The Big Bet Games: No-Limit
Hold ’em and Pot-Limit Omaha
These two forms of poker complete the set of games in the 8-game mix. They are also the most unique since instead of limit betting, they follow no-limit and pot-limit betting respectively. This difference in betting rules leads to no-limit hold ’em being more similar to pot-limit Omaha than its limit counterpart.
In limit poker, the “size” of the game is entirely determined by the size of the betting limits. $10-$20 limit hold ’em is twice the size of $5-$10. But in the big bet form, the size of the game is also determined by the effective stack sizes. $5-$10 with deep stacks could play higher than a $10-$20 with shallow stacks, and this leads to additional complications. Specifically, unlike the limit games, it’s doubtful that we are going to come to any hard and fast opening round rules.
Correct play in short stacked no-limit hold ’em is similar to its limit counterpart. In limit, my advice for players outside the big blind is to raise-or-fold; in short stacked no-limit you should play raise-all-in-or-fold.52 This is because winning the blinds uncontested is a relatively large coup compared to your stack size, and that by moving all-in preflop, you shrink the effective size of the game — reducing your opponents’ options and their ability to make high-EV plays.
As we increase the relative stack size, the usefulness of certain preflop rules decrease. With large implied odds on offer, hands like eight-seven suited can afford to make speculative preflop cold-calls which they would be unable to make in a shallower structure. “Postflop playability” adds to this effect. So in some ways, the
is an easier hand to play well postflop in a deep game than the
The pair of fives will usually only commit significant money after improving to three-of-a-kind, whereas the ace-king offsuit will usually improve to no better than top pair, top kicker — a relatively poor holding to put many blind size bets in even though it will usually be the best hand.
In big bet games, cold-calling with a wide range of pocket pairs and suited connectors is not nearly as bad informationally as in limit games. With this range, there will almost always be a way for us to have some incredibly strong hands on the flop, either with two pair, sets, straights, or flushes. With deep stacks, the mere possibility of sometimes having the nuts can scare an out of position player into submission. This is in stark contrast to limit poker where the marginal benefit of holding a straight over a hand like top pair, top kicker could be just one big bet. Therefore, the most important thing in limit poker is how strong your range is on average. But in deep stack no-limit, much more emphasis is placed on the strength of your very best hands.
Another of the reasons we had for never cold-calling in limit is less relevant in big bet poker: the irresistible pot odds on offer to the big blind. If we cold-call in limit, the big blind will be getting odds of at least 5-to-1. But assuming an open raise to three times the initial bet, the big blind in no-limit will be receiving shorter odds of about 3.5-to-1 (after we cold-call). Additionally, card domination is a much greater issue in big bet. Hands such as the
are much worse in big bet where making a second best hand could cost you a lot of chips (particularly when out of position), than in limit where it might cost only one additional bet. So combining shorter pot odds with the greater disadvantages of being out of position with a poor hand, the big blind should play a much tighter range after a cold-call than in limit poker. This means that since the cost of letting other players enter the pot is less, a much greater flexibility exists with cold-calling raises in no-limit relative to limit.
In addition, the cost of getting four-bet is much greater in deep stack no-limit. For instance, in limit, if you 3-bet a middle position raiser while holding a pair of sevens, then you can always call a cap. However, in no-limit, the initial raiser can make a large enough reraise that you are forced to either fold or make a call that costs lots of EV.53 So in no-limit, your reraises should be largely comprised of two regions: hands strong enough to welcome a reraise, and hands that will gladly fold and won’t get trapped into making a second best hand. Therefore, hands such as 77, AKo, and AQs, that have enough value to play against a raise, but not enough to play a really large pot, should cold-call (unless a moderate reraise gets them all-in).
Playing many hands the same way is the logical way to be deceptive in limit poker. Deep stack poker raises the possibility of utilizing the other mode of deception: playing the same hand different ways. This is Dan Harrington’s no-limit cash game strategy.54 He recommends calling first-in in addition to raising first-in with all of his playable hands in a certain position, raising his strongest hands with a higher probability than his weakest hands. Harrington is very aware of the dangerous information that playing this way could potentially give off, so he stresses the importance of taking each possible course of action with as wide a range of hands as possible. He also recommends using a random number generator to truly randomize your play, otherwise, you might find yourself too often playing your hand the “standard” way, and not utilizing the deceptive strategies enough, such as cold-calling a raise with pocket aces55 which is probably an important balancing play in optimal no-limit preflop play. If they are never played this way, then you would be susceptible to a squeeze play, where a third player makes a large reraise after a first-in raise and a cold-call.56 But by playing aces this way, you can then four bet, usually getting a large fraction of your stack in the pot, and also credibly rebluff on occasion, which in combination will make the squeeze play less profitable for your opponent.
Getting the most out of this style of poker also requires an awareness of your “multiple changing images.”57 Due to the random fluctuations of the cards, different opponents at the same table can have hugely diverging perceptions of how you play. People will usually overweigh the significance of hands they play against you relative to your overall play (at the table). So knowing what each player thinks will allow you to pick the best option to exploit them.
Of course, this process of exploiting a fluctuating image is much easier to implement live than online. Playing on six six-handed tables at a time will result in up to 30 fluctuating images, far too many to keep track of, and playing across this many tables favors the other method of deception — playing many hands the same way.
After covering the opening round, the next logical round to discuss is the last round of betting: the river. Taken together, these two rounds circumscribe the hand; by playing well at both the beginning and the end, the rest should become easier to play. These two rounds are also the simplest to analyze.
In hold ’em, each player will initially be dealt one of 1,326 starting hands, and if we ignore suits, there are only 169. The complexity mushrooms as soon as we reach one of the 22,100 different flops, and play on the earlier streets is also complicated by the fact that we must always be looking ahead to the streets to come. For example, when playing the turn, we can’t do so in isolation since we need a coherent plan on various river cards. However, this intricacy isn’t present on the river since it’s the final round. So once we have analyzed the river, it will be much easier to step back in the hand and examine the middle streets.
Due to this relative simplicity, the majority of poker models that game theorists analyze are simplifications of the final betting round. In The Mathematics of Poker, Bill Chen and Jerrod Ankenman analyze 12 increasingly complex models of river play before considering two multi-street games.58
The most common model of final round play is the U[0,1] model which has been discussed by (among
others) Chen and Ankenman,59 Chris Ferguson, Tom Ferguson, and Cèphas Gawargy,60 and John von Neumann and Oskar Morgenstern.61 Poker hands are replaced by a random number drawn between zero and one; depending on convention, one will represent either the absolute nuts or the very worst hand possible. This mirrors the transitivity of poker hand rankings on the river since if Hand A beats Hand B, and Hand B beats Hand C, then Hand A must also beat Hand C.
This model is less useful for the earlier streets when hand values are imperfectly ordered. This can shown by a famous hustle: The sucker is asked to pick a hand from 22, AKo, and 76s, and the hustler then picks from the two remaining hands before a community board is dealt. If the sucker picks 22, the hustler can get the best of it by picking 76s; if the sucker picks AKo, the hustler can win by picking 22; if the sucker picks 76s, the hustler can win by picking AKo. Because the hand values are not transitive, the hustler can win by picking second, no matter which hand the sucker picks.
