Monday, June 29, 2009

Pascal's wager and decision theory

I think Pascal's wager could be seen as a way of destroying most of standard decision theory in the case of many agents. The reason for this is that just about any significant choice one makes will have the property that according to some religious views, that choice affects the probabilities of getting an infinite payoff, and unless the agent has a way of assigning zero epistemic probability to that religion, these infinitary considerations will swamp all the finite considerations. Thus, one wonders to oneself: "Should I self-flagellate?" There is an obvious answer: "No, because it hurts." But because there are religious views according to which such self-flagellation helps attain an infinite payoff, then unless one assigns zero probability to these views, the infinitary considerations swamp the finitary considerations coming from the fact that it hurts. One ends up having to compare the increased probability that one will get an infinite payoff if one self-flagellates on religious views that are pro-flagellation with the decreased probability of an infinite payoff on anti-flagellation religions, and the apparently relevant consideration that it hurts just drops out by the wayside (unless the infinitary considerations end up being perfectly balanced).

One might think one can dismiss the infinitary considerations because of problems with weighing infinities. But those can be solved fairly easily by adopting an appropriate version of non-standard arithmetic.

Maybe what this is, though, is not so much a reductio of standard decision theory, as a way of showing that practical rationality requires that one assign non-zero probability to at most one religious view (or maybe one moderately narrow family of closely-related religious views). Dogmatic atheists and dogmatic religionists would like this conclusion. And I am a dogmatic religionist, after all. :-)

23 comments:

Matthew said...

Decision theory really has it's problems:
Consider the following:
- Action A will give me infinite payoff
- Action A will give me infinite punishment

And choices become diffucult somehow.

Alexander R Pruss said...

I think that a case is only a really big problem for decision theory if there is some other account of what one should do in that case that works well. But in the case you mention, I think nobody has any idea what to do!

Mr Veale said...

Take Sober’s "Betting against Pascal". He wants to separate p(G) – god exists - from p(P) – that a particular theology is true. That seems fine.
Then he considers p(x) – a theology that states that God damns believers. For our purposes we could say that God only gives infinite rewards to believers that self-flagellate.
Say (P) is Fundamentalist Protestantism.
So our expected utilities given belief in Theism
G&P=Heaven -G&P=Rot G&X=Hell -G&X=Hell
And our expected Utilities on being Atheistic are
G&P=Hell -G&P=Rot G&X=Heaven
-G&X=Heaven

If Pr(P) < 1, then U(theism) > U(atheism) if and only if Pr(G&P) > Pr(X).The important point here is that Pr(G&P) can be less than Pr(X) without violating the assumption that Pr(P) is quite high.
Now I’m actually quite happy to accept that P(G) should be great enough so that P(G&P)to overcome any absurd theology (Y).
But (G&X) may be a contradiction. (G) should at least include Most Worthy of Worship. (Otherwise (G) does not refer to God). The absurd theology needs to be compatible (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? Wouldn't that also apply to (G&Y), where (Y) is a theology where God only gives infinite reward to those who self-flagellate?

Or, to be fair, doesn’t Sober owe us a proposition or set of propositions that makes (X) coherent with (G)? If there was a logical problem of evil, then I can’t see that (G&X) doesn’t have to answer some serious logical problems before it is assigned any probability at all. Doesn't the same apply to (G&Y)?

Sober suggests that an Atheist could believe that (X) theology is true on prudential grounds. That is, if God exists then (X) theology is true, and this would be believed by the Atheist to bring peace of mind. There may be occasions when it would be prudent to believe a contradiction – if a madman says he’ll kill you if you don’t believe black is white, you might give it a go. But is it ever permissible to bet a whole "way of life" on a proposition with a probability of 0?

Graham Veale
Armagh

Mr Veale said...

