In its classical formulation, Pascal’s Wager contends that we have something like the following payoff matrix:
God exists | No God | |
Believe | +∞ | −a |
Don’t believe | -b | c |
where a,b,c are finite. Alan Hajek, however, observes that it is incorrect to say that if you don’t choose to believe, then the payoff is finite. For even if you don’t now choose to believe, there is a non-zero chance that you will later come to believe, so the expected payoff whether you choose to believe or not is +∞.
Hajek’s criticism has the following unhappy upshot. Suppose that there is a lottery ticket that costs a dollar and has a 9/10 chance of getting you an infinite payoff. That’s a really good deal intuitively: you should rush out and buy the ticket. But the analogue to Hajek’s criticism will say that since there is a non-zero chance that you will obtain the ticket without buying it—maybe a friend will give it to you as a gift—the expected payoff is +∞ whether you buy or don’t buy. So there is no point to buying. So Hajek’s criticism leads to something counterintuitive here, though that won’t surprise Hajek. The point of this post is to develop a rigorous principled response to Hajek’s criticism entailing the intuition that you should go for the higher probability of an infinite outcome over a lower probability of it.
A gamble is a random variable on a probability space. We will consider gambles that take their values in R*=R∪{−∞,+∞}, where R is the real numbers. Say that gambles X and Y are disjoint provided that at no point in the probability space are they both non-zero. We will consider an ordering ≤ on gambles, where X≤Y means that Y is at least as good a deal as X. Write X<Y if X≤Y but not Y≤X. Then we can say Y is a strictly better deal than X. Say that gambles X and Y are probabilistically equivalent provided that for any (Borel measurable) set of values A, P(X∈A)=P(Y∈A). Here are some very reasonable axioms:
- ≤ is a partial preorder, i.e., transitive and reflexive.
- If X and Y are real valued and have finite expected values, then X≤Y if and only if E(X)≤E(Y).
- If X and Y are defined on the same probability space and X(ω)≤Y(ω) for every point ω, then X≤Y.
- If X and Y are disjoint, and so are W and Z, and if X≤W and Y≤Z, then X+Y≤W+Z. If further X<W, then X+Y<W+Z.
- If X and Y are probabilistically equivalent, then X≤Y and Y≤X.
For any random variable X, let X* be the random variable that has the same value as X where X is finite and has value zero where X is infinite (positively or negatively).
The point of the above axioms is to avoid having to take expected values where there are infinite payoffs in view.
Theorem. Assume Axioms 1-5. Suppose that X and Y are gambles with the following properties:
- P(X=+∞)<P(Y=+∞)
- P(X=−∞)≥P(Y=−∞)
- X* and Y* have finite expected values
Then: X<Y.
It follows that in the lottery case, as long as the probability of getting a winning ticket without buying is smaller than the probability of getting a winning ticket when buying, you should buy. Likewise, if choosing to believe has a greater probability of the infinite payoff than not choosing to believe, and has no greater probability of a negative infinite payoff, and all the finite outcomes are bounded, you should choose to believe.
Proof of Theorem: Say that an event E is continuous provided that for any 0≤x≤P(E), there is an event F⊆E with P(F)=x. By Axiom 5, without loss of generality {X∈A} and {Y∈A} are continuous for any (Borel measurable) A. (Proof: If necessary, enrich the probability space that X is defined on to introduce a random variable U uniformly distributed on [0,1] and independent of X. The enrichment will not change any gamble orderings by Axiom 5. Then if 0≤x≤P(X∈A), just choose a∈[0,1] such that aP(X∈A)=x and let F={X∈A&U≤a}. Ditto for Y.)
Now, given an event A and a random variable X, let AX be the random variable equal to X on A and equal to zero outside of A. Let A={X=−∞} and B={Y=−∞}. Define the random variables X_{1} and Y_{1} on [0,1] with uniform distribution by X_{1}(x)=−∞ if x≤P(A) and X_{1}(x)=0 otherwise, and Y_{1}(x)=−∞ if x≤P(B) and Y_{1}(x)=0 otherwise. Since P(A)≥P(B) by (7), it follows that X_{1}(x)≤Y_{1}(x) everywhere and so X_{1}≤Y_{1} by Axiom 3. But AX and BY are probabilistically equivalent to X_{1} and Y_{1} respectively, so by Axiom 5 we have AX≤BY. If we can show that A^{c}X<B^{c}Y then the conclusion of our Theorem will follow from the second part of Axiom 4.
