Category: mathematics
Associated thinker(s): Multiple contributors; David Chalmers (infinite-expectation extension)
Original publication or approximate period: Modern literature has several versions; Chalmers’s linked extension published 2002
Difficulty: Advanced
Type: puzzle
Themes: conditioning; expectation; priors
Related experiments: st-petersburg-paradox; monty-hall
Two envelopes contain positive amounts, one twice the other. You receive one at random and may exchange it. If your envelope contains x, a tempting calculation says the other contains either 2x or x/2 with equal chances, giving an expected 5x/4. That appears to recommend switching regardless of the envelope you hold.
But the equal chance of receiving the smaller envelope before opening it does not generally imply equal conditional chances after a particular observed amount x. Those chances depend on how the smaller amount was generated. Treating the observed value as fixed while retaining an unconditional probability can manufacture the apparent advantage.
When is an expected-gain calculation licensed by the information and probability distribution actually given?
The problem teaches a distinction between symmetry before observation and conditioning afterward. It also reveals that infinite expectations require special care: familiar rules about averaging conditional advantages can fail when the relevant signed expectation is undefined. A complete problem statement must specify both the information available and the amount-generating procedure.
Finite-expectation analysis preserves the symmetry before opening: random assignment alone creates no unconditional advantage. After observing an amount, a proper prior can make switching sensible in some cases and unwise in others. Infinite-expectation analysis recognizes distributions for which conditional calculations remain peculiar; Chalmers’s extension warns against unrestricted dominance and expectation principles. These are distinct mathematical regimes, not a proof that probability is inconsistent.
“I do not know which envelope I have” does not justify every proposed 50–50 conditional distribution. The letter x must refer to the same random quantity throughout a calculation. One should not subtract infinity from infinity or rearrange a conditionally problematic expectation as though all sums were finite.
Chalmers’s St. Petersburg variant independently fills the envelopes using an unbounded coin-toss distribution. It isolates the difficulty with infinite expectations rather than merely repeating the ordinary finite-prior mistake.
The author-hosted 2002 paper explicitly distinguishes ordinary prior-based reasoning from infinite-expectation extensions. No single inventor is assigned to the many historical envelope formulations.
Verified on 2026-10-30. Source coverage: [1] Ordinary and infinite expectation versions, conditioning and prior distributions. Reflection questions and illustrative numerical examples are original editorial material.