## Saturday, August 13, 2005

### Expecting Infinity

Whenever I hear about "Pascal's Wager", the phrase "worst philosophical argument ever" inevitably springs to mind. One of the main reasons for this close association is described here (in short, the argument idiotically assumes that Christianity and atheism together exhaust the theological possibilities). But there's also another problem with it, which I mentioned only briefly in that past post, and that is the following bit of reasoning:

Since Expected utility = Reward * Probability, and infinity multiplied by any nonzero probability is still infinity, we conclude that there's an infinite expected utility to believing in God [assuming that there is some non-zero probability (no matter how small!) of receiving an infinite reward for doing so].

Now, as Alan Hajek neatly pointed out, this argument "proves too much". For suppose I decide to flip a coin, and will believe in God only if it lands on heads. This process also has a non-zero chance of obtaining infinite utility (by the Pascalian assumption). So the expected utility of this course of action is, again, infinite. Or suppose I decide to believe in God only if I win the lottery next week. Again, non-zero probability => infinite expected value. Because infinity is involved, the lower probabilities make no difference at all. The maths implies that it would be just as rational to choose a one-in-a-million chance of infinite reward as to choose a guaranteed infinite reward. But that's clearly absurd!

Getting back to Pascal: whatever I do, there is presumably some (miniscule, but non-zero) chance that I'll end up believing in God because of it. So, if Pascal's wager is sound, we've just proved that going down to the pub for a beer has infinite expected utility. A drink sounds pretty good right about now. ;)

1. > the argument idiotically assumes that Christianity and atheism together exhaust the theological possibilities

But the argument only compares atheism with christianity.

For example if I was to say "2>1" saying "the argument idiotically assumes 2 & 1 exhaust all the possibilities" is only valid in the most vague of senses.

2. No, the argument compares belief in God with disbelief in God. If it's possible [i.e. non-zero probability] for a non-standard deity to give atheists infinite rewards, then Pascal's argument fails on its own terms. I explained all of this in the linked post. Further discussion of this point should continue there instead. I'd rather keep this thread on the topic of infinity.

3. BTW the first argument is only "clearly" applicable to (prodestant) christianity amongst the major religions because most other religions don't work via faith.

As for the second part of the argument if you had a beer you would have no reason to believe it would improve or decrease your likelyhood of believing it might work in the opposite direction. But I dont really think that was the main point.

Your argument to absurdity doesnt counter the argument it merely points out awkward implications of it (which would just have to be taken into account).

Anyway actions can only be compared with potential alternative actions and while 2 x infinity is infinity when you divide it by the same infinity to solve the equasion you will (I presume) get 2. Then you have lots of confusing equasions but no more confusing than the usual utilitarian ones.

"Or suppose I decide to believe in God only if I win the lottery next week. Again, non-zero probability => infinite expected value."

And these are not valid comparisons it is analogous to "suppose I decided to shoot myself if I did not win lotto"
"infinite potential loss"
then started debating the size of the bullets you would put in your gun.

4. Ok well we are discussing a certain interpretation of the argument then.

Your argument clearly still falls over regardless though, but we won't go into that.

5. I don't think you understand the argument I'm making. Let me try to clarify it. Pascal's Wager depends on something like the following principle:

(P) If an action A has a non-zero probability of yielding an infinitely valuable reward, then you rationally ought to A (or, more accurately, there is nothing else that you ought to do more than A).

This principle arises from applying the standard "expected utility" formula with infinity as a parameter, thus yielding an infinite expected value. From there one adds the general principle of rationality that one ought to maximize expected value. And obviously you can't get any more 'maximal' than infinity.

Now, the problem I'm pointing to is that every action A has a non-zero probability of yielding an infinite reward. Thus, applying principle P, we find that there is nothing else we ought to do more than A, for every A. In other words: it doesn't matter what you do, every possible choice has the same expected value: infinite. This is no mere "awkward implication" -- it completely undermines the Pascalian argument. (The conclusion that all possible actions are equally rational to undertake is about as absurd as you could possibly get!)

(NB: I'm assuming that no possibilities of infinitely negative utility are entering our calculations here. It isn't clear how to deal with them, i.e. whether they can 'cancel each other out', or what.)

Anyway, to avoid unnecessary complications, let us focus on the most obvious absurdity: the expected utility of flipping coins (or waiting to win lotto) before believing in God is calculated to be equal to the expected utility of believing in God straight off. Both are infinite. The difference in relative probabilities, because merely finite, makes no difference at all to the expected utility calculation. So obviously the formula is in error.

(You nonsensically assert that these are "not valid comparisons". I can't imagine why you would say such a thing. I'm just applying the maths to another possible situation: there's nothing one could legitimately object to here.)

6. As lovely as that argument is (I really like it), I don't see how the very last premise is correct: how can we assign a probability, even a miniscule one, to me coming to a belief in god from doing X where X has absolutely no relationship to belief in god? Absent some known causal relationship between beer-drinking and theism I think we'd have to say that we're unable to say whether beer-drinking has a non-zero probability of bringing me to a belief in god or not.

In sort of bastardized economic terms, this is the difference between risk and uncertainty: a calculable probability (normally those two terms are used about danger) is a risk, whereas an incalculable probability is an uncertainty, and uncertanties can't be used to determine expected utility.

