Middle ground

We have looked at the optimum strategy for the opening stages of a sit&go and concluded that unless you were very confident with your post-flop play, you should stick to playing premium hands.

Unsurprisingly, this tactic can’t be used throughout. By the time you get to the middle phase you are going to have to get more imaginative.

I identify the start of this phase as when the blinds reach 50/100, but that is based on a chip stack that has not veered too far in either direction from the 1,500 starting stack. Generally speaking, you’re going to have to change gears if the blinds total around 10% of your stack, and 50/100 blinds represent 10% of a 1,500 stack.

If you were on the losing end of a big pot previously and have only 600 chips left, you will need to play differently when the blinds are 25/50, rather than waiting for the 50/100 level.

The middle phase ends when the bubble bursts and only three players remain. Those three players will each then be guaranteed a payday, and play shifts to the late phase. Just as the best strategy in the early phase is to begin with very tight play and transition into more aggressive play towards the end, the best middle-phase strategy is also to transition gradually into more aggressive play.

This claim is at odds with conventional advice, and I will justify it below. First however, I must introduce a few key concepts.

EXPECTED VALUE

The first thing to understand is the nature of expected value (EV) in a sit&go. I have already introduced this to some extent in the discussion of the real value of a $10 chip when the early phase begins.

Chips are not actually worth the same as their chip value. Their actual value is relative to the equity they represent in the prizepool. The ramifications of this are that plays that would be +EV (i.e. have a positive expectation) in a cash game may not be in a sit&go and vice versa. How does one know when these cases exist?

Several years ago, some friends and I were discussing this distinction between chip value and real-money value and how it might impact decisions, particularly at the bubble. We decided that figuring out the mathematically correct decision on the spot would be impossible, but were unsatisfied with simply relying on the notions of pot odds and implied odds that you’d use in a cash game.

These obviously mattered, but the real-money ramifications were difficult to properly quantify. At the time, we decided to rely on educated guesses, which was also unsatisfactory but the best we could manage. As it turns out, the maths is indeed complex and advanced, but is represented in what is called an Independent Chip Model (ICM – covered in more detail in last issue’s ‘Robot Wars’ feature).

The maths behind ICM can get very technical as the number of players increases, but it can be explained sufficiently to the layperson. Put simply, your chip stack represents your equity in the prizepool. Assuming there is equal skill among the remaining opponents, your share of the remaining chips equals the probability that you will win the first- place prize money.

You also have equity in the second- place prize money, but determining this requires a regressive analysis in which the probability of each of the other opponents winning first place is calculated, and then on each scenario calculating the probability that you will win second place.

This would be repeated for third place. If there are five players remaining, you can see how it would be impossible to run through all of these calculations in the time you have to make a decision.

Determining all this even now under no pressure would be rather tedious and unlikely to matter much, except to maths nerds like myself.

EQUITY MANAGEMENT

As such, I suggest we simplify things and present a heads-up blinds confrontation for a clear example. I’ll assume we have a standard payout structure of 50% for first, 30% for second and 20% for third.

Let’s say you have a chip stack of 2,000, and your opponents have 1,000 on the button and 12,000 on the big blind. The blinds are at 300/600, the button has folded and you have 7-5 offsuit on the small blind. You estimate that the big blind will call a push with the top 40% of her hands. Now, if we were playing a cash game, we might reason as follows regarding a push. We have a 60% chance of winning $900 but a 40% chance of being called and winning 4,000 or losing 2,000.

Against such a wide calling range, we will be called by a lot of hands that are only slight favourites. In fact, in a cash game, or simply from a chip-equity perspective, pushing with any two cards for these values has a positive expectation. Yet very few sit&go players would push in, because they see that the button has only 1,000 chips left and will likely be eliminated quite soon with 900 chips in blinds to pay on the next two hands.

This insight is also taken into consideration by ICM, which would recommend a fold in this situation. Thus, a +cEV (chip expected value) does not entail a +$EV (real-money expected value). In this case, the value of gaining those chips from pushing does not outweigh the risk of lost equity in the prizepool. While it is important to play for first-place prize money, you still want to pick spots where your overall real-money expectations are positive.

Now, let’s take a look at a slightly modified example that would be part of middle-phase play. We now have four players left. Your cards and stack and the button’s are unchanged, but the cut-off position has a 2,000 stack, while the big blind has a 10,000 stack. Let’s assume it’s been folded to you, but the blinds are now 400/800.

Again, the all-in play is +cEV, but surprisingly it’s -$EV. Many players will push in this spot because the blinds represent enormous percentages of their remaining stacks, but this would be a mistake.

What if we change the hand to K-6 offsuit? Many players will fold all but the biggest hands here, hoping to creep into the money. The button is desperate and may very well push in and bust. The cut-off is also in bad shape.

Nevertheless, pushing with K-6 offsuit is +$EV, and you are passing up a profitable situation if you fold. Folding may get you into the money, but with little chance at first place and 50% of the prizepool. The case would be even stronger if the cut- off’s chip stack were something like 4,500 and the big blind’s 7,500, with you and the button having the same amounts as before. This is an important result.

BREAKING CONVENTION

Conventional tournament wisdom argues that you should play in two stages. First survive to the payout, then play for first. There is no money for fourth place, hence finishing out of the money represents a waste of your time and buy-in. Even if you crawl into third place, you make something, and something is better than nothing.

I am partial to conventional wisdom, I must confess, but the maths is the maths, and +$EV is the deciding factor. Consider the following scenarios. In the first you play the conventional strategy and finish in the money 50% of the time (this is a good figure, by the way), but you creep in there with little chance at better than third place most of the time, so of those 50% finishes, you place third 80% of the time and first 20%.

In other words, for every ten sit&gos, you place third four times and first once. Now let’s say you are playing $10+$1 ten-person sit&gos with a standard payout structure of $50, $30, $20. You will invest $110 for every ten sit&gos and will win $130 (=50+20+20+20+20), thus resulting in a $20 profit.

Now consider a second scenario in which you play more aggressively in the middle phase of play, according to ICM. Let’s assume that you finish in the money 35% of the time or 3.5 times for every 10. You win two of those and the remaining finishes are split evenly between second and third.

Over ten sit&gos you can now expect to win $135 (=50+50+20+15) for a profit of $25. Thus, even though you finish in the money 15% less often, you can still earn more money, because first place is worth so much more than third.

In other words, both strategies are profitable, but the aggressive path is definitely more so.