Return of the two envelope paradox

[TheFallen]

Okay... I *think* I've come up with a way to highlight the error in Hashi's thinking.

Here's the set-up - and it is identical mathematically speaking to the original set-up.

I am the gamesmaster. I have a sealed box. This box has a rubberised seal on it, allowing entry of a hand and arm, but at no point can anything inside the box be seen.

I place a $10 bill and a $20 bill inside the box, put the box in front of Hashi and invite him to put his arm into the box, grab one of the bills and then stop.

Bear in mind that Hashi has no clue as to the actual value of the bills I am using in any given game - he simply and solely knows that of the two bills being used in any game, one is twice the value of the other.

Ok so Hashi does exactly as instructed... he puts his arm inside the box, grabs a bill and stops.

Now at this point it is indisputably true that the bill that Hashi HAS NOT grabbed is either half or twice the value of the one that he HAS grabbed.

I then ask Hashi if he'd like to stick with the bill in his hand, or swap to the other ungrabbed bill.

(See where I'm going with this yet? Let me carry on).

Because this scenario is literally identical to the original "two sealed" envelopes scenario, Hashi's reasoning MUST NECESSARILY tell him that there's an advantage to discarding the bill he's originally grabbed and instead taking the bill he did not originally grab.

So Hashi informs me, the Gamesmaster, that he'd like to switch. On that basis, I tell him to remove his hand still holding the first bill he happened to grab, crumple it up and throw it into a trashcan. I then tell him to stick his hand back in the box, grab and take out the remaining bill and pop it in his wallet.

The above is an EXACT MIRROR of what happens in the originally posed problem if you switch envelopes.

Okay, so I start the exact game again with Hashi. I again place a $10 and a $20 bill in my sealed box... Hashi grabs one, instantly pulls his hand out, crumples the bill up and tosses it in the trashcan (because he maintains that switching is advantageous). He then retrieves the remaining bill and pops it in his wallet.

In fact, I run through the game 100 times with Hashi - every time he discards the bill he first grabs and every time he places the remaining bill in his wallet (because his reasoning assures him that there is a measurable advantage in switching).

So... after 100 plays of the game, how much is in Hashi's wallet? And how much is in the trashcan?

Answer: BOTH amounts will be approaching $1,500. And the more one plays the game, the closer the average amount put into Hashi's wallet AND tossed into the trashcan will EACH approach $15 per game.

Ergo.... there is NO advantage in switching. The statement in bold red italics above, although entirely true, is equally entirely irrelevant to any valid modelling of this problem... and this problem is mathematically identical in every way to the originally posed problem.

QED (please God!)

[Vraith]

TF: reasonably close/similar thought I woke up with, and seems to get at the same point similar way, but less words:
Original setup exact.
Before choosing, does anyone think the odds of picking the x or 2x are anything but equal? [[I don't think anyone's argued that]].
Now you choose one.
Does anyone think the odds have changed yet? There is still a 50/50 chance your envelope is either, and 50/50 the other envelope is either.
At that point, the thought seems to be there is a return on investment that is better if you switch.
BUT: the equation H was the first to write out ALSO applies to the envelope you have, not only the one you don't. [[I've said all that before, I know]].
The extension:
Have 100 people playing the identical game. Say it's 20 and 40. You don't know the amounts, it doesn't matter, I'm just sick of x and 2x.
50 of those people choose to switch, 50 choose to keep what they have.
The result:
25 of the switchers get 20, 25 of the switchers get 40, 25 of the stayers get 20, 25 of the stayers get 40.

Z: I think I previously commented onthe most problematic part.
Just like you don't believe I understand the question here, I don't believe you don't know the attitude, and didn't intend it.
Would "Watch how you speak here." Or "Be careful how you address people" Be different and less offensive than don't be a prick? I don't think so. And it would be SO "PC"...and you hate PC.
Here's the thing/my reason for my shitty mod-ing:
Such are irrelevant...they don't advance the topic/discussion [in fact derail it]. They're unnecessary. They're contrary to the timbre and purpose of this field of dispute/discussion/commentary.
And I'd still let it pass if such had some fun, some humor, some entertainment value. But it mostly just annoys people. I like some good smart-assery as much as anyone...more than most, probably. But not assery. [even with the fairly high risk that the former can be misconstrued as the latter].
This isn't like the tank, it never has been as long as I've been around...these things are rare here, and I'd like it to stay that way.
Can we let this go, and just avoid nastiness now?

