Pascal's mugging
A stranger promises you an astronomically huge reward if you hand over your wallet. It is almost certainly a lie. But if the reward is big enough, the maths says pay.
- Reading time
- 4 min
- Difficulty
- Intermediate
- Branch
- Decision
- First posed
- Eliezer Yudkowsky, 2007; developed by Nick Bostrom, 2009; descended from Pascal's wager, 1670
The puzzle
A stranger approaches you in the street. He has no weapon. Instead he says: 'Give me your wallet. In return, tomorrow I will use my magical powers to give you 1,000 quadrillion happy days of life.'
You are sure he is lying. But are you absolutely certain? Surely there is some tiny chance, one in a trillion trillion, that he is telling the truth. And the reward is so huge that even after multiplying by that tiny probability, the expected value of handing over your wallet is greater than the value of keeping it.
If he senses hesitation, he can simply raise the stakes. However small your probability, he can name a reward big enough to outweigh it.
This scenario, sketched by Eliezer Yudkowsky in 2007 and developed by Nick Bostrom in 2009, is called Pascal's mugging. It seems to show that standard decision theory, which tells us to maximise expected value, can be exploited by anyone willing to make sufficiently outrageous promises.
Why it matters
Expected value reasoning is the backbone of economics, insurance, public health and much of effective altruism. If it breaks down when probabilities are tiny and stakes are huge, we need to know where and why.
The problem is practical. Decisions about existential risks, speculative technologies and long-term futures involve exactly these features: low probabilities of enormous outcomes. Pascal's mugging asks whether we can take such risks seriously without being exploited by them.
Key terms
- Expected valueThe average result you would get if you repeated a gamble many times, found by multiplying each payoff by its probability and adding.
- Pascal's wagerPascal's argument that, since the possible gain from believing in God is infinite and the cost finite, it is rational to believe even if God's existence is unlikely.
- Fanaticism (in decision theory)The property of a decision rule that lets any tiny probability of an enormous payoff outweigh any sure but moderate gain.
- Bounded utilityThe view that there is an upper limit to how good any outcome can be for a person, so that no promised reward can be large enough to outweigh every tiny probability.
- Decision theoryThe formal study of how to choose rationally under uncertainty, usually by weighing outcomes by their probabilities and values.
Proposed solutions
Bounded utility
There is a limit to how good any outcome can be for you. Past some point, more of a good thing adds almost nothing.
Ignore tiny probabilities
Below some threshold, probabilities should be treated as zero for the purposes of decision.
Bigger claims are less believable
Your probability that the mugger is truthful should shrink as fast as his promise grows. Extraordinary claims need extraordinary evidence.
Fanaticism is correct
If the expected value really is higher, you should pay. The mugging shows how strange rationality is, not that it is wrong.
Each solution, tested
Bounded utility
- How it resolves the paradox
- If value is bounded, no promise can be large enough to outweigh a tiny probability. This matches ordinary psychology: a billion happy days is not experienced as a thousand times better than a million. Economists already use diminishing marginal utility for money, and the same idea can be applied to all value.
- Why it doesn’t fully work
- It seems wrong when applied to other people. Saving a billion lives is surely about a thousand times better than saving a million, not barely better. Bounded utility can also produce odd verdicts, where what happens in distant galaxies changes what you should do here.
Ignore tiny probabilities
- How it resolves the paradox
- In practice, people ignore extremely unlikely possibilities, and they are right to. Decision-making that took every one-in-a-trillion-trillion chance seriously would be paralysed. Setting a threshold, perhaps around the probability of the most unlikely risks we already accept daily, protects against muggers without abandoning expected value elsewhere.
- Why it doesn’t fully work
- Any threshold seems arbitrary. And it can lead to inconsistency: many individually negligible risks can add up to a significant one, as with tiny risks from each of millions of cars.
Bigger claims are less believable
- How it resolves the paradox
- The prior probability of someone being able to deliver a reward of size N should fall at least as fast as N rises, so that the expected value of the offer never gets large. A claim to have magical powers capable of delivering a quadrillion happy days is far less likely than a claim to be able to deliver ten.
- Why it doesn’t fully work
- There is no obvious rule that guarantees probabilities fall fast enough for every possible offer. And the mugger can always describe an outcome whose size grows faster than any reasonable penalty on its probability.
Fanaticism is correct
- How it resolves the paradox
- Hayden Wilkinson argues that rejecting fanaticism, the willingness to be moved by tiny probabilities of vast outcomes, forces you into other, worse violations of reasonable principles of choice. Every alternative has costs, and accepting fanaticism may be the least bad option, even if it feels absurd.
- Why it doesn’t fully work
- Most people regard handing over a wallet to a stranger with an absurd promise as a clear case of irrationality. A theory with that verdict seems to have gone wrong.
Who argued what
- Bounded utility Some decision theorists, following Daniel Bernoulli
- Ignore tiny probabilities Bradley Monton, Some approaches to risk regulation
- Bigger claims are less believable Eliezer Yudkowsky, Robin Hanson
- Fanaticism is correct Hayden Wilkinson
Variations
Pascal's wager
Blaise Pascal argued in the 1600s that if there is even a small chance God exists and offers infinite reward for belief, then belief is the rational bet, since infinite reward times any positive probability is infinite. Critics pointed out that many incompatible gods could make the same offer. Pascal's mugging strips away the theology and keeps the structure.
The existential risk version
A charity says that a donation has a one-in-a-billion chance of preventing a catastrophe that would destroy trillions of future lives. Is that a mugging, or a sensible priority? Part of the debate about long-term risks is about whether such reasoning is legitimate.
Go deeper
Why the numbers run away
Expected value multiplies each outcome's value by its probability. Ordinarily, unlikely outcomes contribute little. But if values are unbounded, a sufficiently large outcome can contribute arbitrarily much, however unlikely. Pascal's mugging, the St. Petersburg paradox and Pascal's wager all share this structure: tiny probabilities multiplied by enormous values dominating the calculation.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: Pascal's Wager
- WikipediaPascal's mugging
- PaperPascal's Mugging, Nick Bostrom
- BookPensées, Blaise Pascal
Open questions
- Is there a probability so small that you should ignore it, whatever is at stake?
- How should we weigh tiny chances of enormous catastrophes?
- Is it rational to be immune to arguments you cannot refute?