In the U[0,1] model, there can be no split pots since there is an infinite range of numbers between zero and one. That is, it’s impossible for both players to draw the exact same hand. Notice that this is a reasonable approximation of most situations in high-only poker, although it obviously fails for board playing situations in hold ’em where the five community cards form a ready made hand, such as a straight or a flush, and other situations where the pot can be split.
One useful feature of the U[0,1] model is the division of hands into “action regions.” The best way to play this game is to subdivide your hands based on rank, taking the same action with hands that are located close together. For instance, in position after a check, you might bluff with your worst hands, check an intermediate region of hands, and bet your best hands for value.
This is also a good way to approach real poker: Imagine the range of hands you can hold in a particular situation, then place your actual hand into the appropriate action region and make the specified play. This can make a number of seemingly “tricky” situations a lot easier to play. For example, you are out of position, and judge your holding is fractionally too weak to bet for value. If this is the case, then your hand must fall in the region of hands that check-call.
Since this is important, let’s address it again. By knowing your distribution of hands, and roughly where your current one falls in it, you can often find the correct play by rejecting certain options, leaving the right play by a process of elimination. Here’s an example: Suppose your opponent has checked, your hand is mediocre, not good enough to value bet, but you have many worse hands you can bluff with. In this case, your hand does not fall into either region of hands that bet, either as a bluff or for value, therefore, you must now check.
The U[0,1] model can in principle be used to analyze the last round in any form of poker. The sequential ordering of the action regions will be the same, but the relative sizes of each region will depend on the bet to pot size ratio, which will depend on the game (with big bet poker typically having a larger bet to pot ratio), and on the particular situation (with heads-up pots typically having a larger bet to pot ratio).
Without further ado, here are the action regions Ferguson, Ferguson, and Gawargy propose for a poker model where the betting is capped at two bets:62
Figure I: Action
Regions for Two-Bet U[0,1] Poker
Player 1 acts first and bets his worst hands (near zero) as bluffs. That’s because the worst hands gain the least from checking and the most from betting. If we check an incredibly bad hand, then even if our opponent checks behind, it’s unlikely to win a showdown. If instead these hands bet, then we may gain even if our opponent folds a fairly weak hand. For a hold ’em example, if we bluff six-high, then betting would probably beat checking if we can get a queen- or jack-high to fold.
Player 1 will check his intermediate hands, calling with the strongest of these and folding the worst. Between these two regions, you can see that he will be check-raising a small sliver of hands. These are bluffs, designed to balance the region of hands immediately next to 1 which are check-raising for value. Theoretically, it’s important to balance every value bet with a corresponding bluff, or else our opponent could exploit us by folding bluff-catcher hands every time. In practice, check-raise bluffing the river is an unusual play; many bad players will unthinkingly call a river check-raise in limit poker with any hand they bet for value since it only costs one more bet to reach the showdown. Against these players, it would be better to never check-raise bluff the river; against strong players, however, it’s an important balancing play to sometimes take.
Player 1 will also bet most of his best hands for value. After betting, there is only one more bet left, so he’s restricted to either calling or folding if Player 2 raises. And when this happens, Player 1 only folds the worst of these hands. Otherwise, Player 2 could exploit him by often bluff-raising if he folds too many hands, and by never bluff-raising if he virtually never folds. And finally, if the pot is large relative to the size of the bets, then Player 1 will bet-call with many more hands than he will bet-fold because Player 2 will be getting high pot odds on his bluff-raise.
(See “Appendix B: Game Theory” starting on page 383 for more on optimal calling and bluffing ratios.)
If Player 1 checks, then Player 2 will bet his worst and his best hands. If Player 1 now check-raises, then Player 2 will fold the hands he was bluffing with and some of the weakest hands that were bet for value.
If Player 1 bets, then Player 2 will fold his worst hands and raise his best hands for value. He’ll also balance his raising range with some bluffs taken from just above the folding range, and call with his other hands.
The location of the hands that raise as a bluff, either by check-raising for Player 1, or, for Player 2, raising after Player 1 bets, deserves some discussion. These raise-bluffs can never expect to win a showdown against a rational opponent who knows your strategy, so we could, in theory, choose to turn any of our folding hands into a bluff. However, by choosing the best folding hands to bluff with, we are poised to take full advantage of an opponent who makes the mistake of calling with a poor hand that he shouldn’t. This can sometimes happen if when playing online, our opponent bets a poor hand as a bluff and after we raise, “misclicks” by calling when he meant to fold. By raising with our best folding hands instead of our worst, we can take maximum advantage of such a mistake.
In practice, such mistakes are incredibly rare, and against an optimal opponent, will simply never happen. However, if there is only a minute chance of it happening, then it does make sense to ensure that you bluff-raise with only your best folding hands. By requiring that your hand fits into this narrow range before bluff-raising, you can also effectively randomize and ration your bluff-raises. On the other hand, if you have a particular read on an opponent, perhaps based on a tell or a suspicious pattern of play, then you can use a weaker hand to bluff-raise. (Additionally, in real poker, you may want to use card removal as a criterion for selecting bluff-raises.)
The pot size will influence the relative size of each action region. As the pot grows, calling will become more frequent relative to folding, and bluffs will become less frequent relative to value bets.63
The betting in real limit poker is generally capped at four bets per round, while in this game, only two bets are permitted. Nonetheless, this two bet game clearly illustrates the core concepts in river play; the graphical representation of a four bet game would include many more muddying action regions. If you can understand the placement and relative sizes of the action regions in the two bet game, you should be able to intuit what the regions would be like in the four bet game.
For instance, if our check-raise gets reraised, then we will need to subdivide our check-raise regions further: Some hands will fold, most will call, and some will four bet, predominately for value, but you should very occassionaly four bet as a bluff. This is illustrated in an expertly played hand by Fatal Fog in an online high stakes match:
Game: $1,000-$2,000 limit hold ’em — 2 players with blinds of $500 and $1,000.
Preflop: durrrr raises. Fatal Fog 3-bets and durrrr calls. There is now $6,000 in the pot.
Flop: A♥T♠2♥
Action: Fatal Fog bets and durrrr calls. There is now $8,000 in the pot.
Turn: 6♠
Action: Both players check.
River: K♦
Action: Fatal Fog checks. durrrr bets. Fatal Fog raises. durrrr 3-bets. Fatal Fog caps and durrrr calls.
Final Pot: $24,000. durrrr shows the K♥2 ♦ (two pair, kings and deuces). Fatal Fog shows the Q ♦9♣ (ace-king-high). And durrrr wins the $24,000.
Fatal Fog’s hand on the river is neither a bluff or a value bet, so he checks. When durrrr bets, Fatal Fog’s hand is fractionally too weak to showdown, so he turns it into a bluff check-raise. When durrrr 3-bets, Fatal Fog decides to turn his hand into a rare bluff cap. Note that durrrr is a good opponent to take this play against (in an exploitive sense) since he’s primarily a high stakes no-limit player. This means durrrr is more likely to put multiple bets in on a bluff or make a tough fold on the river. Fatal Fog’s play is an example of a move that we should all theoretically be doing some tiny fraction of the time in order to balance our strategy, but few ever do.
