HeadsUp Push/Fold Nash Equilibrium

[Talking Poker]

[Talking Poker]

Here is a good explanation that I found on 2+2, .

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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.

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.


SO bottom line:

the 1st chart tells you that a certain hand is a profitable shove below the given stack size.

the second chart tells you the stack size below which a certain hand is a profitable call, if your opponent shoves at you assuming the opponent is playing according to the first chart.


** the charts assume you are playing a game where the rules are push all-in, or fold. no other options. This is probably very relevant for the last few hands of a tournament where the blinds are high and only two players remain. It becomes less relevant in other situations.

*** game theory works out that these are "optimal" strategies. "Optimal" is a technical game theory term that means that no strategy performs better against an opponent playing an optimal counter-strategy. It also means that no strategy your opponent employs could decrease your expectation in the game. It does NOT mean that this strategy is maximally profitable -- it is not optimal in the dictionary english sense -- against a weak opponent. If your opponent does not play a game-theory-optimal strategy, you could potentially do better than the game-theory-optimal strategy.

**** no limit holdem: there's always exceptions! there's some weirdness here that can mostly be ignored, where certain hands are good to shove with at one stack size, but at a slightly different stack size the opponent's calling range is slightly different so the hand isn't a good shove any more. If you drew the chart according to stack sizes instead of hands, some hands appear and disappear off the shoving and calling ranges, rather than just appearing at some stack sizes and staying on the "green light" list. The chart shows the most significant cases of this: 63s, 53s, and 43s, which are shoves at some stack sizes in a discontinuous way.

[Talking Poker]

After reviewing these charts, I've come to the realization that I am a little bit too tight (especially calling), as effective stacks get low late in HU SNGs.

Granted, I understand that this is assuming my opponent is playing optimal strategy (ie, also using these charts) and that 95% of my opponents are NOT doing such a thing. I also understand that the worst my opponent's skill level (especially if he's too tight), the less willing I should be to "gamble" and wait for a better spot instead... but in general, I think I am too tight late in matches, and especially against skilled opponents.

Interesting stuff.