianity is 5/2 times higher (0.003I = 0.0005 * 6I, compared with 0.0012I = 0.0002 * 6I) than that of believing in the other religion. It is not hard to see from this example that, given a finite number of possible religions, each with equally good heavens and equally bad hells, we ought to believe in the one that's most probable. If the heavens and hells are not equally good and bad, hyperreal numbers allow us to adjust our analysis accordingly.
The above reasoning runs into the following objection. A critic might say, "Your hell only has pain of -5*I, but I can show you a religion whose hell involves pain of -10*I or -15*I or higher." Indeed, tounge-in-cheek followers of Wonko, the Magic Elf claim: "The flames [of Wonko's pool hall] are ten times hotter than [the Christian] hell. The demons are ten times nastier, and THE ETERNITY IS TEN TIMES LONGER!" Of course, the Christian could respond: "Oh yeah? Your hell is only -5,000*I, but my hell is actually -I^(I^(I^(I^I))). So there!" But this playground-type dispute would never stop, since we could always appeal to bigger infinities.
Alan Hajek, in his "Waging War on Pascal's Wager" (JSTOR link), points out another problem with this approach: If I + 1 > I, and if heaven is worth only I, then salvation is not the best possible thing. But this is at odds with Christian doctrine, which holds that spending eternity with the Lord is the greatest joy of all. As Otto Zimmerman writes in the Catholic Encyclopedia:
God is the self-existing, uncreated Being whose entire explanation must be in Himself, in Whom there can be no trace of chance; but it would be mere chance if God possessed only a finite [or, we might add, limited infinite] degree of perfection, for however high that degree might be, everything in the uncreated Being -- His perfections, His individuality, His personality -- admit the possibility of His possessing a still higher degree of entirety.
As Hajek says, the problem with hyperreals numers, surreal numbers, and the rest is that they try to apply a "finite-looking gloss" to something that is, in some sense, infinitely infinite. What is needed, Hajek maintains, is an infinity that is reflexive under addition and scalar multiplication. Otherwise, we have to ask why heaven only has value (infinity), rather than (infinity) + 1 or 2*(infinity).
Probably the most widely used examples of infinities that are reflexive under addition and scalar multiplication are the cardinal numbers of Georg Cantor. These numbers refer to sizes of sets; for instance, "aleph-null" refers to the size of the set {1, 2, 3, ...}. With this number system, (aleph-null) + 1 = (aleph-null) and 2*(aleph-null) = (aleph-null). So cardinal numbers are reflexive under addition and scalar multiplication (roughly speaking). However, this does not mean that cardinal numbers are all the same size; for instance, the size of the set of real numbers is strictly bigger than the size of the set of counting numbers. In fact, Cantor showed that the size of the power set of any set S is strictly bigger than the size S itself. So yet again, we seem to have the issue of "My infinity is bigger than yours."
Yet Cantor did not see this as a problem. A devout Christian, he wrote in 1896 to a Dominican priest, "From me, Christian philosophy will be offered for the first time the true theory of the infinite" (qtd. in Bruce A. Hedmen, "Cantor's Concept of Infinity"). Cantor did not associate God with any particular transfinite number, but with the entire collection of them, which he denoted by the letter tav. According to the Wikipedia entry,
As Cantor realized, this collection could not itself have a cardinality, as this would lead to a paradox of the Burali-Forti type. Cantor instead said that it was an "inconsistent" collection which was absolutely infinite.
The Wikipedia article on absolute infinity explains:
The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object.
Hedmen's article discusses this idea further:
Cantor [distinguished] between transfinite numbers, which exist in the human mind, and Absolute Infinity, which is beyond all human determination, and exists only in the mind of God. [...] Cantor thought of the infinite ascent of ever-increasing transfinite numbers as an appropriate symbol for the absolute.
