On the first hand of a heads up sit and go, your opponent moves all-in. You estimate your hand to have 50% equity against this player’s range. Should the fact that this would result in an expected loss for the tournament after the rake is accounted for nudge you into folding?

No, it shouldn’t. The rake (and indeed the buyin) is a sunk cost. Sunk costs should not be factored into rational decision making. There is a prize pool and that’s all you’re interested in. All either things being equal you should want the highest probability of winning it. If that happens to be 50%, you should take it.

But wait, you surely couldn’t apply this overall? If your long run winrate was 50% you’d eventually go bust due to the rake. So what’s the difference? The rake in future SNGs are not sunk costs. Since they have yet to be incurred, you can account for them in your decision making.

So how do you do so? Getting back to basics: your goal in poker is to make as much money as you can. (You can certainly have either objectives, such as fun or playing better opponents for the sake of learning in order to hopefully make more in the future, but those are for you to evaluate). There is a finite amount of time that you can play poker, so maximising your hourly expectation will maximise the amount of money you make. (Note that there is nothing special about the use of hours, it’s just the standard way of measuring earnings over time).

**Every decision you make in a SNG should be evaluated on its effect on your hourly expectation.** Rake matters because it reduces your expectation on future SNGs. Notably, it matters in the exact same way as you playing a particularly fishy opponent. In both cases, you have a higher expectation from trying to keep the current tournament going than you than you would starting a new one against a random opponent. Thus, I will evaluate both of them together.

Let’s start with a simple example. When should you chop a HUSNG? For this example, we’ll have a $200 + 10 SNG, assume an even chop and ignore rakeback. (In reality, you must account for rakeback and any other bonuses arising from playing as fully as you would anything else. Note that the value of a VPP is not the average benefit of its effect on your VIP status, but its marginal benefit. If you are planning to play a lot of poker anyway, you can’t count it for much. If you’re struggling against the clock to rack up VPPs, you might rate a VPP quite highly).

What is your opportunity cost for playing the SNG to completion? It is your hourly rate for one table divided by the expected time the SNG will take. Note that it is your expectation for games played now, not “overall”. If it is a fishy time of the day and you’re playing well, your expectation will be higher than an off-peak time in which you’re on your C game.

Let’s suppose for divisibility’s sake that your hourly rate per table is $60 an hour. If the expected time for the SNG is 15 minutes, by playing the SNG you are foregoing $15 in expected profits and must make at least that in the current one to justify continuing playing.

Currently, you are being offered $200 to end the SNG right now. Therefore, you must expect to win at least $215 by playing the game out. This equates to a 53.75% winrate. At anything less, you should chop. Interestingly enough, this winrate is large enough to beat the rake. Even someone +EV to play against after the rake can be still be so far from being the most +EV that it’s worth taking even money to free yourself up to play someone else. In fact, the rake in this example wasn’t important. It was your hourly rate, which all the rake did was presumably to reduce it. Rake here isn’t distinguishable from any other characteristics that might serve to change your hourly rate.

(Uneven chops are possible if both players acknowledge a reasonable skill gap existing. However if done properly they would result in players only concerned about money chopping 100% of the time. Poker is mutually a waste of time for two people only interested in money, since any money exchanges must be at the expense of the other. Both players could be made better off if they proposed an uneven chop.

The fact that games are not chopped show that at least one player overestimates his/her edge and/or they are not playing for direct monetary reasons (which are outside the scope of this post)).

Let’s continue to evaluating specific implications of this to poker hands. Since the chopping example already highlighted both players getting all-in, I’ll move straight ahead to uneven stacks.

Suppose the pot is 200 chips, you have 900 chips, your opponent has 1900 chips and has just moved all-in. How much equity does your hand need to have against his or her range in order to justify calling?

You’re looking for what has the highest expected value over time. To calculate this you need to determine the probability of winning the SNG under each distribution of chips that could arise minus the loss of EV that occurs from the time it would take to play out the rest of the tournament.

**Expected value of calling:**

**((Probability of winning after calling and winning * Prize Pool)-(Hourly Rate * Expected time in hours for SNG to take after calling and winning))*Probability of winning when called + ((Probability of winning after calling and losing* Prize Pool)-(Hourly Rate * Expected time in hours for SNG to take after calling and losing))*Probability of losing when called**

**Expected value of folding:**

**(Probability of winning after folding * Prize Pool)-(Hourly Rate * Expected time in hours for SNG to take after folding)**

(I’ve ignored the possibility of a chopped pot for simplicity of calculations but you can calculate it in the exact same format).

So let’s plug some numbers in. Assume you’re playing a $200 + 10 SNG, with an hourly rate of $60 an hour 1-tabling. If you call and win, you’ll have 2000 chips and figure you’ll win according to your chip equity, 66.67% of the time, with an average game length of 10 minutes. If you call and lose, you’ll win 0% of the time in no time (after the hand is finished). If you fold, you’ll have 900 chips and figure you’ll win 30% of the time of the time with a slightly smaller game time (due to a smaller stack) of 9 minutes. So how often do you need to win when you call?

