Note: While reading a book whenever I come across something interesting, I highlight it on my Kindle. Later I turn those highlights into a blogpost. It is not a complete summary of the book. These are my notes which I intend to go back to later. Let’s start!

Poker Maths

  • Mathematical expectation is the amount a bet will average winning or losing

  • Let’s say you are betting a friend $1, even money, on the flip of a coin. Each time it comes up heads, you win; each time it comes up tails, you lose. The odds of its coming up heads are 1to-1, and you’re betting $1-to-$1. Therefore, your mathematical expectation is precisely zero since you cannot expect, mathematically, to be either ahead or behind after two flips or after 200 flips.Your hourly rate is also zero. Hourly rate is the amount of money you expect to win per hour.

  • You have the best of it when you have a positive expectation, and you have a positive expectation when the odds are in your favor. You have the worst of it when you have a negative expectation, and you have a negative expectation when the odds are against you. Serious gamblers bet only when they have the best of it; when they have the worst of it, they pass

  • A person writes down a number from one to five and bets $5 against your $1 that you cannot guess the number. Should you take the bet? What is your mathematical expectation?Four guesses will be wrong, and one will be right, on average. Therefore, the odds against your guessing correctly are 4-to-1. Chances are that in a single try you will lose the dollar. However, you are getting $5-to-$1 on a 4-to-1 proposition. So the odds are in your favor, you have the best of it, and you should take the bet. If you make that bet five times, on average you will lose $1 four times and win $5 once. You have earned $1 on five bets for a positive expectation of 20 cents per bet

  • A bettor is taking the odds when he stands to win more than he bets, as in the example above. He is laying the odds when he stands to win less than he bets. A bettor may have either a positive or a negative expectation, whether he is taking the odds or laying them. If you lay $50 to win $10 when you are only a 4-to-1 favorite, you have a negative expectation of $2 per bet, since you’ll win $10 four times but lose $50 once, on average, for a net loss of $10 after five bets. On the other hand, if you lay $30 to win $10 when you’re a 4-to-1 favorite, you have a positive expectation of $2, since you’ll win $10 four times again but lose only $30 once, for a net profit of $10. Expectation shows that the first bet is a bad one and the second bet is a good one

  • Poker plays can also be analyzed in terms of expectation. You may think that a particular play is profitable, but sometimes it may not be the best play because an alternative play is more profitable. Let’s say you have a full house in five-card draw. A player ahead of you bets. You know that if you raise, that player will call. So raising appears to be the best play. However, when you raise, the two players behind you will surely fold. On the other hand, if you call the first bettor, you feel fairly confident that the two players behind you will also call. By raising, you gain one unit, but by only calling you gain two. Therefore, calling has the higher positive expectation and is the better play

  • When you go into a poker game, you should try to assess what you think you can earn per hour. If you are playing draw lowball and you see three players calling $10 and then drawing two cards, which is a very bad play, you can say to yourself that each time they put in $10 they are losing an average of about $2. They are each doing it eight times an hour, which means those three players figure to lose about $48 an hour. You are one of four other players who are approximately equal, and therefore you four players figure to split up that $48 an hour, which gives you $12 an hour apiece. Your hourly rate in this instance is simply your share of the total hourly loss of the three bad players in the game

  • You should almost always try to make the play that will maximize your positive expectation or minimize your negative expectation in order to maximize your hourly rate

  • You should try to assess most poker games in terms of your expected hourly rate by noticing what mistakes your opponents are making and how much these mistakes are costing them. Don’t sit in a game with an insufficient hourly rate projection unless you think the game will become better either because you expect some weaker players to arrive soon or because some good players in the game have a tendency to start playing badly when they are losing. If these good players jump off winners, you should quit if you can

Fundamental Theorem of Poker

  • Every time you play a hand differently from the way you would have played it if you could see all your opponents’ cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.

