Pascal’s wager and infinity
July 6, 2009 — 9:55

Author: Alexander Pruss  Category: Afterlife Atheism & Agnosticism Hell Religious Belief  Comments: 14

(Cross-posted to my own blog.)

Some people, I think, are still under the impression that the infinities in Pascal’s wager create trouble. Thus, there is the argument that even if you don’t believe now, you might come to believe later, and hence the expected payoff for not believing now is also infinite (discounting hell), just as the payoff for believing now. Or there is the argument that you might believe now and end up in hell, so the payoff for believing now is undefined: infinity minus infinity.

But there are mathematically rigorous ways of modeling these infinities, such as Non-Standard Analysis (NSA) or Conway’s surreal numbers. The basic idea is that we extend the field of real numbers to a larger ordered field with all of the same arithmetical operations, where the larger field contains numbers that are bigger than any standard real number (positive infinity), numbers that are bigger than zero and smaller than any positive standard real number (positive infinitesimals), etc. One works with the larger field by exactly the same rules as one works with reals. This is all perfectly rigorous.

Let’s do an example of how it works. Suppose I am choosing between Christianity, Islam and Atheism. Let C, I and A be the claims that the respective view is true. Let’s simplify by supposing I have three options: BC (believe and practice Christianity), BI (believe and practice Islam) and NR (no religious belief or practice).

Now I think about the payoff matrix. It’s going to be something like this, where the columns depend on what is true and the rows on what I do:

BC 0.9X-0.1Y 0.7X-0.3Y -a
BI 0.6X-0.4Y 0.9X-0.1Y -b
NR 0.4X-0.6Y 0.4X-0.6Y c

Here, X is the payoff of heaven and -Y is the payoff of hell, and X and Y are positive infinities. I assume that the Christian and Islamic heavens are equally nice, and that the Christian and Islamic hells are equally unpleasant. The lowercase letters a, b and c indicate finite positive numbers. How did I come up with the table? Well, I made it up. But not completely arbitrarily. For instance, BC/C (I will use that symbolism to indicate the value in the C column of the BC row) is 0.9X-0.1Y. I was thinking: if Christianity is true, and you believe and practice it, there is a 90% chance you’ll go to heaven and a 10% chance you’ll go to hell. On the other hand, BC/I is 0.7X-0.3Y, because Islam expressly accepts the possibility of salvation for Christians (at least as long as they’re not ex-Muslims, I think), but presumably the likelihood is lower than for a Muslim. BI/C is 0.6X-0.4Y, because while there are well developed Christian theological views on which a Muslim can be saved, these views are probably not an integral part of the tradition, so the BI/C expected payoff is lower than the BC/I one. The C and I columns of the tables should also include some finite numbers summands, but those aren’t going to matter. A lot of the numbers can be tweaked in various ways, and I’ve taken somewhat more “liberal” (in the etymological sense) numbers–thus, some might say that the payoff of NR/C is 0.1X-0.9Y, etc.

What should one do, now? Well, it all depends on the epistemic probabilities of C, I and A. Let’s suppose that they are: 0.1, 0.1 and 0.8, and calculate the payoffs of the three actions.

The expected payoff of BC is EBC = 0.1 (0.9X – 0.1Y) + 0.1 (0.7X – 0.3Y) + 0.8 (-a) = 0.16X – 0.04Y – 0.8a.

The expected payoff of BI is EBI = 0.15X – 0.05Y – 0.8b.

The expected payoff of NR is ENR = 0.08X – 0.12Y + 0.8c.

Now, let’s compare these. EBC – EBI = 0.01X + 0.01Y + 0.8(b-a). Since X and Y are positive infinities, and b and a are finite, EBC – EBI > 0. So, EBC > EBI. EBI – ENR = 0.07X + 0.07Y – 0.8(b+c). Again, then EBI – ENR > 0 and so EBI > ENR. Just to be sure, we can also check EBC – ENR = 0.08X + 0.08Y – 0.8(a+c) > 0 so EBC > ENR.

Therefore, our rank ordering is: EBC > EBI > ENR. It’s most prudent to become Christian, less prudent to become a Muslim and less prudent yet to have no religion. There are infinities all over the place in the calculations, but we can rigorously compare them.