Indeed, in an 1886 article, Constantin Gutberlet endeavored to defend Cantor's transfinite numbers on religious grounds:
But in the absolute mind the entire sequence [of transfinite numbers] is always in actual consciousness, without any possibility of increase in the knowledge or contemplation of a new member of the sequence. (qtd. in Joseph W. Dauben, "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite," p.100--JSTOR link)
Perhaps readers aren't particularly fond of transfinite cardinals and set-based notions of infinity, which is fine. The essential idea is merely that spending eternity with God would represent, intuitively, "the biggest type of infinity possible." Of course, atheists can invent scenarios involving "absolute infinity," too, but they're often ad hoc and insincere. In contrast, absolute infinity is the focus of the Christian philosophy, a philosophy that many people believe for reasons that have nothing to do with abstract mathematics!
At this point, readers may notice that our discussion has come back to where it started: If we don't use hyperreals or other "finite-looking" representations of infinity, then how can we compare 0.0005*(absolute infinity) against 0.0002*(absolute infinity)? Both just equal absolute infinity! For this reason, Alan Hajek suggests, Pascal's wager doesn't lead to the conclusion that we must believe in God. Instead, we could decide to flip a coin, and believe in God only if it lands heads. Since (1/2)*(absolute infinity) = (absolute infinity), there is no difference in expected value between these two options.
But think about the situation practically. Suppose your teacher is going to give you a multiple-choice test tomorrow. If you pass the test, your teacher will give you an absolute infinite amount of utility. Would you decide not to study because there's some chance that you would get all the questions right by guessing? No! You would spend the entire evening studying intensely in order to increase your probability of passing. (The same would still be true even if there were a tiny chance the teacher would actually give the test, which is perhaps a more appropriate analogy for wagering on the existence of a God who judges people.) This illustrates the intuition behind the principle that George Schlesinger advocates in "A Central Theistic Argument," part of Jeff Jordan, ed., Gambling on God: Essays on Pascal's Wager: "In cases where two acts yield distinct probabilities for the same prize (or prizes of equal value), we ought to prefer the act associated with the higher probability." While this may seem intuitively obvious, some have criticized this principle as "ad hoc" (see Roy Sorensen, "Infinite Decision Theory," also in Gambling on God).
However, P. Bartha's "Taking Stock of Infinite Value: Pascal's Wager and Relative Utilities" develops a robust framework of maximizing expected "relative utility" that has Schlesinger's principle as an elementary consequence. As Bartha notes, "Schlesinger's response is essentially correct. We [Bartha] have simply provided a systematic way to incorporate his principle into decision theory" (26). As for Hajek's requirement that heaven be the best outcome possible, Bartha's formulation allows that as well: "There might be many (even infinitely many) jumps in utility, but it is perfectly consistent to suppose that salvation is absolutely maximal" (29). Of course, we needn't regard Schlesinger's principle as dependent upon Bartha's specific framework; presumably other decision-theoretic approaches, not yet discovered, could yield the same result.
To extend Schlesinger's principle to consideration of negative infinities, we might decide to maximize some quantity f(A; x,y) = x*Pr(+absolute infinity if we take action A) - y*Pr(-absolute infinity if we take action A), where we decide on x and y based on how much we value happiness versus suffering. For instance, x = 0 and y = 1 would correspond to trying to minimize the probability of enduring absolutely infinite suffering. In the case x = y, we get the first part of what Nick Bostrom calls the "Extended Decision Rule" on p. 19 of his "Infinite Ethics." (The second part of his rule is to break ties by considering differences in their finite values--or, perhaps, between their smaller infinite values.)
Applied to Pascal's wager, an analysis using x = y = 1 might look like this. Let the probability of Christianity be 0.0005 and let the probability of an anti-Christian god (one who punishes believers and rewards atheists) be 0.0001. Without loss of generality, we can ignore other religions for the moment. Then becoming Christian has the value of 0.0005 - 0.0001 = 0.0004. If another religion has probability 0.0002 and its corresponding "anti-religion" has probability 0.00005, its difference is only 0.00015 < 0.0004. The reason we could neglect other religions in the above was that they would either punish or not punish both Christians and anti-Christians in the same way. (For a relaxation of this assumption, see "Why Christianity? The Pascalian Argument from the Probability of Hell.") For instance, Allah would punish both Christians and anti-Christians, while a different god might save everyone, including Christians and anti-Christians; switching from atheism to Christianity would have no effect in those cases. Of course, one can imagine more complicated scenarios: e.g., there might be a god who punishes 2/3rds of Christians but only 1/8 of anti-Christians. But these are all contrived and one feels that they should "cancel each other out": e.g., there might also be a god who punishes 1/8 of all Christians but 2/3 of anti-Christians. Unlike the comparison between Christianity and anti-Christianity, there seems to be absolutely no reason to prefer one of these scenarios to the other.
