Reapplying HU Push/Fold Nash Equilibrium to Hyper Turbo 6-maxes Discussion

If you don't know what the title meant (or can't figure it out by reading below), aren't a decent player, or just want to spam a post for 5 points, don't comment here.

If any of you guys aren't exactly familiar with Nash push/fold equilibriums, basically the chart is based on an imaginary game where the first player to act only has two options: fold or shove preflop -- no calling, no raising less than all-in.

Imagine you're playing this games with huge stacks: 1 million big blinds or something. You'd fold everything preflop except pocket aces. Your opponent would always fold to your bet unless he also had aces. (it would be a dull game).

Now imagine raising the blinds in this game. Once the blinds get high enough, it becomes profitable to bluff at a few pots once in a while to pick up those blinds. You could do some math and figure out the best hands to bluff with, assuming you'd only be called by aces. Turns out, smart mathematicians have.

If the blinds get high enough (which is the same as the effective stacks getting smaller, relative to the blinds), it becomes worthwhile to call with something besides aces. Again, mathematicians figured out the best hands to call with, knowing the hands an optimal player will be bluffing with.

Keep increasing the blinds or reducing the stacks, recalculate, and you find new shoving hands and new calling hands. Do lots of math -- use a computer -- and you get those charts. (I have no idea how to do this math.)

The charts actually display the problem backwards, telling you what stack size it becomes profitable to shove (first table) or call (second table).

Above about 20 big blinds, the charts assume you're not playing shove/fold any more, and the chart just says these hands would be profitable shoves if you were forced into push-fold mode by the rules, but you probably aren't playing them that way in real life.

Here's a look at the chart which can and should be adjusted for players playing too tight or loose:

http://www.holdemresources.net/h/poker-theory/hune.html
Push-or-Fold Nash Equilibrium solution for two player No Limit Hold'em.

Pusher
A K Q J T 9 8 7 6 5 4 3 2
A 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+
K 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 19.9 19.3
Q 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 16.3 13.5 12.7
J 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 18.6 14.7 13.5 10.6 8.5
T 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 11.9 10.5 7.7 6.5
9 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 14.4 6.9 4.9 3.7
8 20+ 18.0 13.0 13.3 17.5 20+ 20+ 20+ 20+ 18.8 10.1 2.7 2.5
7 20+ 16.1 10.3 8.5 9.0 10.8 14.7 20+ 20+ 20+ 13.9 2.5 2.1
6 20+ 15.1 9.6 6.5 5.7 5.2 7.0 10.7 20+ 20+ 16.3 * 2.0
5 20+ 14.2 8.9 6.0 4.1 3.5 3.0 2.6 2.4 20+ 20+ ** 2.0
4 20+ 13.1 7.9 5.4 3.8 2.7 2.3 2.1 2.0 2.1 20+ *** 1.8
3 20+ 12.2 7.5 5.0 3.4 2.5 1.9 1.8 1.7 1.8 1.6 20+ 1.7
2 20+ 11.6 7.0 4.6 2.9 2.2 1.8 1.6 1.5 1.5 1.4 1.4 20+

Caller
A K Q J T 9 8 7 6 5 4 3 2
A 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+
K 20+ 20+ 20+ 20+ 20+ 20+ 17.6 15.2 14.3 13.2 12.1 11.4 10.7
Q 20+ 20+ 20+ 20+ 20+ 16.1 13.0 10.5 9.9 8.9 8.4 7.8 7.2
J 20+ 20+ 19.5 20+ 18.0 13.4 10.6 8.8 7.0 6.9 6.1 5.8 5.6
T 20+ 20+ 15.3 12.7 20+ 11.5 9.3 7.4 6.3 5.2 5.2 4.8 4.5
9 20+ 17.1 11.7 9.5 8.4 20+ 8.2 7.0 5.8 5.0 4.3 4.1 3.9
8 20+ 13.8 9.7 7.6 6.6 6.0 20+ 6.5 5.6 4.8 4.1 3.6 3.5
7 20+ 12.4 8.0 6.4 5.5 5.0 4.7 20+ 5.4 4.8 4.1 3.6 3.3
6 20+ 11.0 7.3 5.4 4.6 4.2 4.1 4.0 20+ 4.9 4.3 3.8 3.3
5 20+ 10.2 6.8 5.1 4.0 3.7 3.6 3.6 3.7 20+ 4.6 4.0 3.6
4 18.3 9.1 6.2 4.7 3.8 3.3 3.2 3.2 3.3 3.5 20+ 3.8 3.4
3 16.6 8.7 5.9 4.5 3.6 3.1 2.9 2.9 2.9 3.1 3.0 20+ 3.3
2 15.8 8.1 5.6 4.2 3.5 3.0 2.8 2.6 2.7 2.8 2.7 2.6 15.0
Suited
Offsuit
Pockets
* 63s: 7.1-5.1, 2.3
** 53s: 12.9-3.8, 2.4
*** 43s: 10.0-4.9, 2.2


Now beyond this, I want to go past this chart and consider the 6max hyper turbos Carbon offers.

This chart of course wouldn't be as useful unless one of the following conditions are in place which occur frequently in these games:
-action has been folded to the blinds and I am one of them (charts would have to be adjusted for raggier cards likely folded in front of me)
-when the game becomes short-handed, especially heads up
-when blinds get really high and people don't adjust to the wider range they should be playing

A consideration is that because these are 6-handed and 2 get paid and in a HU SNG, 1 player takes it all, makes the objective of rationalizing a shove a little a bit more unclear as far as if it is +EV or not, plus ICM blurs that line even a bit more. I'm not sure exactly how this would affect ranges, but would likely make them at least slightly tighter as your stack grows and in 3+ players left, would have to be changed slightly due to you relative stack compared to everyone else's. Plus, ranges will be slightly wider than usual right before a blind increase if you are shorter and slightly tight before one if you are a bigger stack. Finding balance between this is rather difficult. These charts do, however, point a direction towards what a good foundation would be.