# Nash equilibrium

• Bronze
Joined: 18.12.2009
I want to know if I understand the theory correctly:

1) If all the players start playing from the p/f phase and everyone is pushing and calling according to Nash, no player will be winning any money in the long run. Is that correct?

2) If a player deviates from Nash, we have to change our play according to the extent of his deviation because we might miss some +EV calls/ pushes and some of Nash pushes or calls will be -EV; and if we adapt accordingly (let us say, as suggested in SNG wizard) our opponent is losing in the long whereas we are profiting. Is that correct?

3) A player who pushes and calls according to Nash is unexploitable in the push or fold phase. Is that correct?
• 14 replies
• Bronze
Joined: 09.09.2010
I think there are more ideal charts than Nash out there

search for Risk Oriented push fold chart
• Bronze
Joined: 16.01.2011
Originally posted by Avatars91
I want to know if I understand the theory correctly:

1) If all the players start playing from the p/f phase and everyone is pushing and calling according to Nash, no player will be winning any money in the long run. Is that correct?

That is correct

2) If a player deviates from Nash, we have to change our play according to the extent of his deviation because we might miss some +EV calls/ pushes and some of Nash pushes or calls will be -EV; and if we adapt accordingly (let us say, as suggested in SNG wizard) our opponent is losing in the long whereas we are profiting. Is that correct?

That is correct also (although DON'T use sng wizard default ranges)

3) A player who pushes and calls according to Nash is unexploitable in the push or fold phase. Is that correct?

He is unexploitable if his opponent also follows Nash equilibrium.
Since this is rarely a case at low limits you have to adjust properly - e.g if players push tighter then Nash you should call tighter.
In general for pushes use at least Nash equilibrium and wider - villain has to prove to you that he is calling wider then Nash for you to tighten up.

• Bronze
Joined: 18.12.2009
3) A player who pushes and calls according to Nash is unexploitable in the push or fold phase. Is that correct?
He is unexploitable if his opponent also follows Nash equilibrium.

Does it not logically follow then that Nash equilibrium is exploitable?
• Bronze
Joined: 29.04.2010
Yes, game theory is applicable to any stage of a tournament (or, for that matter, to all strategic decisions in poker). However when we try and work out an equilibrium outcome with deep stacks and many players the maths becomes impossible.

When we enter the push/fold stage of a tournament (Harrington uses M<5, other authors use slightly different criteria) then we only have one decision to make when we look at our cards. For simplicity these situations often have Hero in SB with all other players having folded and BB yet to act. If we push our opponent may fold, and we add fairly significantly to our stack, he may also call and we may win and double or lose and bust.

Given that stacks are small and there is only one available decision we can begin to approach a GTO (game theory optimal) solution to the p/f stage of a tournament. Using a static analysis, for example we assume villain calles with 40% of his hands, we can calculate an optimum pushing range. However our opponents may adapt to our push frequency and adapt and so it is possible to use a dynamic analysis where villain's calling range changes based on our pushing range (as will realistically be the case). By plotting reaction curves for hero and villain we can find a Nash Equilibrium, where neither player will change their strategy if they assume their opponent is playing rationally. In this case both players are playing unexploitably.

In the case where SB is too tight the BB is in a +EV situation because the SB will fold very often giving him blinds + antes. If the SB were only pushing AA then ~ 209/210 times the BB gets the blinds. When the SB does push villain can happily fold (unless he too has AA or he is getting better than 4/1). If the BB is calling too tight then the SB can shove more frequently and make profit.

Kill Everyone has a pretty good explanation/graphs to explain further.
• Bronze
Joined: 29.04.2010
Does it not logically follow then that Nash equilibrium is exploitable?
No, because if we know that our opponent is playing unexploitably theres no profitable strategy we can devise to beat him.
• Bronze
Joined: 17.06.2010
It looks like this is in the wrong forum. A preflop push/fold strategy is not recommended for cash games when the effective stack depth is deep. You might want to post this in a tournament section, or perhaps focus on short stacks in a cash game.

Originally posted by Avatars91
2) If a player deviates from Nash, we have to change our play according to the extent of his deviation because we might miss some +EV calls/ pushes and some of Nash pushes or calls will be -EV; and if we adapt accordingly (let us say, as suggested in SNG wizard) our opponent is losing in the long whereas we are profiting. Is that correct?
No and no.

First, in many situations you do not have to change, it's just a profitable opportunity. Let's suppose you are heads-up at the end of the tournament. If your opponent calls too tightly, then playing the Nash range will still passively exploit your opponent. You will be better off than if your opponent made all of the correct calls. Against an opponent who calls too tightly, perhaps you should have pushed ATC, and by folding 72o you are missing a profitable opportunity. However, your pushes with K4o will be more profitable than normal. You are not hurt by your opponent's decision to play differently. You have an opportunity to do even better, but you are not required to adjust.

If an opponent is calling too widely when you are risk-averse, this may hurt both of you, and then there may be no way to get as much equity as you would have in the Nash equilibrium however you adjust. For example, if your read is that a player will call you 100% of the time when you have equal stacks on the bubble, it is as though your opponent has pushed, and you can only push the hand which would be calls against a 100% range on the bubble. This might change hands like 87s from an easily profitable push to a clear fold, and even AKo may become marginal because you aren't actually that happy to get all-in as a 65-35 favorite on the bubble.

If there are 4 equal stacks on the bubble at 200/400 and you tighten up from the Nash 100% range to the few hands worth pushing against someone who will call 100% of the time (66+ A9s+ ATo+ KJs+. Does this help or hurt your opponent? I believe he is actually better off than in the Nash equilibrium because he gets so many walks, even though some of his calls are terrible. It can be valuable to commit yourself to something which looks stupid if the reactions of other people will help you.