[TheFallen]

Vraith yes.

What you are not acknowledging is how incredibly tempting (but just plain wrong) it is to go down Hashi's reasoning route.

Using my sealed box exemplar as per above:-

Now at this point it is indisputably true that the bill that Hashi HAS NOT grabbed is either half or twice the value of the one that he HAS grabbed.

The above is completely true. Period.

However, despite the above...

Having decided to term the bill in his hand as having a value of M, Hashi CANNOT then validly assume that the ungrabbed bill has a 50% probability of having a value of M/2 and a 50% probability of having a value of 2M. That is NOT an accurate representation of the probabilities in this scenario AT ALL.

It looks like what Hashi thinks is completely valid and so basic as to be pointless even discussing. Let's face it, it's basic 8th grade probability... but it's completely fallacious modelling, if applied in this scenario.

But it's still extremely tempting because it looks so obviously true.

[Hashi Lebwohl]

But it's still extremely tempting because it looks so obviously true.

That is because the way I am thinking about the problem is true. That fact that you just don't get it...well, I can't help you with that.

Two envelopes, one contains x, one contains 2x, you are given one of them. The optimal strategy is to switch envelopes. Once you switch you open the envelope, take the money inside, and walk away. Done. Anything else is extraneous and is to be discarded as chasing rabbits.

Okay, *now* I am done with this particular thread. The problem was presented, I solved it, so there is nothing else to discuss.

[wayfriend]

In fact, I run through the game 100 times with Hashi - every time he discards the bill he first grabs and every time he places the remaining bill in his wallet (because his reasoning assures him that there is a measurable advantage in switching).

There's an important point to be made right at this point.

Hashi will draw $10 half the time, and $20 half the time.

But that's not what was presented in the original question! According to the original question, Hashi needs to draw the same amount every time - $20 - in order to produce the 1.25X average results that were advertised. And if he does, what remains is always $10, and so he always loses. because he always switches.

It only works when Hashi always draws $20 -- and half of the time the $10 is transformed into $40.

Obviously this cannot actually happen.

[TheFallen]

Hashi, whereas I admire your certainty, and whereas you can indeed "be done" with this to your heart's content... you're the only person here who doesn't spot the fallacy.

Of course, it is possible that you're the only one marching in time and that the rest of the world is completely out of step...

...but since we're talking probabilities here, what are the chances of that?

WF to be fair, the 1.25 results weren't exactly advertised within the originally posed question. I first proposed the 1.25 scenario as a possible answer to the originally posed question. I absolutely knew it must be wrong... but I just could not for the life of me see where and why it was wrong. It looked too obviously true.

Thanks to your and Zee's and Vraith's help, I do now understand where and why it was wrong.

What I personally don't have is enough of a command of the language of maths to concisely describe the false assumption. It's probably something like "You cannot choose and then formulaically extrapolate from an expression denoting a circumstantially exclusive conditional variable", but what do I know?

[wayfriend]

"I have a 50% chance of ending up with x/2 and a 50% chance of ending up with 2x".

This is the original wording. Upon further review, this sentence is false.

There is no set of circumstances where the chance of getting 2X is non-zero and the chance of getting X/2 is non-zero for the same value of X.

Holding $20, if there is a chance of getting $10, there is no chance of getting $40.

[Zarathustra]

Z: I think I previously commented onthe most problematic part.
Just like you don't believe I understand the question here, I don't believe you don't know the attitude, and didn't intend it.

So you're policing attitudes, not actual words. Ok. If you won't point out what I said that was wrong and why, I submit it's because you can't. Just like you can't show your math. Because I didn't say anything wrong. And you're not doing any math.

