What is the two-envelope problem in simple terms?

One envelope has twice as much as the other. The math seems to say switching always pays, but after switching the same math says to switch back, so you'd switch forever.

So should you switch or not?

If you haven't opened your envelope, the two are symmetric and switching gains nothing. The paradox comes from wrongly assuming the other envelope is always 50/50 to be double. Once you see the amount, the right call depends on how the amounts were chosen.

Two-Envelope Problem

🔬 Natural Sciences By CampusDown Wiki Editorial Team Last updated October 27, 2026 Views 6

The two-envelope problem is a paradox in which you pick one of two envelopes, one holding exactly twice as much money as the other. If you reason that the other envelope is equally likely to hold half or double your amount, its expected value is 1.25 times yours, so switching always seems better. But the same reasoning applies after you switch, implying you should switch forever. It has been debated for decades in probability theory, decision theory and the philosophy of probability.

Contents

1. Structure and history
2. The flaw in the expected-value argument and Bayesian resolutions
3. Infinite expectations and randomized switching
4. FAQ
5. References

Structure and history

The paradox traces back to the necktie paradox presented by Kraitchik (1953), in which two men bet on whose necktie cost more, and each concludes the bet favors him. Gardner (1982) popularized it, and Nalebuff (1989) introduced the modern two-envelope version in an economics journal. Say your envelope holds A dollars. The other holds either A/2 or 2A, and if each has probability one half, its expected value is 1.25A. But this calculation applies equally whichever envelope you picked first, which contradicts the symmetry of the setup. Zabell (1988) analyzed it as a problem that arises when symmetry arguments are misused.

The flaw in the expected-value argument and Bayesian resolutions

Many scholars point out that the assumption that the other envelope is twice as large with probability one half, whatever A is, cannot hold. For it to hold for every amount, the prior distribution over amounts would have to be uniform across all scales, and no proper probability distribution is like that. Christensen and Utts (1992) showed that with a proper prior and Bayes' theorem, whether switching pays depends on the amount observed, and the paradox disappears. Clark and Shackel (2000); Albers, Kooi and Schaafsma (2005); and Katz and Olin (2007) analyzed several flaws in the argument philosophically, including the confusion of letting the same letter A stand for different amounts in different situations.

Infinite expectations and randomized switching

Broome (1995) showed that even with a proper prior, one can construct distributions under which switching is favorable whatever amount you see. All such distributions have an infinite expected value, giving them the same root as the St. Petersburg paradox. Chalmers (2002) called this the St. Petersburg two-envelope paradox and showed that under infinite expectations the principle of maximizing expected utility clashes with the dominance principle. Jeffrey (2004) took up the problem from the standpoint of subjective probability. Meanwhile, Cover (1987) offered a surprising result: if you compare the amount in your opened envelope with a randomly chosen threshold to decide whether to switch, you can raise your chance of ending up with the larger envelope above one half without knowing the prior, and McDonnell and Abbott (2009) analyzed this in detail. Like the Monty Hall problem, it is a classic illustration of the pitfalls of intuitive probability.

FAQ

What is the two-envelope problem in simple terms?: One envelope has twice as much as the other. The math seems to say switching always pays, but after switching the same math says to switch back, so you'd switch forever.
So should you switch or not?: If you haven't opened your envelope, the two are symmetric and switching gains nothing. The paradox comes from wrongly assuming the other envelope is always 50/50 to be double. Once you see the amount, the right call depends on how the amounts were chosen.

Sources reviewed: 2026-10-26

References

Kraitchik, M. (1953). Mathematical Recreations (2nd ed.). New York: Dover.
Gardner, M. (1982). Aha! Gotcha: Paradoxes to Puzzle and Delight. New York: W. H. Freeman.
Zabell, S. L. (1988). Symmetry and its discontents. In B. Skyrms & W. L. Harper (Eds.), Causation, Chance, and Credence (Vol. 1, pp. 155–190). Dordrecht: Kluwer.
Nalebuff, B. (1989). Puzzles: The other person's envelope is always greener. Journal of Economic Perspectives, 3(1), 171–181.
Christensen, R., & Utts, J. (1992). Bayesian resolution of the "exchange paradox". The American Statistician, 46(4), 274–276.
Broome, J. (1995). The two-envelope paradox. Analysis, 55(1), 6–11.
Clark, M., & Shackel, N. (2000). The two-envelope paradox. Mind, 109(435), 415–442.
Chalmers, D. J. (2002). The St. Petersburg two-envelope paradox. Analysis, 62(2), 155–157.
Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005). Trying to resolve the two-envelope problem. Synthese, 145(1), 89–109.
Katz, B. D., & Olin, D. (2007). A tale of two envelopes. Mind, 116(464), 903–926.
Cover, T. M. (1987). Pick the largest number. In T. M. Cover & B. Gopinath (Eds.), Open Problems in Communication and Computation (p. 152). New York: Springer.
McDonnell, M. D., & Abbott, D. (2009). Randomized switching in the two-envelope problem. Proceedings of the Royal Society A, 465(2111), 3309–3322.
Jeffrey, R. C. (2004). Subjective Probability: The Real Thing. Cambridge: Cambridge University Press.

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This article was last updated on October 27, 2026. It is based on widely recognized original works and textbooks; when citing it in a paper, please check the original sources listed in the references.