Envelopes!

[lorinda]

My strategy gives higher EV even if the guesser knows my strategy [img]/forums/images/icons/laugh.gif[/img]

I put low envelope as
$2......128 times
$4.......64 times
$8.......32 times
$16......16 times
$32.......8 times
$64.......4 times
$128......2 times
$256......1 time

I actually multiply by 1.9999 not 2 so the guesser knows he can change with an edge (in case he screws me on level game)

So what happens, guesser opens envelope with $32 in it.
He knows this is the low envelope 4 times, and the high envelope 7.9999 times.
He gets 2-1 his money on a swap and he is getting favourable odds, so he should swap.
So on a non-$2, non $512 envelope, he swaps.

$2 envelope will be opened 64 times from the 255 games.
0 EV 25.1% of the time

$512 envelope will be opened 1 time in 510 games.
0 EV 0.2% of the time

The other times we have a $66.67 EV, which is 74.7% of the time = $49.80
I think this tends towards $50.00 the more envelopes we are allowed (Maximum stuffing of 1,000,000 say)
This would make logical sense as we have the difficult end of this bet so near 50% would be our best result.