This thread is done. Vraith, you're wrong. Hashi, you're wrong. This might shock you all, but I'll say it proudly: Wayfriend was the first one to nail this solution. He got it exactly right (though I think--after the help of Wikipedia--TF and I have helped clarify it).

Now I'm done with your little kingdom, Vraith. I'll just post stuff like this in the Close and circumvent you. Screw your self-importance and illusion of power. [Seriously, it's a logical issue, not a tech issue. It should have been in the Close from the beginning, instead of "Vraith's forum."]

[TheFallen]

To be completely accurate here, the original problem is solely and simply stated thus (as initially posted at thread start):-

There are two sealed envelopes and one contains twice the amount of money than the other. That's all you know.

You are given one of the sealed envelopes.

You are told that you can either keep the envelope you've got... or exchange for the other sealed envelope. Is there anything you can do in order to maximise potential gain?

But when initially posting that problem, I went on to lay out my confusion thus:-

I say "switch" on the apparently logical basis that if I decide to term the unknown amount I am holding x and I switch, then I have a 50% chance of ending up with x/2 and a 50% chance of ending up with 2x. To me, switching then produces an average result of ending up with 1.25x, so must be a good thing.

As stated before, I knew this mathematical model MUST be wrong... but I could not spot where or why. It seemed to fit the circumstances exactly and it looked like it had to be true... but it made no sense.

And yes, WF nailed it absolutely bang on, even if it took him a few goes for me to be able to se things correctly (my fault, not his).

And below follows the neatest and most concise statement so far as to why the "1.25x" model is completely fallacious:-

There is no set of circumstances where the chance of getting 2X is non-zero and the chance of getting X/2 is non-zero for the same value of X.

Holding $20, if there is a chance of getting $10, there is no chance of getting $40.

Hats off to you WF... that is beautifully, succinctly and elegantly put. Much respect.

[Vraith]

Hats off to you WF... that is beautifully, succinctly and elegantly put. Much respect.

I don't want to argue that WF is wrong, nor that the explanation isn't succinct.
Just that there is a supplement, the same thing I've returned to.
It's this: even IF you make the mistaken assumption WF is talking about [impossible reality], or even far more random/absurd things...
You can STILL come to the right answer by noting the description of envelopes A and B contents are the SAME when they are just lying there and no ones even playing as they are if/when someone chooses one. They have an equal chance of being more/less than each other.
When you choose one...the description does not change, at all, for either.
If you're told on contains a trapped pet dragon waiting to be released, the other a formula for a chemical that will grant you eternal youth, choosing either envelope, it is equally likely you HAVE either, equally likely that switching or staying will yield one or the other. Getting dragon or formula are both 1/2. They stay 1/2.
two things equal to the same thing are equal to each other.

If A and B are equally likely {which they must be, given the chooser is ignorant of contents} to contain a billion bucks or nothing...
When you choose one, the one you have AND the one you don't are STILL equally likely to be billions or nothing.
The impossibility WF notes is true...but it doesn't matter if you avoid the error by correctly assigning the description to BOTH sides.
[[in these particular case, of course, if you get to open your envelope before deciding changes the outcome, cuz the values are absolute not relative, that's not so when they're relative like x and 2x].

[TheFallen]

It's this: even IF you make the mistaken assumption WF is talking about [impossible reality], or even far more random/absurd things...
You can STILL come to the right answer by noting the description of envelopes A and B contents are the SAME when they are just lying there and no ones even playing as they are if/when someone chooses one. They have an equal chance of being more/less than each other.

Vraith, that is all absolutely correct... but despite that, you're kind of missing my point on/issue with the whole thing.

Maybe this is a question of personality type. For you, it (entirely reasonably) seems to be enough to conclusively work out a provably accurate and valid mathematical way (or model) of viewing the original problem as posed. Doing that then *must* preclude any other contradictory view/model as inevitably being inaccurate and faulty. And that's clearly enough to satisfy you...

...but not me.

Now I'm no natural mathematician or scientist. At start I kind of had a sketchy grasp of the correct view/model... but that didn't stop me also considering a completely contradictory view/model EVEN THOUGH I knew this second model MUST be wrong... a) because it flatly contradicted the view/model I kind of knew to be absolutely accurate and b) because it further led to a clearly absurd result (infinite switching).