7. "infinity multiplied by any [something] is still infinity"

As someone who once seriously considered becoming a professional mathematician (I have the degree and everything), this one really bothers me. "Infinity" is not a number. You cannot multiply anything by it at all. It's as nonsensical as saying "13 multiplied by frogs are pancake hooves." This part of the justification for Pascal's wager is a semantic error, not a logic error.

A number is not just some random thing written on a page; it has a specific definition and meaning(*). Multiplication is a specific and well-defined operation. I'm not going to attempt to define either one here, a) I don't have the appropriate reference books at my fingertips and don't trust my memory that well, and b) don't want to be typing all night.

When we talk about multiplying two numbers and don't specificy otherwise, we usually mean multiplication as defined in the field of real numbers, by which we mean the set of real numbers R together with the addition and multiplication operations as you were taught in school. (Subtraction and division are just special cases of addition and multiplication, of course). The real numbers, R, is a set containing all of what we normally think of as numbers, such as 0, 1, 1/3, -1, pi, the square root of 3, and so forth. So, unless you specify otherwise, it only makes sense to multiply numbers that are members of the set of real numbers R. There is no member of R called "infinity".

"Infinity" is the nouned form of the adjective "infinite", and means "the property of not being finite". A set may be finite, meaning, in non-technical terms (**), that if you started counting its members you would eventually stop, having counted them all. A set is infinite if it is not finite, i.e. if, if you started counting it's members, you would never count them all, ever. The real numbers R is an infinite set; {1, 2, 3} is finite. Even if you tried to use the word "infinity" to mean the size of an infinite set, this is still a mistake, since there are many different sizes of infinite set. One of the most famous proofs in mathematics is that the real numbers R is strictly larger than the natural numbers N = {1,2,3,...}, even though both are infinite sets.

It is possible to consider a set R+, which contains all of the real numbers plus two more elements called 'Z' and '-Z', with multiplication extended so that for any x in R+:
- if x>0 in R, x * Z = Z and x * -Z = -Z
- if x<0 in R, x * Z = -Z and x * -Z = Z
- 0 * Z = 0 and 0 * -Z = 0
- Z * Z = Z, Z * -Z = -Z, and -Z * -Z = Z
Some people are even reckless enough to do so, but use the names 'infinity' and '-infinity' instead of 'Z' and '-Z', a nomenclature that I consider at best dangerously confusing. This is the only situation in which I have ever heard of a set with a member called infinity on which multiplication was defined. There are serious problems with R+ though (for instance, Z does not have a multiplicative inverse 1/Z such that 1/Z * Z = 1), which are very hard (impossible? I don't remember) to resolve. Certainly, such a set should never be used for any sort of economic or probabilistic modeling without an extremely good reason.

It would be a real (if small) benefit to the general quality of human thought if none of you who read this ever used the word "infinity" again, instead structuring your thoughts and sentences to refer to things as infinite.

(*) Philosophers (and mathematicians) used to argue about the meaning of numbers and what exactly was meant by "two" and so forth, but in the 20th century mathematicians came up with pretty good working definitions that, as far as I'm aware, satisfy all the philosophical requirements for the concept of numbers.

(**) The technical definition, as I was taught, is that a set is finite if there exists a bijection from it to one of the natural numbers N. This relies on a particular definition of N, namely that 0 = {}, the empty set, and n+1 is the set that contains n and all the elements of n. So 1 is {{}}; 2 is {{},{{}}}; 3 is {{},{{}},{{},{{}}}}, and so forth. The important thing about the definition is that the set called 0 has 0 elements, 1 has 1 element, 2 has 2 elements, and so forth. This stuff fascinates me, but I'll shut up now, really. :)

8. The key problem seems to be that you understand infinity.

> Anyway, to avoid unnecessary complications, let us focus on the most obvious absurdity: the expected utility of flipping coins (or waiting to win lotto) before believing in God is calculated to be equal to the expected utility of believing in God straight off.

No it isn't. Im strugling to think of a way to explain to you how it works since when one is fighting both lack of understanding and self jsutification at the same time it gets pretty hard. It is very similar to arguing with a creationist about evolution.

You probably went to maths class and were told that infinity divided by infinity is not 1 which is largely true, but that is because infinity obscures information - infinity is more of a class of numbers rather like "real numbers" than an actual number.

thus infinity times 2 is infinity but it is not THE SAME infinity. In maths it may be next to imposible to identify these infinities and thus we will probably say the answer is intdeterminate (or sometimes we could cancel them) but in the case of a hypothetical we can measure them against eachother.

If I was to divide the sum of all positive real numbers by itself I get 1 right?
if I was to divide the sum of positive real numbers by the sum of whole numbers I might still get infinity.

here is a link that explains it a bit

http://mathforum.org/dr.math/problems/parkinson10.26.html

infinity minus infinity or divided by infinity is undefined because not enough information is supplied. In maths or physics one may cancel infinities that are the same just one has to consider the other possibilities.

9. There is a non-zero probability that I have a direct line to God and can put in a good word for you. After I put in this good word, your chances of going to heaven will increase by a non-zero amount. My fee for doing this is a mere \$1000 dollars, very reasonable under the circumstances. Please pass this on to anyone you know who thinks Pascal's wager is a valid argument.

10. Haha Nigel,

1) My money is already fully utilized involved in other non zero probabilities
2) I am not fully rational anyway.
3) I have reason to suspect you will use that money for satanic purposes.

However I will spread the word eh?

