Hájek summarizes the Wager in the following terms:
1. Rationality requires you to give a positive probability to God's existence.
2. There is a decision matrix. In this matrix, if you wager for God and God exists, you get infinite utility; if you wager for God and God does not exist, you get a finite utility; if you wager against God and God exists, you get a finite utility; if you wager against God and God does not exist, you get a finite utility. 'Wagering for God' and 'wagering against God' are understood to be contradictories.
3. Rationality requires you to perform the act of maximum expected utility, when there is one.
4. Therefore, rationality requires you to wager for God.
Hájek argues that the Wager turns out to be invalid: to get its result the utility of salvation must be ∞ reflexive under addition (i.e., adding something to it does not result in something with greater utility); but if it is ∞ reflexive under multiplication it cannot distinguish between pure wagering for God and what Hájek calls 'mixed strategies'. An example of a mixed strategy would be wagering for God if a flipped coin lands on heads. There are, of course, a great many such 'mixed strategies'. So the Wager is invalid; and, what is more, Hájek argues, it looks like no reformulation is possible:
For I see no prospects for characterizing a notion of the utility of salvation that is reflexive under addition without being reflexive under multiplication by positive, finite probabilities, or reflexive under multiplication by numbers greater than 1 without being reflexive under multiplication by positive, finite probabilities. Yet it seems that nothing less will salvage Pascal’s reasoning. So we are left with a dilemma. If the utility of salvation is reflexive under both addition and multiplication by positive, finite probabilities (as in Pascal’s original argument), wagering for God will be just one of many equally rational courses of action, and our choice among them will be arbitrary. If the utility is not reflexive under either addition or multiplication by positive, finite probabilities (as in my reformulations of the argument), salvation will be so far from being the best thing possible as to be unsuitable for Pascal’s theology. I wager that any future version of the argument will succumb to this dilemma.
When we treat a historical argument formally, however, and get the result that the argument turns out to be invalid and, what is more, irremediably flawed, the first and most important question to ask is whether, in isolating our formal considerations, we have interpreted the argument correctly. And I think there is plenty of reason to doubt whether decision-theoretic accounts like Hájek's actually do justice to the Wager. Now, it's clear that the Wager does touch on issues that are dealt with in decision theory; but this concession is a long way from saying that the above reconstruction of the Wager is at all accurate. A few points may be sufficient to suggest this.
(1) Are mixed strategies really a problem for Pascal? No. In the fragments we have discussing the Wager, it's clear that there is a sort of dialogue going on between the presenter of the Wager and an agnostic; and the argument as we possess it is opened by the agnostic, who holds that there are no speculative reasons helping us to decide whether God exists or not, or, if He does, what His nature may be. So the agnostic concludes that when Christians believe the things they claim to believe, they are being irrational. They should instead suspend judgment. Pascal's whole reply is that this is false. Not only is the agnostic's claim about speculative reasons false (and this comes up again at the end of the Wager), but, more immediately (and more importantly for actually discussing the matter with the agnostic), the agnostic is overlooking what we might call strategic reasons. Mixed strategies don't affect this main argument; in fact, they require that it be made, just as much as a pure strategy does.
(2) Further, Pascal in his whole discussion never commits himself to the claim that infinite utility alone is sufficient for the decision. In fact, the Wager is presented not all at once, as decision-theoretical interpretations tend to present it, but in stages. At the first stage, he merely points out that the agnostic is being unreasonable in criticizing people for deciding without proof (the focus is on Christians because of the context, but the point here works for making the decision either way), given that he says that there is no proof-based way to decide. The agnostic replies that they could suspend judgment; Pascal denies that there is really any tertium quid here, and opens the second stage of the argument, one that assumes that a decision must be made.
If you assume that a decision must be made, the natural thing to do is to determine which decision would be least in your interest; and this is precisely what Pascal considers next. There are two alternatives to choose from: for God and against God. There are two things you have to lose: the good and the true. There are two things at stake: knowledge and happiness. There are two things we are trying to avoid: error and misery. Because the agnostic has denied that speculative reason has any ability to help us, the focus is naturally on misery, happiness, and the good; the others are (again, at this stage of argument) of uncertain status in this dispute. So the question Pascal raises is: What is the gain and loss of happiness in this wager? If you gain, you gain everything; if you lose, you lose nothing. So, Pascal concludes, it's reasonable to wager for God.
But the agnostic is not satisfied. Gain and loss are not the only things involved in wagers; there is also the question of what is being wagered. So the third stage of the argument has to consider: what happens when we factor in what we are wagering? And Pascal makes the point, so key to decision-theoretical interpretations, that what we are wagering is finite, whereas what is gained (if we wager for God and God exists) is infinite. So, again, it is reasonable to wager for God.
The agnostic is still not satisfied, which brings us to the fourth stage of the argument, in which the concept of risk is introduced. The agnostic insists that, even if the argument so far is fine, it leaves out our risk. In other words, it is not certain whether we will gain anything; what we are wagering is at risk in the wager. And, what is more, the agnostic tries to turn the tables on Pascal by trying to introduce a defeating infinity, one that blocks the conclusion Pascal has reached up to this point. There is, the agnostic claims, an infinite uncertainty about our gain; therefore, he says, it is unreasonable to wager on it. Pascal does not take the bait, however. Everyone who faces the decision is staking something reasonably certain and definite; and we are weighing this against a finite uncertainty of gain. By the agnostic's own principles, we are not in a position to say that the uncertainty of gain is infinite; he has already committed himself to saying that, for all speculative reason can tell us, the two alternatives are indifferent; that is, for all we know, either might be the right course. Given this, however, we are not dealing with an imbalance of probabilities, which is what the agnostic is really trying to introduce: by his own principles the agnostic cannot say that there is an infinite probability, or even a greater probability, against God's existence. The risk is finite; the risk of loss and the risk of gain can't be treated as different on our information; the possible gain is (on one side of the wager) infinite.
