The equiprobability that I am talking about is the posterior equiprobability between the two possible contents of the second envelope: either X/2 or 2*X. — Pierre-Normand

Okay, tell me if I'm doing this wrong.

Let S be the smaller of the two values, M be your envelope, and N the other. This is certainly true:

$\small \begin{align} E(N) &= (2S)P(M=S)+(S)P(M=2S)\end{align}$

And so is this:

$\small \begin{align} E(N\mid M=a) &= (2S)P(M=S\mid M=a)+(S)P(M=2S\mid M=a)\end{align}$

Let $\small p=P(S=a\mid M=a)$; then $\small 1-p=P(S=\cfrac{a}{2}\mid M=a)$. Now we can say this:

$\small \begin{align} E(N\mid M=a) &=2ap+{a\over 2}(1-p) \\&= a\cfrac{3p+1}{2}\end{align}$
Possible values of p:

$\small \begin{align} p&=1\to E(N\mid M=a)=2a\\p&=0\to E(N\mid M=a)=\frac{a}{2} \end{align}$

So far so good. But we cannot do this:

$\small p=\cfrac12\to E(N\mid M=a)=\cfrac{5a}{4}$

anymore than we can do this:

$\small p=\cfrac13\to E(N\mid M=a)=a$

There are only two possible values for N, given an observed value for M, and the only two values for p that produce possible values for N are 0 and 1.

How am I to interpret that result?

I agree with all of those.

*** You might have (3) and (4) a little wrong but I can't judge. The McDonnell & Abbott paper makes noises about the player using Cover's strategy having no knowledge of the PDF of X.

One interesting point about the Arbitrary Cutoff strategy is that Never Switch and Always Switch can be seen as the degenerate cases: Never sets the cutoff to 0; Always to, well, "infinity".

(Btw, good find on the article, @Jeremiah. Addressed my bafflement over how assigning variables leads to trouble.)

The formula just explains why the gain from the strategy is 0.25; the expected value of the other envelope is 5Y/4. — Michael

Not all Sometimes Switch strategies produce an expected gain of Y/4.

None of them calculate their expected gain using your formula.

If that expected value calculation is correct, then Always Switch should produce the expected gain, shouldn't it?

What, in that formula, suggests that a Sometimes Switch strategy is correct?

But that argument, that "calculation", is not based on using any particular strategy. It's just this:

E(U)=.5(2Y) + .5(Y/2)

Do you believe that the success of the various switching strategies available shows that this expectation is correct?

There are Sometimes Switch strategies that work, so far as I can tell.

Do you believe that shows that your original argument, which concludes that the value of whatever envelope you don't have is 1/4 more than the one you do have, is valid?

Unfortunately, we don't (and can't) know the probabilities that remain. For some values of v, it may be that you gain by switching; but then for some others, you must lose. The average over all possible values of v is no gain or loss.

What you did, was assume Pr(X=v/2) = Pr(X=v) for every value of v. That can never be true. — JeffJo

My first post in this thread three weeks ago:

You're right that seeing $2 tells you the possibilities are {1,2} and {2,4}. But on what basis would you conclude that about half the time a participant sees $2 they are in {1,2}, and half the time they are in {2,4}? That is the step that needs to be justified. — Srap Tasmaner

But I still need help with this.

Yesterday I posted this and then took it down:

$\small \begin{align} O(X=a:X=\frac{a}{2}\mid Y=a)&=O(X=a:X=\frac{a}{2})\frac{P(Y=a\mid X=a)}{P(Y=a\mid X=\frac{a}{2})} \\&=O(X=a:X=\frac{a}{2}) \end{align}$

Bayes's rule in odds form, which shows that knowing the value of the envelope selected (Y), provides no information at all that could tell you whether you're in a [a/2, a] situation or [a, 2a].

I took it down because the Always Switcher is fine with this, but then proceeds to treat all the possible values as part of one big sample space, and then to apply the principle of indifference. This is the step that you claim is illegitimate, yes? Not the enlarging of the sample space.

Essentially, we include the impossible values that may come up in calculations in the range, and make them impossible in the probability distribution. — JeffJo

Is this the approach that makes it all work?