HOWEVER, I wanted to be shown WHY and WHERE the second clearly wrong view/model was in fact invalid. I NEEDED to be shown and thus understand the error - because I could not see it and the wrong view/model superficially at least seemed to be equally compelling and equally applicable to the problem as originally posed.

It wasn't enough for me to come up with - or rather, be shown - just the correct model. The statement "Model A is right and here's how - so you don't need to worry about or even consider anything else" did not satisfy.

As has been noted, looking at the volume of writing all over the place on this seeming paradox, including from people far more able than me - I am clearly not alone in needing the "wrong" explained as well as "the right". And surely having the "wrong" explained as well as the "right" is bound to be more useful and educational?

[Vraith]

Doing that then *must* preclude any other contradictory view/model as inevitably being inaccurate and faulty. And that's clearly enough to satisfy you...

And surely having the "wrong" explained as well as the "right" is bound to be more useful and educational?

On the first...not really. Though sometimes I undertand why those I'm in dialogue with assume that...and it's sometimes my faulty communication that causes it. [[If you really think I'm in that strict school, you should look around and see how often...almost EXCLUSIVELY...I link in the new/strange/unexpected in here.]]
I only want the contradictory model to be accurate about the particular question at hand AND explain equal or more OTHER outstanding but similar problems. [[which some of the explanations did precisely NOT..couldn't really explain the envelopes nor be correct about an even simpler coin toss]].
which flows into the second...
There are quite a number of mathematical/logical problems that have more than one method of solution. The "best" mathematical one is the one that leads/connects to other things.
Sometimes the "best" one, pragmatically, is different [if one is an engineer doing a particular thing...or whatever.] But it only lasts so long/so far.

[[aside...AFAICT, the wiki on this problem, most of the words are dedicated to alternate versions or alternate/purely abstract model universes where there MIGHT be some advantage...which can lead to knowledge, but won't lead to a change in the envelopes in this universe in this problem for these purposes.
The Baysian one is the hardest and most interesting...PROBABLY because Baysian work is pretty closely bound/connected to the psycholigical/epistemoligical]]

Another aside::: the primary reason I kept repeating [in slightly different ways] versions of the same thing was not [and rarely is, anywhere] to insist my way is right...though obviously I, like anyone, have SOME commitment to what I say [[though don't anyone shit that back at me when I'm playing absurd/ironic/sarcastic/satirical]]...it's that MAYBE it will spark in someone ELSE a new thought---whether positive or dispositive. I enjoy the new and different. I'm ecstatic when it's new, different, and right with implications...

Last aside:::though things went elsewhere, I think the full explanation [true under ANY description] of ONE of your initial "questions" or "problems"---the Infinite Switching problem---is nothing more and nothing less than the fact that the "choosing" happens in sequence, instances. But the actual math describes both sides all the time...people think of it as "I'm choosing, then choosing, then..
But REALLY you're rolling two dice at the same time. [mostly].

[Zarathustra]

Vraith, I agree with TF in saying that you seem to be missing the point. WF's sentence, "There is no set of circumstances where the chance of getting 2X is non-zero and the chance of getting X/2 is non-zero for the same value of X," PRECISELY sums up the solution to this problem. It just might be the best sentence he's ever written. It's perfect. Elegant, mathematically relevant, and thoroughly complete in capturing the essence of the error or apparent paradox. If you don't see *this*, I believe you're missing the point.

The problem isn't to find the right answer. We all know (well, most of us know) that there is no advantage to switching. The problem here is to show why the compelling logic used by Hashi and others is wrong. Because it seems plausible.

Think of this like a magic trick. What you're doing is the equivalent of saying, "It's not magic, it's a TRICK!" Whereas what we're doing is explaining how the trick works, the mechanics of it. Big difference. Everyone knows it's a trick. The task here is explaining why it's not magic.

