VP$IP

 
 

===================

click edit-me
double click blogger

dashboard

chart MM2

 

chart-MM1

990 ways

More on hand

This model was placed into an excel file. The rounding
errors were eliminated. Here are the new values.
x=.681
y=.751
EV=126.92
EV-b=76.92
-------------------
Our hero should use the R/R row strategy .681
part of the time. On a river situation he would
randomize. This being a flop situation he does
better by raising his stronger draws.
Hero should raise all in with JhTh, AhKh, AhQh,
and AhJh. Also a small portion of the QhJh hands
to bring his using R/R row to .681 of the time.
The call column EV was 128.88 and the fold column
EV was 126.64.
===============================
The hero's range of hands is expanded to 20. They
now include 77, 99(3), AhKh, AhQh, AhJh, KhQh, QhJh,
JhTh, AhTh, Ah8h, Ah6h, Ah5h, Ah4h, Ah3h, Ah2h, KhJh,
KhTh, and QhTh. This should closely resemble a set
of hands by a real player.
Ran the calculations through an Excel file. Now the
hero's proper frequency for playing row R/R is is .261.
Opener calling frequency is .775. The EV for the game
is 78.1. Again subtract 50 from the EV if you wish to
back it up to the flop decision point.
The semi-bluffing frequency was dropped to about a
quarter of the time.
Amazing this .775 calling frequency is higher than
S/(S+b), where S is the size of the pot(150 in this
case) and b is the size of the bet(200). S/(S+b)
was the optimal calling frequency in river cases
where the cards have already determined the winner.
Against unknown and aggressive players opener should
call higher than S/(S+b) part of the time. As [i]p[/i]
appoaches 1, opener should never call. Against passive
players who rarely bluff the best calling frequency
is zero.

--------------------

hero....\____ opener
______\	call	fold
R/R	57.22	150
R/C	85.47	52.67

x=.261
y=.775
EV=78.1
EV-b=27.1

Decided to delay posting charts until I find an easier way to post them

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

game theory in poker

2.2 The three card deck: general case.

______\____BILL
ADAM	Call	Fold
B/B	1/2(4-2)	2
B/CH	1/2(4-0)	1/2(2+0)

This was the 2 X 2 matrix from the special case.
Row B/B means Adam always bets.
Row B/CH means Adam bets with the best hand and
checks when he misses.

2.2 The general case.

This general case will be more like a hold'em situation.
Take three card deck from the virtual world. Again
Bill is dealt the middle card. This time the high and
low cards will not have equal probability. High cards
are like outs for Adam. Low cards are misses.

S = the size of the pot.
b = the size of the bet.
p = probability Adam hits one of his outs.
(1-p) = prob Adam misses.
Now the 2 X 2 matrix looks like this.

______\_______BILL
ADAM	Call	Fold
B/B	p(S+b)-(1-p)b	S
B/CH	p(S+b)-0	pS

How often should Adam choose row B/B and row B/CH?
Adam will choose row B/B x part of the time
where x is between zero and 1. Adam will choose
row B/CH (1 - x) part of the time.

Again the value for Bill calling and folding will
be made equal. The algebraic steps will appear at
the end of the post.

equation 2.2.1

p(S+b)x-(1-p)bx + p(S+b)(1-x)=Sx + pS(1-x)

	p	b
x = --	------	-----
	(1-p)	(S+b)

Row B/B is in effect Adam bluffing. x is the
chance Adam is bluffing. Adam's bluffing
frequency is a function of p, the probability
of his making the hand, the pot size and the
bet size.
The higher Adam's prob of making, the more often he
should bluff. The large bet relative to the pot size
increases his bluffing frequency.
Adam should bluff that fraction of the time.
Adam only missed (1-p) part of the time, not
100% of the time.

	pb
bluff freq=	------
	(S+b)

Adam's bluffing to betting made hands frequency is
b/(S+b)

How often should Bill call? Bill should call y
part of the time where y is between zero and 1.
Bill will fold (1-y) part of the time.
The value for Adam's two row strategies should
be made equal.

equation 2.2.2

p(S+b)y-(1-p)by + S(1-y)=p(S+b)y + pS(1-y)

	S
y = --	------
	(S+b)

Bill's calling frequency is a function only of the pot
and bet sizes. When the bet is small, Bill should
call often. Bill should make fewer calls against
larger bets.
In limit one should call the river bet much more often
than in no limit.
For Adam p is the probability of making the hand.
That's outs divided by remaining cards.
Given a bet, p is often player dependent. So for
Bill p is conditional on his opponent. There was
division by (1-p) on both sides of the equation.
When p = 1, (1-p) = zero. Division by zero is
undefined in math. When p equals one, don't call.
If there is no chance Adam is bluffing Bill should never
call. This can be because Bill has a lousy hand or because
Adam is a total predictable rock who never bluffs.
Even if Adam bluffs too infrequently Bill should never
call. This occurs when the value of the 'B/B-Call' cell
is greater than the value of the 'B/B-Fold' cell.