Balancing
Your Checking Range
Balancing your out of position checks is a very important lesson that can be learned from this game, and it applies on any street. When Player 1 initially checks, he responds to a bet from Player 2 by utilizing all his strategic options. In a large pot (with small bets), he will most often check-call, but he will also check-fold and check-raise as a bluff and a value bet. A common mistake for many players, on any betting round, is to check in first position with a defined range of hands that is weighted far too heavily towards either calling or folding.
For example, you raise first-in from the small blind in a ring game, the big blind calls both preflop and on the flop, and you then check the turn. A lot of players will virtually always either check-fold or check-call in this spot. But if you almost never check-raise, then the big blind can easily exploit you by always betting the turn, and then giving up an appropriate amount on the river.
It’s really important to balance your weaker checking hands by check-raising an appropriate amount. Good hands to check-raise on the turn are: strong hands that are too weak to bet-reraise and hands that hold key-cards on wet boards. An example of the former would be top-pair, medium kicker, and an example of the latter would be top-pair with a good flush-draw on a three-flush board. (Holding a flush card makes it less likely a weaker hand can outdraw you if he checks behind.)
Balancing your checks is easy when your opponent has been in control of the betting; checking a balanced range of hands to the raiser is only natural. However, it’s much more difficult to check a balanced range when you are the one with the initiative. Continuation betting comes naturally; checking is often somewhat of a last resort.
Additional board cards can affect the strength of hands in subtle ways, and checking a balanced range is a great tool to have in your repertoire. Occasionally check-raising is a great way to turn the defensive play of checking into an offensive weapon, and it can also earn you free cards and cheaper showdowns to the benefit of many other hands.
Here’s a good spot to go for a check-raise with the initiative. You reraise preflop and the first-in raiser calls. The flop comes dry and ace-high, and he calls your bets on the flop and the turn. Go for a check-raise on the river with a pair of aces or better! It’s very unlikely he has a draw, most likely he has a pair below aces that he is willing to showdown or has an ace with a weak kicker, but will likely bet if you show weakness.
It’s also important to balance your checks when the board is extremely coordinated by the river — most commonly with either a four-straight or a four-flush. On these scary boards, a number of hands that were betting for value but haven’t improved, will now check. So in order to balance these checks, you will need to check-raise some of your high straights and flushes, and the perfect hand to do this with is the nut flush on a four-flush board. Here’s an example hand:
Game: $30-$60 limit hold ’em — 6 players with blinds of $15
and $30.
Preflop: Hero is the big blind with the A♣T♦.
Action: The first two players fold. The cutoff raises, the next two players fold, and Hero calls. There is now $135 in the pot.
Flop: J♣T♣5♣
Action: Hero checks. The cutoff bets. Hero raises and the cutoff calls. There is now $255 in the pot.
Turn: 3♦
Action: Hero bets and the cutoff calls. There is now $375 in the pot.
River: 2♣
Action: Hero checks.
Given that Hero holds the best flush, it’s unlikely that he will be able to get three bets in the pot with a bet-3-bet line. Therefore, check-raising can maximize the expected number of bets and balance his overall strategy.
On a four-straight board, the only hand strong enough to check-raise might be the top end of the straight. In addition, in the two situations, with the nut straight or the nut flush, it might be best to play a mixed strategy, both sometimes betting and sometimes check-raising. This way your opponent won’t get too suspicious of your checks, and he’ll be inclined to payoff your check-raises since you won’t be doing it too often (with the nuts) in this spot.
Asymmetric
Ranges in Practice
An implicit assumption in the U[0,1] model is that the players’ ranges are symmetric: the average strength of each range is equal. This is a strong assumption. Players will begin the hand with an identical hand, on average; as the hand progresses, the player who initiates the most bets is likely to hold the stronger range.
The U[0,1] says that Player 1 should check some hands and bet some others. But what if the ranges are asymmetric? If Player 1 holds the stronger range, then he should bet more often than in the case of symmetric ranges. And if his range is sufficiently strong, he may even be able to bet all his hands. On the other hand, if Player 2’s range is stronger, then Player 1 should be checking more hands than before. Against a sufficiently strong range, Player 1 should check all of his hands initially, choosing to check-raise when it suits him.
Asymmetric ranges allow us to explain a lot of what goes on at the poker table, especially since players will tend to “check to the raiser,” who will then usually bet. If the raiser’s range of hands is sufficiently strong, then this can be rational behavior. If, either correctly or incorrectly, a player will bet every time it’s checked to him, then with rare exceptions you may as well initially do so. Here’s an example of asymmetric ranges in practice. Playing heads-up limit hold ’em, your opponent raises preflop, you call, check-raise a low flop, and bet the turn.
Now the river is an ace, a card that hits your opponent’s range of hands much stronger than yours since you refused to reraise preflop, which removes many ace-x hands from your range. You also check-raised a low flop which is consistent with hitting a pair, a draw, or better, and your opponent refused to raise on either the flop or turn, which is consistent with a weak showdownable hand.
Notice that it would have been much better for your hand range if the river had been another low card. In that case, you would have been able to bet many hands for value. But with the ace hitting, a lot of those hands are weakened and will now prefer to check and call. On the other hand, the best strategy is to now check some of the strong hands you intended to value bet, such as two pair or better, and go for a check-raise.
Here’s another example. In heads-up limit hold ’em you reraise your opponent’s button raise. Then on a low flop, you bet, your opponent raises, and you call. You also call his turn bet and the river is then an ace.
This time, the ace on the river hits your range much harder, and if you check, your opponent should know just how much this card improves your range, and will likely then check a significant number of hands behind. So your solution is to bet out frequently. This should probably be done with a pair of aces, some stronger hands (such as aces up), but you can also “merge” your range by value betting some holdings weaker than aces — doing so in the knowledge that your opponent will seldom raise. In addition, you should balance these value bets by bluffing some of your worst hands.
Here’s an example of asymmetric ranges in practice:
Game: $25-$$50 limit hold ’em — 3 Players with blinds of $12.50 and $25.
Preflop: Hero is the big blind with the J♠7♥.
Action: The button raises, the small blind folds, and Hero calls.
There is now $112.50 in the pot.
Flop: T♠8♥5♥
Action: Both players check.
The button is a passive player, so his check on the flop indicates he probably didn’t improve his hand. Against a tricky player, this assumption would not be as reliable.
Turn: J♥
Action: Hero bets. The button raises and Hero calls. There is now $312.50 in the pot.
The button now raises. There are a lot of draws, so an aggressive player could be easily semi-bluffing. However, since
he is passive, the J♥ probably gave him a made hand — most likely a pair of jacks.
River: T♥
Action: Hero bets and the button calls.
Hero bets out on the four-flush river as his range contains many more hearts than his opponent. (If the button holds a jack, it cannot be a heart since the J♥ is on board.) As it happens, the button did have a made hand on the turn, but it was a relatively unlikely straight and not a pair of jacks.
Final Pot: $412.50. Hero shows the J♠7♥ (flush, jack high). The button shows the Q♦9♣ (straight, queen high). And Hero wins the $412.50.
