This is my formal reconstruction of Harrison’s formulation of Pascal’s Wager in his previous series of blog posts:
P1. If epistemic reason cannot decide between belief and nonbelief, then one should instead choose belief, or nonbelief, if it has the highest expected utility.
P2. Epistemic reason cannot decide between belief and nonbelief.
SubC. One should instead choose belief, or nonbelief, if it has the highest expected utility.
P3. Belief has a higher expected utility than nonbelief.
C. One should choose belief.
Alan Hájek has written a trenchant critique (“Waging War on Pascal’s Wager”) of Pascal’s Wager, arguing that the Wager is deductively invalid because it overlooks the possibility of “mixed strategies,” or courses of action determining whether one believes in Christianity or not that can’t be neatly categorized as either belief or nonbelief. In other words, there are actions (other than believing and not believing) whose expected utility may be at least as high as the expected utility of belief. Consequently, even if belief has a higher expected utility than nonbelief, that doesn’t mean it has the highest expected utility out of all the relevant choices one might make. So the conclusion doesn’t follow from the premises.
What are these mixed strategies? Hájek gives the example of tossing a fair coin and believing just in case the coin lands heads. In this example, there is a 50/50 chance of ending up believing. Since the action may well end either with one’s believing or with one’s not believing, it qualifies as an alternative to straightforward belief and nonbelief – that is, as a mixed strategy. The expected utility of this action, if you do the math, is the sum of half an infinite value (assuming the expected utility of straightforward belief is infinite, due to the infinite reward of an eternity in heaven) and half a finite value. Seeing as half of infinity is infinity, and infinity plus anything is infinity, the expected utility is infinite. As such, even granting to Pascal that the expected utility of belief is infinite and thus higher than the expected utility of nonbelief, there is reason to deny that the expected utility of belief is the highest. Mixed strategies have infinite, and so equally high, expected utilities. At best, the expected utility of belief is among the highest attainable expected utilities.
If mixed strategies were limited to special cases like this one, Hájek’s objection would be much less devastating to Pascal’s Wager than it in fact is. But Hájek generalizes the idea behind the coin toss example to expose a much bigger problem, namely that every action apart from straightforward belief or nonbelief is a mixed strategy. No matter what you do, there is some chance, big or small, that belief will be the outcome. For instance, if you’re currently a nonbeliever, there is some chance that you will become a believer by the time you finish reading this paragraph. The chance may even be way less than ½, but if you do the math right, you see that this doesn’t change the expected utility. So no matter what you do, your action has an infinite expected utility, like belief does. But this means you can always be acting so as to maximize your expected utility, due to the chance that you will end up believing in Christianity, even if you always deliberately put off actually believing. So considerations of expected utility don’t narrow down the practically rational courses of action here at all. You can live however you want.
How does Hájek support the claim that there is always some chance that belief will be the outcome of one’s action, no matter what the action is? Aside from mere intuition, he gives an argument along the following lines: it is reasonable to reserve probability assignments of 1 for logically necessary truths and, by the same token, to reserve probability assignments of 0 for logically impossible propositions. But surely there is nothing logically impossible, or contradictory, about an arbitrarily selected action’s having belief as an outcome. So it is reasonable to suppose that there is a nonzero chance of an arbitrarily selected action’s having belief as an outcome.
That is what I take to be the gist of Hájek’s argument for the invalidity of Pascal’s Wager, as Pascal and Harrison formulate it. Hájek goes on to suggest reformulations of the Wager that he concedes are valid. I will not discuss those reformulations here, but suffice it to say, they all face a serious dilemma raised by Hájek in the same paper. However, that dilemma may not be so troubling unless one leans toward a theology like Pascal’s.
Hájek, Alan. “Waging War on Pascal’s Wager.” The Philosophical Review, vol. 112, no. 1, 2003, pp. 27–56. JSTOR, JSTOR, http://www.jstor.org/stable/3595561.