The manuscript presents a three-envelope variant of the exchange paradox with envelopes containing $X, $2X, and $4X where X is drawn uniformly from {1, ..., 100}. While the specific worked example (observing $60) is mathematically correct, the paper's central generalization — the claimed "universal 2/3 rule" — contains a significant error that must be addressed before publication.
The authors claim that the 2/3 switching advantage holds "regardless of the observed amount." This is incorrect under their own stated bounded prior X ∈ {1, ..., 100}.
The 2/3 result requires all three possible X values (X = y, X = y/2, X = y/4) to be valid integers within {1, ..., 100}. At boundary values, fewer than three scenarios exist:
| Observed Amount | Valid X Values | P(switching helps) | 2/3 Rule Holds? |
|---|---|---|---|
| $1 | X=1 only | 1 (certainty) | No |
| $2 | X=1, X=2 | 1 | No |
| $60 | X=15, X=30, X=60 | 2/3 | Yes |
| $200 | X=50, X=100 | 1/2 | No |
| $400 | X=100 only | 0 | No |
The fundamental mistake is conflating two different results: (1) the conditional probability given a specific interior observation, and (2) a universal claim across all possible observations. The authors appear to have over-generalized from the $60 example without checking boundary cases where X/2 or X/4 falls outside {1, ..., 100}.
Specifically, the claim fails for any observed amount y where either y > 100 (eliminating the X = y case), y > 200 (eliminating the X = y/2 case), or y is not divisible by 4 (eliminating the X = y/4 case). The "universal" rule only holds for amounts y that are multiples of 4 satisfying 4 ≤ y ≤ 100.
This is a common error in probability theory: assuming a conditional result generalizes without verifying boundary conditions. The authors' argument is mathematically sound for interior values but fails to account for the bounded support they explicitly define.
Reviewer Verdict: The $60 example calculation is correct and pedagogically valuable. However, the "universal 2/3 rule" generalization is mathematically false and must be either corrected or removed. I recommend the authors add the boundary analysis and present the corrected result: P(switching helps | y) depends on how many of the three possible X values fall within the support.