Here is another hand where the cutoff’s turn check gives him a weak distribution on the river card:
Game: $50-$100 limit hold ’em — 4 Players with blinds of $25
and $50.
Preflop: Hero is the big blind with the A♠Q♦.
Action: The cutoff raises, the next two players fold, and Hero calls. There is now $225 in the pot.
Flop: T♠8♦6♦
Action: Hero checks. The cutoff bets and Hero calls. There is now $325 in the pot.
Turn: 9♥
Action: Both players check.
River: 2♥
Action: Hero bets and the cutoff calls.
When the cutoff checks the turn, his best possible hand is now a small pair, such as ace-six or a pair of fives. Despite the gutshot four-straight board, large pairs should be strong enough to bet forvalue. We can also expect the cutoff to have a lot of high cards in his range, meaning the 2♥ is one of the blankest rivers in the deck.
Therefore, Hero should value bet this river because not doingso makes the cutoff’s river play far too easy. If Hero checks, then his opponent can safely value bet his ace-king and any small pair,while checking his weaker high cards.
The reason Hero can make a thin value bet here is partiallybecause the cutoff’s range is so weak that he can’t possibly raise. When a raise isn’t possible, the out of position player maximizes EV by making a lot of marginal bets, forcing his opponent to put a bet in with weaker hands. In addition, another advantage of value betting thin is that it allows you to optimally bluff more hands.
Final Pot: $525. The cutoff shows the A♦5♦ (ace-ten-high). Hero shows the A♠Q♦ (ace-queen-high). And Hero wins the $525 pot.
The range of hands that your opponent takes to the river after a particular betting sequence will depend on their playing style, and it’s important to understand how non-standard play on his part can significantly alter the shape of his distribution on the river compared to a more typical player. A great example of this is a habitual slowplayer: someone with a fixation for waiting until the river to raise his best hands. You have to appreciate that the distribution of hands that they play passively on the early streets is much stronger than what a more aggressive player would play this way.
If an aggressive player just calls on the flop and turn, then his range on certain river cards can be very weak. Since aggressive play is the norm in tough games, it becomes natural to frequently value bet the river when your opponent takes a passive line.
The habitual slowplayer, however, will call with a much stronger range of hands than is the norm. Hence, you’ll need to significantly alter your river play against him. If you are acting first, it’s best to check with a lot of hands that you would ordinarily make thin value bets with. To balance out these check-calls, you should also go for a check-raise with some of your strong value betting hands.
So assuming your opponent will often bet after you check, playing in this style will have many satisfactory outcomes. Your weak value betting hands will still often get one bet in, instead of it coming through the sequence bet-call, it will come from the sequence check-bet-call. Beneficially, these hands will now never be faced with the difficult decision that arises from being raised.
But your stronger check-raising hands will still do well being played this way since they will often succeed in getting multiple big bets in. Moreover, your overall strategy will be well balanced and hard to exploit (assuming he doesn’t stop slowplaying).
Returning to Real Poker:
Card Removal and Split Pots
The U[0,1] model is a useful representation of poker, but it does have some limitations. There is an infinite range of numbers between zero and one; 0.99 is a strong hand, but not as good as 0.991, or 0.99101. This means that there is a zero probability of ever having a split pot. Additionally, each players’ hand is entirely independent of what the other is holding.
This is obviously not a completely accurate approximation of real poker where the hands are created from the building blocks of a 52 card deck. For instance, if the A♠ is held by me, then I know, categorically, that my opponent cannot have it. Also, if both players hold seven-deuce offsuit, and the suits are not relevant, then the pot will assuredly be split at showdown. But this can never happen in U[0,1] poker since no two players can ever have exactly the same number.
Furthermore, although card removal is an effect in all poker games, its significance grows as the number of holecards increases it’s much more relevant in draw poker and Omaha than in hold ’em. As an example, in Omaha, the concept of bluffing with “blockers” is well known with the most prominent example being “the blush” — bluffing with the lone ace of suit on a three-flush board.64 And while not hole cards, the upcards in seven-card stud can dramatically impact strategy in this game.
The concept also applies in hold ’em, and it can influence which hands you bluff-raise with. And as already seen, the U[0,1] model says we should turn the best hands we are folding into bluff-raises. We could also choose to bluff-raise with the hands that make it least likely your opponent will hold a bluff catcher. The most common spot would be holding a lone suited card on a three-flush board. But in limit hold ’em, the lone ace of suit will often be enough to call down.
It’s best to go for check-raises with the initiative when your hole cards do not decrease the likelihood of your opponent holding a second best hand. For example, in hold ’em, an overpair is an excellent hand to attempt to check-raise with since your hole cards do not duplicate the board in any way meaning that there are more one pair hands in your opponent’s distribution that he can bet if you check.
Split pot games, such as seven-card stud/Omaha eight-or-better can be modelled using a version of the U[0,1] game where each player gets dealt two separate hands.65 In hold ’em, split pots are most significant in board playing situations: when the five community cards form a ready made hand such as two-pair with an ace-high. In this scenario, a bluff-catcher can only hope to win half the pot and should therefore call less often than in an identically sized no-split pot.
This considerably alters play. Due to the caller’s shorter odds, you should bet with a higher proportion of bluffs relative to value bets than normal. (See “Appendix B: Game Theory” starting on page 383 for more on bluffing ratios.) Also, you should never bet or raise if you don’t have any hands in your range that can beat the board. For example, if the board reads a king-high straight and you have no aces in your range, then your only decision should be whether to check-call or to check-fold. Similarly, if your opponent can never beat the board, then you may as well bet all of your hands since in the worst case scenario you will get your bet back at showdown.
In practice, what this all means is that the player with the strongest range can’t make many big mistakes by often bluffing at these pots. Here’s an example:
Game: $25-$50 limit hold ’em — 2 Players with blinds of $12.50
and $25.
Preflop: Hero is on the button with the Q♦T♦.
Action: Hero raises and the big blind calls. There is now $100 in the pot.
Flop: 7♦6♦6♣
Action: The big blind checks. Hero bets and the big blind calls.
There is now $150 in the pot.
Turn: 7♠
Action: The big blind checks. Hero bets and the big blind calls.
There is now $250 in the pot.
River: A♣
Action: The big blind checks. Hero bets.
You should bluff at this river much more often than usual.
Your opponent would most likely have raised with a seven or six by now, and your range contains more aces than his due to the preflop action. By calling, he can only win half the pot, and it’s likely that you can hold either a seven, a six. or an ace (or a pocket pair higher than sevens). Another advantage of bluffing is that it allows you to gain more value with your strong hands; otherwise, your opponent could exploit your river bets by frequently folding.
Here’s an example of the concept on a five-flush board where the button’s play removes a lot of higher flushes from his range: Game: $50-$100 limit hold ’em — 2 players with blinds of $25
and $50.
Preflop: Hero is the big blind with the K♦7♥.
Action: The button raises and Hero calls. There is now $200 in the pot.
Flop: Q♣7♣2♣
Action: Hero checks. The button bets and Hero calls. There is now $300 in the pot.
