# SAGE system vs. Nash equilibrium

• Bronze
Joined: 03.04.2009

Now, as I understand it, the strategy according to the Nash equilibrium is the optimal strategy during heads up play when effective stack size drops below 15BB. The article I referenced to shows two tables with all kinds of numbers, which I don't know how they're exactly calculated, but it greatly has improved my game. I used to win heads-up play around 1/3rd of the time, now I manage to win heads-up play around 2/3rds of the time.
As I understand from that article, SAGE strategy is a mathematical simplification of the Nash equilibrium strategy and that's why I understand PokerStrategy is offering SAGE strategy to its silver members and Nash equilibrium strategy to its golden members. I think that it is wrong.

The only advantage of the SAGE strategy seems to me to be that it can be used during live play, contrary to the Nash equilibrium strategy, since you can't carry around such charts in a casino. And that's where it ends, in my opinion. The advantage of the Nash equilibrium strategy is that it is more accurate and that you can easily look up your hand on it if you have it near your computer. After a while you'll intuitively know where your hand is on the chart and from that moment it's bye bye to SAGE strategy during live play.

Strategy based on Nash equilibrium is very easy to understand and it shouldn't be kept away from silver members, it might even be a good idea to offer it to bronze members.

Of course that is unless I'm missing something, that SAGE truly is a more efficient way to deal with heads-up play, but I really doubt that.

I do have some questions though. In the table in the article I referenced to, there are multiple entries for 63s, 53s and 43s, why is that?
And if I'm playing on PokerStars with antes, how should I relate that to the amount of big blinds. Do I simply add the ante to the big blind, or should I add a third from the ante of the small blind as well? Even though that becomes a rather small number in comparison to the big blind. Or something different?

• 8 replies
• Bronze
Joined: 17.06.2010
When you play online, you should use a Nash chart instead of SAGE. There is no need to use the SAGE approximations to the Nash equilibrium.

The main factor which determines how well you do when you are heads up is how many chips you have to start the heads-up phase. In the short run, you might have great success or many losses in a row, but you can't expect to get a large advantage at the end of SNGs by outplaying your opponents. The blinds are too large, and sometimes you get heads up with 10% or 90% of the chips, or AQ is dealt against 88 and all of the money is going in regardless of what chart you use.

The effective depths at which you should push hands are not just intervals. Most of the time you can approximate them as such, saying that you push each hand up to some number of blinds. However, this is just a simplification. The hands 43s - 63s are slightly unprofitable pushes when your opponents are calling with almost everything. You aren't pot-committed when you post the small blind at a depth of 5 bb, and if your opponent is calling with almost everything, then 43s - 63s lose too much. If your opponent is calling less often, then it can be right to semibluff with these hands because they have a decent amount of equity against tighter ranges.

I haven't studied the effect of antes on the Nash equilibrium, but I usually add 2/3 of the total antes to the big blind and 1/3 of the total antes to the small blind. This is not perfect, but it is not wrong by much. Antes are more significant if you have many players at the table who each contributed the ante rather than just 2. For example, if there are 8 players left at 100/200/25, then there are 2.5 bb in the pot to start the hand instead of 1.5 bb. If you are heads-up at 100/200/25, then you start with 1.75 bb in the pot.
• Bronze
Joined: 03.04.2009
Thank your for your answer. Regarding antes, I understand that they become more important with more players, but these charts only apply to heads-up, right?

You say that you can't expect a large advantage at the end of SnGs by outplaying your opponents, but do you agree with me that your chances are much bigger than the size of your stack normally would indicate if you stick to such charts when you're heads-up against an opponent who doesn't really know which hands to push, call or fold?

And lastly, do you agree with me that this information should be made available to PokerStrategists at an earlier point and, if no, why not?
• Bronze
Joined: 17.06.2010
Originally posted by luitzen
Thank your for your answer. Regarding antes, I understand that they become more important with more players, but these charts only apply to heads-up, right?
The Nash charts are for heads-up play. Normal antes do not matter much in heads-up play.

There are times when you can use the heads-up Nash charts as a guide for blind-versus-blind play when you are not heads-up, as I mentioned in my video, "The Golden Rule."

You say that you can't expect a large advantage at the end of SnGs by outplaying your opponents, but do you agree with me that your chances are much bigger than the size of your stack normally would indicate if you stick to such charts when you're heads-up against an opponent who doesn't really know which hands to push, call or fold?
That depends on what you mean by "much bigger." You suggested that you went from winning 1/3 to winning 2/3 by learning to play well in the heads-up phase of SNGs. That is not the type of improvement anyone can expect. If you reach the heads-up phase with 50% of the chips on average, then winning 53% is doing very well.

I have an extremely high rate of finishing first, but a review of my database shows that my 9-player wins are due to accumulating chips before I get heads-up.

Much of the improvement possible comes from actively exploiting opponents' play, not passively exploiting it by using the Nash ranges.

And lastly, do you agree with me that this information should be made available to PokerStrategists at an earlier point and, if no, why not?
I haven't reviewed the content available at each stage. There are clear reasons not to make too much available early, and if a decent survival strategy is given at the bronze level, then I don't think a Nash chart needs to be included. If the current material gives no hint of how to play heads-up, then perhaps this should be added.
• Bronze
Joined: 03.04.2009
I have to admit that I base my estimates on a rather small sample size, but you must understand that I was an idiot not understanding anything about what you should do in heads-up play. Since I'm playing micro SnGs, I am playing against idiots like I was and I noticed that my win chances have significantly improved. You are probably playing against other people who know what to do.
• Bronze
Joined: 19.04.2008
Originally posted by pzhon
The main factor which determines how well you do when you are heads up is how many chips you have to start the heads-up phase. In the short run, you might have great success or many losses in a row, but you can't expect to get a large advantage at the end of SNGs by outplaying your opponents.

This is true vs good players.. but you have a huge edge on most players imo
• Bronze
Joined: 17.06.2010
Originally posted by viewer88
Originally posted by pzhon
The main factor which determines how well you do when you are heads up is how many chips you have to start the heads-up phase. In the short run, you might have great success or many losses in a row, but you can't expect to get a large advantage at the end of SNGs by outplaying your opponents.
This is true vs good players.. but you have a huge edge on most players imo
At all levels, I play against bad players. I base my observations on how dominant winning players do against bad players, including my own play crushing microstakes games, and from reviewing tournaments and databases for my students who are at the top of the Sharkscope leader boards for microstakes and low stakes games. I am not overlooking that there are bad players, I am addressing how much of an advantage you can hope to get over bad players.

It is a very common but very wrong idea that having more second places than firsts means you need to work on your heads-up play, or that having more firsts than seconds means you are playing well heads-up.

If you have a large sample of 9 player SNGs, please filter your database to heads-up play, and report the number of tournaments, number of hands, chips won, BB/100, the EV BB/100 (luck adjusted). While I expect there to be a reporting bias, you will probably be surprised to find few hands per tournament, and a nice-looking EV BB/100 which translates to a low BB/hand. This means there is a low number of tournaments converted from second to first due to outplaying your opponents in the heads-up phase of the tournament, even if you are up against bad players.
• Bronze
Joined: 17.06.2010
Here are the results for a recent sample of tournaments from one dominant low stakes player (who does a lot better than just playing the Nash push/fold game):

10,237 tournaments