Pascal's Mugging
Expected-value reasoning under astronomical stakes
First published: N. Bostrom, "Pascal's Mugging: Tiny Probabilities of Vast Utilities", *Analysis* 69 (2009): 443–445.
A stranger demands your wallet. Refuse, he claims, and he'll inflict 3↑↑↑3 days of agony. Expected utility says hand it over — but obviously don't.
Bostrom's problem: a stranger threatens you with astronomically vast harms (e.g., torturing 3↑↑↑3 people) unless you comply. The stated harm is so large that even a tiny credence in the threat's being real gives compliance high expected utility. Generalising: any bounded credence times sufficiently large utility wins over reasonable refusal. The puzzle reveals that expected-utility decision theory cannot handle unbounded utilities responsibly. Standard responses appeal to bounded-utility axioms, anthropic considerations, or skepticism toward extreme-tail scenarios.
Formulation
Mugger says: "Pay $5 or I will inflict 3↑↑↑3 person-days of suffering." Credence p > 0. Expected disutility of refusal = p × 3↑↑↑3, which exceeds the $5 cost for any p > 5 / 3↑↑↑3. Naive EU: comply. Intuition: ignore. Conclusion: EU theory misbehaves under unbounded utilities.
Dimensions Engaged
Observer
Observer · Agency under extreme uncertainty: how should agents reason about vanishingly small probabilities of astronomical stakes?
Information
Bears on the structure of priors: how should credences distribute over scenarios of unbounded utility?
Responses — How Schools Engage
Affirms / takes the bait 1
A useful cautionary tale for simulation-style arguments: when the conclusions are sensitive to tiny priors over vast quantities, the reasoning may be brittle.
Denies / rejects the premise 1
A claim with no operational way to verify or falsify (3↑↑↑3 days of torture) has no meaningful empirical content; the puzzle dissolves once EU is restricted to meaningful claims.
Reframes the question 3
Practical rationality should be bounded; agents should refuse muggers on Bayesian-prior grounds (priors over absurd claims fall faster than utilities grow).
Reasonable agents discount extreme-tail scenarios for non-decision-theoretic reasons: such reasoning is exploitable, undermines institutions, and is not how intelligent agents actually decide.
Anthropic and multiverse reasoning often involves astronomical-utility considerations; the Pascal's mugging structure may already infect them.
Holds it inconclusive 1
A live foundational problem for decision theory. Bounded-utility responses, leverage penalties, and reference-class-based priors all have defenders.
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Further reading
- Bostrom (2009), op. cit.
- MacAskill, *What We Owe the Future* (2022), Appendix
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