Two-Envelope Gambit

This question presents a two-envelope setup where two players each hold an envelope, one containing twice as much money as the other. The twist is that the problem is asked in two variants: with envelopes initially unopened, and with both players allowed to peek before deciding whether to swap. The candidate is pushed to reason about whether there is any rational advantage to switching, what each player's expected gain from switching would be, and whether the decision changes after switching once. It probes understanding of how information (or lack thereof) and symmetry affect expected value, and why superficially attractive "always switch" arguments can be misleading.

Solving it leans on careful use of conditional expectation, symmetry arguments, and the role of prior distributions when reasoning about unknown quantities. The interviewer is watching whether the candidate can spot and articulate the hidden assumptions behind naive expected value calculations, avoid double-counting states, and clearly distinguish between ex ante and ex post perspectives. Strong answers balance intuition with precise probabilistic reasoning, handle the infinite or undefined-prior aspect cleanly, and explain why incentives to switch can appear to change (or not) after observing values or after an initial switch.

What it tests

The core structure of this problem class is the interplay between symmetry and information in expectation calculations. When two options are constructed to be indistinguishable (such as two envelopes, one with `m` and one with `2m`), and you lack any information to break that symmetry, any calculation that suggests a preference must be scrutinized for hidden assumptions. The paradox arises when you attempt to assign probabilities to unknowns without a well-defined prior, leading to double-counting or misrepresenting the possible outcomes. The principle is that expected value calculations require a valid probability distribution over all possible states, and when the distribution is undefined or infinite, naive expectation manipulations can yield paradoxical or nonsensical results. This is why, in symmetric and information-free setups, switching cannot rationally improve your position: the system is constructed so that both choices are identically distributed from your perspective.