Game 2. 2-player 2bet game

Sara is the SB, posts 1 chip.
Bob is the BB, posts 2 chips.
Each is dealt a card from uniform [0,1].
Low values are best.

Sara may fold, limp, min-raise, or raise 3XBB.
Bob may fold, call, or raise the pot.

2a. Limp variation.

First bet.

Sara risks one chip to win three. Therefore she
should call 3/4 of the time.

Solve by calculus.

SB EV payoff matrix
Used braces for {SB} SB EV
"<" causes confusion in HTML.
0*(1-s)*1 SB folds
3*s*(1-s) SB limps & wins
1*s² SB limps & ties, return blind

3s(1-s)+s²
3s-3s²+s²
3s-2s²
/ds=3-4s and set to zero
s=3/4

{SB}=3s-2s²=3*3/4-2*3/4*3/4=9/4-9/8= 9/8
net{SB}= 1/8

Second bet.

Was only able to determine Bob's bet range by trial and
error. Bob bluff frequency was 1/2 of his value bet
frequency. Bob's value bet value frequency was 1/4
of Sara's limp frequency.
This game replaced the split pot area and part of the SB
limps/wins area. The new payoff for the split area and
for the overlap area is 1.

After SB limps, BB may bet 4.
0*(1-s) SB folds
3*s(1-9s/8) SB limps and wins
1*s*s/8 + 1*s² Bob's bet region

3s - 27s²/8 + 9s²/8
3s - 18s²/8
1st dev set to zero
/ds=3 - 9s/2
s = 2/3

3s - 9s²/4
{SB}
2-1=1

The line vector strategies are
SB limps =< 2/3
BB bets =< 1/6
BB bluffs [2/3,3/4]
SB calls =< 1/3

Net {SB} = 0