Strategy
Introduction
All poker games can be considered under the conceptual framework of game theory (“Two Schools of Poker”); involve incomplete information (“Information Hiding”); have an opening round (“Rules for Preflop Limit Hold ’em,” “Appendix to Rules for Preflop Limit Hold ’em: Polaris’s Non-Standard Preflop Plays,” “Analyzing the Opening Round in Other Popular Forms of Poker”); and a final round (“End Play”).
With these broad brushstrokes, we have painted a large part of the tapestry of poker. All that remains is to fill in the details by considering the intermediate rounds of play (“Checking Back the Flop,” “Middle Game Concepts”), made easier by the application of concepts introduced in the earlier chapters. “The Anatomy of a Hand” puts all these concepts together with some sample hands.“Man Versus Machine” summarizes some interesting findings from the field of artificial poker intelligence. And finally, “No-Limit Hold ’em: Applications and Extensions,” looks at how these ideas can be applied to the most popular form of poker today.
This is a “top-down” approach to poker: starting off with a view of the game as a whole, gradually working the way down toward finer concepts. It’s also possible to go “bottom-up” by starting off on the level of individual hands and situations, and then working your way up to an overall strategy. In fact, this is the approach of many poker books which start off by recommending individual hands to play and then proceed to the rest of the hand.
Although many poker players successfully use a strategy that is bottom-up in nature, this style makes for an inefficient mode of instruction: It tends to compound the complexity of poker instead of simplifying it. Furthermore, bottom-up hand recommendations frequently date extremely quickly; as the game evolves, so does admissible play (along the tight/loose, aggressive/passive dimensions). If you play poker in an essentially bottom-up way, I serve my style as a natural complement to your game — not as a substitute.
A lot of the strategy is presented with examples from limit hold ’em — ideally, this shouldn’t reduce the usefulness of these chapters for players of other games. That’s because limit hold ’em is a relatively simple form of modern poker that shares a number of similarities with other variants, so this should make it a good pedagogical tool for the student of poker.
Unfortunately, this doesn’t apply for more structurally complex games such as pot-limit Omaha where, for instance, a study of how to evaluate various wrap around straight draws doesn’t have any meaningful comparison in limit hold ’em. But by looking at a more simple form of poker, we can derive insights that will potentially apply in the more complicated variants.
However, this process does not operate in reverse. To fully study a structurally more complicated form of poker, we would need to apply lessons from the simpler version and also analyze the unique features that make the other game more complex.
Finally, playing limit hold ’em well does not require an intimate knowledge of game specific probabilities. (For example, most draws are getting laid enough odds by the pot.) But in shorthanded situations, the most important factor is knowing how to correctly play a wide range of hands over a number of decision points — useful skills for any poker player.
LOOK AROUND. There are idiots.
— Larry Summers
Successful poker players are often members of two competing philosophies: exploitive and game theory optimal (GTO). The exploitive school is by far the oldest, surely dating back as long as poker itself. Essentially, this approach views our opponents’ strategies as fixed in a moment of time. Thus, the goal of the exploitive player is to observe how the other players approach the game, and then to construct counter-strategies that maximize expected value (EV).
An example would be to fold a bluff-catcher on the river against an opponent who is perceived to almost never bluff. This example illustrates what is known as Level 1, exploitive poker in action, and the opponent who very rarely bluffs is acting on Level 0. He doesn’t really think about his strategy, but just plays in a way that feels natural. So by operating on the level of reasoning above the unthinking opponent, the exploitive player can gain an advantage.
Against a more sophisticated player, the reasoning should be more complex. Suppose your opponent knows that his image at the table is one of a player who has not been bluffing. He should understand that the natural response of a Level 1 thinker will be to fold marginal hands against him, therefore, he will elevate his reasoning to Level 2 and start bluffing more! This means that in order to properly exploit this player, you will need to reason up to Level 3, and call down more.1 So as can be seen, success is achieved by operating one level above your enemy, but no more. Mistaking a Level 0 thinker for a Level 2 thinker can obviously lead to grave errors.
Exploitive Poker
and the Beauty Contest Game
Experiments have been conducted that attempt to deduce how many levels of reasoning people are capable of. The most famous of these is the so called “beauty contest” game.2 A group of players is told to pick a number between zero and a hundred, the winner of the game being the person whose guess is closest to two-thirds of the average guess. A prize (often a significant one), is offered to the winning answer to incentivise entrants to take the game seriously.
A Level 0 player would pick a number randomly which would, on average, be 50. A Level 1 player would know this and pick the number that is two-thirds of 50 (33). A Level 2 player would take this one step further and pick 22 which is two-thirds of 33. And a Level 3 player would pick 15. This process can continue indefinitely until the number zero is reached, which represents the game theoretically optimal way to play (or an infinite level of exploitive reasoning). Conducting the game among a large group of expert game theorists should result in everyone picking zero, with the prize then being shared equally between all entrants, and running this game as an experiment represents a great opportunity to find out how many steps of reasoning people are capable of going through while in a relatively simple setting.
An empirical analysis of how people play this game shows that their answers correspond with what we would predict from people who are thinking somewhere between Levels 0 and 3 (see Figure I for a graphical illustration, and see how few people pick zero). The most frequent answer is 33, corresponding with Level 1 thinking. The second highest spike is in the low twenties, corresponding with Level 2 thinking.
Notice that most people are capable of making a couple of steps of strategic analysis, but are unable to iterate their reasoning in order to reach the theoretically optimal answer. The game has been played amongst many diverse groups such as students, CEOs, economics PhDs, and investment portfolio managers. In all these groups the modal answer is no lower than in the mid-20s —corresponding with Level 2 reasoning. Even the economics PhDs, who all know basic game theory, reason only about one level higher than most other populations. This also implies that in poker, against most opponents, we do not need to think at too high a level in order to play exploitively.
Also notice in Figure I that a significant proportion of the answers are above 67, which is the maximum possible winning answer, and that is only in the case where everyone picks 100. There is no logical reason to provide an answer above 67 since even if everyone else did answer 100, 67 would always have a higher chance of winning than an answer of 68 or more. What could they be thinking?
Figure I: An Empirical
Distribution of Answers to the Guessing Game
This evidence that people are unlikely to go through more than a couple levels of strategic thinking, in a game much simpler than poker provides strong support for the exploitive school. If most recreational players are unlikely to operate much above the second level, then getting inside their heads and outplaying them should be an achievable goal. If a number of people are just randomly picking options (that often have no chance of winning), then exploiting their play will be easier still.
If you are ever asked to participate in the beauty contest game, (it has been played in some major newspapers, and is a favorite of experimental economists), then you should not give the correct answer, zero. Any fairly low number will have a better chance of winning provided there are some “bad” players participating.
The beauty contest game has also been played in a two-person format, and here the evidence is even more striking against rational behavior.3 In this game, picking zero is a dominant strategy because the number that is two-thirds of the average will always be the lowest out of the two answers, in which case zero is better than any other number. (This is because the two answers are first averaged, taking the midpoint between them. That average is then multiplied by 2/3, and this result will always be closest to the lowest number.)
So in this version of the game, unlike the n-person game where you want to operate exactly one level more than average,4 zero is the only rational answer to give. And when this game was played with students, under 10 percent of the players gave the optimal choice of zero, and in a sample of economics and psychology professors, less than 40 percent did. Therefore, since so few people managed to give the correct answer in quite a simple game, this indicates an exploitive approach will generally do well in the more complicated game of poker.
The Nash Equilibrium
Success in the guessing game and in exploitive poker is achieved by operating exactly one level above your opponents. But what happens if the other players are also attempting to operate one level above you? If the guessing game is repeatedly played against a group of such players, then their average answer should decrease every round as each player attempts to outmaneuver the other. Soon enough, their answers should converge on zero, at which point nobody will have an incentive to change their guess.
This acknowledgment lies at the heart of GTO play. Whatever exploitive strategy we pick, a clever opponent will have the incentive to use the strategy that operates one level above ours. On the other hand, the GTO player is seeking to preemptively cut out this entire process of leveling and out-leveling, and by playing in a certain “optimal” way, there will be nothing an exploitive player can do to gain an advantage.
Evidence from the guessing game shows that, as the game is repeatedly played, peoples answers do converge towards the GTO answer. And even though most people are unable to think through the number of steps of reasoning required to pick zero initially, they are drawn to it as they learn more by playing. They may not be able to reason that the optimal answer is zero, but they are drawn towards it as the game is repeated. If poker is like this, then the play between two expert players should approximate GTO play. Even if people are not consciously attempting to play optimally, it’s likely that, over time, the play at the highest levels will approximate optimal play as the “worst” players gradually lose all of their money and the best players are left with a strategy that approximately breaks even with each other.
The theory of how to optimally play two person zero-sum games5 such as heads-up poker has been known for a long time,6 but the application of these ideas was incredibly sparse. Few people knew how to approach poker in this way until the 2006 publication of Bill Chen and Jerrod Ankenman’s The Mathematics of Poker provided us with the GTO players’ manifesto. For example, in order to play optimally on the river, they show that our proportion of bluffs to value bets should exactly mirror the odds our opponent is receiving to call. Betting $10 into a $90 pot, our opponent receives odds of 10-to-1: Therefore, we should bet ten times as many hands for value as we should on a bluff.7 (Betting in this ratio equalizes our opponent’s EV between calling and folding a hand that can only beat a bluff because the chances of winning the pot with a hand like this are identical to the offered pot odds. If both options are equally good, then there is no potential for exploitation by picking one over the other.) It has been conclusively proved that there will be at least one optimal way to play any finite game involving any number of players, and when each player is following his optimal strategy, the game is in a Nash equilibrium. 8 The concept is akin to the concept of equilibrium in chemistry, whereby two opposing chemical reactions cancel each other out to create a stable chemical mixture. The definition of a Nash equilibrium is that each player is maximizing his payoff given the actions of the other player. For this reason, there is no opportunity for either to gain EV by attempting an exploitive move as each player is already optimizing his payoff. If only one person is playing GTO, then the game is not in a Nash equilibrium — both players have opportunities to implement a higher payoff strategy, thus the game is not in a steady state.
Many zero-sum games are also symmetric, which means that the rules are the same no matter who is playing. Obviously, if the rules are the same for each player, then in order for there to be a Nash equilibrium, each player must be following the same strategy and have zero-EV (before any rake is removed from the game). Due to the rotating blind structure, poker is a symmetric game. This means that in the long run, players in a Nash equilibrium should break even against each other (even though on any individual hand, the player with position will be plus-EV at the expense of the other, but on the next hand the situation will reverse for net zero-EV).
In many simple zero-sum symmetric games, a player following his optimal strategy can do no better than break even. For example, in the children’s game Rock, Paper, Scissors (where scissors beats paper, paper beats rock, and rock beats scissors), the optimal strategy is mixed randomly between the three options with equal probability. This guarantees the optimal player will not lose, but it also guarantees he will never win either, no matter how bad a strategy the other player chooses. For instance, against a player who plays rock on every move, the optimal player will have no score on (rock, rock), will win on (rock, scissors), and lose on (rock, paper), leading to a break even score over time.