Even if we concede this, however, it would be nice to have more information than the agnostic is letting us have. Pascal reminds the agnostic that we can get more information, i.e., we can look to see if there's anything in the world that is (as it were) an inside tip to help us make a better bet. And Pascal notes that if the agnostic is still not able to decide, he knows (because of the argument) that this has nothing to do with reason. So it must be due to the passions; and the way to handle recalcitrant passions is to discipline them by things that make passions more manageable. And that's more or less the whole Wager as Pascal presents it.
Note that nowhere in this is there a claim that the decision can only be made in terms of infinite utility, or even maximum utility (that is, he never claims (3)). Pascal says (1) in stage one that it is rational to make a decision in this case, and what he says does not favor either decision over the other; (2) in stage two he says that it is rational to wager for God on the basis of what might be gained; (3) in stage three he says that it is rational to wager for God on the basis of comparing what is being wagered with what might be gained; (4) in stage four he says that on agnostic principles we cannot wiggle out of these conclusions by rigging the probabilities. He then concludes by suggesting that the agnostic look again at the speculative reasons he rejected out of hand. The decision does not seem to rely on infinite expected utility; a decision must be made so (1) it is rational to make a decision in any case; (2) it is especially rational to wager for God when we consider what is in our interest; (3) what we are wagering against this possibility of gain is not significant in comparison with the gain itself; and (4) the agnostic has stripped himself of the resources that could be used to avoid this conclusion.
(3) Note as well that in stage four Pascal is not committed to (1), either. All he has to argue in stage four is that the agnostic can't wiggle out of the Wager by assigning God's existence infinitesimal or zero probability. And he's certainly right there, because the agnostic is already committed to saying we don't know what the probability is. The probabilities for any course of action play no role in the Wager at all. All Pascal notes is that on the agnostic's own set-up, we have two possibilities (God exists and God does not exist) and we have nothing that biases the case in either direction. It is this that is usually treated as if Pascal were giving each a probability of 1/2. In a sense he does this, but he does it in the way probability theorists in the seventeenth and eighteenth centuries did it, not in our way. That is, all it says is that there are two cases, and wagering for and against are each one of them. And for the purposes of the Wager, the only thing beyond this is that we can't (for the agnostic's own reasons) say anything beyond this -- this is the maximum information we have to go on, as far as determining probabilities goes.
(4) Given this, one can also see that most of the other objections brought against the Wager by other people are red herrings. As an argument against agnosticism, it is not affected by the possibility of other Gods who don't reward in the way Pascal supposes (the many gods objection), nor is it affected by the possibility of different ways of wagering for God (the mercenary objection), nor is it affected by the possibility that the matrix differs from person to person (the predestination objection), nor is it affected by the suspectness of infinite utility. One might suggest that the St. Petersburg Paradox still causes some problems; but it does not appear to be the case. The St. Petersburg Paradox arises when we assume that it is rational to enter into a wager when the price of entry is smaller than the expected value, and the expected value is infinite; then any price of entry is worth the game, although such games don't seem actually to be worth more than a small price of entry. But, of course, in the Wager we are not talking about just any finite price of entry (it's not necessary to assume that every finite price of entry is being considered); we are just talking about one finite price -- a life. If we weren't dealing with an agnostic, of course, it might be reasonable to wonder whether even this is too much -- but the agnostic is not in a position to say that it is too much. Of course, just as many of the objections against the Wager are irrelevant, so are many of the 'uses' of the Wager completely different arguments that should not be conflated with the one Pascal actually presents.
(5) What is less often noticed is that it is entirely possible -- and nothing Pascal says rules it out as a possibility -- to run an atheistic Wager against the agnostic as well. All Pascal is doing in his Wager is handling an agnostic who starts out by denying that it is rational to do anything but suspend judgment in this case. An atheist can as easily avail himself of a Wager to argue against this as Pascal could. Indeed, if he were too lazy to make his own, an atheist could as easily avail himself of Pascal's own Wager against the agnostic, as long as it is supplemented by an argument for parity (i.e., that wagering against God is more or less parallel) and by the same appeal Pascal makes to look at the possible 'inside tips' again! Of course, the atheist will want to direct the agnostic's attention to different 'inside tips'; but the argument would work the same way. The theist would start out with the advantage; but as the 'many gods objections' show, atheists likely have the ingenuity to nullify this apparent advantage. The locus of argument then moves to where it should be, which the agnostic dismissed out of hand: the 'inside tips', i.e., arguments and evidences for and against.
So that appears to be the lay of the land: Pascal's Wager is perhaps the strongest argument against the rationality of agnosticism that has been formulated. To suspend judgment the agnostic has to ignore strategic reasons. Pretty much everyone who is not an agnostic, however, agrees that there are strategic reasons for making the decision, even if they don't entirely agree about which of those strategic reasons is worth the most. Given the strategic reasons for making a decision, however, it becomes more important than the agnostic had previously allowed to look at the 'inside tips' or speculative reasons; they need to be investigated and evaluated much more carefully than the agnostic's original dismissal allowed. If the 'inside tips', the speculative reasons, turn out to be as useless as the agnostic had originally thought, of course, things get trickier; but Pascal, as I pointed out, never commits himself to the claim that rationality requires only picking out the maximum utility, but only that it is rational to make a decision (and the gain favors one side of the decision, even if not absolutely). So it appears that, contrary to the agnostic's claim, there is reason to think that it is more rational to decide than to suspend judgment. But all Pascal really needs of this in context is that the agnostic is wrong in claiming that Christians are being irrational in believing, even if the speculative reasons fail to shed light on the situation.
