2b. Min-raise variation

SB min-raise or fold.

First bet.

SB post 1 chip, BB post 2 chips
SB can raise to 4 or fold
BB risk 3 to win 3
BB calls =< 0.5

{SB} payoff matrix
0*(1-s)*1 SB folds
3*s*(1-b) SB raises to 4 & BB folds
1*b² SB raises/BB calls and ties, blinds returned
-3*(s-b)b SB raises/BB calls; SB loses 3
BB calls 2 to win 6. b=3s/4. s=4b/3

3(1-b)s + b² - 3(s-b)b
3s-3bs + b² - 3bs + 3b²
3s-6bs + 4b²
4b-8b²+4b²
4b-4b²
/ds=4-8b
b=1/2
s=2/3

min-raise variation after SB's open.
{SB} = 4b-4b²= 2 - 1 = 1
net{SB} = 0
The min-raise was equal after only SB opening raise.
BB's raise will be EV- for SB.

net{SB}= -5/32
The 3XBB raise was -5/32 after SB's opening raise.
Which seems to indicate the larger SB's opening bet
the worst it is for SB's EV.

Second bet.

min-raise variation; BB can raise the pot after
SB min-raises.

6 chips in the pot after SB min-raises.
BB can pot raise with 2 to call plus 8 to raise.
8 chips for SB to call. SB should call 0.5
A simulation game suggested SB calling at 0.472
produce higher EV than calling at 0.5.

{BB} was +0.53125 or 17/32.

{SB} payoff matrix after BB raise option.
0*(1-s)*1 SB folds
3*s*(1-s) SB raises to 4 and BB folds
s² this area was solved by trial and error.

This is the graphic picture of the payoff matrix
for BB using net numbers.

The EV was 17/32 in favor of BB.
{SB} = 1-17/32 = 15/32
BB calls 2 to win 6. b=3s/4. s=4b/3

3s(1-s)+ 15s²/32
3s-3s²+ 15s²/32
3s - 81s²/32
/ds=3 - 81s/16 and set to zero
s = 48/81 = 16/27
b = 4/9

min-raise variation after BB's raise option.
{SB} = 3s-81s²/32= 3*16/27 - 81/32*16/27*16/27 =
16/9 - 8/9 = 8/9
net{SB} = -1/9

Line vector strategies are
SB raises =< 0.592 else folds
BB calls =< 4/9
BB reraises =< 1/9
BB bluff raises [0.571,0.592]
SB calls =< 0.21

The geometric expansion of the pot seem to favor BB.
SB can play fewer hands while min-raising than when
she limps. SB has a small negative EV by min-raising,
while limping was an even game with zero EV.