Two Envelopes Problem
Updated 2026-08-05
INTRODUCTION
English translation pending.
CORE DEFINITION
A puzzle in probability and decision theory in which one envelope holds twice the amount of the other. After opening one and finding a sum, a calculation suggests the other has an expected value higher than what you hold, which would imply you should always switch. The derivation implicitly assumes a uniform prior over all positive amounts, which cannot be normalized. The core claim is that expected-value reasoning depends on a legitimate prior distribution. The qualification is that with any proper prior the apparent advantage disappears.
SCAFFOLDING EFFECT
Reduce cognitive load
- Use Prior Check: State the distribution you assume over possible amounts before computing any expectation. - Use Condition Distinction: Separate what is true before opening from what is true after you see the amount. - Use Bound Setting: Impose a realistic upper limit so the expectation calculation stays well defined.
Anchor fast decisions
The switching argument treats the amount in the other envelope as equally likely to be double or half of the observed sum regardless of its size, which is only coherent if all magnitudes are equally probable. That prior is not normalizable, so it cannot describe any real process, and the computation therefore has no valid interpretation. Once a proper distribution with an upper bound is specified, the expected gain from switching becomes zero or negative.
MINIMUM ACTION
In progress 0/1Practice this model in one real situation:
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Source support: Explicit
- en.wikipedia.orghttps://en.wikipedia.org/wiki/Two_envelopes_problemverified
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