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Preflop raising decisions can be very complex. What stakes are we playing? Is it a live game or online? How strong is our hand? Which limpers will call? Will we be in position postflop?
Sometimes our decision is obvious: 'I have aces, so I'm going to raise for value here.' Sometimes it's subjective: 'I think I can force the limper to fold here.' But we should also consider what a mathematical analysis of preflop raising can teach us. When is a raise our most profitable action for the moment? When might it be the most profitable action for the hand?
Las Vegas $1/$2 NLH Stats
In order to calculate the expected value for our preflop actions accurately, we need to know certain key frequencies that are typical for our game.
Figure 1 below contains some of the most useful stats for these EV calculations for both online NL100 and $1/$2 NL live games in Las Vegas, most of them published here for the first time. (Check out Donkey Poker Volume 2: Postflop, Section 7.5 to see how these stats were generated.)
Here I define 'EP/MP' as the first four seats after the big blind on a nine-seat table. Since the vast majority of Vegas $1/$2 players are position-dumb, we can combine the results from these seats into a single stat in order to make the stat more reliable. The hijack and cutoff stats are similar to the EP/MP stats.
VPIP and PFR are the primary stats useful to determine an opponent's playing style. '35/6' means that the average Vegas player voluntarily invests in 35 percent of his dealt hands, while raising only 6 percent of them. This player would be considered a 'loose-passive' player in an online game, but he is average in a Vegas $1/$2 game.
Calling a Preflop Raise (CPFR) is a statistic that applies only when a player calls a raise after having already invested in the pot, which applies to limp-callers as well as to the blinds. This is not the same thing as Cold Calling Preflop (CCPF).
Compared to an online game, a Vegas preflop raise is fraught with more risk. Not only is our average raise size larger, our Vegas opponents are more likely to call it. Although their calling range is weaker, we often face multiple callers, which is rare online. Consequently, it is extremely important to have a solid idea of how these stats influence the profitability of our preflop action.
A Simple Example: A Big Blind Hero vs. a Single EP/MP Limper
Let's consider a situation which is relatively easy to calculate. Suppose an average player limps from one of the EP/MP positions. It folds around to Hero in the big blind who can check his option or raise.
Figure 1 indicates that that Mr. Average has a range of 35 percent, but he normally raises with 6.1 percent of this range. So we can assign his limping range as [6.1–35]. The specific card combos within his range depend on the ranking system we use. Although there is no single ranking system that is always correct, I will use the Flopzilla NLHE ranking in this column, since it is easy for anyone to access.
Suppose Hero has and decides to check. We can estimate Hero's 'Momentary Expected Value' (MEV) as...
MEVCheck = Pot0 × EWC = $5 × 0.474 = $2.37
...where Pot0 = $5 and EWC (Showdown Equity When Called) is 47.4 percent for Hero's pocket deuces when facing Villain's range. I call this his 'momentary' EV since it applies only at the moment he checks.
Suppose Hero decides to raise to $10 with the same . Figure 1 shows that an average villain will call a preflop raise after limping in EP/MP about 56 percent of the time (Row 8). This would be a Flopzilla range of [6.1–22.7].
Hero's Fold Equity is FE = (1 - 0.564) = 0.436. His showdown Equity-When-Called is EWC = 0.454. Notice that even though the villain has folded nearly half of his range, Hero's EWC is only slightly worse than when he checks. This suggests that the villain's actual limping and calling ranges are not critical in establishing Hero's MEV.
After deciding to raise, we can calculate Hero's MEV as...
MEVRaise-$8 = (FE × Pot0) + ((1-FE) × (EWC × Pot1 - Raise)) = $2.27 + $0.28 = $2.55
The first term is just the average amount earned when the villain folds. The second term is the amount earned when the villain calls. ' Pot1' is the size of the pot after he calls, minus the rake. 'Raise' is the size of our raise ($8 in this case). Since three-betting is so rare in Vegas $1/$2, and limp-three-betting is even rarer, we can ignore it for this calculation. Again, we consider this our momentary EV since it only applies at the end of the preflop action.
So Hero's raise appears to be more profitable (by $0.18) than checking. It gains nearly all of its profitability from fold equity and a just little from showdown equity. This is generally the case for heads-up battles.
However, there is a serious issue with these estimates. MEV uses showdown equity, which essentially assumes that the hand will be checked down postflop. This rarely happens in a real game, which makes its usefulness imperfect. We haven't considered such factors as skill and position, which can increase or decrease our expected profit in the hand. Hero's inferior position in this scenario suggests that his actual 'Hand EV' (HEV) will be lower than his calculated MEV. Even Hero's superior skill will not likely overcome his inferior position.
