Wagering a Life:
Pascal’s Wager and the Rational Gambler (Pt. 3 of 3)

Let’s assume that you have just finished eating your favorite snack, a chocolate brownie, when you realize that you have nothing to drink. I notice your predicament and kindly offer you a glass of milk but warn that I had put arsenic in the drink. Although we’re only acquaintances you know the following information about me: I’m psychotic, I have access to arsenic, and that I tell the truth 67% of the time. While you’re extremely thirsty and tempted to drink the milk, you aren’t quite ready to die yet. What should you do?

In order to determine the most rational choice, you have to determine the expected value for each outcome. The expected value (EV) would be the probability of an outcome multiplied by the payoff less the costs of that choice: EV = (Probability x Payoff) – Cost.

Based on my propensity to make honest claims, you assume the probability of the milk being poisoned to be 67%. The payoff -- the benefit of drinking the milk –- is a bit more difficult to determine. Let’s assume that you gain no other pleasure in life other than eating brownies and washing them down with milk. We can measure this pleasure in units of “brownies and milk” (BM) and use that as our basis for a payoff. Drinking this glass of milk would provide you with 1 BM of pleasure while the cost would be the outcome if the milk is poisoned.

For this example we’ll pretend that you are a 50 year old male who enjoys this snack once a day. Based on a life expectancy of 72 years, you can expect to have 8,030 BMs of pleasure in your lifetime (22 years x 365 days x 1 BM). Plugged into this formula we get the following two outcomes:

Assuming the milk is poisoned -- (67% x 1BM) – 8030 BM = - 8029.33

Assuming the milk is not poisoned -- (37% x 1BM) – 0BM = .37

Since the cost of drinking the milk far outweighs any pleasure you might receive, the rational choice would be avoid the milk (and to stay as far away from me as possible).

There is, however, a 37% chance that the milk is not poisoned. In the short term you could get lucky and “beat the odds”, receiving 1 BM of pleasure from the drink. But if the example were repeated you would, more often than not, end up with a serious stomachache before falling over dead. A rational gambler, therefore, would always refuse the milk.

The French mathematician and philosopher Blaise Pascal claimed that we are making a similar bet when it comes to God. By the way in which we live our lives we are either betting that there is a God or that there is not. Since there are no third options, we are either making the decision based either on w rational choice or willful ignorance.

Like Unwin, Pascal believed that there is no overwhelming evidence that can remove all doubt about which choice we should decide. Practical reason may help us determine which is more probable but it cannot ultimately decide the matter one way or the other. What we can do, according to Pascal, is make a rational gamble.

For the sake of argument, let’s assume (as Pascal does) that the evidence will lead us to choose between Christian theism and atheism.* Each choice will result in different expected values based on the unique payoffs and costs.

In order to determine the costs, let’s make the distinctions as clear as possible for our rational gambler by bringing in a cruel dictator to compel him to make his choice. The tyrant not only forces the gambler to bet but vows that if he selects Christianity he will be instantly martyred. If he bets on atheism, however, he will have a life of ease and pleasure – a daily ration of brownies and milk.

To quantify this choice, let’s assume that one unit of pleasure (BM again) is derived for every day of life that is lived. If the 50-year-old rational gambler chooses Christianity, he will lose 8,030 units of pleasure. In contrast, by choosing atheism, he will incur no cost at all. The cost part of the equation can be framed as C = 8,030 BMs and A = 0 BM.

But what about the payoff? What does the rational gambler gain if his bet on Christianity turns out to be the right one? While we cannot know all the details, we know that the belief entails the promise of everlasting life and eternal joy. However we choose to quantify this benefit, the payoff would be infinite. The atheist’s payoff would be limited to the finite benefits that he would receive in the remainder of his life, a figure we calculated at 8,030 BMs.

Now let’s plug these figures into our formula to determine the expected value of each choice.

Christianity: (50% x Infinite) – 8,030 BMs = Infinite BM
Atheism: (50% x 8,030 BM) – 0 BM = 4015 BM

Obviously, the rational gambler would be wise to bet on Christianity. In fact, even if we were to reduce the probability of Christian theism being true to .000001%, he would still end up with a better expected value than he would by betting on atheism.

Pascal’s argument differs from my example in many respects but the point is still the same: practical reason should lead us to act as if Christian theism is true. But we can’t just act as if something were true that we don’t really believe, can we? As Pascal answered his rhetorical objector:

"Yes, but I have my hands tied and my mouth closed; I am forced to wager, and am not free. I am not released, and am so made that I cannot believe. What, then, would you have me do?"

True. But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavour, then, to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith and do not know the way; you would like to cure yourself of unbelief and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness. "But this is what I am afraid of." And why? What have you to lose?

Such an argument is unlikely to persuade someone who is already convinced that atheism is true. For those who have rejected or refused to objectively examine the evidence for Christian theism, living as if the belief were true simply to test its veracity is hardly a compelling option. Pascal, who believed that the passions rather than reason were the root of every denial of God, was most likely aware of this when he proposed the wager. The strength of the argument, however, lies not in its ability to convince but in what it removes. Pascal’s wager essentially removes any appeals to practical reason that the atheist might have been tempted to fall back on and shows that when forced to make the rational gamble on God, the atheist chooses the most irrational choice.

(Sources: Blaise Pascal, Penses; Thomas V. Morris, Making Sense of It All)

*Obviously, Christian theism is not the only possibility since any theistic belief that entailed a God and an afterlife of eternal, blissful existence would also fit into this framework. For now, though, I’ll set aside any claims of a false dilemma and save explanations for why Christianity is the best of all possible theistic choices for another time.

Update: Jeremy Pierce has some additional thoughts on Pascal and epistemic obligations.