Two Plus Two Poker Forums Has The Basic Simplest NL Toy Head Up Game Been Solved
 Register FAQ Search Today's Posts Mark Forums Read Video Directory TwoPlusTwo.com

 Notices

 Poker Theory General poker theory

06-19-2012, 12:01 AM   #16
Pooh-Bah

Join Date: Aug 2007
Location: ithaca
Posts: 5,013
Re: Has The Basic Simplest NL Toy Head Up Game Been Solved

Quote:
 Originally Posted by muuuuuuufasa We have a similar situation on the top side of his range; one only wants to bet hands above z where z satisfies: (z-x)/(1-x)*2+(1-z)/(1-x)*-1+z=z
this should just be
(z-x)*2+(1-z)*-1+z=z
z = (x+1)/2

and then you have P2's indifference at x:
EV(fold) = EV(call)
0 = 2 * y/(y+(1-z)) + (-1) * (1-z)/(y+(1-z))
0 = 2y-(1-z)
y=(1-z)/2

and then, solving, x=5/9, y=1/9, and z=7/9

 06-19-2012, 02:57 AM #17 enthusiast   Join Date: Mar 2005 Posts: 91 Re: Has The Basic Simplest NL Toy Head Up Game Been Solved If the money amounts were continuous rather that discrete dollar amounts I can't help thinking that the solution to the simple no limit game reduces to some beautiful 3D bifurcation diagram with the number you're dealt along the x axis, amount to bet on the y axis and betting frequency along the z axis. Given your number n on the x axis you can pick any point inside the region of the graph at x=n so you will get a discontinuous 2D region of possible strategies for every n.
 06-27-2012, 01:56 PM #18 adept   Join Date: Jul 2009 Location: Canada Posts: 1,002 Re: Has The Basic Simplest NL Toy Head Up Game Been Solved Ok, so I simplified the original problem a bit. The goal isn't to diverge from it, but more to use the simplified one as a building stone toward understanding what's going on. 1) Player 1 can only bet 1\$ (or check obv.). Of course, it changes the whole dynamic of the problem, but still, I guess we could learn something. 2) I clustered the real numbers to simplify my very simple simulation. I don't think it alter the nature of the problem. So [0; 0.1[ --> 0, [0.1; 0.2[ --> 1, ..., [0.9; 1] --> 9 So now, knowing that there 1\$ of EV in every pot, what's 1st player GTO strategy? And what his 2nd player GTO response? I found out this: - If 1st bets a (0,1,6,7,8,9) range and checks (2,3,4,5), he will make on average at least 52 cents per hand, no matter how 2nd responds. - Based on 1st betting range, 2nd can minimize 1st's EV (to 52 cents) by having this calling range: (6,7,8,9). Note that it doesn't matter if 2nd calls or folds with (2,3,4,5), his EV will remain 48 cents.
 06-27-2012, 02:25 PM #19 Pooh-Bah     Join Date: Aug 2007 Location: ithaca Posts: 5,013 Re: Has The Basic Simplest NL Toy Head Up Game Been Solved mufaasa and i solved the case when only a potsized bet is allowed above. the bettor bets the top 2/9 of his range and the bottom 1/9. the bluffcatcher calls the top 5/9 of his hands.
06-27-2012, 03:24 PM   #20

Join Date: Jul 2009
Posts: 1,002
Re: Has The Basic Simplest NL Toy Head Up Game Been Solved

Quote:
 Originally Posted by yaqh mufaasa and i solved the case when only a potsized bet is allowed above. the bettor bets the top 2/9 of his range and the bottom 1/9. the bluffcatcher calls the top 5/9 of his hands.
Had errors in my simulation. It's actually top 2/10 (20%) and bottom 1/10 (10%) for the better and top 5/10 (50%) for the caller.

EV(better) = 55.5 cents
EV(caller) = 44.5 cents

06-27-2012, 04:17 PM   #21
Pooh-Bah

Join Date: Aug 2007
Location: ithaca
Posts: 5,013
Re: Has The Basic Simplest NL Toy Head Up Game Been Solved

cool, is it easy to increase the number of hand divisions and see if it gets closer to the analytical result?

Quote:
 Originally Posted by yaqh the bluffcatcher calls the top 4/9 of his hands.
fmp

06-27-2012, 04:42 PM   #22

Join Date: Jul 2009
Posts: 1,002
Re: Has The Basic Simplest NL Toy Head Up Game Been Solved

Quote:
 Originally Posted by yaqh cool, is it easy to increase the number of hand divisions and see if it gets closer to the analytical result?
I guess so. For now, I only have a little 10x10 matrix in Excel with some basic equations (lazyness prevails!), but I'll see what I can do. Might not be soon though as I'm busy these days, but will let you know.

 06-29-2012, 08:51 PM #23 journeyman   Join Date: Nov 2011 Location: Southwest USA Posts: 270 Re: Has The Basic Simplest NL Toy Head Up Game Been Solved http://www.math.ucla.edu/~tom/papers/poker1.pdf Here is a paper by Chris Fergusson on this subject.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are Off Pingbacks are Off Refbacks are Off Forum Rules

All times are GMT -4. The time now is 01:43 PM.

 Contact Us - Two Plus Two Publishing LLC - Privacy Statement - Top