Since Hero's inferior postflop position makes it difficult to realize the full MEV of either action, it's difficult to say which action would have the higher HEV. Hero's best decision could depend on whether the villain appears more or less likely than the average villain to fold to Hero's raise. We might also consider that if both actions result in a similar HEV, the check does so with less risk.
Figure 2 shows the results of these calculations applied to some of the hands listed in the Donkey Poker starting hands chart. The solid symbols represent Hero's MEV when raising to $10, which is positive for each of these combos. 'JTs-' denotes 'suited connectors JTs and smaller.' Meanwhile the '+' symbols represent the difference in MEV between raising and checking.
When MEV is positive, raising is more profitable than checking, which is true for a range including approximately 22+, A2+, KT+, and QJ (i.e., any pocket pair, any ace-x hand, any two Broadway cards). This represents 27.6 percent of all starting hands.
Since we are out of position in the scenario, our HEV is probably lower than these values, so checking is probably preferable when the Delta MEV is close to zero.
Button Hero vs. One Limper
The previous scenario is rare in Vegas $1/$2 games, but it serves to illustrate the principals used for our analytical EV calculations. A more common and useful example is when the Hero is on the button with and facing a single EP/MP limper. In this case Hero can fold, limp, or raise, where folding has an EVfold = 0.
This calculation has additional complexity because we have the blinds behind us, which means we must consider their propensity to check behind or complete and their propensity to call a raise. (We will ignore their negligible three-betting frequency here.)
Suppose Hero limps and the two blinds raise 4.8 percent and 3.4 percent of the time, respectively. If the small blind completes with [4.8-50] and the big blind has a range of [3.4-100], we will have a four-way pot about 92 percent of the time.
Let's suppose the other 8 percent of the time Hero limp-folds after one of the blinds raises. (This is plausible for those hands with which Hero would actually limp.) We can also assume that the EP/MP limper has a range of [6.1-35.7]. All of these values are based on the actual Vegas stats listed in Figure 1.
We then have...
MEVLimp = 0.92 × (Pot0 × EWC-Limp) - (0.08 × Limp) = 0.92 × ($8 × 0.222 - $2) - (0.08 × $2) = -$0.38
In other words, limping on the button with is not immediately profitable.
Suppose we raise to $10 with on the button. Using the above frequencies, everyone will fold 23.1 percent of the time. A single villain will call 47.0 percent of the time, two villains will call 25.7 percent of the time, and everyone will call 4.2 percent of the time.
It turns out our EWC is nearly the same no matter which villain calls, so we can treat each villain as interchangeable. Thus, Hero's MEV is approximately...
MEVRaise = (0.231 × Pot0) + (0.47 × (EWC1 × Pot1 - Raise)) + (0.257 × (EWC2 × Pot2 - Raise)) + (0.042 × (EWC3 × Pot3 - Raise)) = $0.63
...where the various values depend on how many players called our raise. We can see that raising with our pocket deuces is much more immediately profitable than limping with them. Furthermore, our Hand EV should be better than this since we have superior position and skill and since we can leverage our profit when we flop a set.
Note that when the blinds are tight and/or passive, they are less likely to call our raise and less likely to three-bet. (Pre-loading tells are very important here.) In these situations, raising a single limper from the button is even more +MEV. And our HEV is even more enhanced since we are less likely to face multiple villains in a raised pot. (It's generally easier to outplay a single villain than three of them.)
Figure 3 depicts the MEV results for various hand types. The symbols represent the MEV for raising to $10 in a Vegas $1/$2 game. Each series represents a descending hand grouping. For example, 'T9s-' denotes 'suited connectors T9s and smaller.'
Bottom line: in low-stakes cash games such as the $1/$2 NL games in Vegas, raising is nearly always more profitable than limping (except for 63s, 53s, 65o and 54o). In fact, 33 percent of all starting combos are immediately profitable. And some of the negative-MEV combos are probably profitable for the hand due to our superior position.
Since most Vegas $1/$2 players are position-dumb, they generally limp and call with a similar frequency from the hijack and cutoff positions. So our very wide button raising range is valid for any single-limper situation when we are on the button.
Button Hero vs. Multiple Limpers
The previous single-limper scenario occurs only a small fraction of the time in Vegas $1/$2 games. We can readily extend this to multiple-limper scenarios, which are much more common. This calculation is even more complex and we are less likely to fold everyone out with a $10 button raise. This time I will spare you the gory math and show you the results below in Figure 4.
Here the top four curves represent the MEV for raising to $10 facing one, two, three, and four limpers. The smooth line represents the average MEV of limping. These curves show that our 'immediately profitable' raising range is 33.6 percent when facing a single limper, and that our range decreases to 13.4 percent when facing four limpers.
We can make several conclusions here:
- The more limpers we face, the tighter our raising range should be from the button.
- When facing one or two limpers, we should generally either raise or fold, since once raising becomes unprofitable, limping is even more unprofitable.
