You are no longer talking about just probability anymore, since you can now sample the distribution you are now engaged in statistics, which is outside the scope of the OP — Jeremiah
You don't know the distribution, you don't know the limits and you only get once chance to switch. — Jeremiah
Statistics is a data science and uses repeated random events to make inference about an unknown distribution. We don't have repeated random events, we have one event. Seems like a clear divide to me. You can't learn much of anything about an unknown distribution with just one event. — Jeremiah
It is also rational to want ice-cream on a hot day. You still don't know anything about the distribution. You are speculating then trying to model your speculations. — Jeremiah
Hey, if you feel lucky then switch, if you think you are close to the cap don't, feel this one out, but you are not going to be able to quantify a positive gain based on the information we have. — Jeremiah
As far as I am concerned I already found the flaw. Take it or leave it, that is your choice. — Jeremiah
So you think you always have a 1.25 expected gain in every case? — Jeremiah
It is rational to want to maximize your expectation even when you only get one single chance to play, and it is irrational to dismiss your expectation merely on the ground that just one of the possible outcomes will be realized. — Pierre-Normand
If you are using the principle of indifference then criticizing people for using the principle of indifference, that is hypocritical. Either accept that as a standard starting point or don't — Jeremiah
You seem to be saying that after picking an envelope I have to go from saying that there's a probability of 0.5 that I will pick the smaller envelope to saying that the probability is unknown that I have picked the smaller envelope. — Michael
But what action does your answer entail? Switching or sticking? If you say it doesn't matter, and so you're being indifferent, isn't that the same as treating it as equally likely that the other envelope contains the larger amount as the smaller amount? And if you're treating them as equally likely then isn't it rational to switch?
Yes, it is rational to switch if you are justified in treating them as equally likely. But if it is axiomatic that they are equally likely, as most statements of the two-envelopes paradox seem to make it, then you must also infer that the prior distribution is uniform and unbounded with all the weirdness that such an ill-defined probability distribution entails. — Pierre-Normand
So what's the rational decision if you know that the prior distribution is isn't uniform and unbounded? There's £10 in your envelope. Should you stick or switch? — Michael
I am fine with acknowledging that there isn't any such thing as the rational decision to make in the vaguely specified case where you merely have a reasonable expectation that the amount of money can't be infinitely large but you don't have any precise idea how very high the distribution might be tailing off. — Pierre-Normand
But this just seems to be saying that there's no reason to believe that it's more likely that the other envelope contains the smaller amount and no reason to believe that it's more likely that the other envelope contains the larger amount and so you're effectively treating each case as equally likely, in which case it would be rational to switch. — Michael
So my takeaway is that if it isn't rational to stick then it's rational to switch. — Michael
Suppose for instance that tomorrow morning you are being called to play this game with real money in the context of some scientific experiment conducted by the psychology department of your local university. You are to play only once and keep the money. Suppose you open your envelope and find $96 in it. Is it rational to expect 1.25*$96 from switching? I am not committed to saying this. What I am committed to say merely is that it is increasingly irrational to expect 1.25*X (or more) from switching as the value that you find in your envelope increases to ever lumpier sums. — Pierre-Normand
Sure, the practical limitations of real life play a role, but I wonder if such limitations go against the spirit of the problem. What if instead of money it's points, and the goal of the game is to earn the most points? There isn't really a limit, except as to what can be written on paper, but with such things as Knuth's up-arrow notation, unfathomably large numbers like Graham's number aren't a problem. — Michael
The average raised expectation for the unconditional always-switch strategy, always is zero. In order that the expectation be exactly 1.25X, whatever X, and therefore also, the always-switch strategy superior to the always-stick strategy, then there ought to be no bound to the prior distribution of possible envelope values, not just an unfathomably large albeit finite bound such a Graham's number. If the upper bound is Graham's number, then the average raised expectation from the always-switch strategy still is exactly zero. — Pierre-Normand
My argument is that given how arbitrarily large the numbers in the envelopes can be (using points rather than money), there isn't really a point at which one would consider it more likely that your envelope has the larger value. If my envelope is 10 then it's rational to switch. If it's 1,000 then it's rational too switch. If it's 10100 then it's rational to switch. — Michael
But what are you rationally to do when you get "average" values from the distribution, which are unfathomably large? — Pierre-Normand
I have no way of knowing that my value is "average". Perhaps the 10^100 in my envelope is a puny value because the upper bound is Graham's number. — Michael
Suppose the expectation is (close to) 1.25X for purpose of reductio. You are being dealt some unfathomably large amount X which is typical from the actual distribution. We suppose that you are generally warranted to switch on the basis of the principle of indifference, and thus the expectation that switching yields the expected value of (roughly) 1.25X. After you've switched, but before you are permitted to look at the content of the second envelope, you are being given to opportunity to switch back. Is it rational for you to switch back? By the very same argument that justified your initial switch, you should deduce that the expectation for switching back is roughly 1.25Y, where Y is the content of the second envelope. But that's an inconsistency. — Pierre-Normand
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