Part 5. GT: Attacker's Inequalities

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

p = prob attacker has the best hand
S = size of pot
b = size of bet
This fraction is attacker's frequency for
choosing the bet/bet strategy line.

This fraction may be the secret to poker.

attacker's inequalities

5.1. pb/(1-p)(S+b) > 1
or
5.2. p/(1-p) > (S+b)/b

When attacker's inequality is true, the defender
should not call. The attacker is either
not bluffing or isn't bluffing often enough.

On the river p is either the probability the
attacker has made his hand or in many cases
the probability the attacker has made his
hand considering that he has bet.
Attacker's tendencies are unknown to defender. Defender
should call S/(S+b) part of the time. Attacker
rarely bluffs. Defender should not call. Attacker
overbluffs. Calling 100% of the time dominates
all mixed strategies. This confirms the correctness
of the crying call in fixed limit.

On earlier streets p is the probability the
attacker is in the lead. The p variable
is a function of the attacker's
aggressiveness. p is very high for passive
attackers. Passive players betting are very likely
to have top pair or better.
For aggressive players where a continuation bet on
a flop is nearly automatic after an opening raise p is very
low. Against these attackers bottom pair
may easily be in the lead. Defender must
call and raise much more liberally.
p is not an absolute constant. It's a relative
term. On any street defender has a fixed linear
hand strength against a random hand. But he has
a variable relative linear hand strength against
aggressor's hand range. Defender must still use
judgment to determine where he stands on a hand.
It will be dependent on the aggressiveness of the attacker.

Part 4. Game Theory Notes.

Chart and formulas are reposted for convenience.

S = the size of the pot.
b = the size of the bet.
p = probability attacker is dealt a higher card.
(1-p) = prob attacker misses.
x = part of time attacker plays row B/B.
y = part of time defender calls.

Now the 2 X 2 matrix looks like this.

______\_____defender
attacker	Call	Fold
Bet/Bet	p(S+b)-(1-p)b	S
Bet/Chk	p(S+b)-0	pS

x = pb/(1-p)(S+b)
y = S/(S+b)

Note 1. Optimal strategy is optimal like Pepsi Free is free.

Note 2. y=S/(S+b) suggests defender should call with high frequency. Look thru the algebra(in part 2)to calculate y. There was division by (1-p) on both sides of the equation.When p=1, (1-p)=0. Division by zero is undefined. If attacker never bluffs, defender should never call.
Note 3. x = pb/(1-p)(S+b) If this value of x>1, defender should never call.
p/(1-p) > (S+b)/b.
When attacker bluffs with too low a frequency, defender should never call. In fixed limit this would be close to rarely bluffing. In nl with pot size bets, defender should not call if attacker bluffs less than 1/3 of the time.
Note 4. In poker optimal strategy does not always dominate exploitive strategies. It's rarely right to call players who underbluff. It's usually right to call players who overbluff. Don't bluff calling stations.
You must know the math to play well. But the math is a guideline, not something to be followed blindly. That's the art part of the game; knowing which math applies.

Two street game.

The two players are X and Y. Each start with
equal size stacks S. Each ante A > 0. On the
first street no cards are dealt. On the 2nd
street each are dealt a card from [0,1]. The
player with the lower card wins. A/2 goes to
the house.
1st street. Y may bet any amount S-A. X may
call or fold.
2nd street. No betting. Best hand is awarded
the pot.
Optimal strategy on the 1st street must be
for X to call any bet. X is 50/50 to win
the game. X loses A if he folds and gets
better than one to one to call and play.

This demonstrates that optimal strategy is
not necessarily a winning strategy. It's
just a lose minimum strategy. As we learned
from the movie War Games not playing is
often the best strategy.

In hold'em the streets are not independent.
But they are not linearly dependent either.
There is only a positive correlation among
the streets. Being ahead on one street
increases one chances of being ahead on the
next street. But there are no guarantees.

That is another reason why optimal strategy
on the flop and turn may not always be the
desirable play option. While optimal strategy
exists on the flop and turn, it may not be
the most desirable +EV action.

Game Theory Limitations

Still not convinced. Game theory is only
clearly useful on the river and the final
half-street of betting. On earlier streets
strategy is influenced by game theory. On
the river there's a showdown winner. Only
changes if winner folds. Half-street is
when someone goes all in. Then it's a
jam or fold type chart.

Two player game.
The pot was P after the flop action.

Matrix A.

You---fold---call----RR----fold---call
Opp
C/R____-1_____M1_____xx_____xx_____xx
fold___xx_____xx____P+2_____xx_____xx
call___xx_____xx_____M2_____xx_____xx
cap____xx_____xx_____xx_____-3_____M3

On this matrix you have been check/raised
on the turn. Only when one player folds
is there a value in the cell. There's
another matrix when both players are still
competing. Most of the cells in the matrix
is empty.
Any time you are willing to fold the pot is
strange money. The C/R-fold cell is -1 because
you lose one chip on the turn by folding. The
C/R-call leads to the river matrix with the
new pot at P+2. The fold-RR you win the pot
plus two chips from opp.
This M1 has appeared in poker books in another
form. You are risking one chip to try to win
pot plus one chip. Ye is your pot
equity. Or your perceived pot equity.

Ye(P+1) > 1
Ye > 1/(P+1)

You can call when Ye is greater than
1/(P+1). Some players think they can outplay
opp on the river and call with less. That's
impied odds.
In a FL calldown situation it's really

Ye(P+1+1) > 1+1
Ye > 2/(P+2)

Which means you need greater pot equity.

With TPTK and a C/R it's a wa/wb situation.
Usually you're wb. There's rarely enough
pot equity to call when you're wb. On a
static board with no flush possibilities
you will even be drawing dead to a hidden
set.

Properties of Variance

All the tight players like to brag about their low
variance while playing cash ring games. In SnGs
it's different.
When I figure out how it's done there will be a
chart. The losing SnGs players have a lower
variance than the winning ones. A player who
never finishes ITM has a variance of zero. Can't
get lower than that.
Average player will win one, one second, one third
and seven out of the moneys. The standard
deviation is 1.67 buy-ins.
The break even player will be ITM 33% of the time.
That's like 100 plays. 11 1st, 11 2nd, 11 3rd and
the rest out of the money. The s.d. is 1.76.
Also the better one does the higher the variance.
In SnGs the better players have higher variance
than the weaker players. This is also true for
MTTs.