I should add that the reader needn't accept the exact formulation of the rule given above. One might, instead, try to maximize a nonlinear function of the probabilities involved. Or one might maximize the ratio Pr(+absolute infinity) / Pr(-absolute infinity). There are numerous other possibilities, but many of them will lead to the same conclusion. In any event, we ought to choose some rule to deal with infinity in order to avoid being paralyzed. For if we don't admit a difference between a 99% chance and a 0.0001% chance of absolute infinity, then oh, how much less do finite changes in our happiness matter!
Finally, I'll mention a technical note that readers may wish to skip. Even though we're dealing with absolute infinities, Pascalian computations using a probability-based decision rule are quite tractable. For instance, we could use the number "1" to represent +(absolute infinity) and "-1" to represent -(absolute infinity), with "0" representing, for the moment, any other outcomes. Bostrom's Extended Decision Rule is then equivalent to maximizing the expected value of these "1" and "-1" numbers across all options provided that there is a unique maximum. If the maxima are not unique, then we could consider the subset of possibilities with highest possible expected values as a new set of choices. We ignore absolute infinities and consider the next level down--whatever we choose that to be--assigning its positive form the number "+1" and its negative form "-1." We continue to proceed like this until, after some iteration, we find a unique maximum.
It would not be inconsistent with Hajek's point to assign nonsymmetric values to +(absolute infinity) and -(absolute infinity). For instance, if we feel that hell would be worse than heaven even though both are absolutely infinite, we might assign heaven "+1" and hell "-3." Hajek's objection does not go through because here we're talking about qualitatively different emotions (happiness vs. suffering) rather than greater or lesser amounts of the same emotion. It would not make sense, for instance, for a religion to claim to have a value +2 while all the others have value +1 because we've defined +1 to be the highest possible value any joy can have. We can still say that hell might have value -3 merely on account of the "exchange rate" between happiness and pain that the decision-maker wants to incorporate into the calculation.
Pascal's wager is just an optimization of an objective function over a space of possible actions; the details arise from how one defines one's objective function and one's probability distribution over the set of possible gods. I find that many people, after first hearing about and considering Pascal's wager, end up living exactly the same way they did before (e.g., in the case of atheists, most remain atheist). But it seems rather strange that people who haven't considered the wager before would, upon solving the optimization problem in a fair-minded way, find that their previous choice of action was actually optimal. If I tell you to choose some real number, and then you encounter a real-valued function you haven't seen before and are asked to find its maximum (assuming it has one), it's rather unlikely the maximum will occur at the number you chose earlier!
One response to this point is that everyone has already seen the function before; they just haven't thought about it explicitly. Even young children have usually heard the basic notions of heaven and hell, for instance. So, the critic might say, people choose their religions in a Pascalian manner even though they're not aware of it. Hence, when they do finally encounter Pascal's wager in a formal setting, it's not surprising to find that their solution to the optimization problem is to stick with their current religion.
This point is probably somewhat valid, but I don't think it suffices to explain why so many people's actions are unchanged by Pascal's wager. Another explanation is that many people (including myself, probably--I don't claim to be able to resist these temptations!) bias the probability distribution and objective function that they input to their optimization problem in order to make the solution come out the way they want.
Objection 2: The probability of Christianity is zero.
It's not mathematically impossible to assign a zero probability to something that could be true. For instance, there's zero probability that a random real number between 0 and 1 will be 1/pi. But the existence of a Christian god is much different: it's something that a third of the world population believes. Can you really say that such a possibility has a probability smaller than 0.0000000001? How about 10^-100? How about 1/(googolplex)?