Crucial to Christianity being favored over Islam was the fact that BC/I was bigger than BI/C: that Islam is more accepting of salvation for Christians than Christianity is of salvation for Muslims. If BC/I and BI/C were the same, then we’d have a tie between the infinities in EBC and EBI, and we’d have to decide based on comparisons between finite numbers like a, b and c (and finite summands in the other columns that I omitted for simplicity)–how much trouble it is to be a Christian versus being a Muslim, etc. However, in real life, I think the probabilities of Christianity and Islam aren’t going to be the same (recall that above I assumed both were 0.1), because there are better apologetic arguments for Christianity and against Islam, and so even if BC/I and BI/C are the same, one will get the result that one should become Christian.

It is an interesting result that Pascal’s wager considerations favor more exclusivist religions over more inclusivist ones–the inclusivist ones lower the risk of believing something else, while the exclusivist ones increase it.

It’s easy to extend the table to include deities who send everybody to hell unless they are atheists, etc. But the probabilities of such deities are very low. There is significant evidence of the truth of Christianity and some evidence of the truth of Islam in the apologetic arguments for the two religions, but the evidence for such deities is very, very low. We can add another column to the table, but as long as the probability of it is small (e.g., 0.001), it won’t matter much.

  • Jacob

    While this is a neat analysis, it neglects context and history.
    You present the chooser as if he lacked a history and was operating outside of a cultural environment. In other words, there aren’t many Hindus that grew up in Appalachia–but there are a lot of Southern Baptists. The choice, if that is even a legitimate way to phrase the process of becoming a believer of some religion, is much narrower than you present.

    July 6, 2009 — 11:37
  • With the availability of the Internet, or just a good library, there are a lot of choices for the Appalachian. The self-interestedly rational chooser will, thus, be able to explore a wide variety of options. That the average person will not in fact engage in such a self-interestedly rational choice does not seem relevant to the question of what one it would be the case that one ought to do were the underlying decision theory correct (which it’s not).
    It’s worth recalling a point Rescher makes about Pascal’s wager: it is an argument aimed at a particular kind of character, a gambler who believes in making decisions in a self-interestedly rational way, and shows that kind of a character that consistency requires him to start going to Mass.

    July 6, 2009 — 13:13
  • Joshua Thurow

    Paul Bartha just read a paper at the Formal Methods in the Epistemology of Religion conference in Leuven, Belgium that attempted to do just this. I was there for it, but I can’t reproduce here all the details. The gist of his conclusion was this: the “many gods objection” is not a successful objection to Pascal’s wager and, just as you said, Alex, exclusivist gods (Bartha calls them “jealous gods”) tend to win out. The paper isn’t available online, but I bet he’d be willing to share it if asked.

    July 6, 2009 — 21:52
  • A many gods objection might carry a bit of weight in special cases, such as where one has multiple options with the same epistemic probability and analogous expected infinite payoffs. But in such a case, the self-interestedly rational thing to do is to choose the religious option that is most congenial in respect of this-worldly benefits; it is not self-interestedly rational to choose atheism.
    I wonder what the payoff table would have to look like for NR to be self-interestedly rational. I think it would have to have enough (with the count weighted by epistemic probabilities) columns with deities who prefer NR’ers to folks who chose the wrong religion. Orthodox Judaism may be like that: Orthodox Judaism holds that it is worse to violate a negative commandment than a positive one; the commandment to believe in no other gods is a negative commandment; the commantment to love God is a positive one; so accepting a religion that counts as believing in other gods is worse than atheism. On the other hand, neither Islam nor Christianity seem to me to be like that.
    My guess is that there is not going to be enough of that phenomenon going around for NR to be self-interestedly rational on any reasonable payoff table, however.