I kept thinking that Michael's mistake was assuming the sample space includes values it doesn't. (That is, upon seeing Y=a, you know that a or a/2 is in the sample space for X, but you don't know that they both are.) But I could never quite figure out how to justify this -- and that's because it's a mistaken approach?

Unfortunately, we don't (and can't) know the probabilities that remain. For some values of v, it may be that you gain by switching; but then for some others, you must lose. The average over all possible values of v is no gain or loss. — JeffJo

Right and that's what I saw above -- the odds X=a:X=a/2 are still whatever they are, and still unknown.

Your last step, averaging across all values of V, I'm just trusting you on. Stuff I don't know yet. Can you sketch in how you handle the probability distribution for V?

If the game is iterated, so that you can accumulate data about the sample space and its probability distribution, then it's an interesting but completely different problem. We can still talk about strategies in the non-iterative case.

The three choices are:

1. Never Switch,
2. Always Switch, and
3. Sometimes Switch.

There is some evidence that Sometimes Switch increases your expected gain. (I've played around a tiny bit with this, and the results were all over the place but almost always positive. I don't really know how to simulate this.) That's interesting but not quite the "puzzle" here.

If you were trying to find a strategy that is nearly indistinguishable from Never Switch over a large number of trials (not iterated, just repeated), then it would be Always Switch. If, for instance, you fix the value of X, so that there is a single pair of envelopes used over and over again, then we're just talking about coin flips. A Never Switch strategy might result in HTHHTHTT... while Always Switch would be THTTHTHH... and it couldn't be more obvious they'll end up equivalent.

So then the puzzle is what to do about the Always Switch argument, which appears to show that given any value for an envelope you can expect the other envelope to be worth 1/4 more, so over a large number of trials you should realize a gain by always switching. This is patently false, so the puzzle is to figure out what's wrong with the argument.

The closest thing I have to answer is this:
If you define the values of the envelopes as X and 2X for some unknown X, you're fine.
If you want instead to use variables for the value of the envelope you selected, Y, and the value of the one you didn't, U, you can do that. But if you want to eliminate one of them, so you can calculate an expectation only in terms of Y or U, you have to know which one is larger. (Note that there is no ambiguity with X and 2X.) Which one is larger you do not and cannot know.

some knowledge of the bounded probability distribution of the possible contents of the two envelopes — Pierre-Normand

I'm having trouble imagining what the source of this knowledge might be.

(Never mind.)

it seems pretty relevant to me. — Moliere

Absolutely. In fact, since posting it occurs to me that the concept of "lying" belongs to one level -- the person level, where we hold individuals responsible for their words -- while "self deception" belongs to another level, where we try to understand how we and others think.

I think that's probably right, but we've become so sophisticated that now we hold people responsible for fooling themselves. Which is not completely crazy -- as I said above, I think there are related norms in play here. Both lying and self-deception are violations; they're just not exactly the same violation of exactly the same norm.

what would make this singular self picture a better picture than a split self picture? — Moliere

For some purposes we ignore what's going on under the hood. You, the single individual person, are responsible for what you say, and for the consequences of your decisions. Looking under the hood provides a more nuanced description, but it's really changing the subject.

If you want your qualitative experience to be the foundation of your knowledge, then I think you need to be able to say something like this eventually:

(F) Because I have experience of A, I have knowledge of B.

The question is how to fill in A and B, and whether one determines the other, so that (F) -- is true? must be true? If is the latter, what's the nature of that necessity?

Lets say I'm at a singles bar looking for a date, and I know that statistically my chances of being successful are low — VagabondSpectre

This is a point that should have been made earlier. Beliefs are almost always best thought of as partial, as confidences. You believe you're unlikely to be successful and that you have a chance of being successful. Alcohol either suppresses the former completely, or just futzes with the numbers, so that your chances look better with every drink. You're not going from believing P to believing ~P or something, because you believe both, partially, from the start. Typical self-deception is deliberately mis-calibrating your confidences.

What's wrong with saying that they believed nothing was wrong, but after all night passing without change in their condition, they began to believe that something was wrong. That is, they changed their belief, as compared/contrasted to misrepresenting it to themselves. — creativesoul

Because that happens too, and it's a different phenomenon. What you're missing is that self-deception is usually strongly motivated and irrational.