I typically check-call a lot of hands on this flop because manyvulnerable made hands (like the K♦ 7♥ ) benefit from seeing a non-flush card on the turn before initiating aggression (by betting the turn). Made flushes and flush draws should probably be played in a similar way. But if a flush card does arrive on the turn, myadvice is to bet some middle flushes but check lower and higher ones.
Turn: 5♣
Action: Both players check.
The button’s turn check removes a lot of clubs from his range. He could hold a small club that he wants to showdown cheaply, but middle and higher flushes should bet. Given Hero’s flop strategy, he should have many flush cards in his range.
River: 3♣
Action: Hero bets.
The fifth flush card creates a board playing situation, but Hero has many more higher flushes in his range than the button does, so Hero should now bet. After the bet, the pot is now four bets and it will cost the button a bet to call, and if Hero is bluffing, the button will only win half the pot. So his pot odds of 2-to-1 are half the 4-to-1 he would receive in a non-board playing situation, meaning Hero should bluff more and the button should bluff-catch less.66
Scare Cards
Many people play badly on river cards that significantly alter the board: Exploiting their mistakes often involves counterintuitive play. One of the important lessons from the U[0,1] game is of balancing your value bets with bluffs. That is, you should bluff in proportion to the pot odds your opponent is receiving to call your bet. And specifically, if the pot is large relative to your bet, your betting range should contain significantly more value bets than bluffs.
When a scare card comes on the river, people will commonly check many hands that they would have bet for value on a more benign card. What’s less likely is for players to retain their correct bluffing ratio by giving up an appropriate amount of bluffs.
In other words, if your opponent does not give up the right proportion of bluffs to go along with his lower frequency of value bets, then it might be much more profitable to call with a marginal hand on a scary card than on a blank. (Bluff-raising also becomes more attractive.) Examples of good scare cards to call down on include an overcard ace (particularly if he check-raised the flop on a low board); a four-straight (bonus points if the straight cards do not correlate with his range such as a low four-straight against a middle position raiser); or a card that completes a flush (particularly if it’s a backdoor flush and his value range is weighted towards one pair hands which will likely now check). Here’s an example:
Game: $25-$50 limit hold ’em — 3 Players with blinds of $12.50
and $25.
Preflop: Hero is the big blind with the K♥2♥.
Action: The button folds. The small blind raises and Hero calls.
There is now $100 in the pot.
Flop: 5♣4♦2♣
Action: The small blind bets and Hero calls. There is now $150
in the pot.
Turn: 6♦
Action: The small blind bets and Hero calls. There is now $250
in the pot.
The small blind should bet the turn with (semi-)bluffs and value hands. Straight draws, flush draws, and overcards as a bluff, and hands like straights, two pair, or top pair for value.
Many other hands will likely check. There isn’t much point in betting a hand like ace-high as it can’t call a raise and showdown unimproved. The small blind should also balance his checking range by check-raising some strong hands and semi-bluffs.
River: A♥
Action: The small blind bets and Hero calls.
An ace comes on the river. Although it’s a bad card for Hero’s hand, it doesn’t really improve the small blind’s range. He could have been value betting a hand like ace-six that has now improved, but this is a small part of his range. His strong value betting hands can still continue, but there aren’t that many combinations of two pair or better. Notice that the A♥ is a bad card for the weaker value betting hands the small blind can hold, and a lot of these hands should now check. In other situations, the small blind would have had a lot of ace-highs still in his betting range, but this is unlikely given the coordinated board on the turn.
Therefore, if he doesn’t balance his value bets and bluffs appropriately, this card makes it more likely his bet is a bluff. (The river play rests on the assumption that the small blind will check many ace-high hands on the turn. If this assumption isn’t true, then the river is much closer to a fold.)
Final Pot: $350. The small blind shows the J♥9♥ (ace-jack-high).
Hero shows the K♥2♥ (one pair, deuces). And Hero wins the $350 pot.
Note that expert players will balance their value bets and bluffs correctly on all types of boards. Therefore, this concept is much more useful against weak opponents or perhaps against players who are not use to playing limit hold ’em.
Incorporating Exploitive Poker
We can easily use the U[0,1] model as a stepping off point for incorporating exploitive poker. The ordering of the regions will stay the same, but the relative sizes will shift in order to take advantage of your opponent’s mistakes.
Against an opponent who bluffs too much, the check-call region will expand: Some hands that were borderline bet-fold, some hands that check-raise-bluffed, and some hands that check-folded will now check-call. The more out of whack your opponent’s strategy is, the more hands can shift to check-call, and the more EV you will gain.
But large exploitive shifts come at a cost: If your opponent now starts bluffing less frequently, at either the optimal frequency or below, then now you will be the one being exploited. Bigger shifts gain the most against opponents who continue to play badly, but they also expose you to greater losses against opponents who are capable of responding to your play.
This is why perfect exploitive poker is not a credible way to play even though it maximizes your expectation in theory. Any small mistakes in your opponent’s play require big shifts in your strategy to maximize expectation, and you must also instantly counter any adaptations in his play. This is not a problem for poker robots. Sonia, for instance, is capable of balancing a lot of base rate information about an opponent’s play, and quickly updating it with any short-term news.
As explained in “Two Schools of Poker” starting on page 6, the best way to play is to use a combination of GTO and exploitive styles. Beginning with a GTO base strategy maximizes your expectation against strong players, and it gives you a much clearer indication of what is, and what isn’t, exploitable. Make small exploitive shifts against poor players, make bigger changes against worse players, and over time against players who don’t seem to counter-respond. As Matt Hawrilenko says:
“If you must play exploitively, do so on the margins. The marginal hands stand to gain more EV from exploitive play than hands where we’d generally think the decision is clear cut. Bluff with the worst hands in your distribution no matter how sure you are the other guy has it. Even when you are right, making big (or even medium-sized) folds just can’t gain you very much edge and can cost a ton. So make the small folds — in limit hold ’em, don’t fold top pair when you’re convinced the guy has the nuts, fold 4th pair; your opponents may be craftier than you think. Similarly, bluff the river when you miss with 43s no matter how sure you are he is going to call. It’s a really important concept and I continue to be surprised when I challenge myself in this way.” 67
In Hawrilenko’s example, four-trey suited will be right at the bottom of your range: in terms of the U[0,1] model, it would be right around zero. If this hand is not a bluff, then you won’t be bluffing at all, and so your opponent could exploit you to the extreme by folding all bluff-catchers.
On the flop in limit hold ’em, the action will most often check to the preflop raiser who must then decide: check or bet? And betting is the most natural thing to do; with position, a strong range of hands, and a relatively small flop bet, the odds are stacked in the preflop raiser’s favor.
For a long time, the standard heads-up play has been for the preflop raiser to continuation bet all his hands. However, there is a great heterogeneity in these sorts of situations. Sometimes, no doubt, always continuation betting can be correct. At other times, it can be correct for the preflop raiser to complement his bets by checking back the flop with a balanced range of hands.
The major factor behind the decision of whether to check or not is the asymmetry of the ranges in play. If the preflop raiser’s range is significantly stronger than the blind’s, then betting every hand is the right thing to do. Say the pot is four small bets and your opponent will receive odds of 5-to-1 to call your flop bet, if 90 percent of your range are clear value bets, then you don’t actually hold “enough” bluffs in your range to reach the optimal bluffing ratio, so you should bet every time.