14 Part One: Strategy
It must be stressed that poker is not like this. Optimal play will be plus-EV against every non-optimal strategy our opponent uses, and the less optimal they play, the greater our expectation will be. (However, if your opponent does make errors, then optimal play will win less than an exploitive strategy.) And the only time an optimal player won’t win is when his opponent is also playing optimally. In this case, the players will each expect to break even in the long run and neither will have an incentive to switch strategy — they will be playing in a Nash equilibrium.
Simplifications
Even though the rules seem simple, poker is an incredibly complicated game, and the optimal strategy in even relatively simple games such as heads-up limit hold ’em is still unknown. But GTO players attempt to work around this problem by making a number of simplifications.
The well known heads-up limit hold ’em robot Polaris 9 approaches this problem by putting hands into “buckets”10 before analyzing its strategy. For example, on certain boards, pocket pairs 22, 33, and 44 might be analyzed and played as one individual hand. Also, A♣A♦ might be grouped together with other combinations of pocket aces, such as A♠A♥ on a board without relevant flush draws.
The group behind Polaris estimate that there are 1018 (the number 10 followed by 18 zeros) possible game states in the game of heads-up limit hold ’em. Using their simplification methods, they have been able to find the optimal strategy for a version of the game with 1012 game states (although still a very large number, 1012 is a million times smaller than 1018). The full game is so vast, they estimate it would take them 100,000 weeks to solve it if given sole access to Google’s computing power.11
An approach that human GTO theorists use is to construct a “model” of poker which will reflect a certain aspect of poker play while removing unnecessary complications. Chen and Ankenman’s approach is to replace each player’s hole-cards with a decimal number between zero and one. This models the strict ranking of poker hands at showdown. For example, the fact that a flush beats a straight which beats two pair could be represented by 0.8 > 0.6 > 0.4, while removing the complicated interaction between hole and community cards in hold ’em.12
The Two Schools are
More Alike than You Think
There is a great divide between the two schools of poker theory. Exploitive players often view GTO players as mechanical, uncreative, and accuse them of leaving money on the table against bad players. Many GTO players see themselves on the brink of a new wave of poker theory, and state that the game is now evolving so rapidly that some old-school exploitive players are now struggling to compete at the highest levels.
It’s my opinion that this debate is somewhat misinformed. The truth of the matter is that working through an exploitive line of reasoning is absolutely vital in the construction of optimal strategies. Likewise, an understanding of optimal play can greatly enhance the efficacy of an exploitive strategy. I see the two styles as mutually complementary, and in today’s increasingly competitive online environment, excelling in both areas should eventually become a necessary condition for success.
Chen and Ankenman explain the process of working out an optimal strategy with the theoretical construct they call the nemesis. 13 Just as it’s theoretically possible to play perfect optimal poker, it’s also possible to play perfect exploitive poker — this is how the nemesis plays. Naturally, perfect exploitive poker depends entirely on the strategy of the other player, and it’s assumed that whatever strategy you play, the nemesis will instantly utilize the most profitable counter-strategy. Therefore, the only way to stop the nemesis from beating you is to play optimally; any non-optimal play will instantly be exposed. Notice that a player skillful enough to play perfect GTO is the only person capable of not losing against the nemesis.
The nemesis is a useful construct in explaining the link between exploitive and GTO play. Playing optimally is very much a process of inverting the standard mode of exploitive reasoning; instead of deconstructing our opponents and out-playing them, we must consider how an all-knowledgeable player could out-play us and then seek to play in such a way as to minimize that player’s EV.
Top exploitive players are very good at outplaying their opponents, all these players need to do in order to become excellent GTO players is to invert the process of exploitation and work at creating a style of play that even they cannot exploit. This will improve the exploitive players results against players who are as good or better than them. Furthermore, in today’s game, there is no use in winning money off the worst player at the table if you will lose most of it back to your other opponents.
GTO players can also gain a lot from a full appreciation of the other style of poker. Game theory players work hard at creating a cohesive strategy immune to exploitation, the corollary of this is that it becomes easy for these players to spot mistakes in their opponents. Once spotted, the GTO player can then find opportune moments to increase their EV by deviating from their standard strategy. The advantage that the GTO player has in this respect is that he should have an excellent awareness of what is, and what isn’t, exploitable. The danger that purely exploitive players face is that of tilting at windmills. They might spot their opponent doing something unusual and attempt to take advantage, but if this unusual play is a part of a well rounded and balanced strategy, then there might be no such opportunity. In this case, the very act of attempting to take advantage will push the exploitive player further away from equilibrium and theoretically cost him money.
These are two equally good ways to become a complete poker player: start off with whichever style of play you are naturally inclined towards, and then gain a full appreciation for the other one. Instead of seeing these two styles as mutually exclusive polar opposites, I think it’s important to appreciate how mutually beneficial they can be.
The GTO player benefits from this process by increasing his win rate against bad players. At the highest levels, it’s becoming increasingly rare to have more than one obviously bad player per table. And when you are at a table with four other professionals and one fish, it’s important to win as much as possible from the fish, which is only possible with strong exploitive poker.
The exploitive player benefits most from this process when facing players at least as good as him. In this case, there will be few, if any, moments of opportunity, and the player should first focus on not losing by adopting techniques from the GTO school. As previously shown, against the nemesis or anyone better at playing the exploitive game, the most that can be achieved is to expect to break even, and so the rational course of action is to attempt to play optimally. This is vital at the highest levels of the game, as even the best players can expect to often be involved against similarly skilled or better opposition.
There’s no question that perfect exploitive poker (the nemesis) is the best possible way to play. Against non-optimal players, this style will win more than any other, which is, after all, the reason we enter poker games. A perfectly optimal player is the only type the nemesis will be unable to beat, and in this case the nemesis will end up also playing optimally, and a Nash equilibrium will be the result. Thus you can see that, no matter what situation, the perfect exploitive player will do at least as good and almost always better than the perfect optimal player. However, in reality, it’s not plausible to play on this level since it requires a degree of intuition and cognition not humanly possible.
Note: The above analysis is not totally accurate in multiway pots.
Interestingly, there is a poker robot, called Sonia, that has been programmed to play heads-up limit hold ’em in this way.14 Her results have been incredibly impressive so far: averaging
in just under 900,000 hands against humanity. She seems to have an edge on everyone, often by taking bizarre strategic options that no winning human would consider. She also seems to have a very slender edge on the 2008 version of Polaris15(remember, Polaris is attempting to play optimally, but has to make a number of simplifying assumptions about the game). Sonia’s approach is an efficient way to build a poker robot: she has no pre-set strategy, so there is no need to specify her play on every branch of the vast game tree, something that makes GTO robots exceedingly hard to program. Her initial play in a match is based on her beliefs about an effective strategy against humans based on a large sample of datamined hands. She updates her play very rapidly against a new opponent, often shifting to a specific, highly exploitive strategy within 50 hands. It’s also interesting to watch what happens when Sonia is involved in a heads-up match versus herself. After many hands, the eventual strategy she settles on will be her best estimate of GTO play (as an equilibrium is defined as two strategies that fully exploit the other).
Sonia’s play creates a lot of interesting talking points, but I don’t think attempting to emulate her is a credible option for humans: her play is simply too far out on the edge for us to copy, even for a relatively simple modern poker variant like heads-up limit hold ’em. Instead, I think a much more practical method for players aiming to reach the upper strata of the modern poker world is to fully integrate the two schools of poker. Doing so will enable you to take full advantage of the increasingly rare profit opportunities at the highest levels, no matter your background. Put another way, the best player in the world can probably afford to use an entirely exploitive strategy. But for the rest of us, combining the two schools is a much more plausible route to the top.
The balance between GTO and exploitive play that you use at the table will depend on which game you are playing. Relatively simple games, such as heads-up limit hold ’em, place more emphasis on knowing and implementing GTO play; whereas more complicated games, such as deep stack big bet, allow more opportunity for exploitive play.
For instance, if we were playing a simple game such as three-card poker, (see “Appendix B: Game Theory,” starting on page 383, for a three-card poker model), then the GTO play would be known by everyone and any attempt to use exploitive poker would result in losses. At the other extreme, the full GTO solution will likely never be known in complicated forms of poker, and so the top players have more scope to exploit weaker players and less of an opportunity to fall back on unexploitable play.
Backwards induction is the method for solving finite sequential games of perfect information such as chess, go, or tic-tac-toe. Starting from the opening move, you draw a game tree —a diagram that represents each possible move a player can make as a single branch. The tree grows with each move until it eventually covers each possible course of play. Starting at the end branches, you prune the tree by deleting each sub-optimal decision in every round, and when you finish this process by finally working back to the opening round, the only branches left will represent the Nash equilibrium.
This method can be easily used to solve simple perfect information games like tic-tac-toe, and in principle it can also be used for more complicated games. However, both chess and go are too complex for today's computers to solve by employing this technique.
See the box on page 23 for the backwards induction solution to “The Entry Game.” In this game, a monopolist firm, the “Incumbent,” owns an industry to itself. A rival firm, the“Entrant,” is considering whether to enter the industry. However, the Entrant is worried that if it does enter, then the Incumbent will respond by aggressively cutting prices, labelled by the strategic option “Fight.” If the Incumbent doesn't fight, then it can choose to allow the rival firm to enter the industry, labelled “Accomodate.” The payoffs are shown in the table below. The Entrant must use backwards induction to predict the Incumbent’s play in order to find if it should enter or not.
However, backwards induction cannot be used to solve poker since it’s a game of incomplete information — each player’s holecards are known only to him. This means you are uncertain about which branch of the game tree play is on. However, the more information about your hand you reveal, the easier it becomes for your opponent to pick his most profitable play since he can eliminate certain branches from consideration, and this is why information hiding is important. (In the example above, if the Incumbent could not observe the Entrant’s play before acting, then the Entrant would be unable to predict that the Incumbent would observe its action of “Enter” and would “Accommodate” as a result.)
24 Part One: Strategy
The classic example of revealing information is through tells— some mannerism that indicates the strength of your hand.16 This is less of a factor in online play, although if your speed of decision is correlated with your hand strength, it can reveal information to an observant opponent.
Your choice of action can also reveal information. Suppose a freeroll qualifier overbets the pot, moving all-in preflop during the first level in the Main Event at the World Series of Poker. Busting out of the first level of this prestigious tournament would be a massive disappointment for this player, so using this information wisely, we can surmise that he must be holding aces the only hand he wouldn’t be afraid to make this play.
As we will see, information hiding is a key concept in all poker variants. The idea is that it’s beneficial to play parts of our range in a consistent manner during early decision points. We do this in order to conceal the strength or weakness of our hand until later decision points where the bets and pots can be larger and thus more meaningful.
As a broad illustration of the concept, suppose we put many more bets into the pot on the opening round with our strong hands than with our weak hands. This is doubly disadvantageous since our strong hands lose the element of surprise, and furthermore, our weak hands are inviting our opponents to put pressure on us during the later streets because they are no longer provided cover by our better hands. Therefore, early on, both sets of hands benefit from being played in a consistent manner.
