## Sunday, November 18, 2007

### Partial SNG book

Introduction to SNGs

Measuring success

In cash games, players quote their profit and losses in terms of big bets per 100 hands, or sometimes big bets per hour. This allows comparison of win rates across different limits. In SNGs, win rates are quoted as a percentage, called Return On Investment, or ROI. This is simply total profits divided by total buyins, including the vig. For example, if you have made a profit of \$660 over 100 \$50+5 SNGs, then your ROI will be 660/5500 = 12%.

Standard prize distribution

In most 10-man SNGs, 50% of the prize pool is paid to the winner, 30% to second place and 20% to third place. This prize structure will be assumed throughout this book.

The difference between tournaments and cash games

In a cash game, if a player stands to either lose \$500 or win \$500 on a given hand, with an equal probability of each outcome, then his expectation is exactly neutral. The axiom underlying this is that a dollar is always worth a dollar, in any context.

In tournaments, chips are not always worth the same amount. Consider a 10-man \$100 buyin SNG where everyone starts with 1,000 chips. In total, the 10,000 chips in the tournament therefore have a value of \$1,000. At the end of the tournament, someone will have 10,000 chips, but they win only 50% of the prize pool. The value of those 10,000 chips has diminished to \$500.

To determine the correct plays in tournament situations (especially in SNGs), we need a way to find out the exact value of chip stacks in a given tournament situation. This will enable us to answer questions like how many chips we should be willing to put at risk to achieve a theoretical gain of 500 chips. Answering these type of questions is the purpose of equity modelling.

Equity modelling

A player’s equity in a tournament is her expectation at a given point at the tournament expressed as a fraction of the prize pool. This is calculated by multiplying together the percentage chance that she finishes in each paid position and the percentage of the prize pool which that position pays, and then summing the resulting numbers.

For example, say a player has a 20% chance of finishing first, a 25% chance of finishing second, a 30% chance of finishing third, and a 25% chance of finishing out of the money. Her equity is:

0.2 * 0.5
+ 0.25 * 0.3
+ 0.3 * 0.2

= 0.235, or 23.50%. The decimal and percentage can be used interchangeably, but in this book we will be expressing it as a percentage.

The total amount of equity in a tournament is constant, because the sum of all players’ equities should always be 100% - the whole prize pool. Therefore, any play which increases one player’s equity must necessarily decrease the equity of one or more other players.

Since your equity is a summary of your money expectation in a tournament, it goes without saying that the goal of every play you make in a tournament should be to increase your equity. You should never “play for first”, or “play for third”. Always aim to simply increase your equity and you will be winning money.

An equity model is a method for estimating players’ equities, given the current chip stacks. In just a second we’ll introduce the Independent Chip Model, or ICM, the equity model which we’ll be using throughout this book.

Why does equity modelling matter more for SNGs than MTTs?

SNGs and Multi-Table Tournaments, or MTTs, are just different varieties of tournament. Equities could be produced for MTTs using the ICM, although the sums would be much more complicated. However, the differences between cash games and tournaments are more pronounced in SNGs than they are in MTTs, for two reasons:

Harsher bubbles in SNGs

The jump from fourth to third place in an SNG is 20% of the prize pool, a huge change in fortune which often has a dramatic effect on the correct play. No MTT has a bubble with that extreme a jump.

Payout structure

The top 30% of players in SNGs are paid. MTTs typically pay out to only the top 10-15% of finishers, with the top 5% collectively getting about 75% of the prize pool. This makes the MTT prize structure a lot more top-heavy. The more top-heavy a prize structure, the more the tournament should play like a cash game. Winner-take-all tournaments play almost identically to cash games.

Conclusion

In MTTs, equity modelling is both more complex to do mathematically and harder to intuitively approximate at the table, because payout structures differ so widely. Because using an equity model often doesn’t affect the correct play in MTTs, most authors of books about no-limit tournaments have chosen to ignore equity modelling and use simple pot odds calculations.

This won’t do for SNGs. Pot odds calculations often give answers which are wildly wrong, especially on the bubble. Using an equity model like the ICM is essential.

The Independent Chip Model (ICM)

The ICM is an equity model that works well for SNGs. This section describes the method used to calculate equities in the ICM. If you would prefer to think of it as a magic box where you put the current chip stacks in and get equities out, that’s fine – skip to the section on ICM tools.

