Indeed. And just as is the case with Newcomb's problem, with the two-envelope paradox also, the dominance principle and the (maximum) expected utility...
Probability is a big philosophical topic. It is quite tightly enmeshed with both metaphysics and epistemology. Michael Ayers wrote a lovely book, The ...
No, that's not what I was saying. I was rather suggesting that, assuming there is some determinate albeit unknown probability distribution of possible...
It does have some philosophical implications. Some of @"andrewk"'s replies raised good philosophical points regarding the status and significance of p...
Yours isn't really a decision tree that the player must make use of since there is no decision for the player to make at the first node. Imagine a gam...
This works if you are treating all the possible lower and upper bounds of the initial distribution as being equally likely, which is effectively the s...
No. The conditional expected values of switching conditional on v = 10, and conditional on 10 being either at the top, bottom or middle of the distrib...
Which is why you have no reason to switch, or not to switch. It may be that your value v is at the top of the distribution, or that it isn't. The only...
But then, if you agree not to disregard the amount of the improbable jackpot while calculating the expected value of the lottery ticket purchase, then...
Unless we are considering that the player's preference is being accurately modeled by some non-linear utility curve, as @"andrewk" earlier discussed, ...
There is another difference between the two methods @"JeffJo" presented, as both of them would be applied to the determinate value that is being obser...
Yes, you are justified in treating the "cases" (namely, the cases of being dealt the largest or smallest envelope within the pair) as equally likely b...
Sure. We can consider a distribution that is bounded on both sides, such as (£10,£20,£40,£80), with envelope pairs equally distributed between ((£10,£...
That's only true if £10 isn't at the top of the distribution. When the bounded and uniform distribution for single envelope contents is for instance (...
Sure, you will never know for sure that the value that you get is close to the top of the distribution. But the main point is that you will have no re...
If the distribution is somewhat uniform with an unfathomably large (albeit finite) upper bound, and you know this, then you can't generally expect to ...
The practical limitations indeed go against the spirit of the idealized two-envelopes problem. That's because if the prior distribution is bounded, al...
Sure, but it is one thing to say that it is rational (or isn't irrational) to switch when you find X in your envelope ($10,000 say) and it is another ...
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 re...
If there exists some bounded and normalized (meaning that the probabilities add up to 1) prior probability distribution that represents your expectati...
What I criticized merely was a failure to draw a logical inference from one particular application of the principle of indifference. The logical infer...
Not at all. I have rather argued that there is an 1.25X expected gain from switching in one specific case of a known distribution {{5,10},{10,20}} whe...
I have chosen a third option, which is to point out the logical flaw in your purported identification of "the flaw". As I suggested earlier, your own ...
This is something I have never disputed. I have never purported to offer an optimal strategy or suggested that there is any way to come up with one. T...
When I say that this can be inferred from that, or that it is fallacious to infer this from that, then I am either right or wrong about it; and in the...
None of my arguments relied on being able to lean "much" about a distribution from one single observation. My arguments rather relied entirely on logi...
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...
It is rather difficult to divorce discussion of probabilities from discussion of statistics. You can't really build an insulating wall between those t...
Of course "only one case is true" at each iteration of the game. Still, the player doesn't know which one is true at each iteration of the game when h...
There is a sort a move in philosophy that is called "proving too much". An argument proves to much when it succeeds in proving the thesis that one pur...
No. You seem to be reaching for an argument that purports to show that your raised expectation from switching, conditionally on having found out that ...
You just said "The envelopes cannot be in both cases at once", followed with the word "therefore...". So it looked like you were making an issue of th...
The argument that purports to show that the expectation from switching is 1.25X doesn't rely on both possible cases (possible consistently with the in...
It doesn't really matter for what? It does matter for invalidating the fallacious argument that purports to show that your expected gain from switchin...
Sure, but that's not what the equiprobability assumption is. What I have been referring to as the equiprobability assumption is the assumption that yo...
Things have blown up long ago. It is precisely the endemic oversight of the logical dependency at issue that is the source of the apparent paradox bei...
I know that. But assumptions regarding the method for filling up the possible envelope pairs (and hence their distribution) entail logical consequence...
OK. Once one is reminded that such an 'uninformed' prior, regarding the initial possible contents of the envelopes, isn't reasonable, then, it follows...
Yes, for sure, but if someone hands me, as a gift, an envelope containing some unknown amount of dollars for me to keep, an 'uninformative' prior that...
I was rather careless in this post where I had made use of Hilbert's Grand Hotel for illustrative purpose. In my thought experiment the equiprobabilit...
The best way to counter that assumption, it seems to me, just is to remind oneself that even though one may not know what the distribution is, assumin...
If you are talking about Don Jr's statement about the meeting (being about adoption, etc.), then it's more than a strong rumor that Donald J. Trump he...
Yes, it is indeed the strange behavior of infinity that generates the paradox. (Edit: Opps, there are some mistakes below that I'll correct later toda...
Yes, yes and yes, but the assumption isn't merely unwarranted, it is impossible that it be true in any real world instantiation of the problem where t...
For sure, I agree, since any non-uniform prior would violate the equiprobability condition that alone grounds the derivation of the unconditional 1.25...
I disagree. Suppose you were to engage in one million iterations of that game and find that the seen envelope contents converge on a specific and roug...
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