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.

The wager makes more sense once you know two things that happened to Pascal in the same year. In 1654 he exchanged letters with the mathematician Pierre de Fermat about the problem of points: how two gamblers should fairly divide the stakes if their game is interrupted before anyone has won. Their answer, to count the ways the game could have finished and share the pot in proportion, is usually treated as the starting point of probability theory. The idea of weighing outcomes by their chances came straight from the gambling table.

Late that same year, on the night of 23 November 1654, Pascal had an intense religious experience that he recorded on a scrap of parchment, in which the word fire stands on a line by itself. He kept the note sewn into the lining of his coat for the rest of his life, and it was found only after his death. From then on he wrote mainly on religion, closely linked with the strict Catholic community of Port-Royal. The wager is the work of a man who had both calculated odds and felt certainty, trying to reach readers who had only the first.

Expected value is easiest to see with ordinary numbers. Suppose a bet costs you 10 units of earthly pleasure and pays 500 units if it wins, and you judge the chance of winning at one in a hundred. The expected gain is 500 times 0.01, which is 5, minus the 10 you paid, so on average the bet loses 5 units and you should refuse it. Raise the chance to one in twenty and the gain becomes 25 minus 10, a positive 15, and the bet is worth taking. With finite prizes, the answer depends on both the odds and the size of the prize.

Pascal climbed this ladder in the text itself. With even odds, he argued, it would be worth staking one life to win two, and still more to win three; and when the prize is an infinity of happy life, the size of the stake and the length of the odds stop mattering, so long as the chance is not zero. In the calculation above, replace 500 with infinity and every version of the sum comes out infinite and positive. That substitution carries the whole argument, which rests far less on Pascal's estimate of the odds than on his claim that the prize is infinite.

Pascal's interest in calculation was practical long before it was theological. As a teenager he watched his father, a tax official, struggle with long sums, and in the 1640s he designed and built a mechanical calculator to help. The machine, now called the Pascaline, added and subtracted using a series of toothed wheels, with a clever mechanism to carry a digit from one wheel to the next.

Pascal built a number of them, and several survive in museums, but they were expensive and delicate and never sold widely. His name lives on in science and computing in other ways: the unit of pressure is the pascal, and a programming language created by Niklaus Wirth around 1970 was named Pascal in his honour, a fitting tribute to an early builder of calculating machines.

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.