In a no-limit texas hold'em hand, you are first to
act with the
river card about to come with $100
behind and the pot after the
turn is $200 (our single opponent has us covered). You hold T
8
on a board of J
9
5
2
and your opponent holds T
T
. Quite obviously we're value-shoving our $100 when a
Q, 7 or club falls on the
river, but how often should we
bluff the
river for optimal play?
First we need to define what we mean by optimal play. When we play optimally in general, there is no way our opponent can
exploit our play. In the described scenario above, our opponent will have two choices at his disposal: to
call our bet, or to
fold. Our goal is to make both of these options have the same EV, in which
case he will have no exploitive
option. Our lives are made easier by the simple fact that the EV of folding is always 0. So now we have to figure out how often we should be bluffing so that the EV of him calling is also 0.
From our
villain's point of view, he will be needing to
call $100 in a pot of $300 after we
shove the
river, getting 3:1 odds. If we are bluffing more often than 25%, he should
call since his 3:1 odds will pull a profit on a
call. If we are bluffing less often than 25%, he should
fold since his 3:1 odds will pull a loss on a
call. But what if we
bluff exactly 25% of the time? Then a
call is breakeven. Zing!
There are 15 cards (nine clubs, 3 more queens and 3 more sevens) that make our hand. If before the
river comes we pick out 5 distinct cards (that aren't a club, a queen, or a seven) that we'll
bluff on, then we will be bluffing exactly 25% of the time, and our
river betting will not be exploitable.