Let X_{2}=A^{c}X and Y_{2}=B^{c}Y. Then P(X_{2}=+∞)<P(Y_{2}=+∞), X_{2}* and Y_{2}* have finite expected values and X_{2} and Y_{2} never have the value −∞. We must show that X_{2}≤Y_{2}. Let C={X_{2}=+∞}. By subdivisibility, let D be a subset of {Y_{2}=+∞} with P(D)=P(C). Then CX_{2} and DY_{2} are probabilistically equivalent, so CX_{2}≤DY_{2} by Axiom 5. Let X_{3}=C^{c}X_{2} and Y_{3}=D^{c}Y_{3}. Observe that X_{3} is everywhere finite. Furthermore P(Y_{3}=+∞)=P(Y_{2}=+∞)−P(X_{2}=+∞)>0.
Choose a finite N sufficiently large that NP(Y_{3}=+∞)>E(X_{3})−E(Y_{3}*) (the finiteness of the right hand side follows from our integrability assumptions). Let Y_{4} be a random variable that agrees with Y_{3} everywhere where Y_{3} is finite, but equals N where Y_{3} is infinite. Then E(Y_{4})=NP(Y_{3}=+∞)+E(Y_{3}*)>E(X_{3}). Thus, Y_{4}>X_{3} by Axiom 2. But Y_{3} is greater than or equal to Y_{4} everywhere, so Y_{3}≥Y_{4}. By Axiom 1 it follows that Y_{3}>X_{3}. but DY_{2}≥CX_{2} and X_{2}=CX_{2}+X_{3} and Y_{2}=DY_{2}+Y_{3}, so by Axiom 4 we have Y_{2}>X_{2}, which was what we wanted to prove.
In the middle sections of his 12th chapter, Sobel goes through a series of adjustments to his deductive argument from evil designed to get around various versions of the Free Will Defense and other tactics attempted by theists. For reasons mentioned earlier, I am not happy with Sobel’s formal treatment of these arguments, so I’m going to reconstruct the substance of the argument somewhat differently. Consider the following:
- If there were a perfect being, it would take a best course of action available to it in creating the world
- If a perfect being took the best course of action available to it in creating the world, the result would be very different from what we observe.
- But the world is as we observe it to be.
Therefore, - There is no perfect being.
The October issue of Analysis is now available on line. It has an article “Fine-tuning is not surprising” by one Cory Juhl. I’d never heard of him so I looked him up on the UT Department website and he’s got good training in HPS.
I only perused the article, but I couldn’t find anything very original upon perusal. Some of it seems like a less-precise rehash of some of Brad Monton’s points discussed previously on the blog. One complaint that I very much share is the lack of any semantics offered for the kind of probability which is supposed to be at work in the argument. I’ve been slaving away at range theories of probability and point-set topology and re-reading Carnap trying to come up with something, but it’s very difficult. Still, I’m less worried about logical probability measures over infinities than I was when I started.
Though I’m personally vexed by this, I’m not sure how big a problem it is for the argument from fine-tuning. I can see one sticking to the intuitive judgements reasonably without being in possession of the mechanics. I’ve discussed this problem with Richard Swinburne and he’s certainly not interested in it. This is what he said in one email: “This notion of evidential probability, like- for example – the notion of cause, is so basic that any philosopher’s attempt to give a precise definition of it in other terms is unlikely to capture its nature adequately.”
He thinks the judgements themselves are intuitively correct. One might say that the intuitive judgements are in fact the standards according to which the theories should be judged (or at least relatively fixed points in a reflective equilibrium). A philosopher wants more, of course, but we might have to live without it.
I’ll be presenting my own criticisms of the fine-tuning argument at the upcoming SLU Religious Epistemology Conference which I’m very much looking forward to.