Two envelopes problem

All the probability math therein, for a problem whose solution is highly intuitive (if you swap, you'd be just as inclined to swap envelopes indefinitely, is all you need to realize), reminds me of this quote:

"The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift." - Albert Einstein

Shameless plug for a blog I like that has little math, yet uses much intuition to solve five of the biggest outstanding problems of physics: http://finbot.wordpress.com

That is a very long article based on flawed argument.

Given no other information, assuming someone gave you 2 envelopes and told you one has $40 vs $20, common sense dictates choose 1 randomly and walk away - with no other information it is illogical to reason any other way.

The chance you choose the lower value is 1/2.

Now, if you are allowed to look inside the envelope (which gets introduced further down) then it becomes a different game.

Get $20...well by swapping you may get $10 or $40...you should probably swap.

Get $2000...well by swapping you may get $1000 or $4000...you should probably swap.

I think this works all the way up...someone with a bit more background on game theory may be able to formalise it, but the realisation that swapping forever leads to $0 nullifies this "paradox"

I don't have a BS in maths, but the important bit, as I read it, is that you don't know the values involved. I.e. if you have 20, there is either 10 in the other envelope, or 40. You have no way of knowing which is the case, so it's in your interest to swap since the benefits outweigh the risks.

This is totally counterintuitive, though, so I'm fully willing to accept I'm missing something! And I don't buy the 'indefinite swapping' argument, since the second swap must surely reverse any advantage gained.

Yes, it's not a paradox it's just seductive flawed reasoning. Yes, at any point EV of picking an envelope at random is 3/4n (n being higher amount of money out of the two). It is all there is to it. The "paradox" is introduced by silent assumption that distribution of amounts put in envelopes is uniform which is impossible (because you can't pick numbers from infite set uniformly even if there was infinite amount of money in "adversary" disposal). The assumption is then used for conditional probability calculations: "if we see 10$ there is 50% chance the other envelope contains 20$" - BEEP, ERROR, THINK AGAIN.

Perhaps good exercise in clear thinking but not really a paradox. Good analogy is this: "If we pick random building and climb to the roof of it there is 50% chance first building we see is higher than the one we just climbed". This is obviously true, now following "paradoxical" reasoning we get: "If we climb a building randomly and see it's the Empire State Building there is still 50% chance first building we see will be higher".

This is exact analogy to reasoning about 2 envelopes problem which is supposed to lead to a paradox.

Interesting.

This reminds me of a bizarre version of the Monty Hall problem: http://en.wikipedia.org/wiki/Monty_Hall_problem