The Nash Equilibrium for the two player push and fold game

When one of two players are playing short stacked. They can play the Nash Equilibrium push or fold game, inwhich you push all in with a certain stacksize and a certain hand with the goal to get the blinds and exploit folding equity. In the table below you see all the hands that can be pushed by 20+ bigblind stacks in green. 15+BB stacks in yellow, 10+ BB in orange and 5+BB in red.

Your heads up push range according to the Nash equilibrium:

Nash equilibrium push all in range

In short you will be able to push all in against one opponent while having the top 40,27% hands and 20+ big blinds. With 45,8% of the hands you can push with 15bb, With 57,8% of the hands  you can push all in while having 10bb+. With the top 72% of hands you can push all in with 5 big blinds. In general you can push almost always under certain conditions.

The heads up calling range of the Nash equilibrium:

Nash equilibrium call all in range

Your heads up calling range is approximately 21,7% while having 20+bb, 28,7% with 15+bb, 37,1% with 10+bb and 62,3% with 5+bb.

How do you memorize all of this? In general you don’t, you just estimate your hand based on the table above.

Having more then one opponent

In theory a similar chart could be made while having two random players sitting behind you, or maybe even 3, 4, or 5. However we will give a shortcut to these all in ranges. In general every Ax combination, pocketpair or suited connector ( 65s+) Has a decent chance even in a multy-way pot. Those type of hands should be pushed with multiple opponents behind you also like calculated before in:

In general the tighter your opponents are the more folding equity you will get. Sometimes you can even push all in from under the gun position, because players will  more likely assume you to have one of the top hands based on your position.

An estimation of a 3 player Nash equilibrium

The idea of the Nash equilibrium is based on the idea that you push with your opponent holding any random hand. The 40% range gives you 59% chance of winning against any random hand. Lets assume this to be a rule for 20+bb stacks.

In the case of having two opponents behind you the average best hand will become the top 50% percentile of hands. Using this handrange to find a new equilibrium you will get a preflop pushrange while two players are behind you of:

18,6% 20+BB

20,4% 15+BB

26,5% 10+BB

32,6% 5+BB

If this table is correct you need to play much tighter when you have 2 players behind you.  However this did not take into account the folding equity you have.

If you take into account the folding equity you will get much more close to 40% as the open range for 20+ BB, even if both opponents may call you with 30% of the hands:

Or the top 20%:

Or the top 10%

Even if you opponents call wit a tight range of 5% you have a positive expectancy:

So the Nash equilibrium may apply as well when having two players behind you. The expectancy is still positive. The worst expectancy is around half your range, which is 20% in this case.

I did the same with 3 players behind you which call 20% of the time:

With 3 opponents the Nash equilibrium gives a negative expectancy. Instead you should tighten up a bit to 25% of the hands according to my calculations.:

Which are the following hands:

This handrange of 25%  applies to going all in with 20bb while having 3 opponents behind you who call you with approximately 12.5% of the hands.

Going all in with more then 3 players behind you isn’t advisable. The handranges may get closer to 15-20% with 4 players and so on. The folding equity will diminish fast with more opponents behind you.

All of the above may not apply in a tournament situation where you are on the bubble (almost in the money). In such case you should try to get in the money first while shortstacked. If you have a bigger stack however, you can exploit the folding equity even more so, because no opponent wants to go out before the bubble.

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