- When facing three or more limpers, limping is generally more profitable than raising once raising becomes unprofitable. So we now have both raising and limping ranges.
- When both raising and limping have MEVs near zero, limping may be preferable since we risk less for the same reward.
- Our slightly negative-MEV combos may still have slightly positive-EV for the hand. Our superior skill and position, coupled with our ability to leverage our big flops should increase our profit for the hand. This is combo-dependent since some combos are more likely to flop big. (22 is stronger than 54o.)
- Another way to state this is that some -MEV combos may have sufficient implied odds play. For instance, we can play a -$0.50 MEV combo if we believe our implied EV for the hand is worth more than this.
The Effect of Raise Size
A typical rule of thumb for these Vegas $1/$2 games is to raise to about 4x BB plus one additional BB for each limper. Thus we would raise to $10, $12, $14, or $16 as the number of limpers increases. The previous analysis kept the raise size constant to see the influence of the number of limpers.
A key question we would like answered is how the villains will respond to reasonable variations in raise size. Unfortunately, that data does not exist for these Vegas games. My general feeling, however, is that most Vegas $1/$2 players are not very sensitive to our raise size as long as we keep it within the table norm. So, to a first approximation, we can assume that the Figure 1 stats do not change very much when we increase our raise size incrementally.
Figure 5 shows what happens to MEVRaise as we increase the size of our raise when facing four limpers. We can see that that our raising range decreases slightly as we increase our raise size. This effect is fairly small and may be counterbalanced by a decreased likelihood that a villain will call a larger raise.
Here the symbols represent the MEV for raising to $10, $12, $14 or $16 in a Vegas $1/$2 game. These combos are sorted in order of MEVRaise.
The important thing to notice in this graph is that we have a much larger MEV when we raise our best hands by the maximum amount that will not result in a villain adjustment. If we have aces and the villains will call a $16 bet as often as a $10 bet, we should bet $16.
On the other hand, we should consider betting smaller with our borderline raising hands. Of course, this sets up the possibility that we could be telegraphing our hand strength. But most Vegas $1/$2 players are not paying much attention to our bet size unless the size is unusual for the table. Here we can make adjustments based on our knowledge of the players at our table.
Hero is in the Cutoff
Putting the Hero in the cutoff makes the analysis even more complex. Making some reasonable approximations, I can make a few generalizations:
- All players, including the button, are very unlikely to three-bet, so we should generally ignore this possibility when deciding to raise. (If our particular button opponent likes to three-bet, we should consider changing seats.)
- The button is less likely to cold call our cutoff raise than an EP/MP player is to call after a limp.
- Being in the cutoff facing three limpers is similar to being on the button facing four limpers. This is equivalent to the button replacing the fourth limper.
- The average button will call our cutoff raise about 22 percent of the time. We will then not have a postflop position advantage on every villain. This means that our average HEV will not be as enhanced from the cutoff as it would be from the button, and thus our cutoff raising range should be somewhat tighter that it would be from the button with one additional limper.
- If we have a tell that suggests that the button intends to fold, we can play the cutoff exactly as we would normally play the button.
Conclusions
We can't yet decide on a specific hand range for each scenario since we have not yet determined just how valuable our superior position and skill is. Nevertheless, it is clear that the more limpers we face, the tighter our raising raise should be.
Also, our raising range should be much wider than the typical Vegas player. We can also play from the cutoff facing X limpers with a similar range as from the button when facing X+1 limpers.
And of course, we should always adjust our ranges depending on the tendencies of the specific villains we face and based on specific tells we observe.
Steve Selbrede has been playing poker for 20 years and writing about it since 2012. He is the author of five books, The Statistics of Poker, Beat the Donks, Donkey Poker Volume 1: Preflop, Donkey Poker Volume 2: Postflop, and Donkey Poker Volume 3: Hand Reading.
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Quick, in your head, think of the top three differences between online no-limit hold’em and live no-limit hold’em strategy.
I’ll tell you mine. In a live game (compared to online), I:
Play more hands
Bet more flops and turns
Overbet more frequently
I do these three things because, in general, I can expect my live game to have one or more opponents that are truly terrible. (In fact, if a live game doesn’t meet that standard, I typically find a different one.)
Now here’s a different question. What are the intrinsic strategic differences between live no-limit and online. That is, what strategic differences should you make because of the format of the game, rather than the players at the table?
I’d argue there are almost no intrinsic differences. Online games tend to play a little shallower on average than live games, so that’s a difference. But it’s not intrinsic to either format, as you can have deep stacks online and shallow stacks live. The rake structure online is often slightly more forgiving, but I don’t consider this a major difference.
Online games are often played six-max, whereas live games are nearly always nine-handed or ten-handed. Ok, that’s another difference, but there are nine-handed online games too.