This concept is most vital for limit games where the cost of giving our hand away on the opening round is seldom equal to the value of getting an extra bet in relation to the size of the pot. In big bet games, particularly with low stack to pot ratios, the concept is less relevant, although, in deep stack play it once again becomes a major factor.
The earliest articulation of this concept that I’ve found is in Nesmith C. Ankeny’s Poker Strategy, originally published in 1981. This book presents a game theoretic approach to five card draw, jacks or better. On the opening round, players are allowed to open with a bet only if they hold a pair of jacks or better, otherwise they must pass. (Once a player opens, you must either fold, call, or raise; if all players pass, then the next hand is dealt.)
Ankeny’s information hiding rule is very simple: never draw two cards. Suppose, in an eight handed game, the under the gun player opens. Due to the rules of the game, we can say that he must hold at least a pair of jacks. However, due to his early position, his minimum opening hand is probably a pair of aces. We raise from middle position, the opener calls, and we enter the draw heads up, at which point we draw two cards. What is our range of hands? We must hold a made hand since drawing two cards to a straight or flush is suicidal, we cannot have two pair, and one pair is not strong enough to raise since our opponent’s hand is at least a pair of aces. Simple hand reading concludes that our only possible hand is three of a kind. In a situation with wider ranges, we might possibly want to draw two cards with a pair and a high kicker. However, these are the only hands that will be motivated to draw two cards.
Ankeny’s solution is to draw only three, one, or zero cards. Drawing three cards admits to holding only a pair before the draw implying a relatively weak distribution of hands after the draw. Drawing one card results in a wider pre-draw distribution of straight and flush draws (without a pair of jacks or better, these hands are unable to open the pot), two pair, three of a kind, and the unlikely four of a kind. This is a nicely balanced distribution of hands that is able to make a range of value bets and bluffs after the draw.
Note that the cost to three of a kind of drawing only one card is minimal. The chances of improving to a full house or better is only 1.9 percent less when drawing one card rather than two,17 18 and the benefit of hiding the hand’s strength in a wider distribution of hands far outweighs any reduction in post-draw expected hand strength.
Another early expression of the concept was in David Sklansky’s The Theory of Poker. In “The Cost of Giving Your Hand Away” he says:
If opponents know exactly what you have, they will never make a mistake except on very close mathematical decisions. The more your play gives away what you have, the less likely it is that your opponents will make a mistake. Yet you want them to make mistakes. Creating mistakes is, in a sense, the whole objective of the game. 19
Poker would be a trivially easy game if every player’s holecards were face up on the table. Uncertainty about what hands other players are actually holding is what makes the game interesting and difficult to play well.
The first usage of the term “information hiding” that I can find is in Bill Chen and Jerrod Ankenman’s The Mathematics of Poker. They say:
Strong strategies play many hands in the same way, making it difficult for the opponent to read the distribution of hands remaining. Hiding information is the most important part of avoiding counter-exploitation. 20
As they explain, the reason for hiding information is to prevent opponents from exploiting us. As was shown in the last chapter, the way to play “game theory optimal” (GTO) poker is to reduce, as much as possible, the profit an all-knowledgeable super opponent (the nemesis) would be able to extract from us through exploitive poker.
The Ankeny example illustrates this concept perfectly. Suppose we draw two cards against an early position raiser. Since our pre-draw hand must be three of a kind, a super opponent could deduce that our post draw distribution is exactly {four of a kind (4.2 percent), full house (6.2 percent), three of a kind (89.6 percent)}.21 With such intimate knowledge about our range of hands, the nemesis would be able to play perfectly on the final round. This would include letting weaker hands go and knowing exactly how many bets to put in with hands that do well against that range, or he might conclude that his starting hand is not strong enough to continue against that range, given the bet and pot size, and fold immediately.
Chris Ferguson, winner of the 2000 WSOP Main Event, is another poker theorist who strongly believes in the virtues of information hiding. In The Full Tilt Poker Strategy Guide he writes:
To conceal the strength of your hand, you need to playdifferent hands the same way. Once you decide you are going to play a hand, make the same bet whether it is the strongest hand you would play in that situation, like AA, or the weakest, like 76. 22
Ferguson’s advice is for the game of no-limit hold ’em, but you can see that it’s exactly the same principle at work as the one behind Ankeny’s rule in five card draw. Playing a variety of different hands the same way benefits every hand since it increases the likelihood of gaining additional action with strong hands, and weak hands are provided cover to win uncontested pots.
Note that playing different hands the same way is a much more efficient way of concealing information than playing the same hand different ways. That’s because in practice, playing the same hand different ways is immensely difficult to do right. Technically, playing a hand different ways is known as a mixed strategy, and in order to play a mixed strategy correctly, we should randomize perfectly between the different options. But unfortunately, humans are bad at predicting (and therefore mimicking) true randomness. For example, when predicting a string of coin tosses, we predict too many reversals and not enough streaks of either heads or tails.23
In many games, the benefits of raising on the opening round, especially first-in, are enormous. Calling allows players either to isolate you with position or the opportunity to profitably play many hands out of the blinds. This means that in many situations on the opening round, in order to play our hands well, and to hide information by doing so, we should use a strategy of either raising or folding, and never calling. Since the majority of our strong hands would much rather raise than call, it makes sense to hide information by choosing to always raise with a playable hand. In the next chapter, I will explain my strategy for preflop limit hold’em where information hiding is the core concept behind my decisions, and in many situations my advice is to play a strict raise-or-fold strategy.
Here’s an example of information hiding that often arises in full-ring limit hold ’em. You join a table and post a blind in the cutoff on your first hand; the action folds to you. The best course of action is to raise every time. Since it costs just one big blind to raise, all except for the very worst hands will want to automatically raise. Since checking would reveal a tremendous amount of information — it will indicate that your hand is terribly weak — the best play is to always raise.24
Later on in his chapter in The Full Tilt Poker Strategy Guide, Ferguson presents three different flavors of his never-call strategy preflop in no-limit hold ’em.25 Chen and Ankenman take a similar approach to first-in play in no-limit hold ’em. They will only raise-or-fold, varying their raise size solely by position.26
Although the most critical demonstrations of the information hiding concept occur on the opening round, it’s also a key factor during decision points later in a hand. This is particularly evident in big bet games, where, potentially, the size of your bet could reveal information about the strength of your hand. In Winning Strategies for No-Limit Hold ’em, Nick Christenson and Russell Fox describe their approach to this problem.27 They vary the size of their bet by a number of factors, including the pot to stack ratio, the number of players, position, board texture, and their opponent’s hand range. Once they have decided on an appropriate bet size in a particular situation, they will bet the same amount with every betting hand. This is a textbook application of information hiding. Since they will be playing a number of different hands in exactly the same way, there should be no additional information an opponent is able to glean from the size of their bet. Put another way, if the size of our bet was correlated with our hand strength, an opponent could exploit us. Instead, by always betting a consistent amount, bluffs will look exactly like value bets, and both types of hands should benefit from being played this way.28
The importance of information hiding is affected by two factors: the number of betting rounds remaining, and the strength of your opponent’s range. In extremely short-stacked no-limit hold’em, there are still four notional rounds of betting. However, effectively, there is only one round of betting as players will either move all-in or fold preflop. In this game, there is no room for information hiding, as players move all-in and call all-ins to maximize EV.
The stronger your opponent’s range, the more information you give away by splitting your betting into two parts — raising some hands and calling with others. Against a range of AA, KK, and AK, the only hand with above 50 percent equity is aces. By raising on an early round against such a tight range, you would completely give away that your hand is aces; it would be much better to play the aces deceptively until a later street. At the other extreme, against someone who is playing every hand, you can raise with some hands on the opening round, and call with others, since the range of hands that you can profitably raise is very wide.
Hopefully, you can fully appreciate the importance of this concept in poker. I’ve shown applications of it in limit and big bet hold ’em, and five card draw, and it’s set to reoccur in the later chapters. Hiding information is an important part of GTO poker, but we can always flip the concept around and attempt to exploit players who fail to play their range of hands in a consistent manner.
There is also another advantage to making our strategy simpler through information hiding rules such as never calling first-in. By doing so, we make the game of poker easier to play since there are many possible scenarios which will never arise once we begin to restrict our strategy. In the first chapter, I explained how even the simplest popular forms of poker are far too complicated for the fastest computers to solve today. But by never doing certain things, we transform the game into a slightly less complicated version which makes the job of playing well a little bit easier for ourselves.
Everything should be made as simple as possible, but not simpler.
— Albert Einstein
The last chapter highlighted the importance of playing many types of hands in a similar manner. This chapter will explore how this concept lies at the heart of my style of playing the opening round in my favorite game: limit hold ’em. My main emphasis will be on tough six-handed or shorter games which are by far the most frequent games online.
The opening round is a logical place to begin our discussion of actual play. The universe of possible situations is much smaller than on the later rounds. And it’s also the only round we are forced to consider every hand dealt.
In addition, limit hold ’em is a useful game to analyze opening round play. It shares an identical blind and seating structure with a number of other games popularly played in 8-game mixed tables: no-limit hold ’em, pot-limit Omaha, limit 2-7 triple draw, and limit Omaha eight-or-better hi/lo.
Compared to the big bet games, limit hold ’em removes the complication of how much to raise. This will make the analysis simpler since at each decision point we will only need to consider three strategic options (call, fold, or raise), instead of a continuum of options (call, fold, or raise to x). So far, in the process of attempting to play unexploitably (“game theory optimally,” or GTO), I have stressed the importance of beginning with simple games — keeping the complexity of the problem on a manageable scale. And once done with the analysis of the simpler games, we can see which features of our strategy will, and which won’t, carry over into the more complicated game.
There are three other types of poker played in 8-game mix: seven-card stud, stud eight-or-better hi/lo, and razz. All of these games follow limit betting, and they all have the basic structure of seven-card stud — the principal difference being the ranking of poker hands. The opening round in these games, called third street, is similar to the other limit games: One player posts a blind bet (called the bring-in, this is in addition to each player anteing), and then, starting from that player’s left, each other player chooses to call, fold, or raise in turn. Due to these similarities, it should be fairly easy to apply a system, akin to my preflop limit hold ’em approach, to third street in these games.
I will not present a discussion of how to play individual hands, nor will there be discussion on whether a particular hand, such as the
is strong enough to play in a particular situation, such as under-the-gun. There are far too many complicated, interacting factors at play to answer such a question. Instead, my focus will be on the level of your overall strategy. In a certain situation, which hands to play will not be addressed; instead, I will be looking at how to play your hands and once my overall strategy is presented, I will leave it to you to answer the question of which hands are strong enough for a certain situation.29 This will of course depend on a number of game specific factors such as the rake and quality of opposition in your games.
Playing many different hands the same way is fairly easy to do in limit hold ’em. At any decision point, there will only be, at most, three possible options. (Call, raise, or fold, usually. If we are in the big blind in a limped pot, we will only have two undominated actions: check or raise. If the pot has already been capped, we can no longer raise and must choose between calling and folding.)
