SAGE advice for heads up play

[EDITOR'S NOTE: The Persistent Irritation is pleased to welcome guest blogger Jeff H.]

***

It’s Friday night. I'm knee deep in an end game against Pedro S., a good, aggressive heads up player. In the big blind, I have Q9 offsuit, a stronger than average hand. Pedro only completes the SB, which looks weak, so I raise three times the BB. He comes back over the top and I suddenly have the sinking feeling that he is slow playing a monster. I either have to commit all my chips or fold.

How would you play it?

After hours of grueling tournament play, going heads up in the finale can be both the most important and confusing part of the game. The difference between 1st and 2nd is often a factor of two in the payoff, so half of your winnings depend on good heads up play. Yet by the time you get to heads up you are tired, you may have a few beers under your belt, and the patience that got you to this point may be waning. It can be hard to know what to do.

Luckily, there is a mathematical solution to help you out. It goes by the name of SAGE (Sit and Go Endgame), the main elements of which are simple enough to memorize or put on a 3x5 card.

Once the smallest stack is 10 BB or less, heads up hold' em can be reduced to a solved math problem, at least with a few reasonable assumptions.

With SAGE, you only make one choice: you shove or fold in the SB. For points of discussion, we'll assume your opponent is going to do the same, so in the BB we either call a shove or fold.

The remaining question is to figure out what hands are shove-worthy. The authors of SAGE determined this by using a little game theory and ranking all 169 starting hands by their equity against random distribution of hands in a showdown.

Here's a simplified version of their results. First, compute the Power Index (PI) of your hand as follows:

1. Identify the power number of each card:

 

2. Determine the Power Index using the following equations:

• Double the power number of your higher card and add this to the power number of your lower card.  Example: AsTd = (15*2) + 10 = 40
• Add 22 for a pocket pair. Example: 4h4d = (4*2) + 4 + 22 = 34
• Add 2 if your cards are suited. Example: Qd9d = (12*2) + 9 +2 = 35

3. Assume R = small stack/BB. Example: If the small stack has $2,100 and the blinds are $150/$300, R = 7.

4. Now simply follow the chart:

* For R between 7 and 10 it is believed that SAGE is not quite optimal, but it is not bad, and I'd use it up to R=10.

An even more simplified version is to shove with any A, K, Q in your hand, any high Jx, or Tx, and any pair. That's pretty much it. So in the previous case against Pedro, calling with Q9 offsuit (PI = 33) would have been the correct move, assuming the short stack was somewhere near 7-8 BB.

SAGE is what is known in game theory as a non-exploitable strategy. It is the analog of random choice, such as that found in each throw of rock, paper or scissors in Roshambo. It is not the optimal strategy if you have a good read on your opponent's weaknesses but it does guarantee that you cannot be exploited, which is often good enough to give you the confidence to shove with J4s in the small blind.

For more details and some refinements have a look at the original article; for more than you could possibly want to know about heads up play, read the FAQ in the 2 plus 2 HU forum.

- JH

Comments

[Ben]

Cheater. ;)

[Sean Carroll]

Oh sure, you'll do guest posts on Chris's blog...

[Jeff H]

Right Sean. Because HE offered me a decent cut of the advertising revenue.

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