Hand from Matros' article

Constructed a sample case based on the hand shown
by Matros from his game theory article in CardPlayer
vol 18 no 20.

Hero has $270. Opener has him covered.
The blinds are 5 and 5. Opener raised to 20. Hero
calls 20. All others fold. The pot size is 50.
The flop is 9h7h7d. Opener bets 50. If hero calls,
the pot is 150. That leaves the hero with only 200
and opener has him covered. This really looks like
a raise all in or call hand. It also simplifies the
matrix. Still there is a family of matrices. Let's
examine just one hand for the opener.
Give the opener AA with no heart. If hero is given a
large range of possible hands the math would be
unmanageable. Limit the set to 10 hands. Raising all in
always with 77 and 99(3 possible). The six drawing hands
are AhKh, AhQh, AhJh, KhQh, QhJh, and JhTh.

The hero's row strategies are raise/raise and raise/call.
Row1: raise both the strong hands and the drawing hands.
Row2: raise the strong hands and call the drawing hands.
The opener's column strategies are call or fold.
This is the 2 by 2 matrix.

hero....\____ opener
_______\	call	fold
R/R	-	150
R/C	-	-

The value of the 'R/R vs fold' cell is 150. Hero always
raised all in and opener folded, therefore hero is awarded
the pot. The other cells' values must to tediously
computed for each of ten possible matchups.

The grudge work follows. Let's jump to the conclusions.



-
-

hero...\____ opener
_______\	call	fold
R/R	119	150
R/C	143	78

Hero should choose row R/R .677 part of the time.
Choose row R/C .323 part of the time
Opener should call .75 part of the time.
Should fold .25 part of the time
The value of the game is 126.75 for hero.

If you wish, you may substract 50 from this number to
show the value when hero must decide whether to call
the first 50 lead bet on the flop. Then the value
of the game is 76.75.

In this example hero's choice of row R/R is higher
than in most normal hold'em situations. 4 of the
10 hands used for hero's set of hands were near nut
hands.

As long as there is a reasonable chance opener has
the better hand, his optimal calling frequency will
remain high.


----------------------
Computations

To compute the 'R/C-fold' cell more assumptions are made
to simplify the calculations. The turn isn't a heart.
Opener bets 200. Hero folds unless he has made the
straight. Turn is a heart. Hero bets all in for 200.

[code]
45 cards remaining.

cards	wins	loses	folds	totals
AhKh	9	0	36	1350
AhQh	9	0	36	1350
AhJh	9	0	36	1350
KhQh	8	1	36	1000
QhJh	8	1	36	1000
JhTh	8+3	*1	33	1912.5
				7962.5

Wins will net 150 as opener isn't calling when a
heart appears on the turn. On the heart ace hero
will lose an additional 200.
For AhKh 9 X 150 = 1350.
The JhTh case creates exceptions because of the
straight and straight flush possibilities.
Three non-heart eights will complete the straight
for hero. Opener will call the 200 making the
pot 350. 4 times out of 44 opener will make a
full house on the river and win. It nets JhTh
300 per case.
When the turn is the heart ace, JhTh can still
win with a river heart 8. Now the loss is reduced
to 187.5
Weighing these 6 cases with the 4 other cases
the approximate value of the cell is 78.

hero...\____ opener
_______\	call	fold
R/R	-	150
R/C	-	78

Now for opener's call columns. Used Sklansky's
poker calc 1.4.1.1 for the figures.

990 chances.
__________wins____loses
77.............989............1
99(3)..........904...........86
AhKh..........337..........653
AhQh..........337..........653
AhJh..........346..........644
KhQh..........307..........683
QhJh..........316..........674
JhTh..........404..........586
totals.......5748.........4152

Wins 350 and loses 200.
5748 X 350 = 2011800
4152 X -200 =-830400
1181400
Divide by 9900 total cases.
The 'R/R-call' cell is 119

hero...\____ opener
_______\	call	fold
R/R	119	150
R/C	-	78

The 'R/C-call' cell. 125.6 from the four
made hands and 17.8 from the 6 drawing hands.
143 is close enough.

hero...\____ opener
_______\	call	fold
R/R	119	150
R/C	143	78

Hero will play row R/R x part of the time.
Will play row R/C (1-x) part of the time.
The call column is set equal to the fold column.

119x + 143 - 143x = 150x + 78 - 78x
65 = 96x
x = .677

Opener will play row R/R y part of the time.
Will play row R/C (1-y) part of the time.
Set the two rows equal.

119y + 150 - 150y = 143y + 78 - 78x
72 = 96y
y = 3/4

Now compute the value of the game for hero.

For the call column it's
119(.677) + 143(.323) = 80.563 + 46.189
Value = 126.752

For the fold column it's
150(.677) + 78(.323) = 101.55 + 25.194
Value = 126.744

Just rounding errors. Value equals 126.75.

Now compute the value of the game for opener.

For the R/R row it's
119(.75) + 150(.25) = 89.25 + 37.5
Value = 126.75

For the R/C row it's
143(.75) + 78(.25) = 107.25 + 19.5
Value = 126.75