My interest is not so much in Pascal's Wager specifically, but in the general problem of scenarios where expected-value calculations prescribe betting on an extremely unlikely outcome simply because the payoff will be absurdly high if it comes true. (Pascal's Mugging is a less fraught demonstration of the problem.) This sometimes comes up in discussions of moral uncertainty or trying to do the right thing for the long-term future, for example. From that general perspective, I think the last three chapters are most relevant, so I'll focus on them in this review. Other chapters discuss some historical, ethical, and practical concerns.

**Jeff Jordan, "The Many-Gods Objection".** Perhaps the most common reply to Pascal's Wager is that it ignores the existence of multiple incompatible religions. We're not just choosing between Christianity and atheism; we're choosing between Christianity, Islam, Zoroastrianism, etc. Jordan responds that even if the Wager can't tell you which religion is the *best* bet, it can still tell you that *any* religion is a smarter bet than being an atheist or agnostic.

What if we consider all *possible* religions though? We can at least *imagine* a god who - quoting from the Sorensen essay I discuss later - "dispatches atheists to heaven (for their spunk) and theists to hell (for their servility)". Adding that into the equation would give atheism infinite expected value, too.^{1} Jordan notes that this assumes "logical possibility is sufficient for an assignment of a positive nonzero probability ... within the domain of subjective probability", and he rejects this assumption. The assumption seems reasonable to me though; having had the experience of feeling certain about something and then later realizing I was wrong, it's difficult to see a principled reason for ever assigning 0 probability (as opposed to some minuscule probability) to anything unless I can immediately and directly verify that it's impossible. (Also, it seems like even assigning some very low positive probability to the statement *all logical possibilities merit positive subjective probability* would itself be enough to require you to assign positive probabilities to all logical possibilities...)

Still, I agree the many-gods objection isn't satisfactory. For those of us who reject the Wager, the core puzzle is why we try to maximize expected value in other decisions but seem not to in this one. Intuitively, the reason needs to be found in the natures of the handful of options we're genuinely considering, not in other options we dream up to make the decision extra challenging. Otherwise, similar logic would undermine expected-value-based reasoning for *all* decisions. For example, the expected value of a lottery ticket is low - unless you consider the remote possibility that winning the Powerball will enable you to buy a bunker in New Zealand where you can survive World War 3 and live into a golden era in which the technology for immortality will be available. Taking the mere *possibility* of that into account makes the expected value of the ticket infinite. But nobody thinks this makes playing the lottery more rational.

**Edward F. McClennen, "Pascal's Wager and Finite Decision Theory".** According to McClennen, postulating infinite values undermines the justifications for maximizing expected value:

Within the last few decades there have been a variety of constructions that establish, as a theorem, that rational choice is choice that maximizes expected value (or utility, as it has come to be known). ... Must we suppose that the rule holds for cases in which some outcome is assigned an infinite value? On the contrary, the constructions in question preclude extending expected-utility reasoning to such cases!

This wouldn't be a satisfying response to the general problem, where payoffs may just be astronomically high finite values. But I don't think it even gets to the heart of the issue with the Wager, either. Suppose Christianity promised merely a finite afterlife of X years. Unless I'm 100% certain that Christianity is false, then there's *some* value of X such that the expected value of belief outweighs the expected value of atheism.^{2} Then, the constructions McClennen refers to would presumably apply, yet I still wouldn't take the Wager. Infinite payoffs have nothing to do with my true reason for rejecting Pascal's argument; they're red herrings.

But McClennen briefly touches on what I think is the real reason the Wager fails:

Much more importantly, however, it needs to be recalled that whatever force the expectation rule has for the habitual gambler, it need not have much force for the casual gambler. And at the limit, in the case of a once and only choice such as that represented by Pascal's wager, it is not at all clear that it should have any force at all. ... In particular, both those who are risk averse and those who are risk oriented will reject the rule.

As a concrete analogy, imagine a lottery where the prize is one hundred trillion dollars, and the odds of winning are 1 in 1000 (not terrible!), but the price of a ticket is all the money you have now plus debt equal to all the money you'd acquire by other means for the rest of your life. Unless you're Jeff Bezos, the expected value of the ticket is really high. But do you feel rationally compelled to buy it? Would you encourage someone you cared about to buy it?^{3} It seems at least reasonable to pay more attention to the fact that a good payoff is very unlikely, than to the fact that that unlikely payoff manages to make the expected value calculation turn out positive.

Some people seem to think there are mathematical proofs that you should maximize expected value if you're trying to adhere to certain plausible principles. These are the "constructions" McClennen mentioned; he apparently doesn't think they really prove that. This is something I want to understand better.

**Roy A Sorensen, "Infinite Decision Theory".** Sorensen gives a rapid, witty overview of a number of considerations and thought experiments related to infinity, including the St. Petersburg Paradox (and Paul Weirich's risk-focused response to it, which - encouragingly for me - involves "repudiation of the idea that prudence is merely the maximization of one's expected utility"); Pollock's "immortal connoisseur who possesses a bottle of EverBetter wine"; Parfit's example of a man whose eternal reward requires a small amount of pain for each happy day of afterlife - to be paid up front; the problem of choosing among infinitely many conceivable options; and how infinities play into theodicies.

There are no firm conclusions, he's only making a "plea for infinite decision theory". Since above I talked myself into thinking infinities aren't the primary issue with the Wager and similar scenarios, I'm less interested in this, though it still seems important. Seeing a connection to the problem of the sorites, Sorensen suggests that a way forward may involve appreciation of the "divergence between addition and enlargement in the infinite realm" (throwing a stone onto an infinite pile adds to the pile without making the pile larger).

One interesting aside in the essay is this theodicy that Sorensen puts forward facetiously:

Now assume the world is infinitely good. ... The fact then that one can easily imagine additional goodness would be compatible with the unsweetened world being the best of all possible worlds. For the reflexivity of infinity, let us separate additional goodness from greater goodness. Reflexivity will also let us acknowledge the existence of infinite evils in the best of all possible worlds. For just as we can subtract the even numbers from the naturals and still have an infinite set, one can subtract an infinite evil from infinite good and still have an infinite good.

...

Despite its technical merits, an infinitely good subsistence world awash in infinite evil seems more the work of the devil.

[1] This is relevant to the more general problem because for any high-potential-payoff option in any choice, you can always make up a story about how there's at least a tiny chance that making a different choice would set off some chain of events that ends up having an equally high or higher payoff.

[2] You might argue that as X gets higher, I should think Christianity is less and less plausible, so the expected value caps out at some point. I doubt this solves the problem. If I initially thought the afterlife was supposed to be a billion years long, and then later learned that it's in fact said to be a trillion years, would my probability estimate really decrease by a factor of 1000? I doubt the exact length was such an important factor in the initial probability estimate.

[3] Of course, you might try to get ~1000 people to buy tickets and pool their winnings. But to keep this thought experiment analogous to Pascal's Wager, we'd have to stipulate that all 1000 people will get the same outcome (they all win or all lose - just as God exists for everybody or for nobody), so there's still only a 1 in 1000 chance that *any* of them win.