    July 6, 2009 — 22:43
  • Antony Aumann

    I wish to apply a standard worry about the Wager to your particular version of it.
    One interesting aspect of your version is that it turns on: (i) how one assesses the inclusivism/exclusivism debate; (ii) how one assesses the this-worldly benefits of C, I, and ND; and (iii) how one assesses the epistemic probabilities of C, I, and A.
    All three of these things are subject to change over time. I’ll focus on (i) and (ii). For example, let’s say I think exclusivism is the better interpretation of Christianity. In addition, in my current cultural context (and here such things *do* matter) BC>BI with respect to this-worldly benefits. On these grounds, I pursue BC. One day, however, two rather ordinary things happen. First, I read a book that rationally persuades me to embrace inclusivism re: Christianity and exclusivism re:Islam. Second, my employer transfers me to a location where BI>BC with respect to this-worldly benefits. If these changes are substantial enough in the relevant respects, following your version of the Wager will lead me to abandon BC and pursue BI.
    A commitment that can be revised this easily, and in response to such ordinary circumstances as moving, seems antithetical to the kind of commitment I take Christianity to require. Thus, if I build my commitment on your version of the Wager, I have a deficient commitment.
    I see at least two plausible responses to this worry. (i) You could claim that the Wager is only meant as a stepping stone to authentic commitment. It prompts you to go part of the way but needs to be augmented by other considerations. (Perhaps the H.S. does the rest?) (ii) You could ramp up your notion of BC such that it requires a point-of-no-return commitment (like marriage). Mundane changes like moving or reading a new book will not affect this kind of BC.
    Do you prefer one of these responses over the other? Do you have some third response? Is my initial analysis flawed?

    July 8, 2009 — 9:10
  • The this-worldly benefits question is easy: Unless the infinities manage to cancel out in the outcome calculations, the differences between BC, BI and NR will be infinite, and hence any finite this-worldly benefits will be irrelevant. It would take either complete symmetry between the cases (which one’s not going to have, because plainly not all apologetic and anti-religious arguments are of equal epistemic worth), or else a conspiracy between the payoffs and the epistemic probabilities, to get a case where the infinities canceled out.
    The question of commitment is very interesting. Note, too, that the same problem afflicts the evidentialist. I think Pascal actually may do a bit better than the evidentialist, thought that once you started going to Mass, using holy water, etc. (his examples), you would become a sincere and devout Catholic–action drives belief. So Pascal’s Wager was a way of turning a rational egoist, to whom these self-interest considerations would have to appeal, into someone who is no longer a rational egoist.
    This is related to David Gauthier’s argument that rational egoism requires that one commit psychologically to morality in such a way that one wouldn’t cease to be moral even in cases where crime would pay (from the egoist point of view). Gauthier thinks that there is no real contradiction in the idea that rational egoism calls on one to cease to be a rational egoist–rational egoism could still be true, even if it called on one to cease to believe it. (The same issue may come up with regard to utilitarianism.)

    July 8, 2009 — 14:04
  • Vlastimil

    Let me to ask again.
    You say (at ) that in some cases: “… infinitary considerations will swamp all the finite considerations. … the apparently relevant consideration … just drops out by the wayside (unless the infinitary considerations end up being perfectly balanced).” This could show that “… practical rationality requires that one assign non-zero probability to at most one … view …”
    Say Jack is a (believing and psychologically committed) Christian and even in fact always behaves accordingly (at least externally, from the third person point of view), as as a Christian.
    There is a positive probability Jack will go to heaven, there’s a positive and SMALLER probability he’ll go to hell, too. So, the infinitary considerations are NOT perfectly balanced.
    So, all the finite considerations are swamped.
    So, Jack should assign zero probability to his going to hell (?). This conclusion seems heterodox.

    July 10, 2009 — 8:40
  • Noah

    Hi Alex,
    I have always wondered at the basic premise of Pascals Wager. To me it appears the presume the ability to will yourself to a belief without supporting evidence.
    Is there a line of thought which simply sees the wager as invalid due to the internal contradictions inherent in attempting to ‘believe’ something you have no rational evidence for?
    To me, the act seems equivalent to trying to actively deceive yourself – ultimately self-contradictory

    July 16, 2009 — 16:13
  • Pascal in Pensees accepts that you cannot will yourself to believe something that you have no evidence for, the wager is about committing to taking certain actions; actions to commit oneself to a certain lifestyle such as, searching scriptures, participating as a member of a religious community, and so on. It is not about beliefs, your presumption as to its basic premise is mistaken.