11. BTW I wrote my post before covaithe posted his. I will take his into consideration for later posts.

12. Hey, how do you add the continue reading thingy? That's cool!

13. Laura - see expandable posts.

Covaithe, you're quite right, of course. You could look at my post as illustrating the absurd results that occur when the Pascalian reifies the infinite so as to use it in expected utility calculations. I was wanting to give an intuitive demonstration that it doesn't work. Mathematicians will prefer more formal accounts, and that's fine by me.

(Though I think it isn't quite enough to say "but infinity is not a number in our mathematical formalisms." One might respond, "so much the worse for our mathematical formalisms -- they've simply failed to capture our intuitive concepts here!" But I won't pursue this point, because I do think our formalisms are right, and the commonsense treatment of 'infinity' as a number is incoherent. I'm just wanting to point out that this would require further argument, if someone disagreed with us here.)

Actually, the Pascalian problem could be restated in more formally acceptable terms anyway. E.g. replace every sloppy instance of the word 'infinity' with the more formal 'limit of n as n tends to infinity'.

Nigel - clever :)

Genius - "when one is fighting both lack of understanding and self jsutification at the same time it gets pretty hard."

Indeed.

Although it is possible in maths to get differently sized 'infinities' (to lapse back into sloppy talk -- sorry Covaithe!), that is not what's happening in my examples.

Consider the following:
(1) The limit of n as n tends to infinity.

(2) The limit of n/2 as n tends to infinity.

These two 'infinities' are of exactly the same cardinality (size). If we put all the 'n's in one set, and all the 'n/2's in another set, we could put the two sets into one-one correspondence (which is the definition of being equinumerous). Odd as it sounds, we can prove that there are just as many odd natural numbers as there are (odd or even) natural numbers.

Here's the upshot: if we have a sequence of numbers that tends to infinity, and we divide each of those numbers by any finite positive amount, then the new sequence still tends to infinity. It's no different. So if you're wanting to take the limits of these sequences to plug into your expected utility calculations, you're not going to get different results, no matter how much you want to.

14. Correction: "These two 'infinities' are of exactly the same cardinality (size)."

This is muddling up the size of the set with the value of its contents. That is, the sets have an equal number of elements. But what we're interested in is the value of the corresponding sequence as n tends to infinity. I haven't shown that the values will be the same. (I'm not quite sure how to do that.) But intuitively, it does seem right to say that they will be.

15. Richard,

I've never found Hajek's take particularly convincing; and I don't think (P) actually captures what most people who accept the Wager are thinking. Pascal-type Wagers are comparative; since Wagers start out from a position of relative skepticism -- as Pascal himself notes, the Wager assumes (at least for the sake of argument) that we are in a position of minimal evidence (we know the options, but we have nothing to tell us which is more likely to be true), and most Wager-users, despite other major departures from what Pascal actually says, follow him in framing the argument in this way -- the only question is which of these options is more rationally preferred over the other given that we lack inside evidence. And this involves much more than what you are considering. The options are compared in terms of possible loss, possible gain, what we stake (which, of course, is not the same as what we stand to lose), and the risk in our staking it; and the option that should be rationally preferred is the one that comes out better when all of these are taken into account. So (P) is missing consideration of what we will be staking, of what we will potentially lose (which, of course, might be more than we are actually staking), and of the probability of our losing it. Likewise, since the argument is practical (it is asking what we should do if we assume, even if only for the sake of argument, that it is necessary make a decision) you have to keep real-life decisions in mind. For instance, the Hajek argument muddles together subordinate and superordinate options; it is pointless to take into account means to an end if you haven't yet decided on the end. So the question of whether we should (for instance) wager on going-to-the-pub-to-get-God's-infinite-reward as opposed to, say, going-to-church-to-get-God's-infinite reward, can only come up as an issue if we have already Wagered in favor of God's existence because of the infinite reward.

And since the ultimate goal is not to make a decision matrix but to make a decision, anything useless to the goal of actually deciding is set in brackets (if the possibility of gain doesn't seem to do anything to contribute to preference one way or another, it is pointless to try to make it do so in one's reasoning about the decision). Hajek makes the mistake of thinking that the point of the argument is a decision-theoretic conclusion; whereas any decision-theoretic conclusions are used strictly to the extent that they're useful for coming to an actual, practical decision.

16. Brandon, when you speak of making a "practical decision" rather than a "decision-theoretic conclusion", do you mean that you reject the formal apparatus of expected utility calculations, and so forth?

If so, how are you now going to get from the theoretical premise (God-belief has non-zero probability of infinite reward) to the practical conclusion (you rationally ought to pursue God-belief)? Do you think you can bridge this gap in a way which won't allow Hajek's lottery ticket to follow?

17. > These two 'infinities' are of exactly the same cardinality (size).

I am aware of this aspect of infinity and it is, strangely enough, irelevant. What you prove is they are both "infinity numbers" and infinity number can be matched in that manner - but that is not what the utility equasion will resolve to so it doesnt matter.

Another way of looking at it is to say that the utility of being rewarded by a god is 1 the utility of being rewarded on earth is limit 1/n n--> infinity ie tends to zero. Problem solved you have the superordinate version.

Or another way to point it out is to just ascribe reward from god a value of lets say 10^220 units (i cant think of any practical use for a larger number but we could change that to 10^220! to cover combinations) and see what choices the answer tends towards. I expect you will find no significant change to 2X being greater than X or X+1 being greater than X. Or you could jsut arbitrarily define infinity as "the largest finite number" forget about the excess and slap it into the equasion.