  • Suppose there is $80 in the pot, and you have two pair. You are playing draw poker, and you bet $10, which we will assume is all you can bet. Your single opponent has a fourflush that is, four cards to a flush. The question is are you rooting for him to call or fold? Naturally you want him to do what is most profitable for you. The Fundamental Theorem of Poker states that what is most profitable for you is for your opponent to make the incorrect play based on complete information about both hands. Since your opponent is getting 9-to-1 odds (his $10 call might win him $90) and is only about a 5-to-1 underdog to make a flush, it is correct for him to call because a call has positive expectation. Since it is correct for him to call, following the Fundamental Theorem, you are therefore rooting for him to fold

  • Just as you are rooting for an opponent to fold when he is getting sufficient pot odds, you are rooting for him to call when he is getting insufficient pot odds. Thus, it is frequently correct to play a strong hand weakly on an early round the converse of your plays in the previous two examples - so that your opponent will make a bad call when you do improve

  • When you have a hand that is rooting for a call, you should not try to make your opponent fold by betting an exorbitant amount in a no-limit or pot-limit game

  • In advanced poker you are constantly trying to make your opponent or opponents play in a way that would be incorrect if they knew what you had. Anytime they play in the right way on the basis of what you have, you have not gained a thing. According to the Fundamental Theorem of Poker, you play winning poker by playing as closely as possible to the way you would play if you could see all your opponents’ cards; and you try to make your opponents play as far away from this Utopian level as possible. The first goal is accomplished mainly by reading hands and players accurately, because the closer you can come to figuring out someone else’s hand, the fewer Fundamental Theorem mistakes you will make. The second goal is accomplished by playing deceptively

Ante and Odds

  • The lower the ante in comparison to future bets, the fewer hands you should play; the higher the ante, the more hands you should play. A different way of looking at it is: The lower the ante, the higher your starting requirements should be, and the higher the ante, the lower your starting requirements should be

  • It is a common fallacy for players to think in terms of the money they have already put in the pot. They make a bad call because they called one or two bets on earlier rounds. However, it is absolutely irrelevant whether you put the money in there or someone else did. It is the total amount, no part of which belongs to you any longer, that should determine how you play your hand

  • The size of the ante in a particular game determines how you play. The larger the ante in comparison to later bets, the more hands you should play. Since there’s more money in the pot, you’re obviously getting better odds, but there are other reasons for playing more loosely. Should you wait to get an extremely good hand in a high ante game, you’ll have lost more than the size of the pot in antes by the time you win a pot. Furthermore, the pots you do win will be comparatively small because the other players, if they are decent players, will notice you are playing very tight and won’t give you much action when you do play a hand. In fact, when you do get action, you’re very likely to be beat

  • If it makes sense to try to win antes right away when they are large, it makes abundant sense not to slowplay a good hand.’ The reason is that if you don’t raise with a good hand on the first round, you are giving an opponent with a mediocre hand the chance to come in cheaply and possibly draw out on you. With a large ante, he is not making a mistake on the basis of the Fundamental Theorem of Poker because he is getting good odds. In other words, if a player is getting 8-to-1 odds or 10-to-1 odds on that first round, it is worth it for him to come in and hope to catch a perfect card on the next round even when he is pretty sure you are slowplaying a big hand. However, when you raise, you wreck the odds he is getting, and he has to throw away his mediocre hand. With almost any good hand, it is not worth letting opponents in cheaply when the ante gets up there. You are satisfied with winning only the antes. On the other hand, when the ante is low, it becomes more reasonable to slowplay big hands in order to suck worse hands in; you want to get more value for your big hands

  • Pot odds are the odds the pot is giving you for calling a bet. If there is $50 in the pot and the final bet was $10, you are getting 5-to-1 odds for your call. If there is $50 in the pot and the final bet was $10, you are getting 5-to-1 odds for your call. It is essential to know pot odds to figure out expectation. In the example just given, if you figure your chances of winning are better than 5-to-1, then it is correct to call. If you think your chances are worse than 5-to-1, you should fold

  • The ability to fold correctly when you suspect you are drawing dead or drawing with too little chance of ending up with the best hand is one attribute that distinguishes a good player from an average one

  • When all the cards are out, your hand is worth a call if you think your chances of winning are better than your pot odds. Before the draw in draw poker and with exactly one card to come in stud games, your decision to call with a hand that needs to improve depends upon these factors:

    • Your chances of improving, taking into account the needed cards already out against you (in stud) and any extra outs you might have
    • Your chances of winning if you do improve
    • The odds you are getting on this next-to-last round of betting, taking into account the possibility of a raise behind you if you are not the last to act.
    • Your expected extra profits on the last round of betting if you do make your hand.This last factor is what I call implied odds. It is the money you expect to win by betting or raising on the last round (or rounds) when you do make your hand
  • When you compute odds on a hand you intend to play to the end, you must think not in terms of the immediate pot odds but in terms of the total amount you might lose versus the total amount you might win

  • Before deciding to go all the way with a hand, you must calculate whether the effective odds you are getting by calling several rounds of betting justify a call now

  • Figuring effective odds may sound complicated, but it is a simple matter of addition. You add all the calls you will have to make, assuming you play to the end, to determine the total amount you will lose if you don’t make your hand. Then compare this figure to the total amount you should win if you do make the hand. This total is the money in the pot at the moment plus all future bets you can expect to win, excluding your own future bets. Thus, if there is $100 in the pot at the moment and three more $20 betting rounds, you are getting $160-to-$60 effective odds if both you and your opponent figure to call all bets. If you know you won’t call on the end unless you make your hand, your effective odds become $160-to-$40. When you think your opponent won’t call on the end if your card hits, your effective odds would be reduced to something like $140-to-$40. If, on early betting rounds, these odds are greater than your chances of making your hand, you are correct to see the hand through to the end. If they are not, you should fold

  • During the early and middle rounds of betting, having to call future bets usually reduces your apparent pot odds considerably, and you have to calculate your real or effective odds. However, there are times when the existence of future bets is the very reason you play a hand. Your immediate pot odds may not seem high enough to justify calling for one more card. But if that card may give you a monster hand that figures to get you a lot of action, you frequently don’t need the initial odds from the pot. You’ll get them later. These odds are what I mean by implied odds

  • Implied odds are based on the possibility of winning money in later betting rounds over and above what is in the pot already. More precisely, your implied odds are the ratio of your total expected win when your card hits to the present cost of calling a bet

  • In general, the larger the difference between future bets and the present bet you have to call, the greater your implied odds. Hence, implied odds become most significant in pot-limit games and in no-limit games, where a future bet can be as large as the amount of money a player has in front of him. In fact, in these games one is almost always considering not how much is in the pot right now, but rather how much can be won on a future round of betting

  • When you estimate your implied odds, you must try to predict how much money you can win if you do make your hand. This prediction depends on three factors:
    • The size of future bets
    • How hidden your hand is
    • The ability of your opponents
  • Implied odds explain situations when your odds are better than they seem. There are other times when you must realize that your odds are not as good as they seem. These situations occur when you have a mediocre hand with little chance of improving, which you think is the best hand at the moment, yet your opponent keeps betting. You think he may be bluffing, and you can beat only a bluff - that is, a hand that is weaker than what your opponent is representing. However, since your opponent is controlling the betting, he will probably back off on later rounds if he doesn’t have you beat. Thus, you are in the position of winning the minimum if you have the best hand but losing the maximum if you have the worst hand. The true pot odds in such situations are much worse than they seem, and so we call them reverse implied odds.For instance, there is $50 in the pot, and your opponent bets $20. You think you have him beat, but you are not sure. You also have little chance of improving. You cannot say, “I’m getting 70to-20 odds here,” because your opponent may come out betting again next round if he has a better hand than yours - or if his hand improves to a better hand but he is likely to give it up if he has a worse hand than yours. You are in a situation where, if you lose, you figure to lose not just the $20 you are calling right now but a total of $60. However, if you win, you’ll probably win only the $70 in the pot right now because once your opponent sees you’re committed to the pot, he won’t bet, further with the worst hand. All of a sudden, then, you’re not getting 70-to-20 odds but closer to 70-to-60.