If you want, you could assign an infinitesimal probability to Christianity. The standard infinitesimal in hyperreal analysis is 1/I = (1, 1/2, 1/3, 1/4, ...). If you assigned an infinitesimal probability to Christianity and assumed that the benefit of avoiding hell and going to heaven would be 6I, then the expected benefit of believing in Christianity would be (1/I)*(6I) = 6, which could be compared with other finite costs and benefits. However, assigning a zero or infinitesimal probability to a highly uncertain question like this one seems to me an example of overconfidence on the part of the decision maker. A number of studies have shown that people often underestimate the likelihood of low-probability high-impact events, especially those that fall outside of their conventional framework of thinking. Nassim Taleb calls such events "black swans," referring to the fact that prior to the 1600s, Westerners thought it impossible that swans could be any color other than white.[2]
Moreover, unlike the case of black swans, many people
already believe in, say, the Judeo-Christian God. The fact that there is serious,
nontrivial debate over the matter should give us pause. Even if we don't
adopt full-blown philosophical
majoritarianism,
it seems prudent to give some weight to possibilities
that large numbers of people strongly believe.[3]
Many disagreements, both about religion and other topics, are due to differing prior probabilities ("priors") held by different people. Formally, a prior represents a degree of belief that an individual holds before ("prior to") learning a particular fact; after the fact is learned, the individual updates his degrees of belief to yield a "posterior" probability distribution. For instance, suppose your friend Joe is about to flip a coin. You think Joe might be pulling a trick on you by flipping a coin with two heads, but you think he may also be flipping a fair coin. You might put 50% probability on each of these scenarios; 50% is then your prior probability that the coin is unfair. Suppose Joe flips the coin 20 times, and it comes up heads each time. This gives you new information that you can use to compute your posterior probability that the coin is unfair (which, in this case, is far higher than the original prior probability, at 0.999999).
Differing priors can cause disagreements. For instance, in the coin-flipping example, if your friend Bert thought Joe was definitely using a fair coin (i.e., if Bert assigned 0% prior probability to an unfair coin), then Bert's conclusion after the 20 flips would still have been that Joe was flipping a fair coin.
A natural question is, Are there other factors, besides differing priors, that can cause sustained disagreements? In 1976, Robert Aumann published an article, "Agreeing to Disagree" (JSTOR link), in which he showed that two Bayesian agents holding the same priors could not rationally disagree if they were fully aware of each others opinions (had "common knowledge"). As Tyler Cowen and Robin Hanson describe in their paper, "Are Disagreements Honest?":
Aumann's impossibility result required many strong assumptions, and so it seemed to have little empirical relevance. But further research has found that similar results hold when many of Aumann's assumptions are relaxed to be more empirically relevant. His results are robust because they are based on the simple idea that when seeking to estimate the truth, you should realize you might be wrong; others may well know things that you do not. [...] One of Aumann’s assumptions, however, does make a big difference. This is the assumption of common priors, i.e., that agents with the same information must have the same beliefs. (pp. 3-4)
What are we to make of differing priors? I'll discuss one answer. In "Uncommon Priors Require Origin Disputes," Hanson argues that "Bayesians who agree enough about the origins of their priors must have the same priors" (p. 1). In particular, "If event E were just as likely in situations where my prior had been exchanged with someone else’s prior, those priors must be the same regarding event E" (p. 5). Hanson elaborates:
Consider, for example, two astronomers who disagree about whether the universe is open (and infinite) or closed (and finite). Assume that they are both aware of the same relevant cosmological data, and that they try to be Bayesians, and therefore want to attribute their difference of opinion to differing priors about the size of the universe.
This paper shows that neither astronomer can believe that, regardless of the size of the universe, nature was equally likely to have switched their priors. Each astronomer must instead believe that his prior would only have favored a smaller universe in situations where a smaller universe was actually more likely. Furthermore, he must believe that the other astronomer's prior would not track the actual size of the universe in this way; other priors can only track universe size indirectly, by tracking his prior. Thus each person must believe that prior origination processes make his prior more correlated with reality than others' priors. (p. 2) [...]