Ignoring the hands we fold, there are only two ways to play preflop: calling or raising. In the last chapter, we explored the benefits of playing many different hands the same way since doing so prevents opponents from reading our strategy effectively on later rounds. Carrying this idea to the limit, the absolute nirvana for hiding information is to play every hand the same way. We therefore have a two-step procedure for playing hands preflop:
1. In a certain situation, decide whether we should carry all of our playable hands to the next round by either calling or raising.
2. Decide which hands can be profitably played in this way.
Note how this simple procedure vastly reduces the complexity of preflop play since we are cutting each decision point from a choice between three available options to a choice between only two alternatives.
The simplest situation is when nobody has yet voluntarily put money in the pot. So in this case, how should we open the pot? Should we raise or call? As it turns out, raising is by far the superior alternative in six-max or short-handed games. The superiority of raising is acknowledged by both Bill Chen and Jerrod Ankenman, and Chris Ferguson, who advocate a strategy of never calling first-in.30 Raising puts the most pressure on players behind us and in the blinds, enabling them to play less hands profitably than if we called. Incidentally, finding players who call first-in is a remarkably successful simple heuristic for game selecting in limit hold ’em. Having one player at your table who regularly calls first-in is sufficient to make any game at middle stakes, where rake is less of a factor than lower down, playable.
Now that we have decided to play this situation by either raising or folding, the only thing left to do is to work out the minimum raising hand from each position, and from earlier positions, we will, of course, raise fewer hands than in later positions. Notice that this makes opening the pot strategy on the first round pretty straight forward.
However, restricting your first-in strategy to raise-or-fold may leave money on the table against extraordinarily bad players. For example, in live low stacks full ring limit hold ’em games, it’s common for many players to use an extremely loose/passive strategy preflop, calling with many hands and hardly ever raising. At such a table, we could increase our expectation by relaxing the raise-or-fold strategy, and call first-in when in early position with some hands that play well multiway. This is because the information hiding argument behind raise-or-fold is largely a defensive one. If we give information away by varying our strategy based on hand strength, opponents could exploit us. However, at a table like this, nobody is going to use the information that we call with some hands and raise with others efficiently — they probably won’t even notice. In this case, it’s better to just focus on playing as many profitable hands as possible and not worry about potential exploitation.
But in the modern game, especially on the Internet, this type of table filled with many bad players rarely exists above the lowest stakes. Thus it would be quite unusual to find more than one or two bad players per table, and so it becomes important to play as well as possible against the other good players.
The second most simple situation is where one player has raised, there have been no callers, and we are sat in any position other than the big blind. Given that we are going to do the same thing with all of our playable hands, should our action be to reraise or to call?
To answer this question, we must consider the importance of the player in the big blind. If we call, then he will be receiving pot odds of at least 5-to-1, and if there was no more betting (after this call), it would makes any hand with at least 16.66 percent equity profitable.
Furthermore, under any reasonable assumption about the hand ranges of the first two players, and what the post-flop betting might be, this means the big blind will be able to profitably play an extremely wide range of hands. In addition, this profit must come from the other players in the pot. Also, the raiser should have the benefit of a strong distribution of hands and the initiative most of the time, he will bet the flop, and get a cheap chance of winning the pot without a showdown. But you, the caller, will have poor relative position as you will have to react first to the initial raiser’s bet on the flop. Therefore, it’s reasonable to assume that most of the cost of letting the big blind into the pot cheap will be borne by you, not the initial raiser. Consequently, my advice in this situation is to reraise-or-fold.
Like my rule for playing raise-or-fold first-in, this rule is based on the assumption of rational play from our opponents —they will attack any weakness in our overall strategy. If this is not the case, then it would be reasonable, in some situations, to deviate from this rule. Specifically, if the players acting after you are bad enough, it would be ok to call with some hands (although your best hands would still gain enormously from reraising).
This situation is most likely to occur when you are in the small blind because there is only one player left to act, and as already noted, it’s unlikely to have a table full of many bad players in today’s game. But if the big blind makes poor decisions both preflop and postflop, then offering him attractive preflop odds is not so bad, and the benefit of being able to play more hands than in a reraise-or-fold strategy might increase your expectation. However, your best hands will probably still benefit from reraising preflop, and you must consider the cost of the information gained by the original raiser when he observes you splitting your hand range into two parts.
The next logical situation to consider is when two players have voluntarily entered the pot, the first raised, the second reraised (since we have decided that, in general, it would be a mistake for the second player not to reraise), and we are not in the big blind. You can probably guess by now that my advice is to cap-or-fold. It’s true that letting the big blind in cheap is less of a concern now that the betting is already at three bets, but I feel that showing aggression here preflop will have a number of benefits postflop. Most importantly, it will make your postflop decisions easier as now you will be the “preflop aggressor,” and will get a cheap shot at seeing how your opponents react to your probable bet on the flop. So in the absence of any compelling reason to call, my choice in this spot is to play cap-or-fold.
So far, things have been very simple as my advice at every eventuality covered is to have a basic game-plan of raising-or-folding. However, the big blind hasn’t been mentioned yet, and this is because my advice here is the mirror image of what we have covered thus far. In the situations we’ve been looking at, the choice was to raise because the majority of our playable hands have benefitted from this strategy. Specifically, raising benefits so many of our profitable hands that it’s the natural umbrella to hide information under. But the situation is reversed when you are in the big blind heads-up against an in-position raiser. Getting immediate odds of 3.5-to-1 on a call, the vast majority of your playable hands will be worth a call, but no more. For example, against a button open raise, hands as weak as the
should have enough equity to call and see the flop. On the other hand, the range of hands we could profitably reraise with is much smaller than the range of hands we can make plus-EV calls with. This means that the logical way to hide information in this spot is by always calling.
Most players will always continuation bet the flop if you call in the big blind and check the flop. In “Checking Back the Flop” starting on page 116, I make the case that it makes sense for a raiser with a wide hand range to check the flop occasionally. But if your opponent always bets the flop, then calling preflop and check-raising the flop will get the same number of bets in as reraising preflop, in which case I definitely prefer to always just call preflop. If your opponent does frequently check the flop behind, reraising preflop can become an option. (Whether your opponent should always bet the flop, or not, will depend on the relative strength of each player’s range — early position raisers will hold stronger hands on average and are therefore more likely to want to always bet.)
I play similarly in the big blind against a raise and a reraise. Here, my strategy is again to call-or-fold. (Of course, we must play a much tighter range against two raises rather than one.) Until recently, my strategy was to cap-or-fold because if there is any chance of getting the original raiser to fold, then capping will be the better option since the benefits of creating a heads-up pot with lots of dead money are massive. However, it has become increasingly rare to see players raise first-in and then fold to two more bets, and this is an example of how the general standard of play has increased in recent years.
In addition, it’s my opinion that a strategy of putting two bets in preflop and frequently folding for two more could be exploitable.31 It’s also worth remembering that when attempting to play GTO poker, we are looking for a strategy that will maximize our expectation under the assumption that our opponents will in turn pick whatever option maximizes their expectation against our candidate strategy. So if raising first-in and then frequently folding for two bets more is a large mistake, then an optimal strategy of ours shouldn’t be expecting our opponents to ever make such mistakes. And if we can’t expect a rational opponent to often fold to a cap, then I like the benefits of calling in the big blind more than raising.32
Another advantage to just calling is that by putting less money in the pot, we can afford to play a wider range of hands and also get to see how the initial raiser reacts when there is a reraise. (Does he cap, or just call? Usually, if he only calls, you can remove the strongest holdings from his distribution of hands, which is useful information for postflop play.) Furthermore, since we are in the big blind and are the last person to make a decision preflop, there is no danger of letting other players in cheaply with just a call.
In summary of the rules discussed so far:
Exploitive Adjustments
Until now, due to the symmetry of poker, I have not discussed how to play after a player at your table has broken one of these rules. If it’s optimal for the player first-in to raise-or-fold, then this is the situation of primary importance for the construction of a GTO second-in strategy.
Suppose these rules do constitute the essence of unexploitable preflop play. If the players who have voluntarily entered the pot so far have all played in accordance with them, then the best you can do is to stick rigidly to the rules. That’s because your opponents have not, as yet, made an exploitable “mistake,” and so your expectation is maximized by playing unexploitably yourself.
On the other hand, if an opponent were to make a mistake, then this creates an exploitable moment of opportunity to potentially increase your expectation by responding in an exploitive manner.33 Of course, not all “mistakes” are equal. In order of importance, calling first-in and cold-calling a raise instead of reraising (or folding) are the biggest errors.
In addition to flagrant errors such as calling first-in and cold-calling raisers, mistakes can also be made in the selection of an inappropriately tight or loose range of hands for a given situation. But this is beyond the scope of this chapter: The complexity of covering the range of potential situations is enough as it is, and discussing individual hand selection at each juncture would be too great a task.
Another point of importance is that hiding information often becomes less of a concern after an opponent has made a strategic error. That’s because we can now focus on exploiting their mistakes instead of being purely fixated on following a GTO strategy, and it can be profitable to split your range of playing hands into two parts. For example, after a raise and cold-call, you could potentially maximize your EV by calling some hands and reraising others since the callers mistake is sufficient compensation for the fact that you are giving information away by your choice of action.
One spot where I still recommend playing a consistent range is when one player has called first-in and you are not in one of the blinds. Raising is an incredibly attractive play here; the opportunity to get heads-up and in-position against a weak player is worth a lot. Therefore, only a very small range of fairly weak hands would prefer calling over either raising or folding. Plus calling overly defines your range, and it signals that your hand is, at best, not that strong. So I think playing raise-or-fold here is best (assuming the players left to act are good enough to exploit the information that a call signals).
A spot that I, for a long time, found difficult was where one player had called first-in and my position was the small blind. After coming to appreciate the benefits of hiding information preflop, and knowing how good it was to raise after a player had called first-in, I tried to play a strategy of raise-or-fold which lead to me raising an extremely wide range of hands. Surprisingly, this strategy didn’t do too badly, for a while. That’s because at that time the average player in the big blind did not call my raise even close to the correct amount — at a cost of only one small bet, he should play nearly every hand — and the player who called first-in would normally play far too tight/passive postflop.
However, I was ignoring the simple mathematics that when calling costs just half a small bet and you’re getting immediate pot odds of 5-to-1, just about any hand is worth at least a call. Yes, raising with your strong hands is important as this charges the bad player for his mistake and can knock the big blind out, but when you can profitably exploit the initial player’s preflop mistake by calling hands as weak as the
you should do so. Furthermore, since calling first-in is such a large mistake in this game, when you are in the small blind you should attempt to maximally exploit it by calling with many hands and raising with your strongest.
Blind-Versus-Blind (BvB) Play
Until now we have not addressed the situation where the action has been folded around to the small blind who must now contest the pot heads-up and out of position against the big blind. Play here can be much more flexible, and I did not want to begin exploring the many shades of grey involved in BvB play until the black and white case of first-in play from other positions had been covered.