  • Reverse implied odds describe situations in which:
    • You’re not sure where you’re at
    • You have little chance of improving to beat the hand your opponent might already have or might make
    • A call commits you to calling future bets all the way to the end
    • Your opponent can back off at any time.In such cases, you must not think you are getting odds according to what’s in the pot and what you have to call right now. You are getting much worse odds - so much worse that it is often better to throw your hand away immediately rather than get committed.
  • Whereas implied odds are based on the possibility of winning more money in later betting rounds, reverse implied odds are based on the possibility of losing more money in later betting rounds. Put another way, when you’re getting implied odds, you’re glad you’re not all-in, for you expect to make money on future bets if your card hits. However, when you’re getting reverse implied odds, you wish you were all-in so you could see the hand to the end without having to call future bets

Disguising hands

  • With weak players, with a large pot, and with large early bets, you need not be so concerned about disguising your hand. A corollary is that the more players in the pot, the less you gain by disguising your hand. You cost yourself too much when you do. You won’t be able to make everybody fold when you bet with a weak hand, and you cost yourself too many bets when you miss a raise with a strong hand. What’s more, when you let many opponents in cheaply, you increase the chances of being outdrawn. Heads-up situations require disguising your hand more than do multi-way pots

  • Let’s look at two early-round betting situations - one in which you don’t care that you’ve given your hand away and the other in which you should use deception. In both situations you have a pair of aces in the hole before the flop in hold ‘em. That is, you have the nuts, the best possible hand at that point. The first game is no-limit. You’ve made a small raise, four or five people have called, and now someone puts in a substantial reraise. You must reraise again even if your play gives away your hand completely. It is worth dropping all disguise because as the pot gets larger and larger, what’s in the pot right now counts more than potential bets on later rounds. With two aces you should put in all the bets you can.On the other hand, with two aces against a good player in a limit hold ‘em game, you should often not put in all bets. A reraise is fine because you could have a variety of hands. However, if your single opponent reraises again, you should probably just call. If you raise one more time, your opponent figures you for two aces. All you have gained is one small extra bet right there, but you may have cost yourself two or three bets later on. In this case, you have lost too much by giving your hand away. You stand to gain more by using deception

  • The general rule is: The better the players and the smaller the pot, the more you disguise your hand when there are more cards to come. The worse the players and the larger the pot, the more you play your hand normally, without regard to giving anything away. Sometimes, though, playing your hand normally may be the best deception of all against very tough players who expect you to be deceptive

  • It is extremely important to disguise your hand against players who put great emphasis on reading hands, though such players may not necessarily be good, and when deceptive play has gotten the super readers confused, they’ve got no chance. This type tends to put you on a hand early, and like a captain going down with the ship, he sticks to his opinion until the end.

  • There are five criteria for using deception to avoid giving your hand away:

    • You are up against good players or super readers
    • The pot is small in comparison to future bets
    • The present round of betting is small in comparison to future bets.
    • You have only one or two opponents against you
    • You are slowplaying a monster hand.The first two conditions are most significant. It is not necessary to meet all five conditions before deception is employed. Three of the five are usually sufficient so long as one or both of the first two are included.Do not use deception against bad players, against many players, when the pot is large, or when the early bets are large. It is especially important to play a good hand strongly if the pot is large. The only exception would be when you have an unbeatable hand and figure you will gain more by waiting a round before making your move.
  • The basis of your decision to play normally or deceptively is simple. You should play each session and each hand of each session in the way that will win the most money and lose the least (except when you intentionally play a hand badly to create an impression for future hands). Always remember from the Fundamental Theorem of Poker that the more your opponents know about your hand, the less likely they are to make mistakes. However, there are situations when deception can be costly and straightforward play is best

  • When the pot is big, you want to win it right away. To try to win it right away, you should bet and raise as much as possible, hoping to drive everybody out, but at least reducing the opposition. You should bet and raise with the best hand, and you should frequently do the same even with a hand you think maybe second best. The fewer opponents you have in a pot, the greater your chances of winning it, even if those chances are less than 50 percent; and when the pot gets large, winning it should be your foremost concern