Okay, a few other responses

(i) Can't we use an evidentially concerned form of the wager, as outlined by Thomis V Morris?
(ii) Wouldn't it be easier to drop the problems cause by inifinities by saying that Heaven and Hell involve "enormous,
though finite, pay-offs"? That would mean that the theist would have to offer *some* evidence, but significantly less than the evidentialist demands.
(iii) And if the pay-offs are finite then can't we actually compare the pay-offs the various theologies offer? Is it so absurd to compare Nibbana with the eternal love of a Deity who died for you? Nibbana seems equal to zero on my estimation. The Dalai Lama probably disagrees, but there you are.
(iv) Does the wager aim at "Belief in God", or "Belief in a Religion"? Can it distinguish between the two?
Because a wagerer could reason that it is prudent to believe that there is a God, and that Infinite reward is available. So they will begin their QUEST for God through a plausible religion. Or maybe just by praying.
The wagerer is betting that God is not static, and that if they search for God, then God can also search for them.
Or at the very least, that they have a better chance of finding God by looking.

Mr Veale said...

By the way your posts on Truth have been very helpful!

Vlastimil Vohánka said...

Alex,

How do we know the infinities are not perfectly balanced?

If they are, then, presumably, the earthly finite values rule. Should we be Christians then? As Lydia McGrew said recently at W4 (in a post on Pascal's wager): "... living as a Christian makes people on the whole happier *in the comfortable West*." Maybe. It seems to me so, too. But it's in contrast with St. Paul's era: "If only for this life we have hope in Christ, we are to be pitied more than all men." (1 Cor 15:19) So, if you're comfortable, be a Christian, if not, don't be?

Secondly, let me an explicatory question. Say a (serious) Christian is a person who believes that Christianity is true and is psychologically committed to follow it. Such a person can go to heaven, yet, he can go to hell, too. And the same holds for a person who is not a Christian. (Or is this heterodox? It doesn't seem so. Christians should not be certain of their own salvation or of damnation of non-Christians.) So, presumably, being a Christian yields a chance of obtaining finite earthly values and disvalues, a chance of obtaining infinite value (heaven), and a chance of obtaining infinite disvalue (hell). Similarly for not being a Christian. What are you suggesting to do now? Are you implying by the last par that (if the subtraction of infinities is not undefined and they are not perfectly balanced) Christians should deny they have a chance to be damned and non-Christians have a chance to be saved?

Mr Veale said...

Vlashtimil

A lot depends on what we mean by "health" and "happiness". We don't need to beg the question by using a Christian, or even Aristotleian, definition. Happiness need not be understood as "satisfaction of perceiveed needs" (Huxley's "Brave New World" comes to mind) but as "a rational judgment that our life has been worthwhile" (Daniel Nettle defends this view in his little book "Happiness". And to go back to Huxley -http://www.huxley.net/bnw/seventeen.html).
So I think that Jeffry Jordan's defence of the wager can stand up to Lydia McGrew's critique.

As for the damnation of believers-well, you certainly decrease your chances by being a believer. Provided belief goes beyong assent, but I think this was what Pascal meant.
But the use of infinities seems to muddy the water here. Can an "evidentially concerned" version of the wager work if we replace inifinity with "enormous yet finite"?

Graham Veale

Alexander R Pruss said...

It seems like it would be too much of a coincidence if the infinities were perfectly balanced. For instance, take the two religion case where each has infinite payoff for practicing that religion and zero otherwise. Then to get perfect balance, we'd have to have p1 I1 = p2 I2, where p1 and p2 are the probabilities of the two religions, and I1 and I2 are the infinite payoffs. Now maybe, without further data, I1 = I2. But there is enough evidence for and against different religions that it is unlikely that p1 = p2 (just as it is unlikely that any given pair of people has exactly the same height). But if p1 is different from p2, but I1 = I2, then p1 I1 is different from p2 I2.

Alexander R Pruss said...

Of course you need non-standard analysis to make sense of that.

Mr Veale said...

Which sort of leaves the apologetic force of the wager in the dust...

Mr Veale said...

I'm also not convinced that a theology that claims that a being worthy of worship would allow non-flagellators to suffer infinite loss is coherent.
It just seems that there is a problem of evil in this theology that cannot be answered by that theology.

GV

Alexander R Pruss said...

I don't know: is it an evil that one does not get an infinite reward?

Mr Veale said...