Suppose in a standard SNG, three players remain - A, B and C – with stacks of 10000, 6000 and 4000 respectively. We assume that all players play with equal skill. The process starts by assigning them a first place finish probability equal simply to the percentage of total chips in play contained in their stack. So:

A: 50%
B: 30%
C: 20%

Now we take each of those possibilites in turn, and mentally eliminate the player from the game, leaving two players. Then we repeat the original process for second place. So taking A first, if we eliminate A from the game, that leaves stacks of 6000 and 4000, with 10000 total chips. That means that along this branch, B finishes second 60% of the time and C 40% of the time. However, we want to know the probability of this finish sequence as a whole. A only finishes first 50% of the time, so to get the overall probability, we need to multiply the B and C second place finishes by 50%, giving:

A, B, C: 30%
A, C, B: 20%

If we had four players, we would need to go down another level, eliminating the second-place finisher from the game, repeating the process for players C and D, then multiplying together all the probabilities we obtained (first place, second place, third place) to get an overall probability for that finish sequence.

Repeating the process for B and C finishing first yields:

A, B, C: 30%
A, C, B: 20%
B, A, C: 21.43%
B, C, A: 8.57%
C, A, B: 12.5%
C, B, A: 7.5%

All these possibilities should sum to 100%, since they represent all the possible orders in which the players can finish.

Supposing we want to determine the overall equity for player A. We need to multiply the chance that A finishes first, second and third by the prizes he receives for each placing (expressed as a fraction of the prize pool).

Equity(A) = 0.5 * (0.3+0.2) + 0.3 * (0.2143 + 0.125) + 0.2 * (0.0857 + 0.075)
Equity(A) = 38.39%

A’s stack of 10,000, according to the Independent Chip Model, is worth 38.39% of the tournament prize pool.

ICM tools

At the time of writing, two commercial tools are available which calculate the correct play in all-in or fold decisions, based on ICM equities. The programs require you to enter hand ranges for your opponent(s).

These programs are:

SNGPT – http://sitngo-analyzer.com/
SitNGo Wizard – http://sngwiz.com/

A free online ICM calculator, which calculates ICM equities based on inputted stack sizes, is available at http://www.bol.ucla.edu/~sharnett/ICM/ICM.html.

Consequences of the ICM – the Bias Against Confrontation

Earlier we discussed the fact that in tournaments, a chip earned is always worth less than a chip already in your stack. One consequence of this is that there is a natural bias against putting any chips in the pot. In a cash game, to call an allin raise, a player only has to be the barest favourite to justify a call. In an SNG, we are going to need to be a more substantial favourite. Thanks to the ICM, we can calculate exactly how much of a favourite we need to be.

Suppose you are playing a SNG where everyone starts with 2,000 chips and the blinds are 10-20. You’re in the big blind the first hand and everyone folds to the small blind, who moves allin. He then accidentally exposes his cards to you. He has the AcKd. You have the 2h2d. Should you call?

If you fold, you’ll have just lost the blind and will have 1980 chips, almost your starting stack. The ICM values this at 9.91%, very slightly less than the 10.00% you started with.

If you call and lose, your equity is pretty obvious: zero. You’ll be out of the tournament.

If you call and win, you’ll have 4000 chips, while your remaining 8 opponents will each have 2000. The ICM values this stack at 18.44%.

Suppose that we wanted to know what our probability needed to be of winning the hand before it would be a breakeven call. Breakeven means that the equity of calling would equal the equity of folding.

Let:

E[call/win] = the equity of calling and winning
E[call/lose] = the equity of calling and losing
E[fold] = the equity of folding
P[win] = the probability of winning

For simplicity, we will ignore the possibility of a split pot. For most hands this minor factor can be safely ignored.

Now:

E[call/win] x P[win] + E[call/lose] x (1 – P[win]) = E[fold]

Since E[call/lose] in this hand is zero, we can ignore that term and end up with:

0.1844 x P[win] = 0.0991
P[win] = 53.74%

According to the ICM, we need to win 53.74% of the time to make this a call. Since 2h2d only beats AcKd 52.34% of the time and ties 0.31%, this would be a fold.

Furthermore, we can calculate exactly how much making this call would cost you. The equity of calling is:

P[win] x E[call/win] + P[tie] x E[call/tie]
= 0.5234 x 0.1844 + 0.0031 x 0.1000
= 0.0994

Subtracting this from E[fold] gives 0.18% of prize pool. In a \$100 buyin SNG, calling here costs \$1.80.