Free cards and Betting

  • When you have the best hand, you do not want to give opponents a free card since you are giving them a chance to outdraw you and win the pot. By the same token, when you do not have the best hand, you want to try to get a free card to get a free shot at winning the pot

  • Giving a free card means checking a hand you could have bet when there are more cards to come. Of course, when you check with the intention of raising, you are giving a free card only when your opponent is so uncooperative as not to bet into you

  • It is almost never correct to give a free card when the pot is large

  • It turns out that it is rarely correct to give a free card with medium-sized pots, even when you know your opponent will fold if you bet. You simply have to be satisfied with what there is in the pot already. One reason you should bet is that generally you want your opponent to fold

  • When you have a chance to bet and you have a decent hand, especially a hand you think is the best one, it is almost always correct to bet. The only conditions that might make it incorrect to bet are the following:
    • The pot is small in comparison to what it might be in the future and you figure to gain more in future bets through deception than by giving your hand away now; this situation occurs most often in pot-limit and no-limit games
    • You think you can get in a check-raise.3. Your hand is so strong it’s worth giving a free card even with a medium-sized pot
  • When you’re trying to decide whether or not to bet your hand and worry about making a mistake, you should keep in mind one very important principle a mistake that costs you the pot is a catastrophe, especially if the pot has become relatively large, while a mistake that costs you one bet is not. When in doubt, make sure you don’t make a mistake that costs you the pot. Checking and giving an opponent with a worse hand a free card may cost you the pot when he outdraws you. However, betting and getting called by a better hand costs you at most just that one bet. Thus, the only time to give free cards with the probable best hand is when your hand is so strong it is in little danger of being outdrawn and your deception sets up the likelihood of larger profits in future bets in comparison to what is currently in the pot

Semi bluff

  • A semi-bluff is a bet with a hand which, if called, does not figure to be the best hand at the moment but has a reasonable chance of outdrawing those hands that initially called it

  • When you bet as a semi-bluff, you are rooting to win right there just as you are when you make a pure bluff. However, in contrast to a pure bluff, you still retain a chance of outdrawing your opponent if you are called. Even when you bet with a legitimate hand, you are generally rooting to win instantly, but when you semi-bluff, you especially want your opponents to fold because one of them may be folding the best hand

  • Betting is particularly important when you’re in first position, in which case you should apply the following rule: If your hand is worth a call or almost worth a call when someone else bets, it is better to bet yourself, especially when you have little fear of a raise and when there is some chance you will win right there by making your opponent fold

  • Both the semi-bluff and betting a marginal hand rather than risking giving a worse hand a free card are cases of the general precept that it is usually better to be betting than calling. By betting as a semi-bluff you have a chance of winning the pot right there, something you are usually hoping to do, and you have shown greater strength than you really have. If you catch scary-looking cards after you have been called, you are still likely to win pots you wouldn’t otherwise have won. When you bet now, your opponent is quite likely to fold. On the other hand, when you don’t improve and are caught in a semi-bluff, that can be of value as an advertisement for the future

  • A semi-bluff is a bet, raise, or check-raise with a wide variety of hands which you believe are not the best at the moment. However, they may win not only right there when your opponent folds but also in a showdown when they improve to the best hand. They may also win when your opponent folds on a later round after you catch a scare card that makes your hand look like the best hand

  • A semi-bluff may be used in any game, but it may be used only with more cards to come

  • Sometimes a hand with which you think you are semi-bluffing is in fact the best hand. By betting, you prevent a worse hand from getting a free card

  • If you have a hand that warrants a call when your opponent bets, it is usually correct to bet yourself, particularly in first position. You thereby gain the chance of winning the pot immediately, and you show more strength than you actually have, which can be to your advantage later

  • Semi-bluffs allow you to be the bettor instead of the caller, which nearly always puts you in a more advantageous position

  • Semi-bluffs are a good way to randomize your bluffs, for you have the added equity of a possible win even when you are called

  • A semi-bluff can frequently be a profitable play in situations where a pure bluff is not. Your extra out of outdrawing your opponent can swing your mathematical expectation from the minus to the plus side

