Pascal’s wager
from 1662 Published posthumously in the Pensées
When one outcome is vastly worse than the others, the decision can be settled by the size of the stakes without settling the probabilities at all.
- Active
- from 1662
- Where
- Paris
- Also
- le pari
- Status
- In use
- Tradition
- Devices and constraints
What it is
Pascal's argument is not an argument for God's existence. It is an argument that the question of what to do can be settled without answering the question of what is true, when the outcomes are sufficiently lopsided.
Stripped of its theology it is the first decision theory: a two-by-two of belief against fact, with the expected values compared. Where one cell is catastrophically worse than the rest, it dominates the calculation whatever the probabilities are, provided they are not zero.
How it works
Set out the options and the states of the world. Fill in the outcome for each combination. Identify whether any outcome is large enough to dominate regardless of probability. If so, choose against it. If not, the probabilities have to be estimated after all.
The third step is where the argument either works or does not, and it is entirely about the shape of the payoffs.
The form of the inference
- There are two options and two possible states of the world
- The four outcomes are not comparable in magnitude
- One outcome is catastrophic and the others are small
- Its probability is not zero
- Therefore choose the option that cannot produce it, whatever the probabilities
The argument
Proposition Install the second bilge pump.
- Two options: fit a second bilge pump on an independent circuit, or do not. Two states: the primary fails offshore at some point in the vessel's life, or it does not.
- Fit and it never fails: the cost is eleven hundred dollars and a morning's work.
- Do not fit and it never fails: nothing, which is the best cell and by eleven hundred dollars.
- Fit and it fails: the second pump runs, and the eleven hundred dollars bought the vessel.
- Do not fit and it fails: the vessel is lost offshore, and possibly the people aboard.
- The last cell is not larger than the others by a factor; it is different in kind, and its probability is plainly not zero — the primary is a mechanical pump in salt water.
Therefore Fit the second pump, and note that no estimate of how likely the primary is to fail was needed at any point.
This is the wager's structure exactly, applied where it works best. Every insurance decision, every redundant system and every cheap precaution against a catastrophic loss is this argument.
What it is good for
Decisions with lopsided payoffs and unknown probabilities — safety systems, insurance, redundancy, precaution. It is the reason you can decide to fit the pump without a reliability study.
Where it fails
Its original application is the weakest one. If several incompatible options each promise an infinite payoff, the argument gives no way to choose between them — the many-gods objection, and it is decisive against Pascal's own use.
Infinite utilities also break the arithmetic. If the payoff is genuinely infinite, then any non-zero probability yields the same infinite expectation, so the smallest possible chance is as good as certainty, and every option with any chance at all ties.
And it says nothing about belief. You cannot decide to believe something because it would be prudent, which Pascal knew — his answer, that acting as if belief will produce it, is honest about the problem and does not solve it.
Applied loosely it justifies far too much. Any speculative catastrophe can be given a non-zero probability and an unbounded cost, and the argument then licenses spending everything on it.
Common misconceptions
That it claims God probably exists. It claims nothing about probability at all — its whole point is that the decision can be made without one.
Where it came from
The Pensées, published in 1670 after Pascal's death, from notes for an unfinished apologetic work. Pascal had already founded probability theory with Fermat, which is why the argument has the shape it does.
What became of it
Attacked continuously for three and a half centuries, and recognised in the twentieth as the first piece of formal decision theory.
Lasting influence
Decision theory begins here, and the precautionary principle in environmental and safety policy has the same structure.
Why it still matters
The theological argument is contested; the decision structure is used every day by people who have never heard of Pascal.
Put another way
A cheap insurance policy against something you cannot survive.
Texts
Blaise Pascal, Pensées
The wager fragment is short, unfinished and less confident than its reputation, which is worth seeing for yourself.
Where to start
Read the fragment, then draw the two-by-two for any precaution you are considering. The structure is the same.
The surprising bit
An argument for religious belief that almost nobody accepts turns out to be the founding document of the mathematics of decision, and its logic is why your car has two braking circuits.
In their own words
If you win, you win everything; if you lose, you lose nothing
Who held it
Pierre de Fermat
Connections
- Relies on
- Probabilistic inference, Constructive dilemma
- Parallel to
- Constructive dilemma
The verdict
A poor argument for its own conclusion and an excellent structure for deciding under uncertainty.