What Is Pascal's Wager? Betting on God With the Mathematics of Chance

The wager is fragment 233 of the Pensees, the notes for a defence of Christianity that Pascal did not live to write, and it is addressed not to the believer but to the worldly sceptic who says he cannot believe. Pascal grants the sceptic everything: God is infinitely beyond comprehension, no argument proves his existence, and the evidence is balanced. Then he changes the question from what is true to what to do. A coin is being spun at the far end of an infinite distance, heads or tails, and you must wager; you are not free to abstain, because to live without God is itself a bet against him. Reason cannot tell you which way the coin will fall, but it can tell you the stakes, and the stakes are not equal.

Pascal's calculation is the first appearance of what decision theorists now call expected value: multiply the value of each outcome by its probability and choose the option whose total is highest. The four outcomes:

• •Believe, and God exists: infinite gain, eternal life
• •Believe, and God does not exist: a finite loss, some pleasures forgone and some time spent in church
• •Do not believe, and God exists: infinite loss, damnation, or at least the eternity missed
• •Do not believe, and God does not exist: a finite gain, the pleasures kept

The argument was answered within a generation and has been answered since in the same few ways. The many-gods objection, put by Diderot, is that the wager works equally for any god who offers infinite reward for belief, and the gods disagree about what to believe, so the bettor cannot know which coin is being spun; a bet on the Christian God is a bet against Allah, Vishnu and every god not yet invented. The second is that belief is not a thing one can choose, and Pascal anticipated it: he advises the sceptic to act as if he believed, taking holy water and having masses said, and expects that the habit will produce the belief, which critics call self-deception and defenders call the way anyone learns anything. The third is that a God who rewards belief adopted for a bet, rather than for love or truth, is not worth believing in, and that an omniscient God would know the difference; William James called it a mercenary faith. The fourth is mathematical: infinite utilities break the calculation, since any nonzero chance of an infinite prize gives an infinite expectation, so that a one-in-a-trillion chance of God ties with a certainty, and a lottery ticket for heaven is worth as much as a life of devotion.

The wager is generally judged to fail as a proof that one should believe, and to succeed at something Pascal may have cared about more, which is unsettling the sceptic's confidence that unbelief is the neutral, riskless position. Pascal's point that not choosing is a choice, and that a person who lives without deciding has decided, is sound whatever the odds. And the machinery of the argument outlived its use: the idea of choosing under uncertainty by weighing outcomes against their probabilities became the foundation of economics, insurance and statistics, and the wager is the first worked example in the history of decision theory, applied, characteristically for Pascal, to the largest stake available.

The form of the argument has been used for many things. The precautionary argument for acting on climate change, that the cost of acting if the science is wrong is finite and the cost of not acting if it is right may be catastrophic, is Pascal's wager with the sign changed. So are arguments about existential risk from asteroids, pandemics and artificial intelligence, and the same objections apply, that small probabilities of enormous outcomes can be manufactured for anything, a pattern that philosophers call Pascal's mugging, after a thought experiment in which a stranger promises unlimited reward for the contents of your wallet. The lesson that has settled is that expected value is a good rule for ordinary bets and a treacherous one where the prizes are infinite, and that is a lesson Pascal's wager taught by getting it wrong.

Pascal's wager, from the Pensees of 1670, argues that since reason cannot decide whether God exists and one must live as if he does or does not, the rational bet is belief, because a finite stake against an infinite prize is always worth risking. It is generally held to fail, because it works equally for rival gods, because belief cannot be chosen for a bet and a God who rewarded it would be unworthy, and because infinite prizes break the arithmetic, but it founded decision theory and still shows that refusing to choose is a choice.