Same Problem Except All Cards Down

[David Sklansky]

That's obviously easier. So someone do it quickly.

[BB King's]

Defining x as Player1’s critical drawing hand.
Defining y as Player2’s critical drawing hand – given Player1 is NOT drawing.

Two equations:

Xy=3/8
(y-x)/(1-x)=(1-x)/2

Solving: x= 0,567381 * y= 0,660931

Why do I bother ???

Exercise: What is Player2’s critical drawing hand – given Player1 IS drawing ? Answer is implied in above.


BB King’s ... Formerly known as the pokerplayer formerly known as Jack

[Darryl_P]

Just wondering, what's the rationale behind the first equation?

I got the same answer as you, but I did it the hard way, ie. I got the 2nd equation which was was the same as in the previous problem and then I made an expression for A's EV in terms of x and found its maximum by taking the derivative. That led to 3-4x-4x^3=0, the solution of which is exactly 0.567381 which is the same as what you got.

[andyfox]

Didn't David used to forbid you from these things because you were too good? [img]/images/graemlins/wink.gif[/img]

[BB King's]

<font color="red"> Just wondering, what's the rationale behind the first equation?</font>

If you solve this …

<font color="red"> Exercise: What is Player2’s critical drawing hand – given Player1 IS drawing ? Answer is implied in above.
</font>
… it should be fairly easy to see the rationale behind the first equation. I’m working on a short explanation !? If you need it ?

<font color="red"> That led to 3-4x-4x^3=0, the solution of which is exactly 0.567381 which is the same as what you got.
</font>
Manipulating my two equation’s I got that equation too. This is the key-equation. You got it - I got it. It somehow proof that our solution is wright.

BB King’s ... Formerly known as the pokerplayer formerly known as Jack

[BBB]

[ QUOTE ]
Just wondering, what's the rationale behind the first equation?

[/ QUOTE ]

I think I can explain this one:

Just as the second equation stems from determining player B's cutoff based on player A's cutoff, the first equation stems from player A's cutoff based on player B's cutoff.

If A draws, B will draw with anything &lt;0.5, and stand with anything &gt;0.5. So, if A draws, then 50% of the time, when B draws, A will win 1/2 the time, while the other 50% of the time, when B stays, B will on average have 0.75, so A will win 1/4 of the time. So when A draws, he can expect to win (0.5)*(1/2)+(0.5)*(1/4) = 3/8 of the time.

When A stands, his expectation is simply X*y (since A will lose whenever Y&gt;y, and will have expectation X when B draws, which he will do fraction y of the time.

So A's cutoff is related to B's cutoff by x*y = 3/8.

[Darryl_P]

Thanks guys, I get it now. (thumbs up icon)