The key rationale for not calling first-in from earlier positions is that doing so lets a table full of players into the pot cheaply. Good pot odds allow them to play a lot of hands profitably, and profitable play for your opponents will frequently be costly to you. But in BvB play, a lot of this doesn’t apply — there is only one player left and the most we can cut his odds down to is 3-to-1. Therefore, whatever we do, it will be correct for him to play a lot of hands preflop. The argument for calling first-in from the small blind is strongest in the two-thirds blind structure. In games such as $15-$30 and $75-$150, the small blind is two-thirds the size of the big blind, and this encourages the player in the small blind to play more hands. And as before, my preference is to hide information by playing every worthwhile hand in a consistent manner. However, if the small blind chooses to raise, then due to the larger small blind he will naturally want to play an extremely wide range of hands.
Now consider the big blind’s situation: If the small blind is raising the vast majority of his hands, then why fold anything getting 3-to-1 odds heads-up in position? But if the big blind should never fold a hand after the small blind’s raise, then much of the argument for raising is lost. There is nothing the small blind can do to prevent the big blind from playing, so why not call instead? Remember, in this structure, calling is extremely cheap.
Against an opponent who never folds, calling also has some benefits. First, it enables you to play more hands since you are putting less money in the pot. Second, if you call-or-fold first-in you get a cheap chance to gain information about your opponent’s hand. Specifically calling gives you immediate odds of 5-to-1, but if your opponent raises, he can cut those odds down to 3-to-1. Consequently, your opponent will want to raise his better hands since he will think there is value in doing so.
Another interesting point is that the information you gain from calling comes when your opponent doesn’t raise. Since his incentive to raise with strong hands is great, if he checks, then a large number of good hands are no longer in his range: You have acquired useful information which will help to win a lot of pots uncontested postflop. However, since you are playing call-or-fold, all of those strong hands are still in your range which helps to keep your opponent in the dark. That is, he will frequently have to give you credit for having the strongest hand, particularly on certain boards (such as ace-high flops).
The argument for calling first-in from the small blind rests on the fact that your opponent shouldn’t be folding any hands if you raise. If he does, in fact, fold at all frequently, then raising is probably the better option. This is another example of choosing an exploitive play over a GTO one, and in a two-thirds small blind structure, I will usually raise first-in from the small blind against bad players and call against good ones.
The usual structure is when the small blind is half the size of the big blind. In this case, it’s hard to tell whether raising or calling first-in is better. The smaller blind means that a raise-or-fold strategy will be playing less hands, in which case the big blind should probably fold a few of his weakest hands to a raise. Hands such as the
and the
are probably not worth a call, although I don’t have any conclusive evidence for this claim. So if the big blind is folding some hands preflop to a raise, then I’m strongly inclined towards the merits of raising. Note also that the informational benefits of calling in the small blind and having the big blind check are less than in the alternate structure. Your initial calling odds of 3-to-1 are identical to the odds you will receive after a raise, so there is less incentive for the big blind to raise quite so many hands after a small blind call. I know some good players who raise first-in form the small blind, and some who don’t, and, as just stated, it’s hard to tell if there’s much difference between the two.34
Big blind versus small blind play also has some key differences to other preflop situations. The stronger your opponent’s range of hands, the more important information hiding becomes as a concept. This is due to the assumption in GTO play that our opponents will always respond to our strategy in a way that maximizes their EV. For instance, we cannot reraise a weak hand “for deception,” or to “level” an opponent into incorrectly folding a better hand. If we reraise a weak hand against a strong range, then our opponent has a profitable counter-strategy he can use. So in order to optimally reraise for value against a strong range, our hand must be at least as strong on average.
If we can also expect the small blind to raise first-in with a wider range of hands than any other player at the table, this makes hiding information much less of a concern for the big blind when he is up against the small blind rather than against a raiser from an earlier position.
Heads-up against the small blind is the only situation where position is working for the big blind instead of against him. If the big blind were to reraise preflop against an early position raiser, then not only should he choose hands at least as strong as the initial raiser’s average hand, but he should select an even stronger hand to compensate for his postflop positional disadvantage. But against the small blind, the big blind can profitably reraise hands slightly weaker than the small blind’s average hand since he now has the advantage of position working for him on every postflop street.
In combination, position and the small blind’s wide range of hands give the big blind sufficient compensation to give away some information preflop by raising some playable hands and calling some others. When splitting your range of playable hands into two, it’s important to hold some hands that are capable of hitting any kind of flop in each range. In particular, make sure to call preflop with some hands that contain an ace since, otherwise, your calling range would be weak on ace-high flops.
Capping
What should we do if our first-in raise gets reraised? Due to the overwhelming pot odds, folding is not an option; our choice is whether to cap or call the extra bet. And in order to profitably reraise against us, our opponent will need a strong range of hands (depending on what position we raised from). This means that hiding information will be an important factor in our play. But we cannot optimally hide information by always capping; much of our range will be weaker than our opponent’s. Thus, the best way to hide information is to never cap heads-up preflop.
As an extreme example, suppose the under-the-gun (UTG) player in a ten-handed game raises, the next player (UTG+1) reraises, and the rest of the table folds. If both players are behaving optimally, UTG should have a strong hand to raise in front of so many players. Knowing this, UTG+1 should have an even stronger hand on average to reraise. Suggested ranges for UTG and UTG+1 are shown below:
Table I: Opening Range for UTG
Table II: Reraising Range for UTG + 1
Due to the strength of UTG+1’s range, when the action folds around to UTG, there are very few hands that could be capped profitably — the only hands with over 50 percent equity are AA, KK, QQ, JJ, and AKs. The extra theoretical fraction of a preflop bet gained by capping is not sufficient compensation for the cost of defining your range so narrowly, and capping such a tight range will also have undesirable side effects for the rest of your hands. That’s because when you refuse to cap, your opponent can surmise that you now cannot hold one of these premium hands, and the range of hands you are left with will be much weaker. Thus your opponent should be able to ferociously bet for value on the later streets.
Table III: Potential Capping Hands
Table IV: Potential Calling Hands
In this example, being out of position against such a strong range of hands made never capping the only sensible option. In practice, your opponent’s range will almost always be weaker than this, providing you with the potential to increase your EV by capping some hands (in such a way as to keep your hand ranges balanced)35
Personally, I follow a strategy of never capping heads-up preflop, in any situation.36 This has the main benefit of making the game much simpler for me to play since I’m always waiting at least until the flop to reshow aggression, which is an easy way to hide information. Note that a preflop cap, by itself, will never be enough to force an opponent to fold — to do so, it will be necessary to follow it up with at least a bet on the flop.
However, I’m not so dogmatic as to believe that this is the only admissible strategy (assuming you are playing against a wide range). When constructing a strategy that utilizes capping, it’s important to retain enough strength in your non-capping range in order to minimize the information you reveal. One way to do this is to cap suited hands while calling with their offsuit counterparts. For example, you could cap hands like AKs, AQs, KQs, and QJs, and call with AKo, AQo, KQo and QJo. Suited hands are dealt only one-third as often as offsuit hands which means that you will be capping infrequently compared to calling, but there are still a lot of strong hands in your calling range. On the other hand, the worst capping strategy you can pursue is to cap a very narrow range of all your extremely strong holdings, making it obvious what your hand strength is when you cap and when you don’t cap.
Continuing with our initial question, notice that the case for never capping is strongest when heads-up out of position. Your opponent will almost always bet the flop after your preflop call, in which case you can check-raise and get the same number of bets by capping preflop and betting the flop. Also notice that due to your positional disadvantage, it will be necessary to cap with a stronger range of hands than if you were acting last.
In addition, calling and then check-raising has a couple of huge benefits over capping. First, if the flop is really bad, you can abort the check-raise and revert to calling. An example would be on an ace-high flop with a pair of kings when both players have tight ranges. Second, calling preflop reveals less information about your hand by playing your range in a consistent manner.
The calling and check-raising strategy works well because almost all opponents will always bet the flop, allowing you to check-raise whenever you want. Notice that when you never cap, your opponent will have position, a strong range of hands, and the flop bet will be small relative to the pot. All these conditions make nearly always betting the flop correct for him. However, if your opponent’s correct strategy involved frequently checking the flop, then this should make you inclined towards capping preflop as your potential flop check-raises would be thwarted by his checks behind.
Usually, cap-or-call decisions will occur heads-up, but these situations can also come up in multiway pots. If you have already put money in the pot and there are four or more players remaining, then at least one of them must have deviated from my rules for optimal play: either by calling first-in or by cold-calling a raise. My advice here is to not worry about information hiding and just cap with your best hands in order to exploit their errors.37
If the big blind follows my advice of calling-or-folding against two raises, then it’s possible to have a cap-or-call decision three-way, even if neither opponent has made a preflop “mistake.” For example, you raise from the button, the small blind reraises, and the big blind calls. My advice here is to vary your play in accordance with your position relative to the reraiser. If you have position on him (such as when you raised from the button), then the best strategy is to cap with some hands and call with others (finding appropriate hands to balance both ranges with). If the reraiser has position, such as when you raised from the cutoff and the button reraised, my advice is to use a strategy of never capping. That’s because in this spot, the reraiser not only has position on you, but the ranges are tighter meaning that a cap would reveal more information.
Heads-up Games
When there are only two players at a table, the blind structure is usually reversed. The small blind still acts first preflop, but postflop the big blind is out of position. The natural consequence of this is that the small blind can now play more hands than in a BvB battle during a ring game. This is one area where specific advice will be given on which hands to play since the number of possible situations is much more limited.
I used to take the idea of playing many hands extremely far, often raising with all of them from the small blind. That’s because the biggest mistake you can make in heads-up limit hold ’em (providing your opponent is playing close to correct) is to fold too many hands preflop (anywhere above 25 percent is suicidal), and this was a mistake shared by many of my most common opponents. Against this sort of player, raising every button can be a good exploitive move. But this is where I became confused. My enemies had seemingly no way of winning against my strategy, therefore, I concluded that my strategy was a close approximation to optimal play.
However, just because these opponents couldn’t come up with a good counter-strategy, it didn’t mean that one did not exist, and I began to struggle against the better players who frequently reraised out of the big blind and went to showdown often. Remember, optimal strategies are unexploitable while exploitive strategies are themselves exploitable by another strategy, even though they can maximize your EV against bad players.
So a good rule of thumb is to raise approximately 85 percent of your hands on the button. This number strikes a balance between being loose enough, but not having too many extremely weak hands in your range. (Hands such as seven-four offsuit, if played, should bluff each street if they completely miss the board, and so getting your preflop hand selection correct by not playing hands such as this one will move you closer to the correct bluffing ratio on the later streets).
Table V: Opening Hands
Heads-Up from the Small Blind
As for the big blind, despite being out of position, he should play more hands than the small blind. That’s because he’s receiving 3-to-1 odds and can close the action with a call —factors which outweigh the positional disadvantage. Therefore, 95 percent is a good approximate number (see Table VI).38 39 The big blind should also be reraising a wide range of hands; the small blind’s range is very wide, so punishing weak hands is more of a concern than hiding information (and reraising reveals less information against a wide range).
