Stathis Papaioannou wrote:

The problem is that you are reasoning as if the amount in each envelope can vary during the game, whereas in fact it is fixed. Suppose envelope A contains $100 and envelope B contains $50. You open A, see the $100, and then reason that B may contain either $50 or $200, each being equally likely. In fact, B cannot contain $200, even though you don't know this yet. It is easy enough for an external observer (who does know the contents of each envelope) to calculate the probabilities: if you keep the first envelope, your expected gain is 0.5*$100 + 0.5*$50 = $75. If you switch, your expected gain is 0.5*$100 (if you open B first) + 0.5*$50 (if you open A first) = $75, as before.

Ignorance of the actual amounts may lead you to speculate that one of the envelopes may contain $200, but it won't make the money magically materialise! And even if you don't know the actual amounts, the above analysis should convince you that nothing is to be gained by switching envelopes.

If the game changes so that, once you have opened the first envelope, the millionaire decides by flipping a coin whether he will put half or double that amount in the second envelope, then you are actually better off switching.

I don't think that's a good counterargument, because the whole concept of probability is based on ignorance--if you were omniscient, for example, you wouldn't have a need for probabilities at all. If someone puts $1000 in a blue envelope and then flips a coin to decide whether to put $3000 in a red envelope or to leave it empty, my expected gain from picking the red envelope should be $1500 dollars--it doesn't make sense to say that from the point of view of someone who saw the envelopes being stuffed, it is already certain whether the red envelope contains the money, therefore *my* expected gain from picking the red envelope should be either $3000 or zero. My expected gain is based on my own ignorance of the outcome of the coin toss, information that I don't have access to shouldn't play a role. Similarly, an external observer who knows the content of both envelopes should play no role in the calculation of my expected gain from switching in the two-envelope problem. If I open the envelope and find $50, I don't know whether the other envelope contains $25 or $100, so that information cannot be used when calculating my expected gain from switching. But as I argued before, if I know the probability distribution the envelope-stuffer used to pick the amount in the envelope with less money, then seeing the amount in the envelope I open will allow me to refine my estimate of the probability it's the envelope with less money, there's no possible distribution the envelope-stuffer could use that would insure that no matter how much I found in the first envelope, the other envelope would have a 50% chance of containing double that and a 50% chance of containing half that.

I don't think that's a good counterargument, because the whole concept of probability is based on ignorance--if you were omniscient, for example, you wouldn't have a need for probabilities at all. If someone puts $1000 in a blue envelope and then flips a coin to decide whether to put $3000 in a red envelope or to leave it empty, my expected gain from picking the red envelope should be $1500 dollars--it doesn't make sense to say that from the point of view of someone who saw the envelopes being stuffed, it is already certain whether the red envelope contains the money, therefore *my* expected gain from picking the red envelope should be either $3000 or zero. My expected gain is based on my own ignorance of the outcome of the coin toss, information that I don't have access to shouldn't play a role. Similarly, an external observer who knows the content of both envelopes should play no role in the calculation of my expected gain from switching in the two-envelope problem. If I open the envelope and find $50, I don't know whether the other envelope contains $25 or $100, so that information cannot be used when calculating my expected gain from switching. But as I argued before, if I know the probability distribution the envelope-stuffer used to pick the amount in the envelope with less money, then seeing the amount in the envelope I open will allow me to refine my estimate of the probability it's the envelope with less money, there's no possible distribution the envelope-stuffer could use that would insure that no matter how much I found in the first envelope, the other envelope would have a 50% chance of containing double that and a 50% chance of containing half that.

Jesse