  • You usually do not semi-bluff when you are sure your opponent will call. However, if there is a possibility that opponent will fold, you should bet - or raise with a semi-bluffing hand, especially as the pot gets larger

  • It is usually better to make a semi-bluff bet when you are first to act; when you are last, you have the opportunity of giving yourself a free card, and you may not want to risk the chance of an opponent check-raising you

  • While calling may be a good defense against the semi-bluff in some situations, remember that normally the correct play is to fold with marginal hands, and if folding isn’t correct, then you should raise. When someone bets or raises but may be semi-bluffing, your decision is one of the trickiest in poker. You must choose whether to fold; raise; reraise; call and bet on the next round; call and check-raise on the next round; call and then check and call on the next round; or call and fold on the next round if the card your opponent catches would make the hand with which he might have been semi-bluffing. Making the correct decision consistently separates the true champion from the merely good player

  • Principal reasons for raising:
    • To get more money in the pot when you have the best hand
    • To drive out opponents when you have the best hand
    • To bluff or semi-bluff
    • To get a free card
    • To gain information
    • To drive out worse hands when your own hand may be second best
    • To drive out better hands when a come hand bets
  • Raising as a pure bluff with a hand that has no chance of winning if called is a tricky play, too risky to be attempted often. It is usually done only when there are no more cards to come, often when you didn’t make the hand you were hoping to make but are trying to convince your opponent you did. Presumably your opponent has a decent hand to bet into you and is reluctant to throw it away when you raise. In limit poker, raising as a pure bluff can succeed often enough to be profitable only against a very tough player who is capable of making super-tough folds. The weaker the player, the more likely he is to call your raise with any kind of hand

  • Raising is often a better alternative than folding, with calling the worst of the three

Slowplaying

  • Check-raising and slowplaying are two ways of playing a strong hand weakly to trap your opponents and win more money from them. However, they are not identical. Check-raising is checking your hand with the intention of raising on the same round after an opponent bets. Slowplaying is playing your hand in a way that gives your opponents no idea of its strength. It may be checking and then just calling an opponent who bets, or it may be calling a person who bets ahead of you. When you slowplay a hand, you are using deception to keep people in for a while in order to make your move in a later round. Clearly, then, a hand you slowplay has to be much stronger than a hand with which you check-raise. Check-raising can drive opponents out and may even win the pot right there, while slowplaying gives opponents either a free card or a relatively cheap card

  • The way you bet or check-raise depends on the strength of your hand in relation to what you can see of the other hands and the position of the player you expect to bet or raise behind you when you check or bet

  • While you generally check-raise because you think you have the best hand, it is frequently correct to check-raise with a second-best hand if the play will drive other opponents out

  • When you check-raise you usually want to reduce the number of your opponents, but when you slowplay you are trying to keep as many players in the pot as you can, expecting to collect later bets from them as a result of your early deception. Obviously, since you are not worried about having many players in the pot and are not particularly concerned about giving them free cards, you must have a very strong hand to slowplay much stronger than a hand with which you would check-raise

  • For a slowplay to be correct, all of the following must be true:

    • You must have a very strong hand
    • The free card or cheap card you are allowing other players to get must have good possibilities of making them a second-best hand
    • That same free card must have little chance of making someone a better hand than yours or even giving that person a draw to a better hand than yours on the next round with sufficient odds to justify a call
    • You must be sure you will drive other players out by showing aggression, but you have a good chance of winning a big pot if you don’t
    • The pot must not yet be very large

Type of Poker players

  • Loose poker players play a large percentage of hands. They have relatively low starting requirements, and they continue in the pot with relatively weak hands. Tight players play a small percentage of hands. Their starting requirements are high, and they are quick to throw away weak hands that don’t develop into big hands. Some players always play loose. Others always play tight. Good players adjust their play to the game

  • You can steal antes with anything (a loose play) much more successfully against tight players, who will fold their marginal hands, than you can against loose players, who are likely to call you with those same hands. However, the principle of playing loose against tight players and tight against loose players is in need of refinement

  • With respect to semi-bluffing, you must play much tighter in a loose game as opponents play loose and tend to call instead of folding