There are a few players who use a never-reraising strategy here. But I think it can be exploited by frequently checking behind on the flop, limiting the number of bets the big blind can get in the pot with his best hands. (See “Checking Back the Flop” starting on page 116 for more on this topic.) However, waiting a betting round to show aggression (when in the small blind on the button) works best when your opponent has a range strong enough to always bet the flop (such as when he has reraised your first-in raise). This isn’t the case in heads-up; the small blind on the button should be raising with a lot of hands preflop, many of which will not want to bluff or value bet on the flop. If when in the big blind your opponent does always bet the flop, then you can use a strategy of never reraising preflop and check-raising the flop, but this is an exploitive move based on your opponent’s mistake of always betting the flop.
Table VI: Playable Hands
Heads-Up from the Big Blind
After the big blind reraises, my preference in the small blind is to never cap. Since the ranges are wide, it’s unlikely a strategy that caps intelligently can be bad. But in this situation, I just prefer the virtue of simplicity that never capping has.
Of course, you can alter these recommended ranges if aiming to exploit your opponent. Against tight players, play more hands from the small blind and fewer from the big blind, while doing the opposite against loose players.
Calling first-in is not nearly as large a mistake in heads-up as it is in ring games. This is because calling in a ring game allows multiple players to profitably enter the pot with a wide range of hands; whereas in heads-up, no matter what you do, your opponent will be able to play most of his hands in the big blind. This may seem like a surprising statement since many of the worst heads-up players do frequently call preflop. But I don’t think it’s the calling per se that causes these players to lose. It’s the likely fact that calling preflop is strongly correlated with a whole host of other errors in their understanding of the game.
Some heads-up robots, such as Sonia, are capable of using first-in calls as part of a strong overall strategy.40 Computers have a huge advantage over humans in this respect, they find it much easier to visualize the whole game tree over different starting hands and board situations, allowing them to extract the most value out of unusual strategic options. For humans, raise-or-fold on the button is the strategy most immune to errors in implementation. And, crucially, the small blind has position in heads-up matches (unlike a ring game), making raising more appealing.
When in the big blind, if your opponent does call first-in, then the first thing you should frequently do is to raise. This allows you to match the pot size with the strength of your hand — a major advantage. However, it’s best to retain a wide range of hands capable of hitting different flops in your checking range, so I would check hands like A2o-A6o and K2o-K7o.
Once you’ve checked in the big blind after your opponent only calls on the button, there are three potential strategies on the flop: always check, always bet, or a mixture of checks and bets. Always checking is a good strategy against a player capable of playing the later streets well. Since you are raising many hands preflop, your preflop checking range is quite weak, and so always checking the flop and frequently check-raising is a good strategy. Another advantage to this strategy is that many opponents will then reveal information by whether they check the flop behind or bet.
Always betting is a good exploitive strategy against many poor players who fold far too frequently on the flop. After betting the flop and getting called, you can then balance your turn play by checking a wide range of hands, balancing across all three options of folding, calling, and raising. However, like all exploitive strategies, always betting the flop is itself susceptible to counter-exploitation.
Conclusion
Returning to the quote I opened this chapter with, I believe my preflop limit hold ’em strategy is “as simple as possible, but not simpler.” Simplicity lends it many virtues:
make. Consistent decisions on the later rounds are made much easier by having the first (relatively simple) round figured out.
Polaris utilizes two strategic options that I recommend against using in heads-up limit hold ’em: calling on the button and capping. There are two reasons for this recommendation:
1. These options are easier to misuse than to add value with, and
2. By not using them, you can easily construct a strategy that elegantly hides information.
Since these options can be used profitably (by Polaris), and we choose not to, then we subject ourselves to a small theoretical cost. However, if anyone can show us how to incorporate these plays into a well balanced overall strategy, then it will be a poker robot. And in July 2008, a group of professionals took on the latest version of Polaris, the poker robot created by the Computer Poker Research Group at the University of Alberta.41 Polaris was the overall victor, winning three matches, losing two, and drawing one.
In this appendix, I will examine how Polaris used these options in its 1,000 hand July 2008 match against Matt “Hoss_TBF” Hawrilenko.42 Due to the simplifications in Polaris’s strategy, it won’t be playing perfectly unexploitable poker, but we can look at the things it does and examine the novel ways that it incorporates different strategic options into a consistent strategy.
When looking at these plays, it’s important to do so in the context of Polaris’s entire strategy, both on the later streets and with the hands it didn’t utilize the option with. (For example, the hands Polaris refuses to cap with are just as interesting as the hands it does cap.)
Calling on the Button
(My comments are in italics.)
It must be noted that Polaris wasn’t playing a purely GTO strategy in this match. Polaris came into the match with five “personalities:” one GTO version and four exploitive variations which were arrived at by adding or subtracting arbitrary amounts from the pot and then attempting to play GTO in the modified game. (See “Man Versus Machine” starting on page 221 for more on this methodology.)
I have reason to believe that most of these hands where Polaris called preflop on the button were played by at least one of the exploitive personalities. For instance, if Polaris believes the pot is larger than it really is when the blinds have been posted, then this will increase the apparent pot odds on offer and will make calling relatively more attractive. Nevertheless, Polaris will still be attempting to play unexploitably in the modified game, so we can expect its preflop calls to be well balanced.
Hand No. 1
Preflop: Polaris is on the button with the T♦8♠.
Action: Polaris calls. Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: A♦9♥4♣
Action: Matt bets. Polaris raises and Matt calls. There is now 4
big bets in the pot.
Turn: 5♥
Action: Matt checks. Polaris bets and Matt calls. There is now 6
big bets in the pot.
River: J♦
Action: Both players check.
Final Pot: 6 big bets. Matt shows the T♥9♦ (one pair, nines).
Polaris shows the T♦8♠ (ace-jack-high). And Matt wins the 6 big bets.
Hand No. 2
Preflop: Polaris is on the button with the J♣2♦.
Action: Polaris calls. Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: A♥6♥4♣
Action: Matt bets and Polaris folds.
Hand No. 3
Preflop: Polaris is on the button with the T♦3♠.
Action: Polaris calls and Matt checks. There is now 2 small bets in the pot.
Flop: A♠5♥4♠
Action: Matt checks. Polaris bets. Matt raises and Polaris calls.
There is now 3 big bets in the pot.
Turn: 8♣
Action: Matt checks. Polaris bets and Matt calls. There is now 5
big bets in the pot.
River: 8♠
Action: Matt checks. Polaris bets and Matt calls.
Final Pot: 7 big bets. Polaris shows the T♦3♠ (ace-ten-high).
Matt shows the Q♦5♠ (one pair, fives). And Matt wins the 7 big bets.
Hand No. 4
Preflop: Polaris is on the button with the 8♦4♠.
Action: Polaris calls. Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: 7♠5♣3♠
Action: Matt bets and Polaris calls. There is now 3 big bets in the pot.
Turn: 6♥
Action: Matt checks. Polaris bets and Matt folds.
Hand No. 5
Preflop: Polaris is on the button with the 9♣6♦.
Action: Polaris calls. Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: T♥3♣2♣
Action: Matt bets and Polaris calls. There is now 3 big bets in the pot.
Turn: 8♣
Action: Matt bets and Polaris calls. There is now 5 big bets in the pot.
River: 6♣
Action: Matt checks. Polaris bets and Matt folds.
Hand No. 6
Preflop: Polaris is on the button with the A♠2♣.
Action: Polaris calls and Matt checks. There is now 2 small bets in the pot.
Flop: 9♣3♦2♥
Action: Both players check.
Turn: J♥
Action: Matt bets and Polaris calls. There is now 3 large bets in the pot.
River: J♦
Action: Matt checks. Polaris bets. Matt raises and Polaris calls.
Final Pot: 7 big bets. Polaris shows the A♦2♣ (two pair, jacks and deuces). Matt shows the J♠6♦ (three of a kind, jacks). And Matt wins the 7 big bets.
Hand No. 7
Preflop: Polaris is on the button with the J♥2♣.
Action: Polaris calls and Matt checks. There is now 2 small bets in the pot.
Flop: A♦A♠8♥
Action: Matt checks. Polaris bets and Matt folds.
The fact that Polaris has some aces in its calling range, it called the previous hand with the A♠ 2♣ , means that it can credibly steal the pot on this flop. There should be relatively few aces in Matt’s range since he will want to raise most of his aces preflop for value.
Hand No. 8
Preflop: Polaris is on the button with the K♣9♣.
Action: Polaris calls. Matt raises. Polaris 3-bets and Matt calls.
There is now 6 small bets in the pot.
Flop: Q♦J♠6♦
Action: Matt checks. Polaris bets. Matt raises and Polaris calls.
There is now 5 big bets in the pot.
Turn: A♥
Action: Matt bets and Polaris calls. There is now 7 big bets in the pot.
River: 2♠
Action: Matt bets and Polaris calls.
Final Pot: 9 big bets. Polaris shows the K♣9♣ (ace-king-high).
Matt shows the T♦8♠ (ace-queen-high). And Polaris wins the 9 big bets.
Hand No. 9
Preflop: Polaris is on the button with the K♠7♠.
Action: Polaris calls and Matt checks. There is now 2 small bets in the pot.
Flop: T♣7♦4♥
Action: Matt checks. Polaris bets and Matt calls. There is now 2
big bets in the pot.
Turn: A♥
Action: Matt checks. Polaris bets and Matt calls. There is now 4
big bets in the pot.
River: 5♣
Action: Matt checks. Polaris bets and Matt calls.
Final Pot: 6 big bets. Polaris shows the K♠7♠ (one pair, sevens).
Matt shows the Q♠4♦ (one pair, fours). And Polaris wins the 6 big bets.
Hand No. 10
Preflop: Polaris is on the button with the J♠2♦.
Action: Polaris calls and Matt checks. There is 2 small bets in the pot.
Flop: A♦T♥4♣
Action: Matt checks. Polaris bets and Matt folds.
By this point, it appears that Matt’s strategy against Polaris’s calls is to either raise preflop and bet the flop, or to check preflop and then always check the flop, then responding to a bet from Polaris by either raising, calling, or folding.
Hand No. 11
Preflop: Polaris is on the button with the J♥2♠.
Action: Polaris calls. Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: A♦K♦9♦
Action: Matt bets and Polaris calls. There is now 3 big bets in the pot.
This is a very loose flop call from Polaris, and this hand is likely played by an exploitive personality which believes the pot is larger than it actually is.
Turn: 6♦
Action: Matt checks. Polaris bets and Matt folds.
Hand No. 12
Preflop: Polaris is on the button with the J♠7♥.
Action: Polaris calls. Matt raises and Polaris calls. There is now 4 small bets in the pot.
Flop: 8♦3♣3♠
Action: Matt bets and Polaris calls. There is now 3 big bets in the pot.
Turn: 8♣
Action: Matt bets. Polaris raises and Matt calls. There is now 7
big bets in the pot.
River: K♥
Action: Both players check.
Final Pot: 7 big bets. Polaris shows the J♠7♥ (two pair, eights and treys). Matt shows the T♦T♥ (two pair, tens and eights). And Matt wins the 7 big pots.
