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)