&gt; boils down to asking &quot;how do I think the person chose?&quot;<p>I agree, but the operative word is &quot;think&quot;. We can&#x27;t talk about how the opponent chose, only about the player&#x27;s expectations of how they chose. And talking about how evidence affects expectations is the problem Bayes exists to solve, is my reasoning.<p>For your second point, that sounds right but assuming a well-formed high-entropy distribution seems like a very arbitrary thing to do. The player might very naturally, say, consider round numbers more or less likely, or at any rate believe that $50 is a more likely value than $50.01, etc, right?<p>And actually, this point highlights why I think this should be considered as a Bayesian problem. In frequentist terms, one talks about the opponent choosing numbers from some distribution, then the player choosing a strategy based on their expectations about the distribution, then they open their envelope and apply the strategy. If nothing else this approach greatly obscures the most important feature of the puzzle, which is that the player gains new information when they open their envelope. More, it seems somewhat bonkers when considered - if the player chooses a switch value of $10, then opens their envelope and finds $50,000, should they really just apply the strategy they chose based on expectations they now know to have been wildly incorrect? Effectively the player winds up deciding whether to switch based on the fact that they thought $10 was more likely than $20, when one would expect them to reconsider their expectations that they now know to have been wildly off-base.<p>In contrast, the Bayesian version is straightforward, and we can completely describe the player&#x27;s strategy without making any assumptions on how it&#x27;s formed. Specifically, suppose the player starts off with some expectation P(T) that that the total value of both envelopes would be T. Then they open their envelope, find a value of x, and <i>and update their expectations</i>. Then, their strategy depends solely on the ratio between P(1.5x) and P(3x). And we can say all this without even considering whether P is continuous&#x2F;discrete&#x2F;etc, or what factors it&#x27;s based on. All we assume is that it&#x27;s defined at the two points that are relevant to a given instance of the puzzle - i.e. that the player can form some expectation of how likely one value is compared to twice as much.<p>I fear I can&#x27;t describe this very well, but does that make any sense?