Game Theory Resolution

[mykey1961]

[ QUOTE ]
Your simplification of the EV equation is incorrect.

Here it is step-by-step, so you can point out specific errors if you disagree:

(3-3x)/2 (1.5x + 3.5 * 1/3 * (1-x) - 2.5 * 2/3 * (1-x))
...
(3-3x)/2 (4.5x/3 + 3.5/3 - 3.5x/3 - 5/3 + 5x/3)
...
(3-3x)/2 * (4.5x + 3.5 - 3.5x - 5 + 5x)/3
...
(1-x)/2 * 6x - 1.5
...
(6x - 6x^2 - 1.5 + 1.5x)/2
...
(-6x^2 + 7.5x - 1.5)/2
...
-3x^2 + 3.75x - .75

The derivative is:
-6x + 3.75
so
6x = 3.75
x = 3.75/6 = 0.625

x = 0.625, a = 0.4375


[/ QUOTE ]

why a = 0.4375

the EV(sb) is the same for a >= 0.0, a <= 0.625
the EV(sb) is worse for a > 0.625

[jay_shark]

Mykey x=(2a+1)/3 which I already explained to another poster .

Just solve for "a" given that x=0.625

I recommend you read this again because you're not understanding this .

[mykey1961]

[ QUOTE ]

x = 0.625, a = 0.4375


[/ QUOTE ]

Ok it took a while but the lightbulb finally flickered.

As far as the BB knows, the SB may or may not play optimal, therefore the BB plays an optimal strategy "x" which locks in a win rate >= 5/64BB per hand.

SB also doesn't know if the BB is playing optimal, so SB plays an optimal strategy "a" which locks in a loss rate <= 5/64BB per hand.

Seems to me that if SB determines that the BB is also playing optimal, then SB should open up the range, and look for the BB trying to exploit the new range. Once that happens, SB should tighen up the range to exploit the BB.. and the race is on.

[jay_shark]

Hey Mykey , let me put you on the spot .

How often should you bluff in this game ?

[mykey1961]

It appears to me that the SB should:
0.0000 <= SB <= 0.1875 Raise
0.1875 < SB < 0.6250 Fold
0.6250 <= SB <= 1.0000 Raise

[jay_shark]

Mykey , I already showed that raising with 0.4375+ is optimal for the sb and that bb should call with 0.625+ .

[mykey1961]

What you did was got an answer from a formula.

That doesn't mean it's the right answer to the question.

[jay_shark]

My answer is consistent with Tnixon's answer . Ignore the algebra error I made in the first post . It's been corrected and verified using simulation .

[mykey1961]

what you haven't shown is why (if it is) your strategy is better than what I presented.

In my opinion ,my strategy for the SB is mathematically equivilant in terms of EV. And it allows the BB to Call more hands without losing EV. That creates a higher variance, which is good for the SB since he is in a -EV situation.

[jay_shark]

Other possible solutions are the following :

raise with [0,0.1875] , raise with [0.625,1] and fold everything else .

BB would always call with [0.625,1]

2) raise with [1,19.75], raise with [0.625,1] and fold everything else .

BB would always call with [0.625,1]

As you can see , there are infinitely many solutions, all of which cannot do better than EV = 0.421875