Lesson · Decision and Rationality · Puzzles of expected value

Pascal's Wager

Pascal argued that one should wager on God's existence, because the reward is large and the cost small. As a decision table the argument turns on a threshold, the reward at which the expected value of wagering reaches zero, and on what an infinite reward does to every act with a positive probability of reaching it.

Open the interactive lesson Practice

wager costs c; God exists with chance p reward M if God exists EU(wager) = p·M − c EU(abstain) = 0 wagering wins when M > c / p M infinite: any p above 0 wins two gods: the argument recommends both

A threshold, and what removes it

Core ideas

What this lesson establishes

Pascal's argument is a decision table with an unusual payoff. Set it out like any other and the argument becomes a calculation.

The wager as a table

Two acts, wager or abstain, against two states, God exists or not. Wagering costs c in every state and pays a reward M if God exists. Abstaining costs nothing and earns nothing.

The threshold

Wagering scores p·M − c, so it beats abstaining when M exceeds c / p. The smaller the probability, the larger the reward has to be, and below the threshold wagering loses.

The infinite reward

If M is infinite, p·M is infinite for any p above zero, however small, and the argument no longer depends on a probability. That is the strongest form, and the one with the worst consequence.

The material

The wager in a table, its tipping point and its limit

A finite reward first, because that is where the arithmetic lives. The infinite case comes last.

Blaise Pascal's wager is an argument that belief in God is the prudent bet. It says nothing about whether God exists. It says that if you are not certain either way, then wagering on God has the larger expected value, because the reward is so great that even a small chance of it outweighs the cost of living as a believer.

Set it out as a decision table. Let p be your probability that God exists, c the cost of the wager, which stands for everything a life of belief gives up, and M the reward if God exists. Scores are measured from abstaining, which is worth 0.

act God exists no God wager M − c −c abstain 0 0

The expected value of wagering is p·(M − c) − (1 − p)·c, which simplifies to p·M − c. It beats abstaining when this is above zero, that is, when M exceeds c / p.

Theorem · The wager's threshold

Wagering has higher expected value than abstaining exactly when M > c / p. At M = c / p the two tie, and below it abstaining wins.

Pascal himself denied that the second row was available. You are already embarked, he said; not to wager for God is to wager against, and there is no standing aside. The table survives the point. Read the second act as wagering against, with the same payoff of 0 in each state, and nothing in the arithmetic changes; what changes is that a chooser who takes it can no longer say she has stayed out of the game. That is the argument in its strongest form, and the objections below are aimed at it.

Take p = 1/1000, c = 1 and M = 1000. The threshold is c / p = 1000, so the wager sits exactly on it: the lab prints a tie, with expected value 0 and the tipping point p = 1/1000. Raise the reward to 1001 and wagering wins, with expected value 1/1000. Lower it to 999 and abstaining wins. One unit either way reverses the advice, so the number to argue about is the reward and not the shape of the argument.

The first objection is aimed at the premise that the probability is worth counting. A probability of one in a thousand is small, and a reader may be tempted to say that nobody lives by chances like that. The table does not allow it. A chance that small is exactly offset by a reward that large, and that is what an expected value is. Whoever ignores small probabilities must say below which figure they stop counting and why that figure.

The second objection is the sharper one. Suppose the reward is infinite, as the wager's defender wants, so no finite cost can outweigh it. Then p·M is infinite for every p above zero, and the argument no longer needs a probability, only that the probability is not exactly zero. That is also its undoing. Any other god, however improbable, who offers an infinite reward on a different condition has the same argument.

Example · Two gods

God A rewards those who wager on A and God B rewards those who wager on B, each with probability 1/1000 and reward 1001, and wagering on the wrong god costs 1 and pays nothing. Wagering on either scores 1/1000. The argument recommends both, and you cannot do both.

The third preset sets this up. The expected values tie between the two wagers and both beat abstaining. With finite rewards the tie is broken by changing a probability or a reward, and the lab shows how much it takes. With infinite rewards there is nothing to break it with, because infinity times any positive probability is the same infinity. The lab cannot take an infinity as input, and the lesson does not pretend that it can.

Three replies are on offer, and each has a cost. One caps the reward, so that the threshold returns and the argument becomes an ordinary comparison of a probability with a price. One denies that a probability can be assigned to claims like these. One keeps the infinity and accepts that it recommends too much. The lab shows what each does to the table and not which to take.

Interactive

A decision table under every rule

the same payoffs, eight ways of choosing, each computed exactly

Find the reward at which the wager turns

The first preset sits exactly on the threshold. Change the 999 to 1000 and the wager wins; change it to 998 and abstaining wins. The tipping point tile prints the probability at which the two acts tie for whatever reward is in the box. In the third preset change the probabilities to 2/1000, 1/1000 and 997/1000 and watch the tie break in favour of A.

Worked example

Acts, separated by commas

States, separated by commas

Payoffs: one row per act, rows separated by “;”

State probabilities (empty for ignorance)

Probabilities given each act, “act: p1 p2; …”

Forbidden acts

Decision rule

Choice—

The rule’s score—

Value of perfect information—

Expected utilities tie at—

Method

Setting up the wager and finding its threshold

Five steps, with the cost counted once.

Fix the baseline

Measure every payoff from abstaining, so abstaining scores 0 in every state. What the believer gives up is the cost c.

Fill in the table

Wagering pays M − c if God exists and −c if not. Write both entries before computing anything.

Score the wager

Weight by the probabilities: p·(M − c) − (1 − p)·c, which reduces to p·M − c.

Solve for the threshold

Set the score to zero and solve for the reward: M = c / p. Above it wagering wins, below it abstaining wins.

Add a rival

Give a second state with its own wager. If its expected value matches the first, the argument recommends both and cannot choose.

Reading the threshold

Score 0 / 0

Choose the answer you can justify.

The threshold, then one unit above it

Take p = 1/1000 and c = 1. The wager pays M − 1 if God exists and −1 otherwise; abstaining pays 0.

EU(wager) = p·M - c M = 1000: 1/1000·1000 - 1 = 0, a tie M = 1001: 1/1000·1001 - 1 = 1/1000 threshold: M = c / p = 1 / (1/1000) = 1000

At 1000 the wager ties with abstaining, and one more unit tips it. Each factor of ten off the probability moves the threshold by a factor of ten, and a reward that grows without bound passes every threshold.

Common mistakes

Thinking a tiny probability can be ignoredAt a chance of one in a thousand and a reward of 1001, wagering scores 1/1000 and beats abstaining. A chance that small is offset exactly by a reward that large. To ignore small probabilities one needs a rule saying how small, and the table then answers to that rule and not to the wager.

Treating the infinite reward as a bigger finite oneA finite reward has a threshold c / p, and the lab shows the wager flipping at it. An infinite reward has none, because every positive probability reaches it, and the same argument supports every rival wager. The two cases differ in kind and not in size.

Reading the lab's verdict as an argument for beliefThe table scores a wager on a probability that you supply. It says nothing about whether God exists, whether belief can be adopted for a reason like this, or whether the reward and the probability are the right numbers. Those are the premises the argument turns on.

Check your understanding

✓

Finish when you can compute the wager's threshold and say what removes it.

Given a probability, a cost and a reward, you should be able to set out the table, compute the expected value of wagering, find the reward at which it ties with abstaining, and describe what an infinite reward and a rival god each do to the argument.

Note: Pascal's own text has several arguments and this lesson takes only the decision-theoretic one. The probability and the cost are inputs the reader supplies; the lab does not think any particular value is right, and no value can be derived from the table.