Since the player with the AcKd is a dog to us, he fairs much worse:

P[win] x E[call/win] + P[tie] x E[call/tie]
= 0.4730 x 0.1844 + 0.0031 x 0.1000
= 0.0875

This is a loss of 1.16%. This hand would cost him \$11.60 in a \$100 buyin SNG.

Alert readers might recall that the total amount of equity in a tournament is constant. But in this hand, both players involved in the hand have lost equity. Where has this equity gone?

The answer is that it is distributed evenly amongst all the other players in the tournament. In an SNG, every player has a stake in every hand. If you have ever been short stacked and heaved a sigh of relief at someone else busting on the bubble, you will be intuitively familiar with this idea. Even though you weren’t involved in the hand, it’s obvious that your equity in the tournament just took a big jump. Similar effects are taking place on every hand of the tournament, albeit usually in a much more minor way.

This net equity loss isn’t limited to hands where the participants are allin. Suppose two players have a confrontation on the first hand of a 2000-chip SNG where they are both exactly 50% to either win or lose 500 chips. Their possible equities after the hand are:

2500 chips: 0.1223
1500 chips: 0.0767

Averaging these gives 0.0995, so each player has lost (on average) 0.05% of the prize pool, or about 50 cents at a \$100 buyin tournament. That money has been redistributed amongst all the other players in the SNG. Having another identical confrontation just makes the problem worse. There’s a 50% chance that they’ll both end up with 2000 chips again, returning everyone’s equity to 0.1. But the other 50% of the time, the stacks will become even more unbalanced:

3000 chips: 0.1438
1000 chips: 0.0524

For the player who started with 2500 chips, his average equity after this hand is 12.19%, a further loss of 0.04%. The player with 1500 chips gets an average equity after the hand of 7.62%, a further loss of 0.05%. Once again, the equity is distributed amongst players not involved in the hand.

Of course, we could alter the victory percentages so that there is a net gain in equity by the two players. For instance, if the short stack was 100% to win the second confrontation, the outcome of the first hand would be reversed, leading to a net equity gain. But then, if the big stack was 100% to win, the equity loss would be proportionally larger again. Over the course of many hands, all things being equal, win percentages will average towards 50%.

This brings us to the theorem of bias against confrontation:

In an SNG, confrontations between players result, on average, in a net loss of equity from those players and a net gain in equity by players not involved in the hand.

To get involved in a hand, you must first be sure that you stand to gain enough chips to overcome the bias against confrontation. The more chips you commit to the pot, the worse the loss of equity will be.

So far, our examples of bias against confrontation have been mild. Before you start to think that the concept is a technical one which doesn’t matter in the real world, here is a hand offering a more extreme example. This hand was taken from actual play in a \$100+9 SNG.

Cutoff: 460
Button: 550
Hero (SB): 13650BB: 5340
Blinds were 300/600. Cutoff and button both folded and our hero raised allin. His hand is not important since he should make this raise with any hand. His opponent in the big blind called with pocket twos. Here are the ICM equities for all possible outcomes:

 Opponent foldsCutoff: 0.1063Button: 0.1267Me (SB): 0.4393BB: 0.3278 Opponent calls and losesCutoff: 0.2502Button: 0.2600Me (SB): 0.4898BB: 0.0000 Opponent calls and winsCutoff: 0.1032Button: 0.1232Me (SB): 0.3733BB: 0.4003 Opponent calls and we tieCutoff: 0.1053Button: 0.1256Me (SB): 0.4328BB: 0.3362

Versus a random hand, our hero’s opponent stands to win 49.39% of the time and tie 1.9% of the time. This gives these weighted average equities for calling:

Cutoff: 0.1749
Button: 0.1899
Me (SB): 0.4312
BB: 0.2041

By calling instead of folding, our hero’s opponent lost a massive 12.37% of prize pool. This represents \$123.70 in a \$100+9 SNG. He lost much more than the tournament buyin in a single hand!

Unfortunately, our hero didn’t benefit from this generosity; he lost substantially as well, although much less than his opponent did. The benefactors were the two short stacks, who were both handed huge amounts of equity on a plate.