That’s because with your opponent receiving pot odds of 5-to-1, the optimal bluffing ratio implies that you would need to be able to value bet 81
percent (five-sixths) of your hands in order to always continuation bet the flop. In addition, the advantage of position makes this figure an upper bound, as this advantage on later streets allows you to increase your proportion of bluffs on the flop.68 Also, if you raised first-in from a position outside of the blinds, then the pot will be bigger due to the extra presence of the small blind. So if your opponent is receiving pot odds of 5.5-to-1, the optimal bluffing ratio implies an upper bound of 84.6 percent value bets to always continuation bet.
The earlier a position you raise from, the more likely it is that your range will be strong enough to always continuation bet. For instance, I would always bet the flop heads-up against the big blind from the under-the-gun seat six-handed (or earlier in a full ring game).
Flop texture also matters. Flops with many broadway cards will tend to favor the raiser’s range over the blind’s. The blind fairs better on low connecting flops, but probably not enough to overcome the raiser’s superior range. Paired flops miss a lot of starting hands, and if the raiser came in from a late position, his range will mostly consist of high cards indicating he might not have enough value hands to always bet.
At least in an exploitive sense, the range of hands your opponent reraises preflop will also matter. If he reraises many hands, his range of calling hands will become weak, and more of your hands will be able to bet the flop. The players that you should be most willing to check the flop against are those who never reraise preflop. (You will recall this is my advice for big blind play in a ring game against an in-position raiser. But if it becomes popular for raisers in late position to start checking back a lot of flops heads-up against the big blind, then this could be a good enough reason to start reraising preflop in the big blind again.)
Checking the flop instead of continuation betting is even more important in big bet poker and should probably be done more often.69 The individual hands that check will be slightly different, but the concept is the same, just differing in its application. In deep stack no-limit hold ’em, you might check the flop with a hand such as top pair because it’s strong enough to call two bets but not three, plus your check might induce a bluff. In limit hold ’em, the hand you choose to do this with might be ace-high, but nevertheless, the concept is the same. In both games, it’s important to balance your checks with a broad range of hands even if there isn’t much overlap in terms of the individual hands checked in each game.
Play on a Three-Flush Flop
I will use the example of a three-flush flop during a heads-up game to illustrate the concept, as the merits for checking back the flop are very strong. Consider a flop of:
If you follow my preflop advice, you will be playing the top 85 percent of your range. (See Table I below.)
Table I: Playable Hands
from the Button in Heads-Up Limit Hold ’em
The trouble with always continuation betting this range is that many of your hands have completely missed the flop. For example, hands without a pair or a flush-draw. Notice that the presence of the three-flush makes any weaker draws nearly worthless which could make check-raising a very profitable play for the blind.
If he check-raises, then your opponent will be receiving immediate odds of 2.5-to-1 on his bluff since the pot is five small bets, and he puts two in to raise. Therefore, if you fold to his check-raise more than 28.6 percent of the time, it will show an immediate profit with any hand. Additionally, your opponent’s bluffs will be semi-bluffs not pure bluffs.
If you bet this flop with your entire range and fold to a check-raise with any hand that doesn’t hold a diamond, a jack, a seven, a deuce, a pocket pair, or an inside straight draw, then you will be folding 32 percent of your range — above the 28.6 percent your opponent needs to show an instant profit.70 Note that ace-high without a diamond is a relatively weak hand on this board.
You could make sure you call his flop check-raise with the great majority of your flop bets — say at least 80 percent. But the trouble with this is that many of your calling hands will still be weak in which case bluffs on the turn and river could also be profitable for your opponent.
This implies that the logical solution is to start checking back the flop with some hands. Note that the arguments used so far for checking have been entirely GTO-based: If we always bet, our opponent could gain a lot of profit by frequently check-raising. When playing GTO, we are trying to create an unexploitable strategy, and we have concluded that always betting this flop with such a wide range could be quite exploitable.
On the other hand, if your opponent doesn’t check-raise frequently, then always betting this flop might be our highest expectation strategy. As an extreme example, some very bad players have the habit of always betting if they hit the flop, otherwise they check. So when this player checks, not only is his checking range weak, but he’s unlikely to ever check-raise, so you should always bet. By the way, I think this explains why always betting the flop has been a mainstay in a lot of limit hold ’em players games’ for so long: If your opponent makes a lot of mistakes, always betting the flop will often maximally exploit him.
However, the game is evolving, and the average player today is much better than a few years ago. Therefore, I think it’s a good idea to spend time working out a defensive strategy of checking a well balanced range of hands (assuming the ranges are roughly symmetric).
Your Range
of Checking Hands
Keeping your range of checking hands well balanced is a vital part of this strategy. You can see that this is related to the general principle of information hiding that I introduced in “Information Hiding” starting on page 21, and checking a narrow range of hands will necessarily reveal a great deal of worthwhile information.
Perhaps the hand that benefits the most from checking back the flop on a three-flush flop is an ace-high without a flush draw. This hand has some showdown potential (assuming favorable turn and river cards), but not a lot. If the hand can see a safe turn card for free, then it will often have enough value to call down. If it gets check-raised on the flop, combining the ways it could be already beaten or out-drawn later, calling down on all streets could be a losing play.
However, if this is the only hand that is checked behind, it can get you into a lot of trouble. Since your opponent will now know exactly half of your hand (the off-suit ace), he will have a good idea of what turn/river cards improve you, and can then go for a lot of thin value bets as well as deducing useful cards to bluff on. So I now give the first rule of checking back the flop:
Check-Back Rule No. 1: Ensure you do so with a broad range of different hand types.
122 Part One: Strategy
You should be checking back with a broad range of hands, and the majority of them can go into two “classes.” The first class, which includes hands like ace-high on the three-flush flop, are the weak showdownable hands. They would like to make it to the end of the hand, circumstances permitting. But if they get check-raised on the flop and then bet at on the big streets, they will often have to fold. So by betting these hands on the flop you are effectively turning them into bluffs since they will usually only win if your opponent folds, but they would fit into your overall strategy better if played as bluff-catchers.
The second class of checking hands are the weak draws. They have some potential, just not a lot. If they are bet, you are also turning them into bluffs; if check-raised, your best strategy will be to frequently fold, giving up on their small amount of potential.
On flops such as the three-flush example, there are already a lot of pure bluffs in your range. And when you have too many bluffing hands relative to value betting hands, it’s usually better to bet the ones with absolutely zero outs, while checking the hands with just a couple outs. (This assumes both types would be forced to fold to a check-raise and that we are last to act). On the flop of the J♦7♦2♦, a hand with zero outs would be the
I’d rather bet that hand and check a hand like the
Checking Back the Flop 123
My second rule is related to the strength of the hand that you check back. On a number of flop-types, the hands that will benefit the most from checking back are no-pair hands: ace-high, king-high, or queen-high. The trouble is, if these are the only hands you check behind, your best hand on certain turn cards will be weak. For instance, on a flop of
and a turn of the 5♣.