It seems to me what you have been doing here is just confusing yourself - it is the equivilent of Zeno saying that archillies will never catch the tortoise.

I also note you have not even tried to adress the argument you have just resorted to noting that you can match infinities - the method used to define infinities.

18. Genius, it isn't clear to me that you have an argument. I'm not sure what it is I'm supposed to be responding to.

The Pascalian depends upon the fact that it doesn't matter how unlikely the divine infinite reward is -- so long as it's non-zero, he can (he thinks) plug it into the EU formula and get his desired result. What I'm saying is that you can do the very same thing with less-probable variations on the same action (e.g. the coin-toss decision procedure).

If the probabilities don't matter, then the Pascalian argument generates these absurd results I've highlighted. If the probabilities do matter, then the original Pascalian argument never would have gotten off the ground.

Now, how does anything you've written impact upon this argument?

19. No offence but you don’t seem to be trying to argue with Pascal’s argument you are just attempting to prove if it is true then the universe is counter your intuition. Well er... no kidding!

But to deal with your "absurd" examples

Most religious people I know easily accept that religion is the most important issue and that theoretically having a beer MIGHT be the best strategy. I am a bit surprised you are finding it absurd furthermore the others are MUCH easier to shoot down.

"Or suppose I decide to believe in God only if I win the lottery next week. Again, non-zero probability => infinite expected value."

Go tell that to an evangelical Christian if he is smart he will say "yes good idea"

Then say

"For suppose I decide to flip a coin, and will believe in God only if it lands on heads. If so I will believe in god."

He will say "fantastic! If it is tails, can we do it again?"

Go for it - your going to make a real happy evangelical Christian!

If you can think of an actual absurd sitaution I will look into it but this falls far short of the ones I get to look at to attack utilitarianism.

20. The condition we're imagining is that I will "believe in God only if it lands on heads." If it lands on tails I will not believe in God. End of scenario. (I have no interest in whether the evangelical is thrilled or not. The question is whether such behaviour would be rational. I do not take an evangelical's excitement to be a reliable indicator of an action's rationality.)

You're really missing the point quite badly. I explained it all in the main post. Look, here it is again:

"Because infinity is involved, the lower probabilities make no difference at all. The maths implies that it would be just as rational to choose a one-in-a-million chance of infinite reward as to choose a guaranteed infinite reward. But that's clearly absurd."

As for your claim that the absurdity is merely (and unproblematically) "counter-intuitive", re-read the second comment I made in this thread. I'm getting tired of repeating myself.

21. > The condition we're imagining is that I will "believe in God only if it lands on heads."

OK i will be clear - yes it is rational. How can you possibly have any issue with it?

22. The coin flip means I'm only half as likely as I otherwise would be to get the infinite reward. That's quite obviously irrational. To make it even clearer:

Do you think it would be just as rational to choose a one-in-a-million chance of infinite reward as to choose a guaranteed infinite reward?

23. > "The maths implies that it would be just as rational to choose a one-in-a-million chance of infinite reward as to choose a guaranteed infinite reward. But that's clearly absurd."

No it isnt absurd. it is just as rational to take 1 million dollars as it is to take 1 billion if it is offered to you. ie any rational person would take both.

Now you come back with "but if i take the million i refuse to take the billion" and I reply with, ok you are not rational then. You have taken an option with negitive return because it includes the refusal to take the billion dollars.

(of course it is your later refusal to take 1 billion that seems particularly strange)

I cant see why this is hard to understand.

24. > The coin flip means I'm only half as likely as I otherwise would be to get the infinite reward.

so you are saying you can resolve the equasion as "if i score tails I with get reward if I get heads or dont flip the coin I will"

Flipping the coin thus has negitive utility.

25. I mean if you get tails you wont get reward

26. I wasn't talking for myself but for most people who use Wager arguments. They don't reject the formal apparatus; their use of it is governed, however, by the supposition that an actual practical decision needs to be made, come what may. And therefore there's a lot that's actually going on outside the formal apparatus. This is actually clear simply from the application of the apparatus itself, for it's generally recognized that one has to construct multiple decision matrices to evaluate the Wager; how these interrelate and precisely how they should be constructed depends on a number of suppositions the formal apparatus doesn't involve. So the actual matter of coming to a decision involves more than constructing a decision matrix or two; it involves coordinating them, and if they are not equally important to the decision, weighting the conclusions, etc. For instance, most people who accept Pascal-style arguments are also Jamesian; that is, they'll only look at options that come up as live options for real-life practical purposes.

The lottery ticket case arises only on the assumption that you've already isolated as an option for practical consideration wagering on God only if you win the lottery; which means (presumably) that a higher-level decision-theoretic inquiry has already been made (if not, then the practical reasons for picking out this as the option would have to be gleaned from the context). It is not a wager about God; it's a wager about how to wager about God (lottery-ticket wise), and would have to be compared to other wagers of this sort (e.g., dice-wise, coin-wise, asking-a-random-person-wise, reading-tea-leaves-wise) in light of the goal of that type of decision-making (which is not the same as in the case of Pascal-type Wagering) and the type of reasons that are both available and relevant to such decision-making (which wouldn't necessarily be the case). So the lottery ticket case doesn't even to be a part of Pascal-type wagering. Indeed, the Pascal-type Wager may be one of the options of this sort of decision; the Pascal-type Wager is one way to wager about God. But the whole decision-matrix for the lottery ticket case becomes otiose if you have reasons to prefer Pascal-type wagering in the first place (and Pascal-type wagerers do have such reasons, e.g., the reasonableness of actually thinking through the matter to see what you might decide on the merits of the options themselves, before going ahead and deciding on the basis of something as dubious as a coin or lottery ticket). Hajek's argument makes the error of assuming that arbitrarily constructed decision-matrices will always be relevant to the practical wagering. Whether they are relevant actually depends on the practical situation and the goals of one's inquiry.