In SNGs it is quite common for poor decisions made by your opponents to cost you money. The flipside is that if your opponents don’t understand the bias against confrontation, then you will be able to sit back and watch equity get shovelled in your direction as your opponents needlessly clash with each other.

You will see throughout this book that the bias against confrontation is a strong influence in SNG strategy.

Limitations of the ICM

It’s important to realise that the ICM is just a tool for estimating tournament equities. It’s a very useful, very important tool, but the equities it gives out are still just estimates. A player’s true equity may differ from the ICM’s calculations. There are three reasons this might occur:

Skill equity

Hidden in the section about calculating ICM equities was an important sentence: “We assume that all players play with equal skill”. If a player’s skill is above average, then an extra amount of “skill equity” should be added to all equity estimates. The amount of this skill equity is highly variable. For example, if I win the hand above where my opponent calls with 22, then it is highly unlikely that my equity is much higher than the ICM’s 48.98%, since it’s impossible for me to do better than 50.00%, which is what first place pays. On the other hand, if my stack is very short relative to the blinds, I won’t have many decisions to make and my skills will be wasted.

To consider the effect of skill on your equity, think about how much your skill will come into play with various possible stack sizes. For example, going back to the AK versus 22 first hand all-in, obviously our skill will have no effect if our stack size is zero. Our skill equity is very unlikely to double with 4000 chips compared to 2000, because the effective stack size will still be 2000, and the blinds are small enough that 2000 is plenty to be able to play effectively anyway. So we will lose more skill equity going from 2000 to 0 than we will gain going from 2000 to 4000, biasing us even more against confrontation.

In general, if you are a good player, you should avoid allins which are breakeven or slightly positive according to the ICM, unless your stack is very small or very large compared to the size of the blinds. Your skill equity loss will make them into losing plays.

Manipulating blind equity loss

A player who is about to face the blinds has a lower equity than the ICM suggests he does, because he is about to be forced to put money in the pot. It is sometimes correct to make plays which will decrease the equity loss which results from taking the blinds. For example, consider the following situation:

Blinds 200/400

UTG (You): 1480
Button: 2840
SB: 2840
BB: 2840

Let’s assume that no matter what you do, you’ll be forced to fold the next two blinds to the buttons each hand, who will steal successfully. The ICM suggests your equity is 17.31%. But if the button for each hand steals successfully for the next three hands, the stacks will look like:

Cutoff: 2840
Button (You): 880
SB: 3040
BB: 3240

Your equity then will be 11.46%, for a loss of 5.85%. But if you steal successfully by pushing UTG for 1480, then the stacks will be:

Cutoff (You): 2080
Button: 2840
SB: 2640
BB: 2440

With an equity of 22.15%. Then after the buttons steal for the next two hands again:

Cutoff: 2840
Button (You): 1480
SB: 2440
BB: 3240

Where your equity is 17.42%, for a loss of 4.73%.

The upshot of this is that if you successfully steal UTG, then your equity loss from moving through the blinds will decrease from 5.85% to 4.73%. Another way to put this is that because the blinds are about to hit you, the larger stack is worth 1.12% more than the ICM suggests. This in turn suggests that if ICM-based calculations have you losing 1.12% by pushing a hand UTG, then the push is actually a breakeven play.

Of course, it isn’t that simple. You won’t always fold your hands in the blind and sometimes by waiting a hand or two, you’ll even get the other players to eliminate each other. The effect of decreasing blind equity loss is not easy to quantify, but it is certainly something which you should take into account when evaluating situations in which the blinds are a large percentage of your stack.

Fold equity

A corollary to the theorem of bias against confrontation is that having the capability to avoid confrontations increases your equity. If your stack is so small that you will be called anytime you go allin, then there’s no way to avoid a confrontation and the resulting equity loss.

Quantifying fold equity is again very difficult, but all experts agree there is a lot of value in maintaining a stack of around 3 to 4 big blinds. Any less than 3 and you will find it quite difficult to ever get the blinds to fold. A very rough guide would be to assign about 0.60% of equity to maintaining a stack that is capable of making other players fold.

The concepts of fold equity and decreased blind equity loss often combine to make the correct play with a very short stack in early position an allin raise, even though a naïve look at ICM equities would dictate a fold.

Conclusion

The ICM is a useful model and you should not ignore what it suggests unless you have a good reason for thinking that it is wrong. But it is only a model and from time to time there are factors which have been left out of the model that you will need to take into account.

