[NMcNasty]

[ QUOTE ]
I don't know if this has been discussed here before.

here's the paradox ...

Suppose you're given 2 envelopes with money in them and you are told that one contains twice as much as the other (let's say they're cheques and therefore weight is not an issue). You pick one and open it up to reveal a $100 cheque. You are now offered the opportunity to switch envelopes.

Is switching +EV?

Argument 1: It's +EV to switch. You had a 50/50 chance of picking the high or low envelope so there's a 50% chance that the other envelope is the high and a 50% chance it's the low. Therefore, EV of switch = 0.5*(+100) + 0.5*(-50) = +25.

Argument 2: It's EV neutral. If always switching was a +EV strategy then it would be more profitable to choose envelope A first and then switch to B then to just choose envelope B and not switch.

Who's right?

[/ QUOTE ]

Argument 1 is flawed because you shouldn't assign the probabilities of the envelopes either being $50/$100 or $100/$200 to be 50/50. It could be that the probability the envelopes are $50/$100 is 95% and the probability they are $100/$200 is only 5%. In that case switching would be a clear mistake.

If there was some premise that says "no matter what envelope you choose the next one will either have twice as much %50 of the time and half as much %50 of the time", then argument 1 would in fact hold up.

However, simply because you cannot logically assign the probabilites of the two possibilites to be 50/50 that doesn't mean that sticking with the envelope you have is always EV neutral. There are some probabilites you should assign to each possibility P($50/$100)= x and P($100/$200) = y. What probabilties you assign have nothing to do logical factors, they are just estimated guesses based on your knowledge of the how the experiment is set up. So if you assign x to be .60 and y to be .40 you should STILL switch envelopes for a +EV.

So in conlcusion, both arguments are flawed. The decision to switch or not is relevant, but its not based on your possibilites being 50/50.