Are You Sage? Getting an Edge in Heads-Up No-Limit Hold'em

A poker tournament is often as much about mental endurance and discipline as it is about the cards and strategy. After all, particularly in no-limit hold'em, a single error can put you on the sidelines, even after playing flawlessly – or brilliantly – up to that point.



But let's assume that you've made it past all but one opponent; you're heads up at the end of a no-limit hold'em tournament. All of your work (and, let's be honest, luck) has paid off; you and your opponent are contesting for the top prize. The cruel irony, of course, is that both of you are mentally (and perhaps physically) exhausted, but you're about to play for the largest single increase in prize money, and this is the portion of the event that scares people the most. Many folks are so delighted to make it down to heads-up play (and are exhausted from the event) that they lower their focus. And, regrettably, lots of players feel completely lost playing heads-up no-limit hold'em. They either guess their way through it or seek a deal with their opponent; they're willing to give up some equity just to be done with the confusing and sometimes terrifying two-player battle. The poker player who can wade into this battle with courage and knowledge stands to win the largest prize in the entire event, not to mention the glory of finishing first.



The good news is that mathematics and game theory can often provide a clear and well-lit path through the foreboding forest of the heads-up portion of a no-limit hold'em tournament.



In this feature, we describe a simple technique that you can use in many heads-up no-limit situations; we call it the SAGE™ System. It is so straightforward that you can write it down on a 3"-by-5" card. With a few minutes' study, you can memorize it and use it anytime you find yourself heads up.

Why do I need the SAGE System?
Some players have said to us, "Well, I think I'm already playing pretty well heads-up." Frankly, this is unlikely, because: Most players play far too tightly in heads-up jam-or-fold situations.



This is not our opinion – it is a mathematical fact. And don't believe for a moment that tournament pros such as Chris Ferguson are not aware of this.



With that said, please understand that this system will not make you invincible in the heads-up portion of a no-limit hold'em tournament. However, against most opponents, who don't correctly adjust for the large blinds and weaker hand values of heads-up play, you should have a 5 percent-40 percent advantage; 5 percent is good, 40 percent is crushing.



You might ask, "Hasn't this material already been published?" Well, yes, but not widely. There are a few places in the online forum world where this information is shared (most notably in the "Two Plus Two" forums – http://www.twoplustwo.com/). But we are not aware of any book or magazine that contains this information, and we believe our presentation is easier to use in practice than anything that has been published in any medium. Dan Harrington has produced a stunning pair of books (Harrington on Hold'em, two volumes) that give a lot of information about playing heads-up. However, the main thrust of his discussion is major tournaments, where the stack-to-blinds ratios are often 100-1 or more (so our system does not apply). And in fact, we believe Harrington gives incorrect advice in one of his smaller-stack examples (more about that in a bit).



When does the SAGE System apply?

Let's be sure that you understand exactly when you can apply SAGE:

1. You are heads up with a single opponent at the end of a no-limit hold'em tournament. The important point is that there is just a single prize left. Note that a one-table satellite into a larger event also has this feature (the single prize is a seat in the larger event).



2. The blinds are large compared to the stack sizes. To put a value on this, if the ratio of the smallest stack to the big blind is greater than about 10-1, our system is of no use to you. However, this small-stack-to-blind ratio is extremely common in online sit-and-go tournaments, and even more so in brick-and-mortar casino sit-and-go tournaments and one-table satellites (thus the acronym SAGE: Sit And Go Endgame System). Please note that our system doesn't worry about the size difference between the stacks. Only the size of the shorter stack matters.



3. The small blind (who is on the button, and acts first) chooses to either move all in ("jam") or fold. As the blinds grow large compared to the stacks, this is a winning strategy for the small blind (SB). Of course, once the SB jams, the big blind (BB) can choose only between calling or folding; our system will tell the BB how to defend optimally.



When does the SAGE System not apply?
1. When there are more than two players left. For example, you may ask: "Suppose that we're threehanded, the button folds, and I'm the small blind. What should I do?" The system we present here cannot answer that question. Here is why: When you are threehanded, there are two more prizes to be contested (second and first; everybody is already guaranteed third-place money). In this situation, the value of the chips is not fixed; that is, the value of a single chip in your stack may be different than the value of that same single chip in an opponent's stack. An exception to this would be a one-table satellite with a single prize: a seat in the target event. In this case, all that matters is the stack sizes.



2. When the stacks are still large compared to the blinds. This is an important point. For instance, if you are in the final event of the World Series of Poker, the stack to blind ratios will be so large that SAGE does you no good. You will just have to (as the pundits are fond of saying) "play good poker."



3. When the small blind limps or makes only a small raise. In these cases, our system provides no guidance to the big blind (BB) player (at least not yet; this is obviously an interesting area for further research).



Now that you understand when the SAGE System does and does not apply, we're finally ready to tell you about the system!



So, what is this SAGE System?

The SAGE System is based on what mathematicians call an "equilibrium strategy," which is a strategy that cannot be "beaten" in the following sense: If either player deviates from the equilibrium, his expectation will go down (and thus, in a two-player game, his opponent's expectation must go up). Studies have shown that for short-stacked, heads-up no-limit hold'em, using an equilibrium strategy makes the outcome of the tournament nearly a coin toss (weighted by the players' relative stack sizes, of course). To put it another way, if you find yourself facing Chris Ferguson heads up, this system will prevent him from using his (presumably) greater poker expertise to get an advantage over you.



