Thursday, January 7, 2016

NASH EQUILIBRIUM Game Theory for Fixed Limit Heads-Up Hold' Em



Transition from No-Limit to Fixed-Limit
Adapting the Nash Equilibrium for Heads-Up LIMIT Hold' Em


Since our concern here isn't Heads-Up No-Limit Texas Hold' Em but Heads-Up Fixed Limit Hold' Em the number of effective BB (Big Blinds) will be different.

On PokerStar the minimum required stack to play
HU NL HE  40 BB
HU FL HE     5 BB

According to this, we need 8 TIMES less money to play Limit Holdem than we need to play No Limit.

So the 20+ BB mentioned above would be reduced to 2.5 + BB in a Fixed Limit game.  Well... ???? I'm not so sure about the correlation between the two.

A more realistic and representative stack is 100 BB in no limit vs. 20 BB in fixed limit as suggested on Pokerstars.

With that being said: 
HU NL HE  20 BB = HU FL HE  4 BB

I have to admit that I feel pretty naked at 4-5 BB.




The following is copy-and-paste general information about the HeadsUp Push/Fold charts and their usage for Heads-Up No-Limit Hold' Em. For the original source, I encourage you to click the links of the source at the end of the explanations. 

Keep in mind that the explanations below were deviced for HU NL HE. So divide the numbers by 5 to give you an idea of what it could be in Heads-Up LIMIT Hold' Em.



General Information

The charts show the Nash Equilibrium solution of the heads-up push-or-fold game. This is a simplified game where the SB is only allowed to go all-in or fold, and the BB can either call or fold when facing a shove.
It is important to keep in mind that this is not the solution for the full game where limps and smaller raises are also available. The push-or-fold solution is generally assumed to be very close to the solution of the full game up to around 8bb. In practice the strategy can likely be used at least until 10bb without being too exploitable.


Usage

The tables show the highest effective stack size in big blinds where a hand can be pushed or called. Values bigger than 20 are only displayed as 20+, which means the hand can be played for any stack size of 20bb or less. (Playing push-or-fold is typically only recommended up to around 10bb, depending on the situation. Using push-or-fold for >20bb is almost certainly a bad idea.)
Some hands have gaps in their pushing strategies, they are marked with * and the details are displayed below the charts. For example 63s is included in the pushing strategy for stack sizes between 7.1 and 5.1bb, and stacks below 2.3bb.


Example

SB (9.0bb): T3s
BB (6.0bb): Q2o

The effective stack size is the smaller of the two, before posting any blinds. So the relevant stack size for both players in this hand is 6.0 big blinds.
To find the strategy for SBs T3s, check the green area (suited hands) of the "Pusher" chart. The value for T3s is 7.7bb, and since that is larger than the current effective stacks the hand is a push in the Nash Equilibrium strategy.
To determine if the BB should call with his Q2o, check the orange area (offsuit hands) of the "Caller" chart and locate Q2o. The value there is 5.6bb. The effective stacks in the current hand are larger than the value for Q2o, so this hand is a fold in the Nash Equilibrium strategy.

http://www.holdemresources.net/h/poker-theory/hune/usage.html
http://www.holdemresources.net/h/poker-theory/hune.html

http://www.thepokerforum.com/joebenik8.htm




I have included the following chart analyzing the game play of Hyperborean and the legendary Polaris.



http://forumserver.twoplustwo.com/29/news-views-gossip/heads-up-holdem-solved-1379043/index5.html

http://en.wikipedia.org/wiki/Polaris_(poker_bot)

http://www.computerpokercompetition.org/index.php/competitions/participants/93-participants-2013?showall=&start=1

http://www.headsupsitandgo.com/

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