[cbloom]

(I wrote this a while ago, HTML)

In cash games it's common to "run it twice". What this means is when the players are all-in before the river, they will
draw the remaining cards twice. If one player wins both times, he scoops the pot. If each player wins once, the pot is
split. The funny thing about how this is usually done is that the drawn cards are not shuffled back into the deck, you
simply draw once, leave those cards in the muck, then draw again.

<P>

This is very bad for the EV of someone with few outs. Consider the case that you only have one out - you can only possibly
win once, because if you win one you have no more cards to hit for the other draw! Let's look at the math in one specific
case, Hold'em, with two players cards up, and just one more card to come, so 8 cards are known and 44 remain unknown. After
one draw there are 43 unknown. The player who's behind has N outs.

<P>

If you just draw once, your EV in units of pots is N/44

<P>

If you draw twice you can either win both : N/44 * (N-1)/43 , or more likely you hit the first and miss the second or
miss the first and hit the second, and if those happen you get half EV :

<P>
<PRE>

1/2 * (N/44 * (1 - (N-1)/43)+ (1 - N/44) * (N-1)/43)

=
[ N * (N-1) + 1/2 * (N * (43 - (N-1))+ (44 - N) * (N-1)) ] / (44*43)

=
[ N * (N-1) + 1/2 * (N * 43 - 2N*(N-1) + 44 * N - 44) ] / (44*43)

=
[ 1/2 * (N * 87 - 44) ] / (44*43)
=
[ N * 87/2 - 22 ] / (44*43)

=
N/44 * 87/86 - 22 * (44*43)

=
(N/44) * 87/86 - 1/86

</PRE>

Well, this is always less than the EV of just drawing once. The difference is ((N/44) - 1)/86. If you somehow had > 44 outs
it would be +EV to draw twice (silly), and the less outs you have the worse it is. In the extreme case of having only 1 out,
you lose 1.1% of the pot in EV by drawing twice !!

<P>

So, the moral of the story is : if you ever get in a big cash pot and someone offers to draw twice - you should refuse
if you're drawing and accept if you have the better made hand. (note that the player drawing may actually be "ahead" in
the sense that if they have more than 22 outs they are favored to win, but it's still a bad move for them to accept the
double run - drawing players should only run once). People often justify running it twice as a way to reduce variance.
It sure is, but if you're playing hold'em your variance is already huge and you should just worry about EV.