THE EARLY GAME

Definition of the Early Game

The early game encompasses any hand in which effective stacks are 27BB or more.

General Strategy

The right strategy early in SNGs is a tight one. There are a couple of reasons for this:

- The bias against confrontation. Consider a 2000-chip SNG, with 20/40 starting blinds. Successfully stealing the blinds gains you 0.27% in equity. Risking big chip fluctuations can cost significantly more than that – for example, getting allin costs you 0.78% in equity if you are 50% to win. You will not successfully steal the blinds every time, so your expectation is much less than 0.27%. If your opponents routinely put up resistance, trying to steal the blinds with modest hands quickly becomes a losing proposition.
- Skill level. Remember that, in the early game at least, being a skilled player exacerbates the bias against confrontation. Note that it is your skill level relative to the other players that matters, which means that the worse they are, the less you should want to play marginal hands. This is especially true if your opponents are ignorant of late-game strategy.

In no-limit, small or medium pocket pairs are often played for set value – called preflop to look for a set on the flop, with the hope of winning a big pot. In SNG, the bias against confrontation means that we need better implied odds. You should never call just for set value for more than 1/15th of your stack. In the following sections “have set value” will be used as shorthand for “can see a flop for less than 1/15th of your stack”.

In cash games, you should often reraise with AK preflop. In early game SNGs, you should usually cold call raises. Why the difference?

In cash games, getting AK allin preflop against a pair (say queens) is a slight loser, but no big deal. In SNGs, thanks once again to the bias against confrontation, the loss is far more severe, so we much prefer not to end up allin preflop with AK. And the bias against confrontation doesn’t just apply to allins – it encourages us to avoid big pots in general.

Reraising AK also causes dominated hands like AJ, AQ and KQ to fold. In cash games, where stacks are generally deeper, there’s not as much to lose from this. Good players will be reticent about putting much of their stack in the middle without the top kicker to go with their top pair. You also stand to lose a lot of money should your opponent be fortunate enough to make two pair. In SNGs, where stacks are shallower, players are more likely to feel committed to going all the way if they flop top pair. Letting dominated hands see a flop can therefore be advantageous.

Postflop play

A discussion of postflop no-limit play is outside the scope of this book. The “Harrington on Holdem” series and “No Limit Holdem: Theory And Practice” by Sklansky and Miller both provide excellent discussions of postflop NL play.

A Simple Preflop Strategy

Playing Against Raises

The guidelines for playing against raises in the early game are identical for all positions.

If it has been raised before you, call with small and medium pairs if you have set value. Call with AK, JJ and TT, unless the raise is for more than about 1/10th of your stack, in which case reraise allin or fold. If the decision between calling and moving in is close, lean towards calling in position and reraising allin out of position.Reraise with QQ-AA, except against very tight players just call with QQ. Against perceptive, strong players, you will need to vary your strategy and sometimes reraise with hands other than QQ-AA.

If it has been raised and reraised before you, fold everything except KK and AA. Whether you call or reraise with those hands depends on the exact situation. Versus loose opponents, you can also play QQ and sometimes AK. JJ and TT will not be playable unless your opponents are bona fide maniacs.

If you raise and are reraised, against typical opponents you should call AK, reraise KK and AA, and either call or reraise QQ depending on the player. All the remaining pairs can be played if you have set value. Against particularly loose aggressive players you might want to play back at them with JJ and TT.

Late Position

When open raising, always try to steal the blinds with these hands: any pair, AT-AK, KT-KQ, QJs. Against tight opponents add these hands: A8s, A9s, K9s, QJs, QT, JT.

After limpers, raise these hands: 99-AA (also 88 if you wish), AJ-AK, KQ. Limp other pairs. You can also limp speculative hands (like suited Ax) behind several limpers if you wish. Against a single limper, raise hands like AT and small pairs.

Middle position

Open raise these hands: 77-AA, AJ-AK, KQ. Limp smaller pairs if you will still have set value if someone raises behind you.

After limpers, raise the same hands as in late position (99-AA (also 88 if you wish), AJ-AK, KQ) but don’t play any other hands.

Early position

Open raise these hands: TT-AA, AQ, AK. Limp smaller pairs if you will still have set value if someone raises behind you.

If someone has limped, guidelines are the same as for opening.

The guidelines for playing against raises and reraises are the same as for late position.

