For the three betting rounds after the flop,
we infer a mean and variance (
and )
of the threshold for the
opponent's observed action and rank all hands according to the
effective hand strength (*EHS*') calculation from Equation
6.4.
Although the relative ranking of all hands is not affected, it is
an optimistic view, because hands with an *EHS*' above
represent more than 20% of all hands (due to the inclusion of *PPOT*
in the equation). Additionally, we use *HS*_{1} instead of *HS*_{n} so
the number of opponents can instead be addressed as part of the
context of the action frequencies (although presently it is ignored in
the interest of a simplified definition of context).

To calculate *EHS*' we must compute both hand strength (*HS*)
and positive potential (*PPOT*). Since *PPOT* is a
computationally expensive operation and we must obtain
*PPOT* values for about 1,000 hands every call to the re-weighting
function, we use a crude but fast estimating function (*PPOT*_{c}).
In terms of cost,
*PPOT*_{c} << *PPOT*_{1} << *PPOT*_{2}.
It is * crude* because it ignores much of the available information
(such as weight arrays) and looks for a few features about the hand,
using heuristics to associate a value to each.
For example, if the board has only 2 diamonds of our 4-card diamond
flush draw, then we have 9 outs (1 out per remaining card),
but if the board has 3 diamonds, each of the remaining cards is
worth 0.5 to 1 out, depending
on the rank of your suited hole card (a Two is worth 0.5 and an Ace is
worth 1). This is because we are more likely to be up against other
diamond flush draws.
*PPOT*_{c} approximates very roughly our winning chances given
each possible card to be dealt).
This function produces values within 5% of *PPOT*_{1}, 95% of the time
(this is from a random sampling of five card unweighted cases).
Since these values are amortized over about 1,000 possible hands,
the overall effect of this approximation is small.

However, the opponent is also likely to play some hands based on
the pot odds required for the action (* i.e.* the decision is not
always based on *EHS*'). For this reason we have introduced an adjustment
to the re-weighting algorithm for the post-flop. When *PPOT*_{c} is
sufficient to warrant calling the pot odds, which is

the weight for that subcase is not reduced. This is a simplification of the betting decision the opponent could be making based on the pot size, however we feel it is sufficient to capture enough cases to prevent hands with low

What value do we use for ?
We chose to use a typical value of
at
(interpolating over the range .3 to .7) and to increase
with smaller
while decreasing it for larger .
This
reflects the tendency for * loose* players (with low )
to
exhibit more uncertainty
and * tight* players (with high )
to adhere more
consistently to the threshold. Hence, we use
.

However, as was the case with pre-flop re-weighting, there is a clear
source of error when
is very low.
Observe that the area of
the re-weighting function is a rectangle of height 1 and width
,
with a triangle between
and
(Figure 7.2). That is,

(7.5) |

The value *r* is calculated to give .
The function now
looks like a rectangle to the right of ,
a triangle
of height 1 - *r* (width )
and a second
rectangle of height *r* below it (Figure 7.4).
The total area of the function is now
the area of the triangle subtracted from 1:

(7.6) |

(7.7) |

(7.8) |