27. Look, it really isn't that complicated. Here's the problematic reasoning:

1) If given an exclusive choice between two options, it's rational to take the option with the highest expected value.

2) The expected value of a guaranteed infinite reward is no greater than the expected value of a 1-in-a-million chance thereof.

Therefore,
3) It's no more rational to choose the guaranteed infinite reward than it is to instead (note that these are meant to be exclusive choices!) choose the improbable one.

Clearly (3) is false. The problem lies in (2). But (2) follows from the Pascalian principle (P) I described in my second comment.

One final point: your recent comments confuse a negative return with the absence of a positive return. These are not the same thing. I suspect this is what's muddling you up.

Now, for goodness sake, just stop and re-read the main post and the comments so far. If you still can't follow the argument, I can't be bothered explaining it any further. This is getting tiresome.

28. (Sorry Brandon, my previous comment was addressed to Genius.)

29. (which wouldn't necessarily be the case)

Whoops; I meant "(which wouldn't necessarily be the same as in the case of Pascal-type wagers)." The point is that (for instance) we are not considering the sorts of gains, losses, risks, and uncertainties relevant to wagering about God; we are considering the sorts of gains, losses, risks, and uncertainties relevant to wagering about how to wager.

30. Brandon, that's an interesting point. I'm suspicious of it, because I would want our formalisms to be universally applicable (such as general expected utility formulas which can applied to any specific situation). But the idea of separating out different types of practical decision, and assessing them against distinct rational standards, is an intriguing one. I guess it could potentially get the Pascalian out of the problem I've raised here.

But how would the Pascalian motivate the original wager, without appeal to some general principle along the lines of (P) above?

31. richard,

Please make up your mind what you want me to deal with and I will deal with it. There is absolutly no lack of understanding - I know what you are saying - it is just you who dont know what I am saying, apparently. And have probably stopped reading the posts properly.

32. Richard, I can understand the suspicion, but it does make sense: we want our formal apparatus to be universal, but not formal in the sense that we want it to cover everything; rather, we want it to apply exceptionlessly to a particular domain. And decision theory is generally recognized to presuppose certain things in any given case (it doesn't tell us how to determine the probabilities, for instance). Also, no formal argument of itself tells you what to do with itself; what we do with a given formal argument is another question, depending on what our purposes in appealing to the formal argument at all are (and there are other considerations, e.g., depending on the risks and utilities, it's possible that decision theory itself might suggest that decision theory is not the most rational way to make a decision in a given case). But in a sense, one doesn't even have to step outside decision theory to give the Pascalian room to maneuver in the face of something like the lottery ticket case; because we can't assume the decision matrix for wagering about how to wager will turn out to be an implicit decision matrix for wagering about whether God exists. It depends on how one evaluates the risks, gains, etc. of wagering about how to wager.

The question of motivation is an interesting one. I assume it varies from Pascalian to Pascalian. It's noteworthy, though, that in the notes we have of Pascal's own version, it's the agnostic who starts the discussion by saying that since reason doesn't tell us whether God exists, Christians aren't justified in believing that God exists. Since all we have are fragmentary notes, we don't know if this is how the dialogue Pascal intended to write would have actually started; but it wouldn't be surprising if it did. So, in Pascal's case, what motivated his argument was what he thought a particular argument was a likely objection to Christian belief in his own culture; and he responds to it in terms that he thinks the objector would be likely to understand. Pascal himself is very clear that Christians are not in the state of minimal evidence assumed by the Wager; while he allows that there are lots of uncertainties, he is very clear that there is 'inside information'. We have two things to lose (the true, which pertains to reason, and the good, which pertains to the will) and two things to stake (knowledge, which pertains to reason, and happiness, which pertains to the will); the Wager only deals with the will, because he's addressing an agnostic, and one of the things he tries to do with it is to give the agnostic a reason to take more seriously the inquiry into whether Christianity is true (i.e., if the agnostic is somewhat persuaded by, but still doubtful about, the Wager, he should seriously and open-mindedly investigate claims to additional evidence that might make the choice easier by suggesting what is actually true).

I suspect that most Pascalians, though, just assume (1) that people are interested in the question of whether God exists or not; and (2) they do not have good reasons for either option. (1) is plausible; (2), depending on the actual context, could be much more dubious.

33. BTW brandon seems to be right (about "pascalians") and I generally support his main point and believe that it is fundimentally superior approach.
Also I note (2) is flawed as I explained a couple of times previously and "one final point" is also pretty intuitively flawed in the context that you are using it - as surely you can tell.

34. Brandon is better at academically explaining his position in philosophy than me it seems - and his name makes people less defensive and promotes less arogance. Having said that I dont mind arogance in people I debate - well not too much anyway.
I could almost accuse brandon of taking toastmasters !

