Pascal's Wager is basically a decision theory argument. You have two choices: believe in God (specifically, the Christian God), or don't. The world has two possible states, there is a god or there isn't. Depending on your choice and the state of the world, there are different outcomes. According to the usual argument, these are the outcomes:
| God exists | God does not exist | |
| Wager for God | Gain all | Status quo |
| Wager against God | Misery | Status quo |
A lot of people say that you can't really choose to believe in God or not, but it's possible to influence one's own beliefs. I'm sure if you made a point to listen to apologetics all the time and avoid discussions of religion on the internet, then you could increase your chances of believing in God. I think some people have this conclusion in the back of their minds, and use it as a reason to avoid questioning their own religion.
With this outcome matrix, it seems that believing in God is a "dominant" strategy. That is, the outcome is always better than or equal to not believing in God. In each column, the outcome in the first row is at least as good as the outcome in the second row. Dominant strategies are better strategies, therefore we should make an effort to believe in God.
But it isn't really a dominant strategy for several reasons. Worshipping God costs time and effort (and if any religious practices were genuinely beneficial, you could do them without believing in God). It would also cost extra if, like I suggested earlier, believing in god means listening to lots of apologetics and avoiding discussions of religion on the internet.
But suppose, for the sake of argument, that believing in God actually costs nothing. Imagine that no worship is called for, and that choosing to believe is as easy as choosing to speak. Believing in God is still not a dominant strategy, because the world has more than two possible states. If you can think of some pathological scenario where people who believe in God are rewarded, then you can also think of a pathological scenario where people who don't believe in God are rewarded. For example, there's the scenario where a different but equally whimsical God exists, and this god rewards people who don't believe in gods.
| Whimsical God #1 exists | Whimsical God #2 exists | God does not exist | |
| Believe in God | Gain all | Misery | Status quo |
| Don't believe in God | Misery | Gain all | Status quo |
It's not really necessary to come up with a realistic scenario where people who don't believe in God benefit. As long as there is any such scenario, believing in God is no longer a dominant strategy. That is to say, it is no longer true that in each column, the first row has at least as good an outcome as the second column.
But strategic dominance isn't the only kind of argument you can make with Pascal's wager. You can also argue based on what you think are the probabilities of the different states of the world, and how much value you place on the different outcomes. Many people see salvation as an infinite reward. If you multiply an infinite reward by a nonzero probability, then it's still infinitely more preferable to any finite outcome. Therefore, as long as there is a nonzero probability that God exists, one should make every effort to believe in God.
There's already a problem with this argument: if it's at all effective, then it's too effective. If it's truly the case that a nonzero probability of an infinite reward is infinitely preferable to any other outcome, then all of us should ignore all finite rewards in favor of slightly increasing the probability of satisfying this whimsical god. More than just spending some time listening to apologetics, you should dedicate your life to brainwashing yourself to maximize the probability that you will believe in God. And if there's a nonzero probability that God infinitely punishes people who eat shellfish or who mix wool and linen, then one should make every effort to appease this unlikely God.
Given that this is not how Christians behave, and very few Christians believe we should behave that way, we should consider if there's something wrong with the argument.
Pascal's Wager is comparable to the St. Petersburg paradox. In this paradox, you play a game where you flip a coin repeatedly until you get tails. If it took N coin flips, then you win 2N dollars.* The expected reward for playing this game is infinite, and therefore it is worthwhile to pay any large, finite amount of money to play the game. And yet, intuitively, it does not seem like a worthwhile investment.
*Dollars have the property that the more you have, the less valuable they are to you. In a more sophisticated version of the paradox, the reward is in "utils" rather than dollars.
Experts have many different opinions on the resolution to the St. Petersburg paradox. But very few experts bite the bullet, arguing that it is in fact worth it to invest any finite amount of money to play the game.
Pascal's Wager captures some of the essence of the St. Petersburg paradox, because it also offers a very small chance of a very large reward. It's a neat trick, harnessing a controversial mathematical paradox to make an argument for God. This way, even though experts agree that the argument is wrong, nobody agrees on exactly why. It allows apologists to look at the disagreement among their opponents and believe that they've won.
Myself, I believe that the probability of an outcome with infinite value is precisely zero. This doesn't necessarily make me a "strong" atheist, it just means that I think it's impossible for any outcome to be infinitely valuable, as impossible as it would be for God to make a square circle.
This makes philosophical sense, because "value" is just what we use to describe what our preferences are. There is no reward whatsoever that I prefer so strongly that I would risk everything else just to have an infinitesimal chance of winning it. Therefore there is no infinite value.
My discipline is physics, so I have to say that it makes physics sense too. Suppose we have someone who claims that they'll give you some payoff, if only you provide the initial investment for them to fly over from Nigeria. If you think there's a million to one chance of getting the payoff, does that mean that they can just offer a million times as much money in order to overcome your doubt? No, because then you would doubt the story even more! As they increase the magnitude of their claims, the magnitude of our doubt increases even faster. Salvation is just the limit of this process of runaway humbug.
There's also the same problem I brought up earlier, that we can also think of whimsical gods which punish rather than reward believers. For instance, if conservative Christians of a different denomination are right, then your denomination of Christianity may go to hell for heresy. Is whimsical god #1 really any more likely than whimsical god #2? I think they are about equally likely, although it's understandable that people are more likely to believe in #1.
I will concede one thing about Pascal's Wager: it may in fact present a problem to a certain set of agnostics. That is, if you think there's a 25% chance that God exists and 75% chance that God doesn't, then Pascal's Wager applies to you. However, my understanding, speaking with agnostics, is that this is not what most self-identified agnostics believe. Most agnostics are not on the fence, probabilistically speaking. Rather, they think gods are unknowable, or that it is inappropriate to assign probabilities at all. Further, while many agnostics may be uncertain about gods, they may feel more certain of the nonexistence of salvation.
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