[NaobisDad]

@BBB: thank you for your reply. You said: "The flaw in argument #1 is that it assumes that, regardless of the amount that we see in the envelope we open, there is a 50-50 chance that the amount in the other envelope is double this amount, as opposed to half this amount. This statement cannot possibly be true for all values that we may encounter in the envelope."

It is possible to make a distribution such that those chances were equal, according to your rational it would then always be +EV to switch. I think this is not right. The reason I think this can be illustrated by making explicit how I expect the game to look.

The way I understand it, you have two envelopes, one containing value N and the other the value 2N. One envelope is picked randomly, and then there is the option to switch or not. Let's say we always switch (for reasons I stated earlier). Most would probably agree that whether you switch is rather trivial.

In the second condition the only change we make is that we flip open one of the envelopes and reveal it's contents. Then we switch again.

Nowhere in the original problem is it stated that the value you're shown influences the probability that the other value will be higher or lower. It simply states that you had 50% chance to pick either N or 2N, and that now that you know one value, you have a 50% chance that you hold either N or 2N. Reasoning on from there they then state that your EV is 50% * value/2 + 50% * value*2.

If argument 1 is wrong, which I feel must be the case, then I think that it is because somewhere a logical step wasn't allowed.

For those who state that sometimes switching might affect EV, do you think that this would still be the case if you were switching blind. And do you think that seeing the value matters?

If not, then you probably agree that agrument 1 is probably flawed, where is it flawed?