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.