Polaris called a total of 12 out of 500 buttons, or 2.4 percent of the time. This is quite a small sample due to this low calling frequency, but we can discern three hand groups that call:
1. Weak hands on the borderline between raising and folding. Hands such as J2o, T3o and 84o are incredibly close towards both options. (I raise J2o and fold 84o. See “Rules for Preflop Limit Hold ’em” starting on page 32 for a graphical illustration of my raising range from the button.) These hands appear frequently in this sample of hands: J2o appears four times, T3o once, and 84o once. Polaris could well limp with other hands of this type such as 94o, 74o, and 43o.
2. Slightly stronger balancing hands such as A2o and T8o. These hands allow its calling range to connect with a variety of flops, giving it more stealing opportunities such as in Hands Nos. 7 and 10 where it wins on an ace-high flop. Notice that Matt, because he checked preflop, will have relatively few hands that hit these kinds of flops.
3. Suited hands capable of playing either a large or a small pot, such as in Hands Nos. 8 and 9. (K9s and K7s, where K9s 3-bet preflop, and perhaps K7s was also ready to.) Calling with these hands prevents Polaris’s opponents from attacking its calls too liberally, and these suited hands do not lose as much as other strong hands by letting the big blind in cheaply.
Again, the sample of hands is very small and it would be interesting to see other hands that Polaris might call with, but we can already visualize how this could be a well balanced and highly nuanced strategy.
But as previously stated, it would be quite difficult for humans to properly implement a mixed strategy like this over many different hands; this is an area where robots have a significant edge over us. And considering that Polaris calls so little (only 2.4 percent in the sample), you really don’t lose much of a theoretical edge by playing raise-or-fold.
Capping
It’s much more common for winning humans to cap than it is for them to call first-in. Nevertheless, a bad capping strategy can also do harm, and there are a few different ways this can come about.
One error would be to cap with too few hands (if it’s almost always your best hands you do this with). In this case, the gains from capping your premium hands would be exceeded by the losses from giving away too much information about your hand. It’s not worth gaining one extra bet preflop at the cost of making your opponent’s postflop decisions easier. Another problem with this is that you also reveal information when you don’t cap, and your opponent will now have a number of easier decisions when he knows you can’t hold a premium hand.
Another error would be to cap too frequently; now you would give away less information when you cap, but at the cost of excessively weakening your calling range (unless you cap with some hands you should normally fold), and you can lose value on your caps if your opponent tightens his 3-betting range.
So never capping avoids both errors, and it performs incredibly well for such a simple strategy. However, perhaps we can do better: by creating a well-balanced capping strategy that doesn’t give information away by either capping or calling, and this is something a poker robot can help us with.
First, although sometimes capping, we should just be calling the 3-bet the majority of the time. This feature is apparent in Polaris’s strategy, in the 500 in-position hands it played against Matt, it was 3-bet 170 times and capped on 23 occasions (16.4 percent). Here are the strong hands it could have capped with but decided to call:
Table I: Strong Hands
Polaris Chose not to Cap
Note how many broadway hands Polaris has managed to retain in its calling range, allowing it to challenge on all types of flops after calling a 3-bet.
Seven out of the 23 hands Polaris capped with preflop checked the flop. This is a really strange strategy to human eyes; all the good players I know who use preflop capping will always continuation bet the flop. But upon further reflection, Polaris’s checking strategy makes a lot of sense: Polaris is capping a very balanced range, so its caps are not that strong on average. (If it did cap really strong hands on average, then it would give too much information away.) Therefore, it makes sense for Polaris to not always bet the flop.43 Here are the hands Polaris capped and then checked the flop:
Hand No. 1
Preflop: Polaris is on the button with the 9♦9♣.
Action: Polaris raises. Matt 3-bets and Polaris caps. There is now 8 small bets in the pot.
Flop: K♠7♣5♣
Action: Both Players check. There is now 4 big bets in the pot.
Turn: 3♣
Action: Matt checks. Polaris bets and Matt calls. There is now 6
big bets in the pot.
River: Q♦
Action: Matt checks. Polaris bets and Matt folds.
Hand No. 2
Preflop: Polaris is on the button with the 3♥3♣.
Action: Polaris raises. Matt 3-bets. Polaris caps and Mat calls.
There is now 8 small bets in the pot.
Flop: K♣5♥5♠
Action: Both players check. There is now 4 big bets in the pot.
Turn: A♣
Action: Matt checks. Polaris bets and Matt calls. There is now 6
big bets in the pot.
River: A♠
Action: Both players check.
Final Pot: 7 big bets. Polaris shows the 3♥3♣ (two pair, aces and fives). Matt shows the 4♦4♠ (two pair, aces and fives). And it’s a split pot.
Hand No. 3
Preflop: Polaris is on the button with the J♦5♥.
Action: Polaris raises. Matt 3-bets. Polaris caps and Matt calls.
There is now 8 small bets in the pot.
Flop: Q♣J♥2♣
Action: Both players check. There is now 4 big bets in the pot.
Turn: 8♣
Action: Matt bets and Polaris calls. There is now 6 big bets in the pot.
River: T♠
Action: Both players check.
Final Pot: 6 big bets. Polaris shows the J♦5♥ (one pair, jacks).
Matt shows the 3♥3♣ (one pair, treys). And Polaris wins the 6 big bets.
Hand No. 4
Preflop: Polaris is on the button with the 9♦9♣.
Action: Polaris raises. Matt 3-bets. Polaris caps and Matt calls. There is now 8 small bets in the pot.
Flop: K♠5♦3♥
Action: Both players check. There is now 4 big bets in the pot.
Turn: 7♦
Action: Matt bets and Polaris calls. There is now 6 big bets in the pot.
River: K♣
Action: Matt bets. Polaris raises., Matt 3-bets and Polaris calls.
Final Pot: 12 big bets. Polaris shows the 9♦9♣ (two pair, kings and nines). Matt shows the 5♥5♠ (full house, fives full of kings).
And Matt wins the 12 big bets.
Hand No. 5
Preflop: Polaris is on the button with the K♦7♠.
Action: Polaris raises. Matt 3-bets. Polaris caps and Matt calls.
There is now 8 small bets in the pot.
Flop: 9♣5♠4♥
Action: Both players check. There is now 4 big bets in the pot.
Turn: Q♠
Action: Matt bets and Polaris folds.
Hand No. 6
Preflop: Polaris is on the button with the J♥T♥.
Action: Polaris raises. Matt 3-bets. Polaris caps and Matt calls. There is now 8 small bets in the pot.
Flop: A♦7♠3♥
Action: Both players check. There is now 4 big bets in the pot.
Turn: 9♠
Action: Both players check.
River: 7♥
Action: Both players check.
Final Pot: 4 big bets. Polaris shows the J♥T♥ (one pair, sevens, ace-jack-kicker). Matt shows the Q♦8♥ (one pair, sevens, ace-queen-kicker). And Matt wins the 4 big bets.
Hand No. 7
Preflop: Polaris is on the button with the J♠5♥.
Action: Polaris raises. Matt 3-bets. Polaris caps and Matt calls.
There is now 8 small bets in the pot.
Flop: Q♥7♥5♠
Action: Both players check. There is now 4 big bets in the pot.
Turn: K♥
Action: Matt bets and Polaris calls. There is now 6 big bets in the pot.
River: T♠
Action: Both players check.
Final Pot: 6 big bets. Polaris shows the J♠5♦ (one pair, fives). Matt shows the K♦7♠ (two pair, kings and sevens). And Matt wins the 6 big bets.
These hands are all similar; each has a modicum of value, ranging from the backdoor flush draw and undercards of Hand No. 6, to king high in Hand No. 5, and then second or third pair in all the other hands. They are also, on the flop, around the middle of Polaris’s hand range, and Polaris would bet its best hands for value and worst hands as bluffs.
Interestingly, Polaris is checking the flop with some fairly strong hands for heads-up poker, such as in Hands Nos. 1 and 4 where it has pocket nines on a king-high flop. I can only assume that it has a relatively high requirement to value bet the flop because the preflop 3-bet from its opponent indicates Polaris is up against a strong range. Notice that, elsewhere in the match, the computer flat called a reraise with pocket nines indicating that Polaris likely mixes on a number of individual hands between calling and capping.
Hands Nos. 3 and 7 show an intriguing strategy of preflop bluff-caps with jack-five offsuit which then improve enough on the flop to get turned into bluff-catchers. So just because Polaris caps, it does not necessarily mean a strong starting hand.
Here are the 16 hands where Polaris capped preflop and continuation bet the flop:
Table II: Hands Where
Polaris Capped and then Bet the Flop
These 16 hands are primarily value bets and bluffs, supporting the idea that Polaris bets its best and worst hands on the flop while checking the middling hands. Hands Nos. 6 and 8 can be considered bluffs, Hand No. 1 seems slightly out of place since it doesn’t appear to function as either a bluff or a value bet, but every other hand is a value bet. Also, notice that since the pot is bloated by Polaris’s preflop cap, it should bet the flop with a high proportion of value-bets-to-bluffs. (See “Appendix B: Game Theory” starting on page 383 for more on bluffing ratios.)
The hands that Polaris caps seem to fit into a few neat categories:
Polaris appears to have all the elements of a good preflop capping strategy. It builds the pot with its premium hands, it caps with suited hands that have good equity, it retains a lot of value in its calling range, and it includes some weaker hands for balance. The preflop bluffs are also interesting in that the bluff has no chance of initially succeeding, but they are there in order to set up Polaris’s strategy for the later streets. I’m not sure why it picks the hands that it does for bluffs; in my opinion hands like the could do equally well, although maybe it wants to hold these hands in its calling range (or it also bluff-caps these hands and I just haven’t observed it in the sample).
Finally, Polaris avoids all the major errors that humans often make when capping, and you would do well to incorporate these elements into a capping strategy if you use one. However, Polaris does cap relatively infrequently compared to calling, 16.4 percent of the hands it could have capped in the sample, and so it’s my opinion that you would not subject yourself to much of a theoretical cost if you chose to never cap. (Additionally, some other robots also choose to never cap, furthering this argument. See “Man Versus Machine” starting on page 221.)
Let’s briefly consider the opening round in the other seven games that are a part of the 8-game mix — the most popular rotation of mixed games today. I will say nothing on the level of individual hands; my focus is on patterns in your overall strategy, and to what extent your decisions will be driven by the concept of hiding information.
Omaha Eight-or-Better Hi/Lo
The type of poker most similar to limit hold ’em in the 8-game mix is Omaha eight-or-better hi/lo. In terms of community cards, number of betting rounds, and betting rules, the games are identical. These similarities suggest that hiding information on the opening round could be of equal importance, as you still need to set up your play for an unexploitable strategy on the later rounds.
You can recall that in limit hold ’em, a reason why I insisted on raising first-in was the presence of the two blinds. Calling allows the small blind to enter profitably with very many hands, and the big blind can play every hand in an unraised pot whilst charging you extra with his best hands. These facts are equally true in Omaha; the game is not so radically different that the lower price of calling could offset these costs. But it’s still limit poker, and so the game is very much based around a struggle for the initial antes (as implied odds are limited).