There are many ways to find the equilibrium strategy; we started by ranking the 169 possible hold'em starting hands according to their overall "power" in heads-up play (unsurprisingly, A-A is first and 3-2 offsuit is last). Starting with this list enabled us to devise the SAGE System so that it is easy to use in practice (it also results in a slight deviation from the "true" equilibrium strategy, but it gives up virtually zero edge). We then modeled the expected winnings of each player mathematically and used a procedure called "minimax" to find the equilibrium strategy. The following table summarizes the results:



Optimal Top Percentage and Cutoff Hands

Here's how to interpret the above table:



• The "R" column is the ratio of the shorter stack to the big blind, after the blinds have been taken. For instance, if the shorter stack is $17,000 and the big blind is $3,000, R = $17,000 ÷ $3,000, or about 6.



• The "SB Top %" is the percentage of your hands that you should jam with in the small blind, given that value of R.



• "SB Cutoff Hand" is the worst hand that meets that criterion. Of course, for this to be any good to you, you have to know the ranking of all 169 starting hands. SAGE makes it unnecessary to know this list.



• "BB Top %" is the percentage of your hands that you should call with, if the SB jams.



• "BB Cutoff Hand" is, again, the worst hand that you should call with, if the SB jams.



• "SB Edge" is the amount that the SB can expect to win, in "big blind" units, for that value of R, if both players follow this system. Note that when R is 7 or greater, the SB has a slight negative expectation if he employs only the jam-or-fold strategy. If R is this high and you're the SB, you should "play good poker" (that is, mix up limping, minimum raising, and jamming). However, if you think your opponent is substantially better than you, you can simply play the equilibrium strategy, secure in the knowledge that you're giving up only a microscopic edge.



Now, if you had a list of all 169 starting hands, you could just find your hand in that list and see if it's above or below the threshold in the table above. While this is straightforward for a computer to do, it's not the sort of thing that most of us could do in a live tournament. The SAGE System simplifies the entire process into two easy steps. As promised, you can keep it on a 3"- by- 5" card with you if you're playing online, and can quickly memorize it for brick-and-mortar casino use.



1. Compute a Power Index (PI)

1. The "power number" of each card is its rank: J=11, Q=12, K=13, A=15 (don't forget that the ace is 15!)

2. Take the power number for your higher card and double it.

3. Add the power number of your lower card.

4. If it's a pocket pair, add 22.

5. If they're suited, add 2.

6. The sum is the Power Index (PI) of your hand.



2. Use the PI

1. Compute the ratio® of the shortest stack to the big blind.

2. Look up the necessary PI for that value of R.

3. If the PI of your hand is greater than or equal to that value, jam (if you're the button/SB) or call (if you're the BB).

3. The SAGE Table

SAGE Examples

Example No. 1:

The blinds are $500-$1,000. After the blinds are taken, the SB has $5,635 in chips and the BB has $2,865 in chips. The SB has pocket threes. So, what's his power index (PI)?



PI = (2 × 3) + 3 + 22 = 31



The BB has J-4 suited. His PI = (2 × 11) + 4 + 2 = 28. The value of R is the smaller stack ($2,865) divided by $1,000, which is close enough to 3 for our purposes.



Looking at the table, the SB should jam. His PI of 31 is much greater than the necessary value of 22. The BB should call; his PI of 28 is greater than the necessary value of 24.



Example No. 2:
The stack sizes and starting hands are the same, but the blinds are $200-$400. Is anything different? The SB's PI is still 31, but R is now $2,865 divided by $400, which is about 7. Looking at the entry for R=7, the SB number is 26, so he should still jam. But the BB, whose PI is 28, doesn't have the value of 30 necessary to call. He should fold if the SB jams.



Conclusions
There's one important thing to learn from this: Heads-up no-limit hold'em is not rocket science when the blinds are high compared to one of the stacks. Note that suitedness counts for almost nothing, and connectedness counts for less. We suspect that you will now agree with us that most players grossly underestimate the need to jam early and often. For that reason, you will often have a dramatic advantage over your opponents when using this system.



This information is correct, and provably so. Yet, even Dan Harrington got one of his examples wrong. On Page 409 of Harrington on Hold'em, Volume II, he describes a scenario in which you are in the big blind with Q-7 offsuit. The blinds are $3,000-$6,000 with a $300 ante, and this is the second hand of the heads-up finale of a tournament. Your opponent has $35,400 and you have $144,600. Your opponent, on the button, jams; what should you do? Dan goes into quite a bit of discussion about why this is a close decision. With SAGE, you simply compute the power index for your hand and the current value of R:



PI = (12 × 2) + 7 = 31.

R = $35,400 ÷ $6,000 6.



You look in the table and see that the value for calling is 29. You have 31. It's an easy call, and you hope that the flop is Q-Q-7.



Last question: Why are we giving this away? First, because it advances the game of poker. No discipline (including a game) can grow and thrive unless its disciples discuss, argue, prove, and disprove. Second, we don't see why just a select few people should have this information. You can bet your entire bankroll that Chris Ferguson and many others have it. Now, you do, too.

We are indebted to Mike Maurer for his review and critique of this system.



Please note that the SAGE System is Copyright © 2005 by James Kittock and Lee Jones, and the terms SAGE System and "Are you SAGE?" are trademarks. You have our permission (indeed, our encouragement) to use it. But you can't take credit for it and you can't sell it.



Lee Jones is the author of the best-selling book Winning Low Limit Hold'em, and is the cardroom manager for PokerStars.com. James Kittock is a math teacher at Mission College in Santa Clara, California, and a serious low-limit hold'em player.