The Power of Pascal's Wager

By Andrew Rogers and Liz Jackson

1. What is Pascal's Wager?
Pascal's Wager is a powerful tool when it is used as a framework for apologetics. The power of the wager comes from the fact that it renders irrelevant all arguments for any worldview with a finite afterlife (including those with no afterlife).

French Mathematician Blaise Pascal suggested approaching the question of religious belief as a gambler. Think of belief in God as "betting on God" and atheism as "betting against God."

If you bet on God and there is a God then you will go to heaven. If you bet against God and there is a God then you will go to hell. If, on the other hand, there is no God then it won't matter which way you bet since atheists and theists will all rot in the ground equally.

The obvious conclusion is that one ought to bet on God. As Pascal said, “Wager, then, without hesitation that [God] is” because “there is here an infinity of an infinitely happy life to gain” and “what you stake is finite.”

2. Objections to Pascal's Wager
In this post, we will go over two important objections to the wager and then we will explain how we believe the wager can still be useful for choosing between religions in spite of these objections.  In particular, we will show how Pascalian reasoning can give us a reason to favor religions with infinite afterlives over those with finite afterlives.

2. 1. The Many Gods Objection
One common objection to Pascal’s Wager is to point out that Pascal’s version of the Christian God isn’t the only God possible; the Gods of other religions need to be included in the matrix.  Many of these religions are mutually exclusive, and believing the truth of one theistic religion will often not give you the payoff of another.  If one adds a Muslim God who sends Christians to hell, then the results become inconclusive.

When there are many religions involved the decisions becomes more complicated. However, it still seems that betting on atheism is a bad bet. This is because the best atheism can do is offer an opportunity to sin for a finite time whereas most religions offer an infinite afterlife of pleasure as reward and an infinite afterlife of pain as a punishment.

However, philosophers Sober and Mougin (1994) argue that atheism can avoid this negative outcome. They suggest the possibility that theists go to hell and atheists go to heaven. This could be either because there is a God who punishes theists and rewards atheists or because the laws of the universe are so structured that atheists will live forever in pleasure while theists live forever in pain (given our extremely limited understanding of the universe it would be premature to say that this isn't possible). Let's call this possibility "Atheism +."

Even if one thinks that this possibility has a low probability, it should not be assigned a zero probability. This leads to a situation where it is not obvious that any decision is better or worse than another.

Sober and Mougin conclude, “grant that there is some chance, however small, that [Atheism+] is true, and prudential considerations lead straight to the conclusion that it doesn't matter whether you are a theist or an atheist.” (Sober and Mougin, (1994), p. 386. They call Atheism+ “Theology X.”)

2.2. The Mixed Strategies Objection
In Waging War on Pascal’s Wager, Hájek argues that the original wager is simply invalid. He contends that even if the many gods objection were somehow addressed, the argument would still fail. He points out that any decision one makes includes the positive probability that one will eventually come to believe in God and therefore has an infinite expected value.

The power of Pascal’s original wager is that no matter how small one’s credence in the existence of God - as long as it is positive - that number multiplied by infinity will be infinity. Hájek turns this around on Pascal and argues that any action which could potentially lead to belief in God, no matter how small the possibility (as long as it’s positive) will be infinity once it is multiplied by infinity. For example, if the probability of eventually coming to believe in God given the decision to tie your shoe is greater than zero, the EV of deciding to tie your shoe is infinite.

As Hájek puts it in his paper,
“Wager for God if and only if a die lands 6 (a sixth times infinity equals infinity); if and only if your lottery ticket wins next week; if and only if you see a meteor quantum-tunnel its way through the side of a mountain and come out the other side ... Pascal has ignored all these mixed strategies - probabilistic mixtures of the "pure actions" of wagering for and wagering against God - and infinitely many more besides. And all of them have maximal expectation. Nothing in his argument favors wagering for God over all of these alternative strategies.” (Hájek, (2003), p. 31).

3. Salvaging Pascal's Wager
To a large extent we agree with the points made by Sober, Mougin, and Hájek; they bring out some serious problems with Pascal’s Wager. However, we think that they prove too much if an implication of their arguments is that we cannot rationally rank one infinite reward over another using contemporary decision theory.  There are many situations where it is clearly rational to prefer one infinite reward to another.  Two such examples are as follows.

3.1. Eternity of ecstasy versus eternity of moderate happiness
Imagine a relatively happy moment of your life: perhaps receiving a good grade on a test or enjoying a decent meal. Now imagine one of the most incredibly joyous occasions of your life: a moment of great love, compassion, glory, creativity, etc.  Now imagine that you have the option to choose between two “heavens.” In the first heaven, the moderately good moment is repeated infinitely for an eternity of moderate happiness.