((0.67*400)-(60*(10/60)))*PrW + 0 = (0.30*400)-(60*(9/60))

(266.67 – 10)*PrW = 120.00 – 9 256.67*PrW = 111

PrW = 0.43246

This is less than the 0.45 equity you would need in a standard chip equity calculation. Despite the fact that you’re not playing against a player who has an edge against you, it’s still worth getting the money in somewhat lighter.

You could plug numbers into the above equations, altering them as you wish to describe different scenarios (like the equity of betting yourself and getting called) and get the right answers. However, it may be somewhat difficult to estimate your win percentage at given stack sizes and the expected time to finish a SNG. One way to do so is empirically – if you have a large enough database of SNGs, you could go through all of them and find out your win percentage/time taken at each stack size against opponents of various skill levels. This is probably unrealistic, however.

One method is using the “Theory of Doubling Up” from the Mathematics of Poker, Chapter 26. If you can estimate what your overall winrate will be against a player, you can estimate your equity for any stack size. If E is the probability of a player winning the tournament, N is the number of times that player will need to double up to win and C is the constant probability of that player doubling up, you have the relationship:

E = C^N

Suppose you know your E to be 55% for an entire SNG against a given player. What is your equity when you have 20% of the chips?

For a full SNG, N is 1, since you have to double up once to win.

0.55 = C^1. C = 0.55.

Note that N for 40% of the chips is not 2.5. To work out N, take a log to the base 2 of the number of times you will need to multiply your stack to win.

N = log2 2.5

= 1.321928

Plug these values into the equation to solve for your equity:

E = 0.55^1.321928 E = 0.45371.

So this model projects that with 40% of the chips, you will win 45.371% of the time against someone that you would beat 55% of the time with 50% of the chips. Note that both proportionally and absolutely this is a higher return. However, it is not so simple to infer that this means you should be willing to take slightly the worst of it in order to get to this spot. This does not take account the effect on time. When you win a SNG from 40% of the chips, it should on average take longer than winning one from 50% of the chips – there is more ground to make up. If you are the big stack, the gap in equity will not be so large that as a short stack, but the average game length will be smaller as you are more likely to eliminate your opponent than vice versa. I am not sure exactly how to quantify this in a model: if someone else wants to propose one, I‟m open to it.

The problem with this model is that it assumes a constant probability of a double up. This is almost certainly not the case. At low stack sizes, the blinds are larger relative to the effective stacks, so it‟s more likely that your chip equity will be closer to your tournament equity at some levels. However, tournaments go faster whenever blinds are relatively higher. This is the major reason for playing turbos and indeed HUSNGs in general (compared to cash). Both of these factors will need to be accounted for.

Nonetheless, we can still infer a number of concepts:

– All other things being equal, against players that you have a positive overall winrate against, you should pass up slightly +EV spots in order to avoid stack sizes from becoming too unequal. Conversely, against players that you are an underdog to, you should be more willing to get into spots with shorter stacks all other things being equal.

You can see the effect of this by plugging small and large numbers in the equation above. With 10% of the chips, your equity is 13.72%, with 90% it is 91.31%. These numbers are not so useful because they include the higher equity that occurs after double-ups have occurred and stacks become larger. It is perhaps better understood intuitively: it is bad to play with effectively small stack because your potential upside is limited to the size of the short stack. Now, a small stack can be accompanied by factors that you might find desirable (such as higher blinds relative to the stacks, which you may find to increase your hourly rate), but that can be modeled separately. Controlling for the blind/stack ratio shows it clearly: you would rather have 1500 chip stacks and 75/150 blinds for someone you have an edge against than a 50% chance of 2700/300 or 300/2700 stacks and 15/30 blinds.

– All other things being equal, you should be willing to make slightly –EV decisions if they result in net favourable blind/stack ratios. Conversely, you should pass up slightly +EV decisions that will result in unfavourable such ratios. The type of tournament you play is evidence as to what this may be: if you play turbos, you‟re presumably doing so because you think your hourly rate will be higher with increased blinds, and the opposite if you play regular speeds. However this is only true on average.

There will be some opponents whom you would have a highly hour rate playing deep stacked, others you would have a higher hourly rate playing short stacked. When combined with the absolute stack size effect above, you have a way to calculate your tournament equity from the chip distributions that arise from a decision.

– When you’re attempting to estimate your probability of winning and/or the expected time the SNG will take, you need to do so taking into account that you will also be making these calculations in the future. You cannot, for instance, reason that “The average length of a SNG that I’ve played against a winning player is 10 minutes. Therefore, the expected length of this SNG is 10 minutes.” This is incorrect because the fact that you’re willing to get the money in slightly bad later will reduce this length. Additionally, you should consider whether your opponent might be aware of these concepts (and also if he or she believes that you are, too). Two winning players who are 50/50 against each other should both be willing to try to shorten game lengths by making slightly -EV raises and calls. But if both are aware of these concepts, they’ll shove slightly lighter for value and call slightly wider against their opponent’s new lighter shoving range. It’s what you might call “implicit chopping.” This should tend to an equilibrium, as the more frequent all-in situations make the SNGs shorter and thus reduce the value of getting the money in wider in the first place.

– Nichlemn