Here's a salient example: relationships. Self-deception often involves manipulation of evidence, but when it comes to figuring out what are other people think and feel, a lot of that evidence is subtle and ephemeral. We're good at picking up on these tiny tells, almost unnoticeable variations in inflection, expression, eye movement and focus, tone of voice -- all of that stuff we process without usually being consciously aware of it. We just know.

My point is this: it's particularly easy to get away with fooling yourself in this context because your "judgment" was arrived at automatically based on "evidence" you probably couldn't articulate. And that makes it all too easy to dismiss. You don't want to believe something's bothering your spouse? No problem: there's not much you could really point to as evidence anyway. (It was just a feeling you had.) But anyone who's ever done this knows they were fooling themselves.

Or, from the other side, want to believe that cute girl in your homeroom, or at work, or making your coffee, is into you? You can probably find something to count as "evidence". For most of us, enough contrary evidence arrives and quickly enough that a restraining order is unnecessary.

Are the quality of an experience and its content related?

having this experience qua this experience cannot be reasonably refuted, even if what my experience is of cannot be known absolutely known (that is, even though I can be certain I am having an experience, it's possible that the content of my experience is false — numberjohnny5

Your experience has a quality and a content: the quality you know ("know"?) infallibly, but the content -- maybe not infallibly? Maybe not at all? Are you sure there's a foundation for knowledge here?

My god, that's brutal. I had forgotten.

Chomsky said somewhere that his life's work was organized around two complementary problems that he called "Plato's Problem" and "Orwell's Problem". Plato's problem is: How do we know so much, given so little evidence? While Orwell's Problem is: How do we know so little, given so much evidence?

It is humanly impossible to knowingly believe a falsehood. — creativesoul

I daresay you haven't had much practice. When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six falsehoods before breakfast.

They're nearly two sides of the same coin, the same virtue and the same fault: falling to aim for truth in what you say, failing to aim for truth in what you think.

Aren't you shifting the focus from what counts as belief to what counts as acting rationally? — creativesoul

I'm insisting on another aspect of the connection between beliefs and actions. Banno has talked about our use of "belief" in giving post facto explanations of our behavior.

I'm just pointing out that we also say, if you want to find your keys and you think they're in the kitchen, you should want to look for your keys in the kitchen, and anyone who didn't must either have some good reason not to or they just don't think the way we do.

I don't see anything on offer that can take up the role of belief (or expectation, or something else in this neighborhood) in such judgments.

Yes, I understand that as far as you're concerned the phrase "lying to yourself" is just a contradiction. But it's a phrase we all use, so what are the options?

• It's just an idiom and the word "lying" is not meant literally.
• People do literally deceive themselves, even though you, and maybe none of us, don't quite understand how that's possible.
• Your criterion for lying is too narrow and leaves out this case and perhaps others, like "lying by omission".

Stage magic and storytelling both include techniques that rely on our capacity for self-deception. Sometimes the magician, instead of trying to hide how a trick is done, can get the audience members themselves to dismiss the solution, and this is much more effective. A movie can present a character that's a little "off" but not make a big deal about it, and the viewers will mostly decide not to worry about him, until the third reel when it turns out he's the killer.

You could say these are cases of deception, but really it's just giving us the opportunity to deceive ourselves and most of us are generally quite prepared to do so.

All else being equal. It is, as I keep saying, a question of our norms of rationality. In my first presentation of the lost keys, I put a man with a knife in my kitchen. Then it's rational not to look for my keys just then.

I remember hearing years ago that it's common for emergency rooms to have a spike in admissions just before dawn. The explanation was people lying awake all night telling themselves "It's nothing" and eventually accepting that something was terribly wrong.

Are you really not familiar with any of these phenomena?

I would have thought that things like "Pp" and "PPp" were our symbols — Banno

So how do you read 3.1431?

Let me put it a different way.

If you can show that we can, using the perhaps unknown value of our selected envelope, get the same answer we get by ignoring that value and doing some simple algebra, then that would count as a solution to the two envelopes problem. So you're offering a solution and I'm really not.

We haven't actually been talking about solutions much in this thread because some participants did not accept that the simple algebraic solution is actually right.