  • In a loose game people are willing to play a hand that is relatively lower in value than the average. Therefore, your own legitimate hands don’t need to be quite as good as in a normal game since your opponents are likely to be staying with you with even worse hands. This becomes especially true when you get heads-up against one opponent.However, because of the action and the participants’ style of play, loose games frequently tend to have multi-way pots. With many players staying in, you would be wrong to loosen up with hands like two small pair or one medium pair. Even though these marginal hands might be favorites to hold up against each of several loose opponents individually, chances are they will lose when there are several opponents in the pot. By the same token, if you bet with these hands, you are much less likely to get two, three, or four opponents to fold, particularly when they are loose players, than you are to get one opponent to fold

  • In loose games, you should tighten up considerably on semi-bluffs but loosen up with legitimate hands. However, you would not play loose with marginal hands like two small pair or one medium pair when several opponents are in the pot

  • In a tight game semi-bluffs increase in value, and even pure bluffs can be profitable since tight players are more likely to fold. Paradoxically, though, legitimate hands don’t have nearly the value in a tight game that they would have in an average or loose game. The reason should be obvious. When you bet a legitimate hand for value in a tight game, you will be called only by players who have strong hands themselves because tight players’ starting requirements are higher. In a loose game an opponent with two small pair at the end will probably call your bet with aces up. But when you bet that same hand in a tight game - especially if both of your aces are showing - and you get called, you cannot feel too comfortable. The caller probably has you beat

  • In a tight game the value of a hand goes down because players who stay in the pot will have good hands themselves better hands on average than players in a regular game would have.In a tight game, then, you loosen up on bluffs and semi-bluffs, but you tighten up on your legitimate hands. Nor would you play as many drawing hands in a tight game, since you’d be getting pot odds sufficient to make it worthwhile less often, and when you did hit, you wouldn’t get paid off as much as you would in an average or in a loose game

  • To use all the poker tools at your disposal, you need to adjust your play according to the game and according to the individual players in the game

  • Never forget that in poker we are trying to win money, not pots. Every decent player wins a fair share of pots, but it is the extra bets you can get into the pots you win and those you can save from the pots you lose that increase your hourly rate and the money won in the long run

  • In a horse race you like being first, in a poker game you like being last

Position and pure bluffing

  • Position is important in relation to the playing style of the other players in the game. You prefer having the loose, aggressive player in the game sitting to your right and the tight, conservative player to your left. Then you can usually decide how to play your hand after the aggressive player has acted, while you don’t have to worry about many surprises from the conservative player behind you. You are also in a better position to control the aggressive player and indeed to trap him into mistakes. Similarly, if there are players in the game who tip off whether or not they are playing a hand, you’d like them to your left so you can use that information when deciding whether to call the first bet yourself

  • A person who bluffs with approximately the right frequency - and also, of course, in a random way - is a much better poker player and will win much more money in the long run than a person who virtually never bluffs or a person who bluffs too much. The person who never bluffs will never get much action. The person who always bluffs will get all the action he wants until he runs out of money. But the person who bluffs correctly keeps his true holdings disguised and is constantly forcing his opponents into tough decisions, some of which are bound to be wrong

  • What is the right bluffing frequency? It is a frequency that makes it impossible for your opponents to know whether to call or fold

  • When there are more cards to come, your bluffs should rarely be pure bluffs - that is to say, bets or raises that have little or no chance of winning if you are called, even taking into account the cards you may get on future rounds. Instead your early-round bets should be semi-bluffs

  • For a pure bluff to work, your opponent or opponents must generally fold immediately

  • Deciding whether to continue with a semi-bluff really depends on how the next card affects your chances and how your opponent’s card seems to have affected his. Each individual round should be evaluated separately