************************

equation 2.2.1

p(S+b)x-(1-p)bx + p(S+b)(1-x)=Sx + pS(1-x)
pSx+pbx-bx+pbx+pS-pSx+pb-pbx=Sx + pS-pSx
*#* pS,pbx, and pSx cancel out
pSx+pbx-bx+pb=Sx
pb=Sx-pSx+bx-pbx
pb=(1-p)Sx+(1-p)bx
pb=(1-p)(S+b)x
pb/((1-p)(S+b))=x

	p	b
x = --	------	-----
	(1-p)	(S+b)

equation 2.2.2

p(S+b)y-(1-p)by + S(1-y)=p(S+b)y + pS(1-y)
pSy+pby-by+pby+S-Sy=pSy+pby+pS-pSy
*#* pSy and pby cancel out
-by+pby+S-Sy=pS-pSy
S-pS=by-pby+Sy-pSy
(1-p)S=(1-p)by+(1-p)Sy
*#* divide both sides by (1-p)
*#* okay as long as p < 1
*#* division by zero is undefined.
S=(S+b)y

	S
y = --	-----
	(S+b)

game theory in poker

2.1. The three card deck

This deck has three cards. The words high, middle and low are written on them. There are two players, Adam and Bill. Each ante one chip. Start with a special case. Bill is dealt the middle card. Adam has equal chances of being dealt one of the two remaining cards. This approximates a common river situation in hold'em. Bill has been betting and a scare card appears on the river. There is only a half a street of action as Bill checks. Adam can bet the pot(two chips) or check. Bill can call or fold.
Questions. Is this a zero sum game? Who does the game favor and what is the expected value of the game?
Adam always bets when he holds the high card. Adam is dealt the low card. Should he bet or check? This is solved by a two by two matrix. Adam uses a mixed strategy. B/B is bet/bet. Bet with both the high and low cards. B/CH is bet/check. Bet with the high card and check with the low card.
With the middle card Bill uses a mixed strategy of sometimes calling and other times folding.

_________BILL
ADAM	Call	Fold
B/B	1/2(4-2)	2
B/CH	1/2(4-0)	1/2(2+0)

Let's do the easy cell first.
Adam bets both the high and low cards. Bill folds. Adam always wins two chips.
B/CH : Fold. Adam bets high and checks low. Half the time Adam wins two chips.
B/B : Call. Adam always bets and Bill always calls. Half the time Adam wins 4 chips and half Adam loses two chips.
B/CH : Call. Adam bets high and checks low. Bill always calls. Half the time Adam wins 4 chips.
The parts within the cells are summed to a single value.

_______BILL
ADAM	Call	Fold
B/B	1	2
B/CH	2	1

optimal strategy
is when you could tell your opponent your plan and there wouldn’t be anything he could do to change your expectation in the game.

How often should Adam choose rows B/B and B/CH? Adam will choose row B/B x of thetime and row B/CH 1 - x of the time.
Adam will randomize between rows B/B and B/CH so that whether Bill calls or folds Bill will have the same expected value. This is done by making the value of the calling column equal to the folding column.
x+2(1-x) = 2x+(1-x)
On the left side of this equation Bill is always calling. On the right he is always folding.x is the first cell on Bill's calling column. 2(1-x) is the second cell. etc.
x+2-2x = 2x+1-x
2-x = x+1
1 = 2x
1/2 = x
Adam choose the B/B line half the time and the B/CH line the other half. By inspection due to the symmetry of the matrix Bill should call the time and fold half the time.
For the value of the game just add the value of the cells.Each cell occurs one quarter of the time. 1+2+2+ 1=6 Divide by 4 = 1.5
This game favors Adam. Adam's expected return is 1.5 chips. He had one chip invested before the betting street. Therefore Adam's net expected return is 0.5 chips.

Additional comments. The math of the matrix supports intuition. If Bill is a calling station, Adam does best by never bluffing. EV of 2. If Bill never calls, Adam should always bet. If Adam always bets, Bill should always call. If either Adam or Bill uses a fixed strategy the other can adjust his strategy to improve on optimal strategy. By using a mixed strategy of properly randomizing their two fixed strategies neither can gain advantage on the other.