Poker related math problem

[Paul2432]

I recently came across this thread (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=348730&page=&view=&sb =5&o=&vc=1) where the problem of what to call with versus an all-in SB, BB and straddler when you are on the button was discussed.

I have come up with an exact solution to the related problem using real numbers from zero to one instead of poker hands. I'll restate the problem here and then if anyone is interested post the solution in a few days.

The problem: Player A post the SB and is all-in, Player B posts the BB and is all-in and Player C posts a live straddle equal to twice the BB and is all-in. You are the only other player and have all players covered. Each player will be randomly given a real number between 0-1. What is the lowerst number you should call with if you wish to maximize your EV per hand?

Just to clarify, there are two questions that I am not asking. I am not asking, "what is the lowest number you can play to break even?" (zero). I am also not asking, "What is the lowest number to play to maximize ROI?" (one).

Paul

[bigpooch]

For simplicity, let the small blind = 1, big blind =2 and
the straddle =4 so that the last side pot is 4 (which is
contested between the straddler and the button).

Suppose the button has some hand value in [0,1) as do the
other players (where the higher number designates the better
hand).

If v is the hand value, then for the last side pot heads up
with the straddler, the expected value is just
+2v -2(1-v) = 4v-2.
Similarly, the chances of winning the three-way side pot
will be v^2 and the expected value for that side pot is
+2v^2-1(1-v^2)=3v^2-1.
Similarly, for the main pot, the expected value is
4v^3-1.
Summing, the EV is 4v^3+3v^2+4v-4. For this to be >0, you
would require v>0.5703999.

Is this the estimate you obtained as well?

Of course, with holdem hands, it's more complicated!

[M.B.E.]

Approx. 0.5704, which is the only suitable solution to the cubic equation

4x^3 + 3x^2 + 4x - 4 = 0.

[Paul2432]

Summing, the EV is 4v^3+3v^2+4v-4. For this to be >0, you
would require v>0.5703999.

Is this the estimate you obtained as well?

Although I used a slightly different method (yours is simpler) I did get the same answer. FWIW, the exact solution to the equation is:

(39/64 + (13/36)(3)^0.5)^(1/3) - (-39/64 + (13/36)(3)^0.5)^(1/3) - 0.25

Paul

[bigpooch]

I wonder how many readers remember that esoteric formula
attributed to Cardano ?

[Paul2432]

I wonder how many readers remember that esoteric formula
attributed to Cardano ?

I sure as heck didn't. I found the method here:

The Cubic Formula (http://www.sosmath.com/algebra/factor/fac11/fac11.html)

Paul

[squiffy]

Maybe Mrs. Cardano would remember, if she is still alive?