    July 17, 2009 — 5:43
  • graham veale

    Dr Pruss
    I’m sorry to return to this. But I’d like to clarify my concerns about the “deviant gods” objection to the Wager. That is, a wagerer needs to consider “deities who send everybody to hell unless they are atheists, etc.”
    I’m reacting to Sober’s “Betting against Pascal”. He wants to separate
    p(G) – god exists – from p(P) – that a particular theology is true. So there may be a theology (X) in which God sends every Theist to Hell.
    But I argued on your blog that(G&X) is a contradiction. (G) should include the property “Worthy of Worship”, or (G) would not refer to God. The absurd theology needs to be compatible with (G). But surely there is no possible world in which (G) involves a being worthy of worship damning worshippers. So shouldn’t we assign assign (G&X) an epistemic probability of zero?
    You seemed to reply that we can alter (G), replacing it with (D), a “deity who is not God”. I don’t think that I made my objection to this very clear.
    Being “worthy of worship” depends on other properties. At the very least it depends on Power, Knowledge and Goodness (PKG).
    Something like the Beatific Vision is required for an infinite reward. That is, a reward that cannot be surpassed. A reward that cannot be surpassed requires (G), a God worthy of worship.
    The reward offered by the wager is infinite. We need a being capable of generating an infinite reward. Now a deity with (PKG) can generate such a reward, having the correct knowledge, ability and motivation to use the knowledge and ability in the best manner possible for humans.
    But I cannot see how a parody of God (D) can produce an infinite reward (a reward that cannot be surpassed). One reason for thinking so – (D) certainly cannot produce a reward on a par with a being “worthy of worship”.
    Another reason for thinking so: there is more to an ifinite reward than a neverending pleasant experience. A (D)that has granted us a neverending supply of sexual partners has done no more than turn us into a boar at an everlasting piggery.
    I cannot see how God parodies save the objection from Deviant Deities. Does my reasoning make any sense here?
    Graham Veale

    July 17, 2009 — 9:21
  • Graham:
    An intensively infinite reward can, perhaps, only be provided by a good God, yes. But an extensively infinite reward can be provided by a sufficiently powerful non-divine being (one with a lot of power, but not an infinite amount of it). To be able to have fun proving newer and more complex math theorems for eternity is an infinite good, because eternity is infinitely long. Likewise, such a being can provide an infinite amount of pain, e.g., by daily electrical stimulation of pain centers forever.
    Now you might be able to argue that an intensively infinite reward had for an infinite amount of time trumps any merely extensively infinite reward (i.e., a finite reward had for an infinite amount of time). Moreover, one might argue that suffering can only be extensively infinite (maybe there is a limit on how much a person can suffer–beyond a certain point, one ceases to be sensible to further increments of pain). If so, then you could make the following rather nice and interesting argument: The reward provided by beings like God trumps anything (reward or punishment) that can be provided by non-divine beings. I like these ideas. Email me, and maybe we can work on them some more together.
    But do notice that these arguments do depend on heavy-duty metaphysics (such as the metaphysics of the beatific vision) and axiology. It was one of the merits of the Wager that it didn’t require metaphysicking, fun as metaphysicking is. All it required was the idea of an infinite reward (or an infinite punishment or both), without further distinctions.

    July 17, 2009 — 10:24
  • Stephen Kirkby

    Dear Matt, Noah and Alex,
    I have wondered whether one’s decision to believe in God or the supernatural could be analogous to amending some kind of scientific or naturalistic evidence; for example: making an error in calculation, realising one’s mistake and correcting it. Would recognising the legitimacy of Pascal’s wager thus be like admitting a state of affairs in a similar vein?
    Any reply would be appreciated.

    August 8, 2009 — 23:38
  • Stephen I am not sure what you are getting at. Pascal’s wager is not about making a mistake and correcting it. It is about a situation where you don’t know which of two options is correct but in practice you cannot sit on the fence. The idea is that in such situations you make a decision based on prudence or self-interest given that epistemic considerations cannot decide the issue.

    August 10, 2009 — 5:32
  • Stephen Kirkby

    Dear Matt,
    Yes, please forgive me, I forgot to clearly state the argument I was commenting on. If you would allow me to quote you from one of your previous posts:
    ‘Pascal in Pensees accepts that you cannot will yourself to believe something that you have no evidence for, the wager is about committing to taking certain actions; actions to commit oneself to a certain lifestyle such as, searching scriptures, participating as a member of a religious community, and so on. It is not about beliefs, your presumption as to its basic premise is mistaken.’
    Many times I have heard the atheistic argument that one cannot will oneself to believe. However, I wonder if religious belief is at all analogous, contra Wittgenstein, to naturalistic, empirical or common-sense corrections one make’s to certain propositional convictions about the world. For example, a child might be convinced that Paris is the capital of Germany. Once her teacher disabuses her of this notion, she would see her mistake, and promptly correct her own belief system. Likewise, could not one recognise that religious belief is preferable, and thus amend one’s worldview, comparably with the child?
    Any reply would be appreciated.
    Thanks again.

    August 11, 2009 — 22:05