In a particularly loose aggressive game, you can choose to just limp TT and JJ. In a particularly loose game, you can limp AQ if the game is passive and fold it if the game is aggressive.

In the blinds

Against limpers, raise TT-AA, AK and perhaps AQ. If only the small blind has limped, raise a lot of your hands – somewhere around 50%.

To play against late position open raisers, you will need to put your opponent on a range of hands and decide what to do accordingly. There are no hard and fast rules since correct play in these circumstances varies a lot depending on your opponents’ play styles.

Summary

You should be tight. Experienced no-limit players might find the tightness of these recommendations surprising. But you are at a full table, often not particularly deep stacked, with the bias against confrontation – aided by your skill equity – to contend with. Stay solid and only get involved when you will have a clear advantage.

The recommendations above are just that, recommendations, and for maximum profitability you should tailor them to meet the game. Against very bad players you might be able to play significantly more hands, and against good players you will want to mix your strategy up a little to avoid becoming predictable.

THE LATE GAME

Definition of the Late Game

The late game encompasses any hand where effective stacks are 12BB or less and 4 or more players remain in the tournament.

General Strategy

The common thread running through all of SNG strategy is the bias against confrontation. In the early and middle games we achieved our aim of avoiding confrontations by simply not playing many hands. In the late game that is no longer possible or desirable. A player who sits there folding all his hands will be rapidly eaten up by the blinds. Additionally, even players largely ignorant of the bias against confrontation understand that it is bad for them to be stacking off when only 4 players remain and the next one out will miss the money. Many in fact play the late game too tight. Curiously, many SNG beginners play the game exactly back to front – splashing around in the early game, and then shutting up shop in the late game, when they need to aggressively steal blinds.

In the late game, the vast majority of the time you should be moving allin or folding. The confrontation against bias should be wielded as a weapon, in a strategy of brinksmanship. A lot of your SNG profits will come from aggressively stealing your opponents’ blinds in this phase, moving allin against them and forcing them to be the ones to back down.

The focus of this chapter is when to move in and when to fold, and what factors you should be looking at when making your decision. The first question to consider is how short a player’s stack should be before he is simply making a choice between open-raising allin and folding.

The 10BB Rule

The 10BB Rule is a rough guide to follow when considering open raising a hand. The rule says that if you are going to raise, then you should raise allin, rather than making a smaller raise, if effective stacks are 10BB or less.

The justification for this is that if you are flat called, then the pot will be roughly the size of your stack and the vast majority of the time you should be betting allin on the flop, trying to take the pot down. If you instead face a reraise, you will be getting at least around 2:1 pot odds, which is virtually always good enough to call. For instance, the matchup A2o versus 22+, A8o+,A6s+,KJ+,KTs,QJs might look like an ugly one, but actually the A2 is almost 35% to win. If you raised 3BB with a 10BB stack, you will be getting roughly the right odds to call.

This justification is given in terms of pot odds, and it’s true that in SNGs we are concerned with ICM equities of different outcomes, not with pot odds. However, this is not the end of the story. Since your aim in SNGs is to avoid confrontations, there is advantage to moving allin straightaway and not giving your opponent the illusion that he can make you fold. Usually this will significantly decrease the range of hands he plays. In general, the more ICM considerations come into play, giving you more incentive to fold a reraise, so too it becomes even more important to keep your opponents’ ranges tight and avoid confrontation if possible, making it more attractive to simply move in. The stack size at which you want to start moving in therefore stays reasonably constant.

Not Pushing With Less Than 10BB

There are two circumstances where you might decide not to push, even though you have less than 10BB:

If you have something like pocket aces, you may want to raise small to induce your opponents to call or reraise. This can be a useful play, but you should be careful not to overuse it. For one thing, you need to stay aware of what constitutes an excellent hand. In the right circumstances, even pocket kings can become a hand that you don’t want action on! Also, if there is a skilled opponent in the small blind who is aware that you are a knowledgable SNG player, and you (for example) minimum raise on the button with an 8BB stack, then you might as well be turning your hand face up. Of course, it might still be worth it if the player in the big blind is bad enough!

Your opponent’s range is both tight and invariant

If your opponent’s range is tight, then if you make a small raise, you might be correct to fold if reraised. However, it will still frequently be better to move allin because it forces your opponent to become even tighter. But if your opponent’s range is also reasonably invariant, meaning that he plays essentially the same range versus both a small raise and an allin raise, then all justification for moving allin disappears and you are better off with a small raise.