35. "the Pascalian problem could be restated in more formally acceptable terms anyway. E.g. replace every sloppy instance of the word 'infinity' with the more formal 'limit of n as n tends to infinity'."

Exactly. :) Your discussion of limits of the sequences n and n/2 is precisely how one should deal with this question, and leaves the rest of your arguments intact. (And convincing!)

I might rephrase one of your sentences as "if we have a sequence of numbers that increases without limit, and we divide each of those numbers by any finite positive amount, then the new sequence still increases without limit." "[Tends|goes] to infinity" is of course a common phrase in mathematics when dealing with sequences; it's how most people, even professors, pronounce the notation "n --> (sideways figure-eight)", but I try to avoid it whenever possible, preferring "n increases without limit". Saying that sequences "tend to infinity" suggests that there's some sort of goal (or limit) that they are approaching, called "infinity", when actually the sequence does not have a limit.

This rephrasing is also relevant to your next comment, wherein you're not quite sure how to show that the "values" of the sequences are the same as they "tend to infinity". Talking about the "value" of the sequences themselves makes it sound like you're talking about some kind of limit or limit-like property, when of course neither sequence has a limit. The property that I suspect you're trying to describe is that neither sequence has an upper bound. When I use the "increases without limit" phrasing, these things all seem much clearer to me.

36. Thanks Covaithe, your rephrasing is very helpful in clarifying these things. :)

37. Genius: No toastmasters, but my inability to say something concisely goes a way toward it - if I'm going to be wordy anyway, I might as well make the best of it!

38. I've been thinking more about your original post and trying to figure out exactly how the expected reward argument falls apart the way you point out. I think I have an idea: this formulation of the expected utility of believing only takes into account the (possibly infinite) reward if that belief turns out to be correct; it ignores the (also possibly infinite) punishment if that belief turns out to be wrong. The expected utility calculation shouldn't just be p*Reward, it should be p*Reward + (1-p)*Punishment.

What follows is a simplified attempt to calculate Pascal's wager this way. I don't think it works, and I'm not sure that any useful conclusion can be drawn from it, but it may be interesting. Or maybe not. :)

Consider a very (very very) simplified universe in which there are only three possible belief choices: religions A, B, and C.

Religion A says that if you believe in its god, when you die you will feast forever with him in the hall of heroes, which is supposed to be a really great party. If you don't believe in it, you will be cast out into the cold emptiness outside the hall to starve and freeze forever.

Religion B says that if you believe in it, when you die you will sing the praises of its god forever, thinking pure thoughts and basking in his radiance, but if you don't believe, you will be cast into a burning hellpit to be consumed by flames forever.

Religion C is atheism, which should be fairly self-explanatory.

Suppose that a person named Fred in this world is trying to decide which religion to believe in. After thinking about the various religions for a while, he decides that the probabilities are as follows:
p(A) = the probability that A is true = 0.3
p(B) = the probability that B is true = 0.2
p(C) = the probability that atheism is true = 0.5
Suppose further that Fred really likes parties, isn't very afraid of the cold, thinks that singing and pure thoughts sounds kind of boring, but is really afraid of the idea of hellfire.

I suggest that Fred might model the expected utility of believing in, say, religion R as follows. First I'll consider what I call the nth partial reward for if R turns out to be true. Obviously this reward is different for believers and nonbelievers. I arbitrarily set PR(n,A,believers), the nth partial reward for correct believers if A turns out to be true, to 100*n. Since Fred doesn't like singing and thinking pure thoughts as much as he likes parties, I'll set PR(n,B,believers) = 50*n. And the reward for atheism is nothing, for believers or nonbelievers: PR(n,C,believers) = PR(n,C,nonbelievers) = 0. Now for the negative rewards. Since Fred is very afraid of hellfire, I'll set PR(n,B,nonbelievers) = -1000*n. He's not so worried about cold and hunger, so PR(n,A,nonbelievers) = -50*n.

Now we can get an nth partial expected utility for believing in a religion X. I'm not going to write out the general expression because without a chalkboard, the notation is horrible. So here's the expression for the nth partial expected utility for believing in A:
Util(A,n) = p(A) * PR(n,A,believers) + p(B) * PR(n,B,nonbelievers) + p(C) * PR(n,C,nonbelievers)
= .3 * 100n + .2 * -1000n + .5 * 0
= -170n
Similarly,
Util(B,n) = .2 * 50n + .3 * -50n + .5 * 0
= -5n
and
Util(C,n) = .5 * 0 + .2 * -1000n + .3 * -50n
= -215n

Then I can take the final expected utility as the limit as n increases without bound of the partial expected utilities. It turns out that with the coefficients I've assigned, all the utilities are infinitely negative. Looks like poor Fred is doomed no matter what he does.

Some observations.

1) No matter what probabilities you assign and how you weight the various rewards, all of your final expected utilities are going to come out infinitely positive, negative, or zero in this model. In retrospect this doesn't seem surprising; we're talking about eternity after all. It's possible to compare unbounded sequences using l'Hospital's rule (which I should have remembered earlier), but in this case that puts us right back at the nth partial utilities. Which, maybe, is where we should be: it's not necessarily unreasonable to talk about comparing eternities by comparing finite slices of those eternities, especially when the eternities in question are supposed to be unchanging. In that case, this model shows that for Fred, he should pick religion B.