If your best checking hand was ace-high, then your best hand on this turn card would be bottom pair, ace-kicker. When your best possible hand is this weak, your opponent can bet the turn with impunity — knowing that your best hand is too weak to raise. So here is my second rule:
Check-Back Rule No. 2: Ensure that your checking back range contains some hands that are capable of improving to at least top pair or better on every possible turn card.
If the turn is an overcard to the flop, this won’t be a problem. If the turn is a low card (such as the 5♣ in the example), you will want to hold some hands that are capable of improving to two pair or better. If you check back with some more hands as well, such as 66-22, 7x, or Tx, then this won’t be a problem.
The reason for always wanting some hands that could improve to top-pair or better on any turn card is that this is often around the minimum strength hand you will need to raise a turn bet for value. But if you can never value raise on a certain turn card, then your opponent will be able to bet a lot more hands without any fear.
Now can you guess what the important corollary to my first rule might be? Here it is:
Corollary: Ensure that your betting range remains balanced.
There are also good reasons to check back with hands like bottom pair and ace-high. However, always checking these hands on the flop will unbalance and weaken your betting range on certain turn cards.
This is a problem, and my solution to it is to vary your play based on kicker size. If my holding is ace-high, middle pair, or bottom pair, I will bet the flop with a good kicker and check with a bad one since high kickers are stronger and worth more as value bets. This is a much easier way to balance my play than pure randomization between the two options.
The range of hands that I’m advocating to check are, on average, fairly weak. There is a good reason for this: when we’re betting every hand, there were too many weak holdings relative to value betting hands. So by removing these weak hands from out betting range, we can move our range of betting hands closer to the optimal bluffing ratio. However, I would really like it if there are some better hands to balance my checking range. It’s not good to check a strong hand just for the sake of it. Thus, I would much prefer to find a strong hand that can actually increase its EV by checking.
In the next chapter, we’ll see that my usual strategy after getting check-raised on the flop is to mostly wait until the turn to raise. A hand that is good enough to raise the turn after checking the flop, but not good enough to raise the turn after getting check-raised on the flop, could fit my requirements of being a strong hand that plays better by checking the flop.
An example of this hand might be the Q♠ 2♠ on a flop of the Q♦9♣5♥. If this hand is not strong enough to raise the turn after flop action of check/bet/raise/call, but is strong enough to raise the turn after the flop goes check/check, then this holding might be a good one to check behind on the flop.
In addition, card removal effects might give us enough cause to check some very strong hands. For example, if you flop top set after raising preflop with aces, then your opponent will know the ace on the flop hits your range, but there’s only one ace left in the deck for him to hold. So if you bet, don’t expect much action except for the rare case where he holds a decent hand. If you check, his good second-best hands will still produce a lot of action, but you should also generate action against bluffs and hands that improve to being second-best on the turn. (This concept is much more prominent in big bet poker where checking a hand that cripples the deck is a common play.)
Analyzing
a Sample of Flop Checks
In this sub-chapter, the hands Matt “Hoss_TBF” Hawrilenko checked the flop with in his two matches against Polaris in July 2008 will be analyzed. This is an interesting match to look at from a historical perspective. Always continuation betting the flop was standard play at the time, and it was the first time a number of aspiring professionals, such as myself, had observed this strategy in use by one of the top players in the world.
Matt is well known for being one of the best GTO limit hold ’em players in the game today. Before the match, he said that he would be playing his basic strategy, his best approximation of unexploitable play in heads-up limit hold ’em.
Matt’s cards were only visible if the hand went to showdown. This creates a biased sample of observed hands relative to the actual strategy he was using since we are more likely to observe the medium strength hands that he checks with. That’s because he’s more likely to bet his best and worst hands on the turn/river, and we will only see these if Polaris is also willing to showdown.
Over the two matches (1,000 hands total), we get to see the hand Matt checks the flop with a total of 25 times. On a further 35 occasions, he checks the flop without a showdown. My comments on each hand are shown in italics:
Hand No. 1
This hand meets the criteria for a weak showdownable hand. The 3♠ on the turn is a great card, unlikely to improve Polaris, and gives Matt a gutshot straight draw. And once Polaris checks the turn, the hand is worth betting “for value” — I don’t mean that Polaris will call with that many weaker hands, more that this is one of the stronger holdings Matt will have in this spot, and is thus worth betting as cover for his bluffs.
Polaris check-raises the turn, which is obviously bad news. However, Matt’s hand is one of his better possibilities in this spot, so folding a hand this strong would make check-raise bluffing incredibly profitable for Polaris. Therefore, he has to call the check-raise and hope to either improve on the river or catch Polaris bluffing. Polaris had the
in this hand.
Polaris checks on the river. Since Matt’s hand is neither a value bet or a bluff, he checks behind.
Hand No. 2
Q♠ 2♣ is a good hand to check the flop. It has a tiny smidgen of showdown value, and some equally tiny draws — 3 outs to second pair; gutshot backdoor straight. If I had bet this hand on the flop, it would be difficult to call a check-raise, so checking is better than turning it into a bluff.
The 9♣ , giving Matt a gutshot, is a good turn card. On various river cards calling, folding, and bluff-raising are all possible plays. The J♣ gives Matt the straight, it’s an easy raise for value.
Hand No. 3
King-high on a paired board is an excellent hand to check. People do like to check-raise these flops with a lot of bluffs; if my flop bet got check-raised, I would be incredibly torn between calling down or folding at some stage. The T♥ and the 2♥ are fairly benign cards and the hand is good enough to call both streets.
Hand No. 4
Matt checks back with a weaker hand this time. Polaris certainly will not fold ace-high, and is likely to call with queen-high to boot. Against a bad player, I would be tempted to bluff either the turn or the river since many players do not call with enough hands after checking the big streets (either once or twice).
Hand No. 5
The 9♠ on the turn gives Matt top pair making the hand an easy raise for value, but the river card is ostentatiously bad. However, on the turn, Polaris’s most likely hand is one pair. If one holecard has paired the board, then Polaris only has one card left in his hand to hold an ace (and Polaris will also reraise many of its aces preflop).
Hand No. 6
This hand is right at the bottom of his flop checking range and is a good hand to turn into a bluff on the turn, and even more so on the river when the 7 counterfeits his holecards. The presence of a backdoor flush-draw would make me more likely to play the hand this way. I also don’t think there’s anything wrong with bet/folding the flop.
Hand No. 7
As noted earlier, it’s important to sometimes check the flop behind with at least a pair so that you can credibly threaten to occasionally have a strong hand on any turn card. Bottom pair, no-kicker is a good hand for this purpose.
Matt checks again on the turn when a king appears. This card will improve a number of other hands in his flop checking range, so in that sense, he has “enough” value hands that he can check this hand again. Against a lot of bad players, I would be tempted to make an exploitive turn bet.
Hand No. 8
King-high checks the ace-high flop back again, and the turn card pairs the board. So when Polaris checks the turn, this hand is near the top of Matt’s flop check range. Therefore, he bets the turn and calls down after Polaris check-raises.
Hand No. 9
Queen-high checks back. The ace on the turn will improve a number of hands in Matt’s checking range, but not this one. Also, his hand isn’t close enough to the bottom of his range to turn into a bluff. Therefore, he checks both the turn and river.