This may occur if one of your opponents is weak-tight and will not try a resteal against you even after he has seen you steal blinds with small raises several times. It may also occur versus good players. If you reach the bubble with a very short stack present, you might be correct to start making small raises (with all your raising hands) against other skilled opponents with shorter stacks than you. Your opponent mostly can’t take the risk that you have a hand that will call a reraise, since that would be disastrous for him, so he will play virtually the same range that he would play against a push, with the plus that you get to avoid taking a hit when he is dealt a big pair. However, if you don’t want your opponent to reraise and don’t think he will fully grasp the risk involved in doing so, then you would be better off pushing to make it clear to him that he will be playing for his stack.

Pushing With More Than 10BB

There are many hands that are strong enough to show a profit pushing for larger amounts than 10BB. The reason it’s not normally right to just move in with those hands is that they can be played even more profitably by raising small. But there are two situations in which that isn’t the case:

When the bias against confrontation is very large

Suppose you are on the bubble in this situation:

UTG: 2200
Button (You): 4800
SB: 6500
BB: 6500

Blinds are 200/400. UTG folds and you have AQ. Earlier in the tournament, it would be best to raise small, inviting a reraise from one of the larger stacks, hopefully with a weaker hand than yours. In this situation, though, the bias against confrontation is intense, and you just want to take the pot down without any fuss, so you should move in, hopefully forcing hands like small pairs to fold.

When you are out of position with an awkward hand

This situation commonly comes up when open-raising from the small blind with a hand like A8. With a stack of 12 BBs, if you raise small and are called then the hand will be awkward to play. Most of the time you will flop no pairs and be forced to choose between pushing allin and checking, neither choice being very attractive. It is better to negate the disadvantage of position by moving in preflop.

Similar situations occur with a good but not excellent hand in early position. If you are dealt AQ under the gun with more than about 5 players remaining, frequently your best option is to move in with anything up to 12 or 13 blinds. Raising small risks being reraised by a small or medium pair, or flat called by someone with position on you.

Introduction to Push/Fold Strategy

Suppose you are playing the bubble of an SNG – it’s 4 handed, everyone with 5000 chips. Blinds have risen to 300/600. You’re in the SB, your opponents are all skilled, and your opponent in the BB plays perfectly – making the most profitable play every time. The CO and button fold to you. What hands should you push with?

If you’re not familiar with late game strategy, it might surprise you to learn that the answer is any two cards.

We’ll start by looking at the hand from your opponent’s point of view. How much of a favourite does he need to be to call? Here are the ICM equities for each outcome:

Fold: 0.2300
Call/win: 0.3833
Call/lose: 0.0000

To break even we want the overall equity of calling to be equal to the equity of folding. Call the probability of winning P(win). Then:

P(win) x 0.3833 = 0.2279
P(win) = 0.2279/0.3833
P(win) = 60.01%

Now assume that your opponent knows you are going to push with any two cards. What hands are 60.01% or better favourites against a random hand?

Not many! These are the hands your opponent can call with:

55+, A9+, A7s+, KJ+, KTs+, QJs

Now let’s assume you actually have the hand that fares worst against this range, which is 72o. Versus your opponent’s range, you will win 26.17% of the time. Should you still push?

The ICM equities for each outcome are:

Push, not called: 0.2688
Push, called/lose: 0.0000
Push, called/win: 0.3833

If you have 72o, your opponent will be dealt one of the hands in his calling range 15% of the time. The overall equity of calling is:

(P(opponent dealt hand) x P(win) x 0.3833) + (P(opponent not dealt hand) x 0.2688)
= (0.15 x 0.2617 x 0.3833) + (0.85 x 0.2688)
= 24.35%

The ICM equity of folding is only 24.02%, so you should push with 72o and therefore with any hand.

You can see now how the bias against confrontation can be used as a weapon in the late game. Your opponent is well aware that you are a thief, but the strength of the bias against confrontation on the bubble renders him powerless. He can’t hurt you without also hurting himself.

Of course, most opponents don’t play ideally. If you have reason to believe that your opponent will call looser than the range given above, then you should push tighter. Most of the time, when calculating whether you should push with a hand, you will start at the point where we assumed you had 72o – that is, you will give your opponent a range of calling hands, and look at whether your hand should push.