2) The probabilities and the weighting of the various and punishments in this model are completely subjective. If Fred hadn't been so afraid of fire, it's quite possible that A would have come out on top.

3) If we only consider religions that reward believers and punish nonbelievers, then it's impossible for atheism to come out on top in this model. Unless there is at least one religion that punishes believers and rewards nonbelievers, atheism will always have the lowest partial expected utilities of the available options. Even if you assign atheism a finite reward (e.g. if you claim the atheist has a more satisfying life or something), this disappears as soon as limits come into play.

If nothing else, this example convinces me that actually trying to calculate Pascal's wager for any actual religion would involve so many subjective, unverifiable numbers that any answer would be pretty much useless.

39. Covaithe! Usually one tries to avoid having to do maths with infinity but you seem to be intentionally spreading it right through your analysis!

Let’s say I have your problem

And I take the option 1 -170n
Option 2 -5n
Option 3 -215n

Now I choose my scale of utility and my point of zero utility. My choice of scale determines the size of the numbers and my choice of "zero" determines their size. I can define zero as lets say "1 billion n" in this case the problem (rounding) looks like this
Option 1 = "negative 1 billion n"
Option 2 = "negative 1 billion n"
Option 3 = "negative 1 billion n"

So the most logical scale in decision matrix would seem to be one where you place the zero in the middle because that is one of the few options that does not obscure the decision (besides it doesn’t offend the theory of relativity). In this case the numbers rattle off in different directions and one can hover around at zero if you want.

You can apply the same theory in reverse to any problem in order to make it unsolvable.

Let’s say I have a choice between eating cheese and eating ice-cream eating ice-cream has a utility of 1 "unit" and eating cheese has a relative utility of -1 "unit". So I will reset the two utilities reversing what we did above and say cheese is 1 "unit" and ice-cream is 3 "units" (still the same incentive to choose ice-cream over cheese) but then we wonder how big a "unit is" we discover that there is an infinite number of up units in it and that a sub unit is just as reasonable as a "unit" as a measurement - so we define it as 1*n as limit n-->infinity and 3*n as limit n-->infinity. Now 1*n infinity can be described as an infinity so we conclude each option has a reward that can be called an infinity and we cant solve for an answer

But one wonders why one would try to make a problem harder to solve, but then just after that you do actually solve it…..

> It's possible to compare unbounded sequences using l'Hospital's rule

Cripes that was easier than I thought – anyway this is another way to conceptualize what I was adressing above from anotehr angle.

> The probabilities and the weighting of the various and punishments in this model are completely subjective.

Indeed

> If nothing else, this example convinces me that actually trying to calculate Pascal's wager for any actual religion would involve so many subjective, unverifiable numbers that any answer would be pretty much useless.

Hmm your arguments don’t seem to have lead entirely in that direction to me but having said that I can see how you could have concluded it and it is a reasonable position Infact it is a pretty bullet proof - but that relies on you successfully becoming a....... err...... what do they call those people who don’t believe anything because they believe it is all useless and not understandable and generally all depressive? Nihilists?

That is unless your point is that you have an objective, easy and superior method for determining what religion is correct in which case that does indeed clearly obsolete the argument - by the way please tell me !!!!

40. The reason I included all the limit stuff in the calculation is that I was trying to model an infinite (eternal) reward/punishment. The original Pascal's wager (as described by Richard) relies heavily on the fact that the reward is infinite. I was trying to include discussion of the punishment too, while leaving the "infinite" part of the argument intact. Perhaps it would have made more sense to call the variable "t" and claim it referred to time, but oh well. Ice cream is, of course, heavenly, but unfortunately it is finite, and unless you're talking about some sort of Zeno's paradox in ice cream, I don't see that it makes sense to subdivide it infinitely.

"but that relies on you successfully becoming a......Nihilist"

No, I'm not claiming that it's impossible to pick a religion at all, just that Pascal's wager is not the way to do it.

"That is unless your point is that you have an objective, easy and superior method for determining what religion is correct in which case that does indeed clearly obsolete the argument - by the way please tell me !!!!"

Hehe, no, I can't help you there. I do suggest, however, that everyone interested in religion should check out the Flying Spaghetti Monster.

41. You have run across the idea that god is a Logical Positivist? This god designed the universe carefully to give no hint of divine intervention and gets angry at people for believing in the unsupportable idea that it was created by a god.

42. As to the theme of this thread, besides the problem of whether the expected value is well defined, I think that what is overlooked is the problem that this is a one-off offer. Expected value reasoning works well when there are repeated cases, but may not be the best tool to get at what you want.

I’d rather take a 1 in 10 chance of winning a million dollars instead of a 1 in 100 chance of winning a billion. (Ignore for the moment that my utility is not linear here—it is not.) Although the billion dollar wager has a expected value 100 times higher, I place more value in merely winning at all. So I’ll take the wage that gives me 10 times better odds.

Having an unlimited reward makes this particular problem worse to the point of making the process meaningless.

43. driftwood,

first post - My main concern with your theory (which is possible) is that it sounds like the main justification is "it is what I do". Believing in the theory is, it would seem, a gross breach of your theoretical god's rules and not very disimilar to concepts like "I am the messanger for god".