Without some basis for believing that the process that produced your prior was substantially better at tracking truth than the process that produced other peoples’ priors, you appear to have no basis for believing that beliefs based on your prior, are more accurate than beliefs based on other peoples’ priors. (p. 6)
Suppose we accept Hanson's conclusion (whether on account of his argument or because of our own intuition about the matter). Now imagine that there are only two people in the universe, Meg and Sam, and that they're discussing two competing explanations of cosmological fine-tuning: (a) the universe is a multiverse and so is bound to contain some regions with physical constants hospitable to life, and (b) the universe's constants were set by a creator who has many of the characteristics associated with a traditional, monotheistic god. These hypotheses are not exhaustive; in particular, (b) leaves out a large subset of the space of possible universe creators. However, I assume for simplicity that Meg and Sam both assign all other hypotheses zero probability.
Meg assigns 99% prior probability to the multiverse hypothesis, and Sam assigns 99% prior probability to the creator hypothesis. Since both hypotheses explain fine-tuning about equally well,[4] these will also be roughly the posterior probabilities that Meg and Sam arrive at. Suppose Meg and Sam don't consider any further evidence. Then, to use colloquial terms, Sam will "believe in God" while Meg will not.
Meg and Sam may not agree on the exact mechanism by which they arrived at their priors, but they probably do agree on the essential assumption for Hanson's theorem: That nature would have been equally likely to have given Sam Meg's prior and Meg, Sam's. But in that case, Meg and Sam should not hold differing beliefs. They ought to adjust their priors until they agree, which presumably means Meg should increase her prior for the creator hypothesis by some amount.
In addition to differing priors, differing perceptions of evidence may cause differing views (in practice, if not in theory). Like people in the tropics who have never felt snow or young children who haven't fully experienced sexual desire, the atheist may regard God as absurd--unlike things he can see and touch--because he has never been in a position to experience God. Four to five percent of Americans are red-green colorblind; nine percent of Americans are atheists (of course, the worldwide percentage is a little higher). Might it just be that some people have a harder time sensing God than others? (This is not to deny the fact that, metaphorically speaking, those who can distinguish red and green still see those colors in different, mutually contradictory shades.) Are atheists willing to believe that their senses, intellect, and understanding of reality are so flawless that they can't possibly be deficient in certain areas where others' are less so?
"But," the atheist may reply, "we can look at the wavelength of light to determine whether it's red or green. With God, we have no such independent verification." When Bertrand Russell was asked what he would say to God if God requested a justification for his atheism, Russell replied, "Not enough evidence, God, not enough evidence." Yet isn't it only perhaps fortuitous that in cases like light and other physical phenomena we have non-direct means of corroborating sensory data? Why should we expect this to be true in general, especially when many people tell us that it isn't the case with religion? As Paul notes in 1 Corinthians (KJV 2:14): "But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned."
Finally, consider this fanciful analogy about paradigms: There are many black holes in the universe, but once a spaceship passes the event horizon of one of them, then no matter how hard the spaceship searches--north, south, east, and west--it won't find any other black holes (barring the possibility of wormholes and so on). Why are we so sure that we've fallen into the right black hole? As William James observed in his lecture, The Will to Believe:
We want to have a truth; we want to believe that our experiments and studies and discussions must put us in a continually better and better position towards it; and on this line we agree to fight out our thinking lives. But if a pyrrhonistic skeptic asks us how we know all this, can our logic find a reply? No! certainly it cannot. It is just one volition against another--we willing to go in for life upon a trust or assumption which he, for his part, does not care to make. […]
Evidently, then, our non-intellectual nature does influence our convictions. There are passional tendencies and volitions which run before and others which come after belief, and it is only the latter that are too late for the fair; and they are not too late when the previous passional work has been already in their own direction.
Conclusion
According to Norman Macrae's 1992 biography, John von Neumann--one of the founders of game theory--converted to Christianity on his deathbed on account of Pascal's wager. If you have found the above argument at all convincing, I encourage you to try believing in the Christian God, too. I myself had been an atheist for most of my life--ever since I was four years old and decided that religion was silly. I enjoyed reading articles by prominent atheists and engaging in debates in which I argued against the existence of God; I viewed religion as a byproduct of the same type of irrational biases that lead people to believe in fortune tellers or cults. Then around two years ago, I began to think about Pascal's wager. Initially this was an intellectual exercise, but as I pondered the question more, I couldn't help worrying whether I was wrong about atheism. I looked for ways to defuse the wager, and I read philosophical criticisms that claimed to render it invalid, but I was unable to convince myself. Finally, I decided to give in. I'm now glad that I did.