3) A player who pushes and calls according to Nash is unexploitable in the push or fold phase. Is that correct?
Not exactly. Using the Nash ranges makes you unexploitable by any one player, but you can be exploited by a team of players who collude against you. For example, one player might make spite calls which hurt him slightly, but which hurt you a lot, transferring a lot of equity to a friend of his who has folded. So, in a multiway situation, using the Nash ranges does not guarantee that you get as much equity as in the Nash equilibrium.
• Bronze
Joined: 29.04.2010

On point 1) if our opponent is calling too tight then we do not have to deviate from our strategy in order to be profitable, but we can deviate from the Nash Equilibrium and be more profitable we can do this by pushing more of our hands.

On point 2) risk aversion is surely a huge detriment to a poker player. It's why risk neutral players have such a huge edge on the bubble, forcing opponents to fold large edges. Also it's very rare to find an opponent who calls to wide in these push/fold situations. The mistake made by many players is to play scared and fold too many hands.
• Bronze
Joined: 17.06.2010
Originally posted by sambucatus
On point 2) risk aversion is surely a huge detriment to a poker player.
Risk aversion in \$ is a handicap you address in part by proper bankroll management. Risk-aversion in chips, within a particular tournament, is an important and correct part of tournament strategy. See my video on the "ICM -- Independent Chip Model" which is a standard tool used by almost all serious SNG players for estimating how much money a stack is worth. If you ignore this rational risk aversion, you will win more tournaments, but you will not get your fair share of lower prizes, and you will win less money over all.

It's why risk neutral players have such a huge edge on the bubble, forcing opponents to fold large edges. Also it's very rare to find an opponent who calls to wide in these push/fold situations. The mistake made by many players is to play scared and fold too many hands.
You may be mixing together situations from MTTs, where most of the risk aversion on the bubble is irrational, and situations from MTT final tables or STT bubbles, where risk aversion is critical for playing well. It is common to tell MTT players to aim for first place to discourage them from overvaluing minimum cashes. However, that's far from optimal even for MTTs, and it is very wrong for STTs.

There are many tournament situations in which casual players call too widely, in part because they do not understand how risk-averse they should be. See my video "Super Turbo Ranges" for data on aggregate calling ranges. On average, players call way too widely against 8-10 bb CO shoves on the bubble.
• Bronze
Joined: 18.12.2009
First, in many situations you do not have to change, it's just a profitable opportunity. Let's suppose you are heads-up at the end of the tournament. If your opponent calls too tightly, then playing the Nash range will still passively exploit your opponent. You will be better off than if your opponent made all of the correct calls. Against an opponent who calls too tightly, perhaps you should have pushed ATC, and by folding 72o you are missing a profitable opportunity. However, your pushes with K4o will be more profitable than normal. You are not hurt by your opponent's decision to play differently. You have an opportunity to do even better, but you are not required to adjust.

By saying "we have to change our play" I imply that we wan't to make all the best decisions in order to profit the most, I can see how our play could still be profitable if we don't deviate, of course.

What I don't understand though: if we don't deviate from Nash equilibrium is our overall play profitable if our opponent calls extremely widely? Let us say that the low end of the Nash range in a hypothetical situation includes 75o and our opponent calls with almost ATC: how can a play according to Nash equilibrium be profitable in this situation?
• Bronze
Joined: 17.06.2010
Originally posted by Avatars91
What I don't understand though: if we don't deviate from Nash equilibrium is our overall play profitable if our opponent calls extremely widely? Let us say that the low end of the Nash range in a hypothetical situation includes 75o and our opponent calls with almost ATC: how can a play according to Nash equilibrium be profitable in this situation?
You have to distinguish between situations where the value of chips is linear and those where it is not.

When the value of chips is linear or close to linear, such as when you are heads up or at the start of a big MTT, then your opponent's loss is your gain. If your opponent calls 9 bb with K2o, which was not a correct call against your range, then he invests 9 bb and gets less than 9 bb back on average. If he gets back 8 bb, then he loses 1 bb on average, and you gain 1 bb on average. When you have AA, you gain more than 1 bb. When you have 75o, you gain less than 1 bb, and in fact you lose. Your range gains on average or else his call was correct.

When you play the Nash pushing range, your opponent's exploitively optimal strategy is to call with the Nash calling range. Each deviation costs him money compared with playing the Nash calling range. Particular hands might not gain from a deviation, but your range gains on average.

If the value of chips is not linear, then your opponent's loss is not the same as your gain, and his spite calls may hurt both of you.
• Bronze
Joined: 18.12.2009
I'm sorry if my inability to understand this instantly is annoying to anyone.

Does this mean that playing according to Nash equilibrium against someone who is not doing the same is always profitable although not optimal?
• Bronze
Joined: 17.06.2010
You have to distinguish situations where the value of chips is linear from situations where you are risk-averse, where doubling your chips does not double your expected share of the prize money. What you said is close to the truth for situations where the value of chips is linear. It is very far from correct where you are risk-averse such as on the bubble of a STT.

It is not always profitable to play your side of the Nash equilibrium strategy against someone who is deviating. There may be nothing you can do to be profitable when a player is determined to get you all-in even if it hurts him. If there are 4 equal stacks on the bubble at 200/400, the Nash pushing range for the SB is 100%. The Nash calling range for the BB is very tight. If the BB instead calls with ATC, then pushing ATC means you are taking a coin flip on the bubble which is a disaster, far worse for you than pushing against a Nash caller. The caller loses, and you lose. This is possible because the caller's loss is not your gain. The other players in the tournament who have folded are the beneficiaries.
• Bronze
Joined: 18.12.2009
Thanks a lot for your input, I finally got it!
• Bronze
Joined: 07.02.2009
Awesome explinations pzhon