The Two Envelopes Paradox
Expected-value reasoning misbehaves
First published: M. Kraitchik, *Mathematical Recreations* (1953); modern: J. Broome, "The Two-Envelope Paradox", *Analysis* 55 (1995).
You hold an envelope containing $X. The other envelope contains either $2X or $X/2. Switching has expected value 1.25X. So you should switch — but the same logic applies after switching.
Two envelopes are presented; one contains twice the amount of the other. You select one and find $X inside. The other contains either $2X (probability 1/2) or $X/2 (probability 1/2), giving expected value 1.25X — so you should switch. After switching, the same reasoning recommends switching back. The paradox reveals subtleties in expected-value reasoning under ill-defined priors; standard resolutions appeal to improper priors over unbounded distributions or to conditioning issues. The case is a clean illustration of how naive Bayesian reasoning can produce contradictions.
Formulation
Envelopes A and B; one holds $Y, other $2Y. Choose A, observe $X. P(B = 2X) = P(B = X/2) = 1/2 (naive). E(B) = (2X + X/2)/2 = 1.25X > X. Hence switch. Same applies after switching.
Dimensions Engaged
Observer
Observer · Agency under uncertainty: when expected-value reasoning self-destructs, what should an agent do?
Information
Tests how prior assumptions implicit in expected-value calculations can produce contradictions when ill-specified.
Responses — How Schools Engage
Affirms / takes the bait 2
A clean demonstration that without operational specification of the prior, "expected value" lacks definite content. Decision theory is operational physics for choice.
Even decision theory — the supposed bedrock of rational choice — can dissolve into contradiction under sufficiently abstract setups. A vindication of skeptical caution.
Reframes the question 2
The paradox is a technical artifact of improper priors over unbounded distributions; with any proper prior, expected-value reasoning is consistent. A useful warning against careless Bayesianism.
In any actual application, the prior distribution is constrained by context; the paradox arises from a context-free idealisation.
Holds it inconclusive 1
Multiple resolutions (Broome's conditioning analysis; Albers' bounded-utility resolution); the case continues to motivate careful decision theory.
Related Experiments
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Further reading
- Broome (1995), op. cit.
- Albers, Kooi, Schaafsma, "Trying to resolve the two-envelope problem", *Synthese* 145 (2005)
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