If, at that point, you postulate a value in an envelope, you need to postulate a probability distribution that covers all possible ways that value could be in an envelope. Even if it is unknown. — JeffJo

I'm just trying to understand the "need to" in that sentence.

Did you miss the part where I said — JeffJo

Don't be that guy. I'm reading your posts and asking about what I don't understand.

(Btw, MathJax is available here, so it's possible to make your equations more readable.)

There is an easier way, but it applies only if you don't know what is in your envelope. — JeffJo

If, even before selecting an envelope, you see that there is no reason to prefer one envelope over the other, what compels you to discard that analysis upon selecting and opening an envelope? Is it no longer true that if you drew X you'd trade for 2X and if you drew 2X you'd trade for X? How could selecting or even looking in an envelope make that false?

Can you give an example of how you'd analyze a typical knowledge claim? Anything about that analysis that makes you uncomfortable?

But confabulation is a little of each.

You know how it's impossible to walk any great distance** if your stride with one leg differs slightly from your stride with the other? Now tell yourself at each step that it's only a little different, and that can't make much difference. It's like that: you relax your cognitive standard just a bit, and indeed it does not make the inferential step you're taking invalid, but if you keep compounding this little compromise you end up in the wrong place. I'd call this a kind of lying to yourself and it's incredibly pervasive.

** in a straight line

Do you have numbers you could share? I've always assumed post deletion and thread deletion were pretty rare, a tiny fraction of the posts submitted and threads created.

It's curious that Plush offers a member reputation system (which I understand there was a decision not to use) but no ability to upvote and downvote threads, which could allow moderators to police policy instead of quality.

How many deletions are for low quality rather than forbidden content? Do we also get a lot of spam?

I guess I'm just wondering if we're really under siege in our little corner of the internet. I remember Usenet becoming, well, unusable, and the internet's response was tools that allowed communities to self-police a bit. Here we're forced to rely on (so far as we know) human moderators. I very much agree with what @TheWillowOfDarkness posted, but I still think allowing posters a way to freely self-sort could be helpful, and it looks like the only tool we have that might reduce the workload of the moderators, whatever that is.

Oh, and "expressed" can be glossed as "becoming perceptible by the senses," like a propositional sign. That's cool.

Well the way I've done this we do get a solid distinction we can work with, between trivial and non-trivial mappings or projections.

It is curious to think of a possible fact actually obtaining as a possibility being expressed. The projection we need, the form of representation, would be a mapping onto substance.

Not sure if the projection is going in the direction W wants though...

Yeah but that misses the whole issue by talking about signs.

Good discussion. (I skimmed a little.)

Here's a point against my suggestion: you would have to say that the projection of the fact onto itself is trivial.

Banno says logical space is populated by propositions. I thought it was populated by possible facts. Propositions have a structure that mirrors the structure of facts, so maybe this makes little difference, in one sense. BUT-- IF a propositional sign, which is a fact that has the form, the structure of another fact, can consist of any sorts of things, not just entities we're accustomed to thinking of as symbols, THEN might we not say the fact itself is in effect a propositional sign, a perceptible expression of a proposition?

That seems slightly crazy, but it dissolves the difference between me and Banno (and it's terribly important we do that). It does mean thinking of propositions as something a bit different from, you know, sentences we assert.

Not clear to me yet. Rest of the book will make it all plain as day, I'm sure.

They deliberately trick themselves into thinking it's ok? — creativesoul

Yes. People, for instance, buy lottery tickets.

To come back to the now canonical example, searching the kitchen for my keys only makes sense if I both want to find them and think they might be there.

Maybe that should be "if and only if".

AND

There's a common asymmetry here: I don't want specifically to find my keys in the kitchen, just to find them; but my (partial) belief about their location has to be more specific to explain the specific action I take. Looking for my keys anywhere and everywhere isn't much of an option.

Oh I don't think *we* should have a homework category-- just an example of a self-sorting option other sites have.

I took another look at the learning center before I posted, but it just doesn't really have a spot for "My First Attempt at Doing Philosophy" posts.

I remember another example, a writing forum I was on years ago, that had categories roughly divided by how harsh you could expect the criticism to be. A sort of "shallow end"/"deep end" thing. It's not that the shallow end means starting from 0, but at least you could avoid the deep end if you wanted to, and others could completely ignore the shallow end.