Accepting the wager is a first step, but it's important, at some point, to move beyond the mere intellectual position that it makes sense to believe in Christianity. God may not look favorably upon people who only claim to believe in him with the hope of avoiding hell. Rather, treat the wager as, in Pascal's words, an opportunity "to incite to the search after God." This may be God's way of starting you along the road to genuinely seeking him.
But how does one begin to bring about faith? Pascal offered the following advice in his Pensees:
Endeavour, then, to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith and do not know the way; you would like to cure yourself of unbelief and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. [...]
Now, what harm will befall you in taking this side? You will be faithful, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognise that you have wagered for something certain and infinite, for which you have given nothing.
Acknowledgements
I thank very much all of those who have taken the time to discuss these ideas with me. Their insights and objections have influenced and refined this essay.
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[1] Why Christianity? Well, this is just an example, but I do think Christianity is more likely true than most other religions. For one thing, it's believed by a large number of people (2.1 billion), including a fair number of scientists and intellectuals, which at least suggests that it hasn't completely collapsed under scrutiny. (I realize I'm being a little ambiguous about the definition of Christianity.
Does that label include, say, old-earth creationists as well as young-earth creationists? We can define it how we like, but for Pascal's wager, it's convenient to consider a definition like the following: Christianity
refers to the set of possible worlds in which exactly those people who refuse to trust in Jesus go to hell. There are certainly many old-earth creationists who hold such beliefs.) In addition to being popular, Christianity includes a number of reported miracles that need to be explained, such as Jesus's empty tomb. Finally, even if Christianity is no more probable than other major religions, I think it's still a good Pascalian choice because of its exclusivity and the infinite duration of its hell.
[2] I should qualify that it's not always
the case that we should account for overconfidence by increasing the
probability we assign to an outcome we perceive as unlikely. As
Eliezer Yudkowsky notes, there's a need for our probabilities to
sum to 1
-- there are simply too many possible hypotheses out
there for us to increase the probability we assign to each of them for
fear of overconfidence; instead we need to take into account both
human overconfidence and the desire-to-dismiss, and also the
temptation for humans to make up silly things with huge consequences
and claim 'but you can't know I'm wrong.'
I think Christianity (and other existing religions) deserve higher probability than some random member H of the space of possible hypotheses if only because the fact Christianity has even been thought of is nontrivial evidence. Indeed, I'd say that any idea that has ever been imagined, at least in our region of the universe, deserves higher probability, ceteris paribus, than H for the same reason. (I say at least in our region of the universe
because it's possible that every hypothesis that can be imagined in finite time has been imagined in some universe.)
[3] This is a rather general statement that needs qualification. It's true that, ceteris paribus, more people believing something should usually cause us to increase our credence in it more than smaller numbers believing it. But it's certainly not the case that every opinion should be given equal weight; for instance, people who tend to be wrong in their predictions in some areas might tend to be wrong more often in their beliefs about other areas, and we can use that information in weighing their opinions. (Thanks to others for emphasizing the importance of this point.)
All of this is really a round-about way of saying the following: Other people's beliefs are evidence that we can use to update our own probabilities, just like any other form of evidence. Larger numbers of people believing something tends to be stronger evidence because it's generally harder to explain how a whole bunch of people were misled than it is to explain how one or two people developed wrong beliefs. Giving more weight to people who make consistently accurate predictions makes sense because it's harder to explain how they came to hold false views on the belief in question. But these are just heuristics -- all we really need is Bayes' theorem applied to the evidence that other people believe what they do.
[4] Maybe this isn't quite true. For instance, many traditional monotheistic gods are benevolent and so might want to choose particular physical parameters that would lead to worlds that don't contain a lot of needless suffering; or maybe, if temporary suffering in the physical world is necessary to bring organisms closer to God, the opposite would be true. One can imagine a host of other considerations that might influence the relative likelihoods of hypotheses (a) and (b). I've ignored these complications for the sake of a simple but concrete example.