Pascal’s Wager

Pascal’s wager, practical argument for belief in God formulated by French mathematician and philosopher Blaise Pascal. In his Pensées (1657–58), Pascal applied elements of game theory to show that belief in the Christian religion is rational. He argued that people can choose to believe in God or can choose to not believe in God, and that God either exists or he does not. Under these conditions, if a person believes in the Christian God and this God actually exists, they gain infinite happiness;

if a person does not believe in the Christian God and God exists, they receive infinite suffering. On the other hand, if a person believes in the Christian God and God does not exist, then they receive some finite disadvantages from a life of Christian living; and if a person does not believe in this God and God does not exist, then they receive some finite pleasure from a life lived unhindered by Christian morality. As Pascal states, “Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.”

American philosopher William James objected to the argument that it supported belief in any religion that promised an eternal afterlife. Others have objected that though the argument does give one a reason for believing in the Christian God, it does not make that belief “rational” in the proper sense.

Pascal’s Wager is a philosophical argument formulated by the 17th-century French mathematician, philosopher, and theologian Blaise Pascal, primarily presented in his posthumously published work Pensées. It posits that individuals must make a practical decision about the existence of God, even though reason alone cannot determine the truth of God’s existence. The argument frames belief in God as a rational choice based on a decision-theoretic analysis of potential outcomes.

Pascal outlines a scenario where there are two possibilities: God exists or God does not exist, and two choices: to believe in God or not to believe. If God exists and one believes, the reward is infinite happiness (e.g., eternal life in heaven); if God exists and one does not believe, the consequence is infinite punishment (e.g., eternal damnation). Conversely, if God does not exist, the believer loses only finite pleasures or comforts associated with religious practice, while the non-believer gains finite pleasures from a life unconstrained by religious morality. Given this, Pascal argues that the potential infinite gain from believing outweighs the finite loss, making belief the rational choice.

Pascal emphasizes that the decision is unavoidable: one cannot simply refrain from wagering, as living one’s life inherently involves a commitment to one of the two options. He states, “You must wager. It is not optional”. The wager is not an argument for the existence of God, but rather a pragmatic justification for belief, asserting that it is rational to act as if God exists due to the asymmetry in potential outcomes.

The argument has been formalized in different versions, including the “super-dominance argument,” which claims belief is always better regardless of probability, and the “expectations argument,” which uses probability and utility to calculate expected value, concluding that belief yields infinite expected utility. A more refined version, the “dominating expectations argument,” holds that even a small probability of God’s existence leads to infinite expected value for belief, provided the reward is infinite.

Despite its influence, the wager has faced significant criticism. The “many-gods objection” challenges it by pointing out that other religions also promise infinite rewards, undermining the assumption that the Christian God is the only relevant possibility. The “intellectualist objection” argues that belief cannot be chosen at will, questioning the feasibility of deliberately adopting theistic belief. Additionally, concerns arise about the authenticity of faith if motivated solely by self-interest, and the use of infinite values in decision theory has been questioned as logically problematic.

Overall, Pascal’s Wager remains a foundational concept in philosophy of religion, illustrating the intersection of probability theory, pragmatism, and existential decision-making.

It’s important always to remember Pascal’s project of trying to provide a defense of Christiantiy. Part of the crowd that Pascal hung around with before his conversion did all the things he later identified as diversion: hunting, partying, but especially gambling. Pascal himself is often credited with inventing roulette. So it’s not surprising that he uses the device of a wager to try to convince people, who are already gambling, to take a chance on God. That is, if you want to make a point to a gambler, put it in the form of a bet. That’ll at least get their attention and disturb their indifference.

Pascal begins by presenting a decision. Either a person will choose to believe or she will not (there are only two possibilities). As for the way things are, either the God whom Christians worship exists or that God does not (there are only two possibilities). Pascal thought that reason by itself cannot help you decide which path to follow. And in light of this, and the two choices listed above, Pascal asks ‘how will you wager?’ And note that we have to wager. We can’t just not play the game at all. Pascal thinks that by the way each of us is already leading our lives, we are all already making a bet one way or the other. As he says, “There is no choice, you are already committed.” He continues, “A coin is being spun which will come down heads or tails. How will you wager? Reason cannot make you choose either, reason cannot prove either wrong.”

If we have to bet, then, how shall we decide? Well, says Pascal, let’s look at the possible payoffs (that’s just what gamblers do, after all).

• If God REALLY exists, and we believe (= bet that God exists), we have an infinite gain (heaven).
• If God REALLY exists, and we don’t believe that, then we have the potential of an infinite loss (hell, or at least eternal separation from God).
• If God really does NOT EXIST, and we believe that God exists, we essentially lose nothing.
• If God really does NOT EXIST, and we believe that God doesn’t exist, we essentially gain nothing.

This can be represented in a table as follows:

There are really two other versions of this wager in Paragraph 418, but they just build on the point made above that if one has to bet (and we do), then it’s always more rational to wager on an infinite gain. An infinite gain will always outweigh even a finite loss or gain. Therefore, it’s always more rational to bet that God exists. As Pascal says, if you wager and win, “you will win everything.”

Here’s how he presents the wager in Paragraph 418:

Let us then examine this point, and say, “God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.

Do not, then, reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. “That is very fine. Yes, I must wager; but I may perhaps wager too much.” Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; where-ever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.

For it is no use to say it is uncertain if we will gain, and it is certain that we risk, and that the infinite distance between the certainly of what is staked and the uncertainty of what will be gained, equals the finite good which is certainly staked against the uncertain infinite. It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from fact that there is an infinite distance between them. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain. This is demonstrable; and if men are capable of any truths, this is one.

“I confess it, I admit it. But, still, is there no means of seeing the faces of the cards?” Yes, Scripture and the rest, etc. “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?

But to show you that this leads you there, it is this which will lessen the passions, which are your stumbling-blocks.

The end of this discourse.–Now, what harm will befall you in taking this side? You will be faithful, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognise that you have wagered for something certain and infinite, for which you have given nothing.

“Ah! This discourse transports me, charms me,” etc.

If this discourse pleases you and seems impressive, know that it is made by a man who has knelt, both before and after it, in prayer to that Being, infinite and without parts, before whom he lays all he has, for you also to lay before Him all you have for your own good and for His glory, that so strength may be given to lowliness.