However, it’s customary to see a lot of preflop calling in small stakes Omaha hi/lo games. This is most likely a function of poor postflop play by others. If you are playing in one of these games, then you can call first-in. But these calls should be restricted to early positions; when it’s folded to you in late position, it’s still usually best to raise to put pressure on the blinds. When playing in tougher games, instead of promulgating multiway limped pots, first-in calls are highly likely to get isolated by in-position raisers, and so you should restrict your early position play to raise-or-fold.
The argument for never cold-calling a first-in raise since doing so allows the blinds to play many profitable hands, applies in this game to a lesser degree. The trouble with cold-calling in limit hold ’em is that the big blind can make many profitable calls for just one small bet. Say the hijack raises first-in and the button cold-calls. These two players will both have fairly wide ranges, and the button is likely to have reraised his best hands (weakening his calling range). The combination of wide ranges and high pot odds allows the big blind to call with many hands, even offsuited possibly dominated hands such as the and every time the blind makes a profitable call, it costs at least one other player and likely both of them some expectation.
However, it’s my belief that the argument for cold-calling is slightly stronger in Omaha eight-or-better. That’s because as a hand becomes more multi-way, the importance of holding a hand that can build the nuts in at least one way (and ideally have some value in the other direction) increases. A hand like the
plays much worse in a multiway pot than seemingly similar hands do in hold ’em (such as the K♦4♣). Therefore, the less costly it is to let the blind in cheaply, the more the other benefits of being able to play hands with a cold-calling strategy come to dominate. This means you should likely raise-or-fold against late position raisers or when extremely short-handed, but tend to call against early position raisers since their range will be the tightest. And the tighter the initial raisers range is, the fewer profitable overcalls the blind can make, and the harder it is for you to make profitable reraises, so the cost of cold-calling decreases.
The information hiding argument still applies against cold-calling, but in Omaha there is a greater heterogeneity of starting hands, and so developing a cold-calling strategy that can hit a range of flops is easier. If the flop comes with three high cards, then the initial raiser has no idea whether you were calling with a hand that strongly connects with the board or not; additionally, many of his starting hands, such as ace-deuce-x-x, will have missed. This is unlike limit hold ’em where strong starting hands retain their value more consistently going into the flop. (For instance, a pair of queens is a good hand heads-up on just about any flop).
Nevertheless, the greatest merit that my opening round approach to limit hold ’em has is the virtue of simplicity, and this can be particularly beneficial when you are playing in a game you don’t know well. The benefit of neglecting certain strategic options is that there are fewer threshold hands that need computing. For example, if you were using a strategy that used all possible options (call, raise, fold) from an early position in Omaha eight-or-better, you would need to split your range of starting hands between the three segments. So A2xx or better could be worth a raise, A3xx a call, and A4xx a fold.44
This can be a difficult task to perform well in a variety of opening round decisions, in a game that you have little experience with, when there is a wide range of potential starting hands. So in this example, by removing the call option, your opening round decisions become easier and this should greatly decrease the probability of making any significant hand selection errors. Hence, it’s my opinion that players starting off in Omaha eight-or-better should use a strict raise-or-fold strategy outside of the big blind.
2-to-7 Triple Draw Lowball
This game is structured identically to limit hold ’em in terms of antes, limit betting, and number of betting rounds. However, one critical difference leads to radically different play: the number of cards you draw reveals direct information about the strength of your hand.
The chief reason behind playing raise-or-fold in limit hold’em is that it allows you to play your weaker hands just like your stronger hands. But that doesn’t apply in triple draw. Say the button raises, you reraise from the small blind, and the button calls. Now you draw two cards and he only draws one. Your opponent will know your post-draw distribution of hands is significantly weaker than his, and this prevents you from being able to steal with a bet on the next round.
Likewise, playing call-or-fold heads-up out of the big blind is no longer an option. Say the button raises, you call out of the big blind, stand pat, and your opponent draws two. There is no way you can now check-raise as he has revealed that his distribution is significantly weaker than yours, and so your strategy will be to bet out on the next round. Given this fact, it’s better to reraise on the opening round before you have revealed the strength of your hand.
This important fact of information revelation means that a number of options, which can be entirely avoided in limit hold’em, all have their proper place in triple draw: cold-calling and capping in any position, and reraising while heads-up and out of position in the big blind. So you will want to put in an extra raise when your hand is strong compared to your opponents distribution, and when this fact will be revealed by the next draw. But it’s best to cold-call a raise when your hand is fairly weak relative to your opponents average hand, yet you still have sufficient pot odds to draw. Specifically, I would cold-call most often in the small blind, due to the cheaper price, and against earlier position raisers since their stronger ranges increase the threshold for reraising.
However, one restriction that I would still use is to raise-or-fold first-in from outside the small blind. That’s because your hand should be stronger than average to raise with two or more players yet to act, its strength will be revealed as soon as you draw, and you don’t want to give players a cheap chance to beat you, especially in a game like triple draw lowball where weaker hands have good chances of drawing out on stronger ones.
But when in the small blind heads-up against the big blind, I would frequently call. The reason for this strategy is that you will want to play a lot of hands, but playing raise-or-fold will often build large pots with a marked weak hand out of position — not a good situation.
So how can we hide information in triple draw lowball? We can’t use a device such as Nesmith Ankeny’s advice to never draw two cards in five card draw45 since ending hand strength is so highly driven by the number of cards drawn. That is, you don’t want to draw more or fewer cards than is optimal.
The exception is snowing: refusing to draw (standing pat) with a bluff, and it’s the key to hiding information in this game. Suppose you raise first-in on the button and the big blind calls. He draws two cards and you stand pat. Since your range is so strong, he will likely check and you will bet. If your hand was never a bluff here, his play would be easy: He could fold bluff catchers, call with plus-EV draws, and raise his best hands. Therefore, in order to balance your strong pat hands, it’s important to stand pat with some weak ones which will then bluff.46
The next chapter highlights the importance of card removal, which means that your bluffs should be with the hands that make it hardest for your opponent to have a strong hand too. A hand like the
fits the bill perfectly, as it removes three of the four best cards from the deck. My play with this hand would be to raise on the opening round, stand pat on every draw, and bet on each round. This might seem like a crazy play to a weaker player, and if it gives people this impression, all the better, but it’s actually an important balancing play which prevents opponents from exploiting your pat hands by folding.
Finally, there are a total of three draws which you can stand pat on, and it’s important to find some bluffs for every eventuality. So you might draw two cards on the first round, one card on the second round, and stand pat on the last round, and a small fraction of these pat hands should be bluffs.
The Stud Games: Seven-Card Stud,
Stud Eight-or-Better Hi/Lo, and Razz
These three games compromise the other forms of limit poker in the 8-game mix, but they share a slightly different structure. Before the cards are dealt, each player posts an ante, the cards are dealt, and the player with the lowest up-card posts a forced bringin which can either be for the minimum bet or for a raise. (The player chooses after seeing his cards.)
The first point to mention is that the bring-in should never use the larger bet;47 the reasoning behind this should by now be familiar. First, it doesnt make sense to voluntarily put a large bet in on a weak hand since up to seven more players are yet to act. And second, strong hands will do better if they can hide their strength until a later juncture, plus, retaining these hands in your distribution will make it less attractive for players to attempt to steal your minimum sized bring-ins by raising from late position. (However, this won’t stop them.)
Players acting after the bring-in can then choose to call, raise, or fold.48 In terms of information, the stud games are halfway between games such as hold ’em and Omaha where no information about starting hands is revealed until the river, and games such as triple draw lowball, where a lot of information about starting hands gets revealed over the course of a hand. In stud, two of your starting cards remain concealed until the river, the other third of your starting hand is on display, and depending on the other cards on show in your opponents’ hands, this can reveal even more about your starting hand distribution (having your up-card shared by other players weakens your distribution). In general, the more information other players know about your hand, the less able you are to represent a strong hand just by raising, and the less utility you will get from using a simple strategy such as raise-or-fold.49 In addition, the greater heterogeneity of opening round decisions in stud than in hold ’em makes it more difficult to give simple rules, as your play must be conditional on the greater amount of available information.
Therefore, in general, you will need to assess how favorable an overall situation is and how much information a certain action would give away. For example, calling first-in with an ace showing from a late position would give a lot of information away, as the situation is perfect for raising so your hand must be very weak or on rare occasions extremely strong. At the other extreme, if you have a low card showing in an early position, then raising might restrict you to too narrow a range, such as large hidden pairs, in which case calling would not give away as much information. However, I would require both poor position and a poor up-card before calling would be my preferred option, so in the majority of situations, it’s probably still best to raise-or-fold.
The structure in a $2-$4 game would often be an ante of $0.25 from each player, plus a minimum bring-in of $1. Six handed, this would result in an initial pot of $2.50, which, relative to the size of the first raise to $2, is larger than in an equivalent size hold ’em game (where an initial raise to $2 would be into a pot of $1.50). However, notice that the bring-in of $1, which the other players can call, is smaller than the lower betting limit of $2. And because of this, many stud games will play smaller than their hold ’em counterparts.
Holding all else constant, a large ante structure will make us want to play a greater proportion of starting hands than in a game with small antes.50 The large ante increases the attractiveness of winning the pot right away with a raise, it also means that if we call, the players yet to act will be getting attractive odds of at least 3.5-to-1 to call. (Hand values do change less rapidly in stud relative to hold ’em, as on the second round of betting only one extra card is dealt to each player as opposed to a three card flop.) If a player at your table calls the bring-in, my preference in tough games would be to raise-or-fold in spite of the attractive pot odds. When a player calls in this type of game, it’s generally best to isolate and attempt to shut out the other players. However, be aware that certain hands like small 3-flushes or a 3-straight with a gap can be profitable for just the bring-in.51
If the pot has already been raised, then much like in limit hold ’em, I advise you to play a game of reraise-or-fold outside of the bring-in. In Seven-Card Stud for Advanced Players, David Sklansky, Mason Malmuth, and Ray Zee give similar advice. If someone has raised from middle position with, say, a ten showing, they advise you to reraise with premium hands, and to raise-or-fold with lower hands (such as a small pair or three-straight) depending on factors such as how live your outs are, how good your kicker is, and how live your opponent’s door-card is.
Like my limit hold ’em strategy for the big blind, when you are the bring-in, my advice is to call-or-fold when heads up against a third street raiser. (When there are other callers in the pot, you should sometimes reraise to force them out.) This play is reinforced by one of the features of stud: The bring-in will have the lowest up-card at the table, and hence a weaker distribution than players with bigger up-cards, and the weaker your distribution, the more willing you should be to slowplay your better hands on the opening round.
And finally, there are a total of five betting rounds, three of which are denominated in large bets. This is one more than limit hold ’em, which has a total of four betting rounds — two of which use large bets. And as the number of betting rounds increase, the more important it becomes to disguise your hand strength on the opening round with rules such as call-or-fold.