In the second version, the moment of peak joy is repeated infinitely for an eternity of ecstasy. However, without a way to compare infinities, we are multiplying a finite amount of happiness by infinity, so the result will be infinity for both. The natural interpretation of the arguments given by Sober, Mougin, and Hajek do not give us a way to prefer one afterlife to the other.  Therefore, it appears like their arguments have proved too much, because it seems rational to prefer the infinity of ecstasy to the infinity of moderate happiness.

3.2. Same happiness; different probability
Imagine that you have two eternities laid before you.  Both “heavens” are infinite, and in both, you will experience the same level of happiness at each moment.  The catch is, neither heaven guarantees you will receive its reward; in each, there is a chance you could be annihilated instead.  In the first heaven, the probability you will get the reward is 0.000001.

In the second heaven, the probability you will get the reward is 0.999999.  Both heavens offer the same payoff, but it is clear that you should prefer the second to the first.  Therefore, simply because two religions offer the same infinite rewards does not necessarily mean they are equal; the probability you will get the reward should also be a part of the equation.

These two thought experiments show that, in many cases, depending on per-time-period payout and probability you will get the reward, it is rational to prefer one infinity to another. We will incorporate this intuition into Pascal’s decision matrix, and utilize it to response to Sober, Mougin, and Hajek’s objections.

4. Using Pascal’s Wager as a Framework for Apologetics
In order to address both objections at once, we propose that one deal with infinity differently than it is dealt with in the standard formulation of the wager. In the standard formulation, the agent’s credences are multiplied by infinity for the religions offering infinite rewards and, as long as the credences are positive, this always leads to an infinite expected value. We want to reformulate how the quantities of infinity are compared.

4.1. Pleasure Per Period
First, we will distinguish the amount of pleasure experienced in the moment from the duration of time for which one gets to experience pleasure.   We will assume it is possible for a finite being to exist for an infinite amount of time, but that it is impossible for a finite being to experience an infinite amount of pleasure at any particular moment.

Hájek (2003) proposes approaching infinities in a similar way; he considers both the idea of using finite utilities over an infinite time period and the idea that humans have a saturation point for experiencing reward.  Hájek points out that, if God could have created beings with a higher saturation point, salvation would no longer be the greatest thing possible.  Pascal would have rejected this assumption, and so Hájek discounts this approach because it is not true to the spirit of Pascal.  However, this seems to be more of a problem for Pascal’s particular theology than an objection to the logic of the reformulated wager itself.

4.2. Ratio in the Limit
The second way in which we want to deal with infinity differently is that we want to focus on finding the ratio in the limit between two (or more) rewards, instead of simply multiplying everything by infinity. In section 3, we explained how it can be rational to prefer one infinity to another. Measuring different infinite rewards using ratios and limits will enable us to capture the intuition that often, one infinite reward is better than another.

Our proposal is to find what the ratio in the limit between the two options would be; instead of multiplying the two finite amounts of pleasure by an infinite amount of time, we propose multiplying them by larger and larger amounts of time until one finds their ratio in the limit.  In our first example of section 3, where one chose between receiving moderate happiness or ecstasy for infinity, suppose the moderate happiness was 1 unit of pleasure/day and the ecstasy was 100 units of pleasure/day. The ratio would be 1:100, and we could rationally choose the second option over the first, even though they are both infinite rewards.  We will also include one’s credences for each religion in the ratio, since our second thought experiment showed that, ceteris paribus, one ought to prefer the religion for which one has a higher credence over the one which has a lower credence, even if they both offer the same infinite rewards.

4.3. Maximizing Expected Value
We should be clear that when we say we are salvaging the wager, we take the important part of the wager to be that it is a decision theoretic apparatus that favors religions which promise an infinite afterlife over those which do not. Using our approach, an infinite religion with a non-zero credence will always beat out any non-infinite religion.

5. The Power of Pascal's Wager

The power of Pascal's Wager is not that it requires one to automatically convert to a specific version of Christianity. Rather the power of the wager is that it renders irrelevant the arguments for all finite worldviews.

Let's say that some atheist John Doe Atheist knew of only two arguments relating to religion: one for Christianity (such as Nathan Conroy's post) and one for a finite atheist worldview. Let's say that John Doe Atheist has a .3 credence for Christianity and a .7 credence for Atheism based on these two arguments. The power of Pascal's Wager is that it eliminates the argument for the finite atheist worldview with the power of infinity.

Therefore, John Doe Atheist will either need to readjust his credences based on eliminating the atheist argument or come up with arguments for something like the Atheism+ worldview where theists go to hell and atheists go to heaven. This is much more difficult for the atheist because he can no longer simply play defense. The Atheist must give a positive argument for the view that there is either a God (Gods) that sends theists to hell and atheists to heaven or for the view that the laws of nature somehow automatically give atheists an infinite life of pleasure and theists a life of infinite pain.

The power of Pascal's Wager is that it drastically raises the bar for any standard atheist worldview and thus consequently lowers the bar for any theist worldview.