[irchans]

>pick a random number from the set {1, 2, 4, 8, 16, 32, 64, 128, 256} and put it in the low envelope.

Many people had very interesting stuffer and guesser strategies. I don't have time right now to comment on all of them, so I will only comment on Manhattan Jac's strategy.

Let's assume that Manhattan Jac publishes his strategy and then plays the stuffer role. What is his expectation?

The selector will always switch unless he chooses the 512 envelope. (For this stuffer strategy, if the selector picks an envelope containing x dollars and x is neither 1 nor 512, then his expectation for switching is 0.5*0.5*x + 0.5*2*x = 1.25 x, so he must switch.)

So if the selector picks an envelope with

$1 in it, the suffers expectation is 0
$2 in it, the suffers expectation is 50
$4 in it, the suffers expectation is 50
$8 in it, the suffers expectation is 50
$16 in it, the suffers expectation is 50
$32 in it, the suffers expectation is 50
$64 in it, the suffers expectation is 50
$128 in it, the suffers expectation is 50
$256 in it, the suffers expectation is 50.
$512 in it, the suffers expectation is 0.

So Jac's stuffing expectation is 50*8/9 = $44.44 against a smart opponent with that strategy if the selector knows the stuffers strategy.

Could someone check my reasoning here?

My solution is different but it has exactly the same stuffer expectation.