Part 3. Applying game theory.

The introduction to the 3-card deck is posted on
8 NOV 2005.

On the turn Bill has been the aggressor and clearly
in the lead. Adam is drawing, possibly to a 4flush.
The river card apparently has not helped Bill. Bill
checks. On the river Adam and Bill flipflop roles.
The aggressor becomes the defender. The chaser
becomes the attacker.
What is Adam's best strategies? Assuming Adam bets,
what's Bill's best response?

The formulas are reposted.

S = the size of the pot.
b = the size of the bet.
p = prob river made Adam's hand best.
(1-p) = probability Bill's hand holds up.
x = part of time Adam plays row bet/bet(B/B).
--bet/bet means Adam bets whether his hand improves
or not.
1-x=Adam chooses the B/CH row.
--bet/check(B/CH)Adam bets a made hand and checks
when he misses.
y = part of time Bill chooses the call column.

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

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

y= S / (S+b)

Now how do we use this info?

Example.

Bill has AsKd. Board on the turn is

[Kh 8h 2d],[4s]

Bill has been the aggressor and Adam has been calling.
The river card is a heart. Bill checks. Adam always
bets whenever he has made the flush. How frequently
should Adam bluff? How often should Bill call? The
answers assume both are playing optimal strategy.

Limit.

The size of the pot is large relative to the size
of the bet. Let's say the pot is 10 bb.
Bill should call S/(S+b) part of the time. That's
over 90% of the time. Bill should almost always call.
Against an unknown opponent don't bother to randomize
just call 100% of the time.
Adam should choose the B/B row

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

part of the time. b/(S+b) is less than one in ten.
46 cards unseen and 9 cards make a flush. Therefore
p=9/46 and (1-p)=37/46. p/(1-p)=9/37~1/4 Adam
should bluff less than one in forty times. The
math supports the CW. In limit Adam should almost
never bluff. For pots of 10bbs or more it's close
to never bluff.

Pot size bets.

S=b. S/(S+b) = 1/2.
Bill should call half the time. This is quite
different from limit where Bill should call nearly
all the time. The larger the bet, the less often
Bill should call.

b/(S+b) = 1/2.
(1-p)=37/46
p/(1-p) = 9/37. Same as in limit.
But now [p/(1-p)][S/(S+b)]
= 9/37 X 1/2 = 9/74 ~ 1/8

Adam should be bluffing just under one in eight times
when he missed. Remember Adam is betting the flush
(9/46) nearly 20% of the time. When playing optimal
strategy a bet by Adam is a bluff 38% of the time.
This is for pot size bets.

When Adam bets he should have the made hand
(S+b)/(S+2b) part of the time.
**explain-Adam bets 100% of made hands.
--[p/(1-p)][S/(S+b)] of missed hands.

-------------
Bluffing

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

37 of the 46 times Adam misses his 4flush.
Adam missed the draw (1-p)/1 part of the time.
What part of the 46 times should Adam be bluffing?

bluff freq= x times (1-p)

The (1-p) cancels out therefore
bluff freq = pb/(S+b)
nuts freq= p

For S=b
bluffs freq = p/2

For S=2b
bluffs freq = p/2

Take the set of hands which are bet. The proper
ratio of bluffing hands to made hands is the
fraction b/(S+b). When Adam bets the proper
ratio of made hands to bluffing hands he is
indifferent to whether Bill calls or folds.

Real world hold'em.

Those numbers are fine in theory. It doesn't work
in hold'em. Adam's drawing to a 4flush. The
river produces a non flush card. It's not a threat
card. If Adam bets, Bill will call. The reality
is Adam should never bluff when he misses the flush.

When should Adam bluff? Sometimes Adam will still
be in the pot drawing to something other than the
flush. A flush card appears on the river. Adam
should bet/bluff as if he made the flush. Adam
must paint a consistent picture of the hand.

Optimal strategy is a fine defensive approach to
poker. The winningest players are whose who know
when and why to deviate from this strategy.

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

In the previous examples Bill, the one with the
better hand, was always the aggressor. Sometimes
the weaker hand is the aggressor.
In these new examples Bill will defending his blind
from a steal. Adam open raised from button or CO.
By the river Bill may feel he needs to improve
before calling. Adam should bluff more often when
Adam thinks Bill is willing to fold.

Example: Adam opens from CO with 8c6c.
Button and SB fold. Bill defends blind with Qs7h.
Flop: KsJdTh
Bill checks. Adam bets. Bill calls.
Turn: 5c
Both check.
River: 4c
Bill checks. What should Adam do?

If Adam thinks Bill was drawing to a straight, he
should bet. If Bill had a jack or a ten, Bill may
have bet the river. Since the turn and river did
not improve Bill, he may be willing to fold.
Adam must attempt to profile Bill. With which hands
is Bill calling? If they are drawing hands which
have likely missed Adam should bluff when Adam can't
beat the drawing hand. With a no pair ace high Adam
should showdown and see whose hand is best. If Adam
thinks Bill is calling with a pair, Adam should only
bet when he can beat that pair. Adam should not
bluff when Adam suspects Bill has a pair and will
call.

4-tier limit

The goal of 4-tier limit is to increase the skill
compotents relative to the luck in hold'em. After
35 years the theorists have been able to analyse
the game. They have been able to reduce the both
fixed limit and no limit to two card poker. The
intent of 4-tier limit is to increase the number
of decisions per hand. 4-tier limit will also
try to make each street have nearly equal effect
on a player's winrate.
In 4-tier limit the betting units would increase
every street. One unit for prf, two units for
the flop, four units for the turn, and ten units
for the first river bet.
First neutralize the effectiveness of playing tight.
Have three blinds rather than two. All blinds will
be big blinds. The button would also post a blind.
Preflop the action will start with the player to
the left of last big blind. He may open for one
or two units. The rest of the betting is like
regular fixed limit.
Always have at least three blinds each deal. If a
blind bust out or leave, move the button to the next
active player on the left, and have the two players
left of the new button position post blinds. New
players to the table may be dealt a hand by posting
a blind and a dead half blind.
On the flop the betting unit is two blinds or two
units.
On the turn the betting unit is four blinds.
The rules change drastically on the river. The
action will now start with the last aggressor.
That's the last player to bet or raise during this
hand. Check/raise will no longer be allowed. Each
player in turn is allowed to bet ten units. Raises
on the river will be twice the size of the previous
bet. In 4-tier most of the big action can only
occur after all the cards are dealt and the winner
is determined assuming he doesn't fold. This big
bet would also eliminate the automatic crying call
of fixed limit.
These changes would create a game more favorable
to strong post flop players.