Generate a random outcome between 0 and 1 to represent a value placed in one of the evelopes (x).

Based on that random outcome create another value that is twice that amount to represent the other envelope (2x).

Generate a second random outcome between 0 and 1 representing the initial choice of the player, where if that outcome is >0.5 the first envelope is chosen and if it is <0.5, the second envelope is chosen.

Subtract the unchosen envelope from the chosen one for the change in wealth.

Loop that a substantial number of times counting each decision and divide the accumulated wealth by the number of decisions.

Should trend to 0.

Seems to me that this would simply be 0.5(x-2x)+0.5(2x-x) = 0.

Note, by the way, that if the distribution is finite, the math works out fine, since you now have values in the envelope in which the probability of 2 times that value is zero. (Any value greater than half the finite limit on the distribution). I leave as an exercise for the reader to show that in this case switching envelopes is a wash.

Quote:jfalk...The reason you can't do an expected value calculation here is that there is no proper (ie integrates to 1) distribution with the feature that the probability of 2x is equal to the probability of x ...

I've read that argument before. You've stated it well, and I agree that is a flaw in the EV=1.25x argument. However, is an understanding of integral calculus necessary to see the light? My gut tells me that there should be a way to debunk the EV argument with just simple algebra.

p.s. Welcome to the forum, I hope you'll stick around (I know this guy, he is very smart).

Quote:WizardI've read that argument before. You've stated it well, and I agree that is a flaw in the EV=1.25x argument. However, is an understanding of integral calculus necessary to see the light? My gut tells me that there should be a way to debunk the EV argument with just simple algebra.

p.s. Welcome to the forum, I hope you'll stick around (I know this guy, he is very smart).

You can also debunk it with third grade arithmetic, or simple common sense, in that if it were possible to add EV by switching, it would also be possible to increase the value of each envelope infinitely by infinitely switching. I would imagine that this intuitive conclusion is mirrored by the GIGO effect that happens when you plug "zero" into one of those complex calculations.

I do have an interesting question (which I am posting on a new thread) that is probably best solved by complex analysis, because the obvious answer seems counterintuitive.

Quote:mkl654321Yes, it does. It means, "X is coming! RUN!!!"

STOP!!!...

--Dorothy

Quote:mkl654321You can also debunk it with third grade arithmetic, or simple common sense, in that if it were possible to add EV by switching, it would also be possible to increase the value of each envelope infinitely by infinitely switching. I would imagine that this intuitive conclusion is mirrored by the GIGO effect that happens when you plug "zero" into one of those complex calculations.

I do have an interesting question (which I am posting on a new thread) that is probably best solved by complex analysis, because the obvious answer seems counterintuitive.

Yes, that makes perfect sense, but it is the easy way out. The question is where is the flaw in the EV=.5*(2x+0.5x)=1.25x arguement?

I look forward to sinking my teeth into your complex analysis problem.

Before you open the envelope pick a random value you would like to win. If the envelope contains at least that amount, keep it but if it doesn't swap for the second one. There are three scenarios.

1) Both envelopes contain more than your target. You never switch but this does not effect your overall return.

2) Both envelopes contain less than your target. You always switch but this does not effect your overall return.

3) One envelope has more than your target and one less. In this case you will (without knowing) keep the 2x envelope if you pick it and swap the x envelope if you pick it (winning 2x every time).

Because some of the time 3) will occur, your overall return on the game should be fractionally better than 3/2x.

Shouldn't it?