Tough isn't it? ;)

Second post - it is very difficult to properly calibrate your utility scale conceptually - it is very tempting ot start talking about two icecreams being twice as good as one or somthing along those lines. I think that is the problem you are having.
It is posible for you to argue that it is not possible to have a utility ten times more than you winning 1 million dollars - I think this is the point you are really making.

furthermore I would have thought a utilitarian should not maximize utility across just his own life he should maximize it across EVERYONE's life. So the numbers are repeated - if it is infinitly unlikely you would require an infinite number of incidents or maybe greater than an infinite number of incidents - but that is possible (I am thinking of physics as opposed to just philosophy).

44. My point was just the simple one that a single number--exected utility--doesn't account for everything that we are interested in. In this "save your soul", I was assuming that each "soul" had one shot at the lottery based on what they believed at time of death. How other "souls" fair is of little interest.

So all I was pointing out is that it seems the odds of a "win" are of more concern than the size of the payout.

45. That's a really good point, driftwood, and may well be why expected utility calculations lead to such odd results.

46. "it seems the odds of a "win" are of more concern than the size of the payout."

I think our intuitions are corrupted by diminishing marginal utility, even when we try to stipulate that DMU won't apply in a particular example. It's just so fundamental to the way we think: the difference between benefits of 1-100 seems of greater significance than the difference between 101-200, even though an appropriate definition of utility guarantees that they are (objectively) exactly the same.

So if one ever came across a magic casino where betting was done in 'utils' rather than 'dollars', it isn't clear to me that there's any rational reason to go against the expected utility calculations (at least when only finite values are involved).

But it's pretty difficult to really conceive of utils properly, so I'm not too sure about any of this. If we can't trust our intuitions either way, what else is there for us to fall back on? (Perhaps generalized theories that are known to work in other domains? Can we trust them to work as well in novel situations?)

47. Richard, I think driftwood's point (if I can be pardoned for putting words in someone else's mouth) is that the size of the reward is not relevant because the experiment cannot be repeated.

Consider if you were walking down the street one day and were given a choice of two games to play. In game A, you have a 50% chance of winning \$1 and a 50% chance of losing \$1, but in game B you have 10% chance of winning \$N and a 90% chance of losing ten million dollars. The catch is that you only get to play once. Which game should you choose to play? I suggest you should choose game A, regardless of the size of N. If you could repeat the experiment, the choice would clearly depend on the size of N, but since you can't, the stakes are too high to play B. (I'm assuming, of course, that "you" can't afford to lose ten million dollars. I know I can't...)

This doesn't necessarily have anything to do with Pascal's wager, at least not directly, just with the relevance of expected value calculations.

48. Yes, and that's precisely what I was disagreeing about. There is no possible value of N for which the utility difference between \$N and \$10m is greater than the utility difference between \$10m and nothing. Extra money gets less and less valuable, the more of it you have. That's what DMU is all about, and that's why our intuitions in such examples as you point to are worthless when considering expected utility (rather than expected dollars).

Suppose an angel gave you a one-off chance to bet with units of your own happiness (supposing that happiness can be quantified in such a way). Would it be worth betting 1000 happiness-units (the loss of which, we may suppose, would leave you feeling quite miserable) for a 10% chance to win a million units (which we may suppose is happier than any human has ever been before)?

It seems to me that we have no grounds for going against expected utility in such a situation. Even if the chance of winning is low, this can be compensated for by appropriate large rewards, so long as said reward (unlike material resources) does not exhibit DMU.

49. Yes, sorry, in retrospect I can see how your earlier comment says that.

After doing some reading on expected utility and diminishing marginal utility (this is a concise introduction to Bernoulli's original expected utility hypothesis with DMU. It's a little confusing, since my browser doesn't render the math symbols correctly, but still readable.), this seems like exactly the sort of dilemma DMU was invented to solve. If I understand correctly (not a great assumption, I admit), you're arguing that *if* DMU doesn't apply, then we should trust the expected value calculation. This seems clear, but it's not clear at all to me that DMU doesn't apply. I don't really have an argument for this, except perhaps that DMU not applying leads to what is to me the wildly implausible conclusion that one should play the angel's game in your example. I admit that this isn't much of an argument.

Aside: the page I linked to above is part of a much longer essay on various modern theories of expected utility. I didn't get much out of later parts of it -- my browser's mangling of the math got to be too much for me once it got into serious proofs -- but I wanted to point to the introduction, which has a very interesting comparison of various schools of thought regarding the definition of probability.

50. Neat, thanks for the links.

And that makes 50 comments! My most productive comment thread yet :)

51. hasn't the discussion become too complicated to understand..well here is my contribution to make it further complicated :)

"The expected value of a guaranteed infinite reward is no greater than the expected value of a 1-in-a-million chance thereof."

Any logical person would definitely choose the formar option though the expected utility is same in both the cases.So where is the problem?

I guess the problem is in the existence of infinite reward. What does it actually mean?? Even if God makes me God and he/she/it chooses to become me, i would not call it infinite reward.
mathematicians warn against using infinite for practical purposes and pascal wager is violating that and hence creating so much of confusion. If we replace infinite with a very very large number, flipping a coin would make utility half, though it would still be very large value but of course lesser than a definite reward and clubbing it with a drink will make it insignificant enough not to claim :)

52. Let me elaborate my point..we know that count of natural numbers is infinite but there is no natural number which is infinite..there is always a number which is greater than any number.
Same way no reward can be infinite, if you call some reward "R" infinite, i would say i am offering "R" + 1 cookie, which is definitely greater than "R". Hence theoritically infinite reward doesn't mean anything.

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