[miloshh]

Well, of course you're right, as is yanowitz. But there is something unsatisfying about this explanation. Intuitively it is obvious, but why is the relative reasoning incorrect?

There is no formal distinction between absolute and relative quantities, and no theorem that says that expected values can only be taken from absolute quantities. There are just random variables, and these have some distributions, and they can be independent or not. Nothing prevents you from taking an expectation of a random variable that is a ratio of two other random variables.

Another angle - I could say that our definition of expected value, based on weighted arithmetic average, is completely arbitrary, and instead define my own expected value G[X], based on the geometric average. Suddenly, the relative approach becomes correct: sqrt(2 * 0.5) = 1, so the expected relative improvement from switching is 1. What the hell is going on?

[scotty79]

> Intuitively it is obvious, but why is the relative reasoning incorrect?

Because in the reasoning symbol A is used to denote "expected value of a random variable representing the amount in the envelope you picked" but later on "expected value of a random variable representing the amount in the envelope you picked provided that you picked envelope with more money" and "expected value of a random variable representing the amount in the envelope you picked provided that you picked envelope with less money".

Error comes from splitting reasoning for two cases and failing to factor in the condition on which you split in your further calculation of the cases.

If you solve some equation and you have to split your reasoning in two (or more) cases, the while reasoning those cases you have to remember the condition that you assumed for given case and factor it in (possibly toss away some solutions).

I'd like to see some day less one dimensional way of writing down mathematic reasoning so one can see how information flows through through the course of a proof and errors like this would show up more easily.

[miloshh]

OK, now I see there are two slightly different possible formalizations of the problem:

1. You are told that the two envelopes contain amounts A and 2A, but you aren't told what A is. After you pick one envelope, you are allowed to open it, and then you're given the choice to switch. Here the optimal move depends on the distribution of A, and if you don't know it, you can't do much other than pick randomly. After some googling, this is the more common formalization, and it is analyzed in several math papers and blogs.

2. (The version I was assuming.) You are told that the envelopes have, say, $100 and $200. You pick one and you aren't allowed to open it yet. Now you're given the option to switch one last time. There is no problem with undefined priors and weird conditional probabilities in this version. However, the freaking paradox still holds! The expected value you get by switching is $150, no question about that. But the expected relative gain you get by switching is 1.25, there's also no question about that! This is the real paradox to me. Taking an expectation of a relative quantity is intuitively wrong, but why exactly?

[scotty79]

Ad.1 If in the reasoning A is a fixed number then you can't assume that you have 1/2 probability that the other envelope contains more money. Probability depends on distribution and if you don't know the distribution it cannot be calculated. Fact that it cannot be calculated does not entitle you to assuming its 1/2.

If you have problem with that imagine stack of cards. You and your opponent pick one card. Whoever picks higher card wins. Until you see your card you have 1/2 probability of winning. But after you see you just picked 3 you see your probability of winning sharply changes (depending on what cards are left in the stack).

Ad.2 When you don't open your envelope you must treat amount in it (A) as random variable. Then all I wrote in post above applies.

It doesn't matter that you did not looked into an envelope. Because whether you got higher or lower amount is significant and you don't know which occurred you must split your reasoning and consider both cases. But upon splitting you must remember that condition you assume for each given branch of your reasoning must be taken into account. You can do this by pruning your random variable A with the appropriate condition and that influences its expected value. So by all means you can sum up expected values but not before appropriately adjusting them with the conditions you assumed for branches of your reasoning.

Consider solving equation x^2 + yx +1 = 0 While solving your may split you reasoning to three cases depending on whether y^2-4 is larger, smaller or equal to zero (even though you don't know what is the value of y). But when summing up solution you must remember that you assumed something about y in your branches so for the x you've found y can no longer be any real number.

[miloshh]

Ad 1 - Not sure what you're trying to say, all I said is that you don't know the distribution of A - if you knew it, you could make an informed decision on whether to switch.

Ad 2 - If you carry out the analysis you're trying to teach me, you'll find that the expected relative gain from switching is 1.25. Now explain that. (You'll also find that the expected absolute gain is 0 - that's not a paradox.)

[scotty79]

Let's denote expected value of money in envelope I picked with A.

If I want to switch I have to consider two cases. Let's label them e1 and e2. e1 means I picked envelope with smaller amount of money. e2 means I picked envelope with larger amount of money. Probability of each case is 1/2.

What is the expected value of money I'll have after switch?

Let's consider first case:

S(e1) = 2 * A(e1)

which mens two times the amount of money I had PROVIDED THAT I PICKED ENVELOPE WITH LESS MONEY.

A(e1) is not equal A If I denote exact amount of dollars in envelope with less money by X then A = 1/2 * X + 1/2 * 2 * X = 3/2 * X but A(e1) = X

Similarly considering case e2 I get expected value of money after switch equal

S(e2) = 1/2 * A(e2)

which means half of the amount of money I had PROVIDED THAT I PICKED ENVELOPE WITH MORE MONEY.

Again A(e2) is not equal A but 2 * X, if you prefer not to use X then A(e1) = 2/3 * A and A(e2) = 4/3 * A

To sum up probability of each case is 1/2 so S that I use to donate expected value of money I could have after switch is equal to:

S = 1/2 * 2 * X + 1/2 * 1/2 * 2 * X = 3/2 * X = A

or if you want without using X

S = 1/2 * 2 * 2/3 * A + 1/2 * 1/2 * 4/3 * A = A

So there is no gain or loss from switching. You can switch zero or more times without changing expected value of money you will get after opening your envelope.

Similar reasoning can be conducted for any amount of envelopes and any probability distribution and I guess it will to same conclusion. I'm not sure if observable symmetry of situation can be taken as proof of this conjecture but I sincerely hope so.

Paradox comes from using same symbol A to denote three different expected values of random variables which I denote above as A, A(e1) and A(e2)

If you have further doubts about above "relative" reasoning I'll gladly try to clear them up.

[miloshh]

You just painstakingly carried out the absolute analysis, which I know works.

Let X be the value in the envelope you have, and Y in the other one (X and Y are both random variables with well-known distributions). Then E[Y/X] = 1.25. That's what I wanted you to explain. You just keep saying that E[X] = E[Y], which I know.

Note that this paradox would not arise if X and Y were independent, since then E[Y/X] = E[Y] / E[X] = 1.

[yanowitz]

You're combining two worlds and then adding up probabilities. But there's no world where all three possible values are in play at the same time. There's only two, and you either picked high or low. It's a 50/50 shot.

[miloshh]

Which three worlds? After you have picked one envelope, there are two possible relative outcomes - 2 and 0.5. The average is 1.25 - there is nothing wrong with the math, the only problem is that taking an average of two relative quantities is intuitively not useful for decision making. Putting the intuition on a formal grounding is not something you succeeded with.