I think in the context of the Pensees, the loss of the infinite reward is pretty much the same thing as infinite punishment. But that's an aside.
To take an infinite reward away from one, for no other reason than not beating onseself - this certainly seems arbitrary and irrational and hardly consistent with a being worthy of worship.
My point is that the "Many Gods" objection assumes that just any Theology (P) is consistent with God (G). But that's to ignore that God is defined as most worthy of worship (maximally perfect, or whatever).
Some of the absurd theologies (God leaving the afterlife in the hands of a devil that punishes believers) just leave us with a Logical Problem of Evil that these theologies are unable to answer. So they should be assigned a probability of zero (in terms of the degree of belief we should place in them).

GV

Alexander R Pruss said...

I don't know that the one I listed is like that. Does it really leave an insoluble problem of evil if, say, God recognizing our sinfulness requires us to punish ourselves by self-flagellation?

Mr Veale said...

Is self-flagellation the *only* way of attaining the infinite payoff on the scenario that you are suggesting?

That would seem capricious, and rather unfair.

Also, how does the wagerer come upon the belief that they should self-flagellate for infinte reward?

I'm also interested - do you think that a version of the wager can run if we drop the infinities for "enormous yet finite?"

Mr Veale said...

Just to be clear - it's one thing to say that God wants us to experience self-discipline, or share in Christ's sufferings.

It's another to say that this is the only way to gain infinite reward.

(Especially if the rest of the Pensees is correct, and that reward is our reason for existing).

Graham Veale

Eric Telfer said...

I think the wager has a very practical function: it focuses on the *decision* that each can make regarding God, as opposed to the state of knowledge, state of belief, certainty of the belief, confidence in the belief, and the rest. That is, each person can make a decision regardless of whether he can clinch the case one way or the other. These types of decisions are common in life. In fact, in life there is typically evidence for and against x. There are many times narrow margins between x and y or x and non-x. And, many times, we need to not just consider the evidence and strength of the evidence for or against x (in isolation or as compared to y), but also the stakes, risks and odds of this or that. Pascal's wager brings out these considerations- or can be used as a stepping stone to such considerations. When I am sick on the gurney and the doctor wants me to make a decision, I typically do not just consider the evidence in terms of its strength, but also the risks and stakes involved. The same is true here, I think, when we discuss God.

Eric Telfer said...

Also note that we sometimes choose a safer course of action, especially when the stakes are higher, even when the evidence is weak for x. For example, I might think it is very unlikely that I have appendicitis. But, if the state of the evidence is such and such, and the risks and stakes are sufficiently high, I might error on the side of caution and decide to have the extra test in question. So much of philosophy has focused on 'knowledge' and 'belief', but we also have to focus on risks and stakes.

Mr Veale said...

I'm pretty much with you on this Eric.
I do think that we get into difficulties whenever we introduce infinite utilities. But enormous, finite pay-offs allow us to use an "evidentially concerned" version of the wager.
And this provides a powerful challenge to evidentialism.

Graham

Eric Telfer said...

I am not sure that I have a problem with 'evidentialism' here, depending on what that term means here. But I do think that the state of the evidence is not the only thing to consider. We also have to consider the risks and stakes. So, it is not just about the state of the knowledge in question, i.e., strong, weak, etc., or the state of belief in question. When we make decisions these are not the only factors involved, even if philosophers have tended to focus mainly on such factors, i.e., justified true belief or not, knowledge or not, certainty or not, reasonable or not, etc. We also have to consider the stakes and the risks involved in being right and wrong. That is, I think, where Pascal's thinking can come into play.

Vlastimil Vohánka said...

Alex,

One more question.

You say 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 ..."

Isn't that proving too much?

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.

Alexander R Pruss said...

By saying that finitary considerations are swamped, I just meant that one can ignore all finite payoffs, not that one can snap probabilities to 0 and 1.

Vlastimil Vohánka said...

But you also said that (maybe) "... practical rationality requires that one assign non-zero probability to at most one ... view ..."

As I suggested, by a reasoning analogical to that of yours, (maybe) Jack should assign zero probability to his going to hell. Am I missing something?