“Correct strategy” (whatever that term may mean to you) in a nine-handed online $5-$10 game played with $1,000 stacks is nearly identical to that in a nine-handed live $5-$10 game played with $1,000 stacks. And yet, if you play the two games, you’ll see stark differences in how people play. Indeed everyone plays differently in the live game, even the good players.
The live game deviations most players make are, strictly speaking, mistakes as compared to a more theoretically sound strategy. This gives you a window to gain an edge, even on some pretty good live players. Here are three mistakes good live no-limit players at the $2-$5 and $5-$10 levels make.
1. They open too many hands.
This one is nearly universal. Theoretically speaking, no-limit hold’em is a fight for the blind money. When you’re under-the-gun (UTG) in a nine-handed game, how frequently do you think you should be trying to attack the blind money?
Poker Strategy Charts
There are seven players outside the blinds. If we were to assume that each of these players would attack the blind money an equal percentage of the time, that would have you opening about 14 percent of the time.
All is, however, not equal. Position matters. When you’re first to speak, you run the risk of any of six non-blind players behind you waking up with a hand. Furthermore, if one does wake up with a hand, you will have to play out of position postflop. Therefore, the UTG player clearly should not be opening 14 percent of hands. Perhaps 10 percent is a better estimate.
The worst hands in a range that includes the best 14 percent of all preflop hands is roughly Q-10 suited or K-Q offsuit or 7-6 suited or 2-2. Theoretically speaking, all four of these hands are likely too weak to open from UTG, yet most live players would open every one of them.
Furthermore, many live players would open to four or five times the big blind (for example, $40 or $50 in a $5-$10 game). This is just plain too much money with too many hands.
This over-looseness carries through nearly all of the early and middle positions. Good live no-limit players play too many hands from the first six or so positions.
If you merely play correctly tight preflop ranges against these players, you will automatically exploit their looseness. (Though it’s possible to play so poorly after the flop that you give back your advantage.)
2. They reraise too many hands preflop from outside the blinds.
It used to be that you’d rarely see anyone make a preflop reraise. When you did see one, you could count on the raise coming from A-A or K-K or maybe A-K or Q-Q.
Now you see it much more frequently, with good players leading the reraising revolution. It’s now not uncommon to see hands like this:
In a nine-handed game, a good live player raises first in to $40 second to act. Two players fold, and then another good player reraises to $110.
If the hand goes to showdown, you’ll frequently see that the reraiser has a hand such as A-Q offsuit or 10-10.
These reraises are too loose. There are two main problems with them. First, when you make this reraise, you’re risking $110 to win the $40 open plus $15 in blind money. So it’s $110 to win $55. You have to win this particular pot often to justify risking $110 to win $55. But there will often be four or five players who can still wake up with a big hand. The chance someone wakes up with a big hand, combined with the chance you get out-flopped makes this a marginal reraise.
The second problem is that reraising these hands depletes the strength of your calling range. If you typically reraise hands like A-K, A-Q, J-J, and 10-10, then it becomes difficult for you to have a strong hand on an ace-high, king-high, queen-high, jack-high or ten-high flop when you just call preflop. The original preflop raiser does not have this problem, as all these hands are in his opening range. Moreover, the vast majority of flops contain one of these five cards.
Good players reraise too much preflop, which causes them to be vulnerable in pots where they just call. And, in some cases, they may be turning otherwise profitable hands like A-Q and 10-10 into unprofitable ones.
3. They don’t defend their checks enough.
Another way to say this is that they bet too many hands. Since good players bet so much, they tend to bet nearly all of their legitimately good hands (as well as a bunch of bluffs). Thus, when they check, they have nothing too often, and they’ll fold too predictably.
Let’s put these concepts into practice. It’s a $5-$10 game. You’re on the button with $1,000 and 9 8. A good player opens to $40 second to act, and three players fold. A good player calls, and it folds to you. You call. The blinds both fold.
The flop comes Q J 2.
1 2 Live Poker Strategy
The original preflop raiser checks. The next player bets $90 into the $135 pot. You call with your bottom-end gutshot and backdoor-flush draw. The preflop raiser folds.
The turn is the 4. The flop bettor checks. You can make a small bet like $100 (into the $315 pot), and you will win far too frequently.
The preflop raiser opened too many hands. This may include stuff like A-5 suited and 8-6 suited that whiffed this flop. He checks. He doesn’t defend checks often enough, so you can count too heavily on him to fold.
The next player bets. His range, however, is depleted in strong queen and jack hands. You call.
He then checks the turn and, even to the small bet, predictably folds. ♠
Ed’s newest book, Playing The Player: Moving Beyond ABC Poker To Dominate Your Opponents, is on sale at notedpokerauthority.com. Find Ed on Facebook at facebook.com/edmillerauthor and on Twitter @EdMillerPoker.