Hand No. 10
Almost an exact replica of the previous hand. Note the paired board on the flop.
Hand No. 11
Matt checks ace-deuce offsuit again — see Hand No. 1 — improving enough to raise the turn.
Hand No. 12
Bottom pair checks the flop again, calling the turn bet. I would bet this river against many players who do not properly balance their river checks. (See “End Play” starting on page 93.) But Polaris is definitely capable of check-raising this river in a balanced manner, so this comment doesn’t apply here.
Hand No. 13
A hand at the bottom of Matt’s flop checking range. He calls Polaris’s reraise on the turn without the required pot odds to hit his gutshot. He must be relying on some combination of pair outs and bluff-raising equity on the river. My preference would be to call the turn bet and bluff-raise the river if one of the 10 outs doesn’t hit.
Hand No. 14
Similar to Hands Nos. 5 and 9.
Hand No. 15
The 6 on the turn propels the hand up Matt’s range making it an easy value bet on both the turn and river.
Hand No. 16
Bottom pair checks behind again. Due to this hand’s strong kicker, I would rather bet the flop and check behind with hands like the 7♠ 3♠ — adding more balance to the betting and checking
ranges. And against the majority of (bad) players, I would bet this river.
Hand No. 17
Ace-king-high checks behind and bets the turn unimproved.
Interesting, since Matt has twice checked bottom pair again on the turn — Hands Nos. 7 and 16. I assume this is a balancing play since he will have bottom pair more often than strong aces.
Hand No. 18
Another weak hand with some high-card value.
Hand No. 19
The ten-five offsuit checks behind again, (see Hand No. 4), bluffing the turn on this occasion since the ace is a good card for Matt’s range (and ten-high is near the bottom of it).
Hand No. 20
Another weak hand with some high-card value.
Hand No. 21
Bottom pair.
Hand No. 22
Another weak hand with some high-card value.
Hand No. 23
King-high bets the turn again, (see Hand No. 8). I assume he’s calling a turn check-raise and I would check this hand down on the river.
Hand No. 24
Another weak hand with some high-card value.
Hand No. 25
Weak hand with high-card value that improves enough on the turn to call all the way down.
Here are some statistics from the match:
folded 11 times, and raised 4 times.
When checking back the flop, you should balance your range over the two possible turn situations: when your opponent checks and when he bets. When he bets, you should utilize all three of your options: folding, calling, and raising. Fold your weakest hands, call with a range of hands capable of taking a variety of actions on certain river cards, and raise your strong hands (balanced with some semi-bluffs).
When he checks, you have only two options: checking and betting. Some weak players have a tendency to bet too many hands in this spot. They figure their opponent has shown weakness by checking after their flop check, and they seek to exploit this. The trouble is that betting the turn too often can be exploited if your opponent starts to check-raise frequently. This is why you see Matt check the turn again with a low pair such as in Hand No. 7 above.
When betting the turn, you should do so with your best hands (value bets) and your worst ones (bluffs, see Hands Nos. 6 and 19). It’s important to check the flop with some low cards that can then bluff the turn. You can see this in Hands Nos. 6 and 19 where Matt checked six-high and ten-high respectively. My preference
when selecting hands to bluff the turn is to pick those with some backdoor value:
on a flop of the
can be thought of as an extremely weak draw in addition to being a potential turn bluff.
Notice that the range of hands that you are value betting will be weaker than the range you value bet on the flop. You can see this in the sample of hands that Matt played. When he bet the turn with ace-high or king-high, he was doing it for value, not as a bluff. When you bet this type of hand on the turn and get check-raised, you have to stay consistent to your game-plan, often getting to showdown with them.
Balancing the turn is harder when you are out of position and your opponent checks the flop. Betting some hands and checking others is easy, but these two groups have to be subdivided into a further three regions: bet-raise, bet-call, bet-fold, check-raise, check-call, and check-fold.
To do this, divide your range up by putting hands into the regions where they fit the best. Strong hands that are vulnerable to free cards should bet; hands with value that are too weak to bet-call should check-call, and hands that check-raise should fare well even if your opponent checks again. You can see this in Hand No. 1 where Polaris check-raises the turn with two small pair, but it’s also evident in the hands that Polaris check-raise-semibluffs the turn. Here are two examples:
Hand No. 1
Preflop: Polaris is the big blind with the 8♥6♠.
Action: Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: Q♦9♠4♠
Action: Both players check. There is now 2 big bets in the pot.
Turn: J♠
Action: Polaris checks. Matt bets. Polaris raises and Matt folds.
Hand No. 2
(This is also Hand No. 8 from above.)
Preflop: Polaris is the big blind with the 6♥5♦.
Action: Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: A♦8♥4♥
Action: Both players check. There is now 2 big bets in the pot.
Turn: 8♦
Action: Polaris checks. Matt bets. Polaris raises and Matt calls.
There is now 6 big bets in the pot.
River: 9♦
Action: Polaris bets and Matt calls.
Final Pot: 8 big bets. Polaris shows the 6♥5♦ (one pair, eights —ace-nine-high). Matt shows the K♣2♣ (one pair, eights — ace-king-high). And Matt wins the 8 big bets.
You can see that Polaris is check-raise-semibluffing the turn with weak draws: a low one card flush draw in Hand No. 1 and a gutshot straight draw in Hand No. 2. These holdings will do well if Matt checks the turn, in which case Polaris gets a free card and can bluff a blank river, but they also have some outs to fall back on if Matt calls the check-raise.
Polaris’s Flop Checks
We can also look at the hands Polaris raised preflop and checked the flop in the same match. Polaris is a strong player and we have already seen how it has quite an interesting preflop strategy in the “Appendix to Rules for Preflop Limit Hold ’em: Polaris Non-Standard Preflop Plays,” and we will look at another element of its postflop play in “Middle Game Concepts” starting on page 152. Furthermore, when we see two strong players independently using a similar strategy, we can make a strong inference that they are onto something.
One benefit of looking at Polaris’s flop checks is that we get to observe Polaris’s holecards every hand, so the sample is without bias. Polaris checked the flop 29 times, just under half as often as Matt, which translates into a flop checking frequency of approximately 10 percent. I will look at these hands by grouping them by type:
Pairs
Hand No. 1
Hand No. 2
Hand No. 3
Hand No. 4
Hand No. 5
Hand No. 6
Hand No. 7
Hand No. 8
Polaris checked top pair twice (Hands Nos. 2 and 7), middle pair four times (Hands Nos. 1, 3, 5, and 6), and bottom pair twice (Hands No. 4 and 8). It’s interesting to see it check the flop with each kind of pair, and it’s my assumption that it would have raised on different turn cards with its top pair hands (although I would have raised the turn in Hand No. 2). Matt checked the turn twice (Hands Nos. 1 and 4), and Polaris bet the turn with its flopped middle pair but not its flopped bottom pair. I would expect Polaris to also check back a lot of fairly weak pocket pairs, such as a pair of fours on the
flop in Hand No. 1, but there were no hands like this in the small sample.
Showdownable High Cards
Hand No. 1
Hand No. 2
Hand No. 3
Hand No. 4
Hand No. 5