[Vraith]

Vraith, I agree with TF in saying that you seem to be missing the point. WF's sentence, "There is no set of circumstances where the chance of getting 2X is non-zero and the chance of getting X/2 is non-zero for the same value of X," PRECISELY sums up the solution to this problem.

And what I'm saying is:
You must do unto one side what you do unto the other.
is equivalent to what WF said.
If one mechanically follows the previous, it will reveal/suggest WF's statement.
[x/2, x, and 2x will never appear or will cancel out if you do unto both, resulting in WF's explication].

[Lazy Luke]

[x/2, x, and 2x will never appear or will cancel out if you do unto both, resulting in WF's explication].

Absolutely.
But don't get me wrong, I like pure mathematics as much as the next person.
Cancelling out, however, returns the paradox to a simple coin toss.

[TheFallen]

Well... as the originally confused party that started all this, I think I am within my rights to say the following:-

Vraith, you may have a godlike and near instant understanding of all aspects of maths...

...but when it comes to perfectly clear and easily understood explanation (given the freely acknowledged limited ability of the enquirer), at least in this example WF has from my POV run absolute rings round you.

[Zarathustra]

And what I'm saying is:
You must do unto one side what you do unto the other.
is equivalent to what WF said.
If one mechanically follows the previous, it will reveal/suggest WF's statement.
[x/2, x, and 2x will never appear or will cancel out if you do unto both, resulting in WF's explication].

No, that's not what WF's statement means. It means this (from Wikipedia):

"Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2." - Although, in the given situation, that claim can never be applicable to "any A" nor to "any average A".

This claim is never correct for the situation presented, this claim applies to the "Nalebuff asymmetric variant" only (see below).

In the situation presented, the other envelope cannot "generally" contain 2A, but can contain 2A only in the very specific instance where envelope A, by chance, actually contains the smaller amount of Total/3, but nowhere else.

And the other envelope cannot "generally" contain A/2, but can contain A/2 only in the very specific instance where envelope A, by chance, actually contains 2*Total/3, but nowhere else.


So the difference between the two already appointed and locked envelopes is always Total/3. No "average amount A" can ever form any initial basis for any "expected value", as this does not get to the heart of the problem.

It's a description of an error in one's perception of probability that comes from thinking things are more variable than they actually are.

It is true that Hashi's point applies to both envelopes, which is why there is an infinite switching problem. But that's another point, distinct from this one. This one identifies the actual flaw in the thinking, not merely showing that an absurd result happens if we accept the (flawed) thinking. So, your argument is a reductio, whereas WF's point identifies precisely where the logic/math goes wrong.

[TheFallen]

Yep, exactly what Zee said.

I could absolutely see that the extremely persuasive "A/2 and 2A both being simultaneously possible" model - which let's not forget is so persuasive that it can demonstrably sometimes fool those with maths degrees - led to an absurd conclusion.

But I couldn't see where and why it was wrong to apply such a model... and that is a) what I was asking and b) what I needed. Merely knowing that it was wrong was not sufficient.

I now see where and why - and this has without a doubt been educational for me. For starters, it'll cause me not so glibly to apply probability models without really thinking about the scenario. And that is therefore a useful piece of gained enlightenment for me.

[Skyweir]

What a very interesting read. Thank you to every contributor ... Vraith, Z, Wayfriend and Hashi and TF.

Good discussion

[Vraith]

Vraith, you may have a godlike and near instant understanding of all aspects of maths...

...but when it comes to perfectly clear and easily understood explanation (given the freely acknowledged limited ability of the enquirer), at least in this example WF has from my POV run absolute rings round you.

On the first...HAH!! No, I absolutely do not have godlike nor instant understanding. I even think [though not sure...] I was befuddling around on the other side of this the first time it came around...[[and I'm 99 and 44/100ths percent sure you don't believe I have any such powers. ]

I wholly support WF's thing.
My klutzier and mechanical is simply from the other direction.
IF one knows WF's rule/statement, then you will write/describe the problem correctly.
But if you follow the mechanics [also a mathematical rule] you will write/describe correctly and can derive WF's rule.

Sry for resurrection, however temporary. Wasn't around for a while.