The Stop-And-Go

The stop-and-go is a play made when facing a non-allin raise out of position in the blinds. The idea is to just call preflop and bet allin on any flop, hopefully increasing your chances of avoiding an allin confrontation. The prerequisites for making this play are:

Your hand must be good enough that you wish to go allin
Your hand should not be so good against the hands that your opponent will fold postflop that you lose from him folding.
There should be a significantly better chance of your opponent folding on the flop than preflop. (Normally the play is made when you think there is no chance of your opponent folding preflop).
You must be first to act postflop.

Rather than betting allin on literally any flop, it is better to check to trap your opponent if you flop a strong hand, such as a set, or sometimes top pair.

The stop-and-go is not very commonly the best play in SNGs. The reason is the difficulty of simultaneously satisfying conditions (1) and (2) above. The bias against confrontation means that your hand must be very good against your opponent’s range before you will want to go allin. If, after the flop, your opponent is so dismayed with his hand that he will fold to whatever bet you have left, then you are likely to be crushing him and would like him to call.

Nevertheless, the stop-and-go is sometimes the right play:

CO: 5800
Button: 5800
SB: 5800
BB (You): 2600

Blinds are 200/400. The CO folds and the button raises to 1000. He is, reasonably, not raising allin so that he can fold if SB reraises – or because he is trapping SB with a strong hand. Suppose you think his raise represents top 25% of hands (22+,A2+,KTo+,K8s+,QTs+) and that he has no intention of folding any of them if you reraise preflop.

You hold 66. If the button had moved allin preflop, you would have a call (+0.6% of equity). Reraising allin preflop is therefore better than folding. Is the stop-and-go even better?

If you call and bet your last 1600 into the 2200 pot on the flop, then if the button calls and you win, your equity will be 26.65%. If he calls and you lose, your equity will of course be zero. If he folds, your equity will be 20.82%.

Therefore, to break even:

P(win) x 26.65% = 20.82%
P(win) = 20.82/26.65
P(win) = 78.12%

You will benefit from the stop-and-go if, on average, you are less than a 78.12% favourite when you make your opponent fold. It is very implausible that you will be that much of a favourite. For example, if your opponent’s folding range on, say, an AK8 rainbow flop is only 22-77, then the stop and go is still the best choice, winning 1.4% in equity compared to moving in preflop whenever he was dealt a hand in that range.

Here are some tips for recognising good stop-and-go opportunities:

The stronger the bias against confrontation, the better. The profit in a stop-and-go comes from the bias against confrontation. If there was no equity gain in picking up the pot without a showdown, it would be a bad play because it gives your opponent the opportunity to see a flop before committing his chips. The stronger the bias against confrontation, the higher your profit will be. Hands on the bubble are likely candidates. Of course, the stronger the bias against confrontation, the less likely it is that you will be willing to go allin in the first place.

Pick hands where your opponent might fold a hand that beats you, and is unlikely to have hands which you are a huge favourite against. The best hands for stop-and-gos are small pairs, because sometimes you can make your opponent fold a hand which beats you (e.g. 7c 7h on an Ad Kd Jd flop) and since you will be checking sets, you’ll never be a big favourite when your opponent folds. Hands like KQ are also quite good, since you can get folds out of small pairs on flops like AJ9, and can check flops containing kings or queens to avoid making your opponent fold hands you are a huge favourite against. Hands such as strong aces (AT, AJ) and large pairs (TT, 99) are never candidates for stop-and-gos, since it is likely that many of the hands your opponents fold will have very few outs against you.

The less chips you have remaining on the flop, the better, as long as there is still a chance your opponent will fold. The downside to a stop-and-go is loss of value from doubling the rest of your chips when you are winning. The less chips you have remaining, the less this matters. In the example of your opponent holding 7c 7h on an Ad Kd Jd flop above, even a 500 chip bet into a 2000 pot will be sufficient to get many opponents to muck their hand in disgust.

Stop-and-gos are rare, and when they do arise, win very small amounts of equity because most of the time the result is identical to moving in preflop. The amount of discussion this play has generated online is way out of proportion to its value. If you are not already an advanced player, forget about stop-and-gos and concentrate on developing flawless basic skills. Once you have mastered the basics, it makes sense to add at least the risk-free stop-and-gos (such as the 22 example in the paragraph above) to your repertoire.