  • When all the cards are out, you obviously can no longer semi-bluff. You have either made your hand or you haven’t. So all bluffs on the end are pure bluffs. They are bets or raises that you do not expect to win if you are called.When you are sitting there knowing you have the worst hand, knowing you cannot win by checking, knowing you cannot win by calling your opponent’s bet, the only question is whether or not to try to bluff. You should not if you think the chances your opponent will call are too great in relation to the pot odds you are getting. You should if you think your opponent will fold often enough for a bluff to show a profit. If there is $100 in the pot, you should make a $20 bluff if you think your opponent will fold more than once in six times. If there is $60 in the pot, you must assume your opponent will fold more than once in four times before you try to bluff. If there is $140 in the pot, your opponent needs to fold more than once in eight times. But, of course, the larger the pot, the better pot odds your opponent is getting to call your bet and the more likely it is he will call with any kind of a fair hand

  • It is rarely correct to try to bluff out two or more people when all the cards are out; your chances of success decrease geometrically with each additional player in the pot. Paradoxically you might have a profitable bluffing opportunity against each of two opponents individually, but not against both of them as a group

  • When you bluff, you are rooting for your opponent to fold because that is the only way you can win the pot. When you bet for value, you are rooting for your opponent to call because you want your legitimate hand to win one more bet from him. It is important to realize that it may be right to bet a fair hand for value, and it may also be right to bluff, but it is almost never right to do neither. If you decide you can’t get away with a bluff on the end when you miss your hand, then you should bet for value when you do make your hand

  • Similarly, when you don’t think a value bet is justified with a fair hand, since your opponent will only call if he has you beat, then if you miss your hand, you should usually bluff. For when you bluff, it is possible your opponent will throw away his fair hands

  • You must, of course, consider your opponent when deciding whether to bet a fair hand for value or to bluff. Against a perpetual caller, obviously you should rarely bluff. However, against such a player you should bet any hand that you figure is a reasonable favorite to be the best hand. In contrast, against a tough player capable of tough folds, you can get away with bluffs more often, but you should be more reluctant to bet your fair hands for value. A tough opponent is not likely to pay you off with his worse hands, and when he does call, he’s likely to show down a hand that beats you

Game theory in Poker

  • When using game theory to decide whether to bluff, you must first determine your chances of making your hand. You must then determine the odds your opponent is getting on that bet. Then you must randomly bluff in such a way that the odds against your bluffing are identical to your opponent’s pot odds.Here’s one more example. Suppose you have a 20 percent chance of making your hand, there’s $100 in the pot, and the bet is $25. Your opponent is then getting $125-to-$25 or 5-to-1 odds if you bet. The ratio of your good hands to your bluffs should, therefore, be 5-to-1. Since you have a 20 percent chance of making your hand, you should randomly bluff 4 percent of the time. (20 percent-to-4 percent equals 5-to-1.) When you bluff in this fashion, you take optimum advantage of the situation.A good, convenient way to randomize your bluffs, as we have seen, is to pick cards from among those you haven’t seen. If, for example, ten cards make your hand and you need a 5-to-1 bluffing ratio, then you should pick two additional cards to bluff with.Here is another example. You draw one card to a spade flush in draw poker, and your opponent draws three cards. Therefore, the chances are enormous that your opponent will not be able to beat a flush, only a bluff. The pot contains $20. The bet is $10. If you bet, your opponent is getting $30-to-$10 or 9-to-3 odds from the pot. Since nine unseen spades make your flush, you should pick three additional cards to bluff with, such as the two red 4s and the 4 of clubs. You now bet with twelve cards creating a 9-to-3 ratio between your good hands and your bluffs. It is not always possible to use cards to arrive at exactly the ratio you need to bluff optimally. However, as long as you are close, you can still expect to gain. You recall that choosing six cards to bluff with in the draw lowball example created exactly the right proportion vis-a-vis the pot odds my opponent was getting; nevertheless, I still ended up with a profit when I bluffed with five or with seven cards whether my opponent called or folded. Of course, the closer you are to the exact ratio, the better, in terms of game theory

  • When using game theory to decide whether to call a possible bluff - assuming your hand can beat only a bluff and assuming your judgment doesn’t give you a hint - you must determine the odds your opponent is getting on a bluff. Make the ratio of your calls to your folds the same as those odds. If your opponent is getting 4-to-1 odds on a bluff, you must call randomly four out of five times to make that bluffing unprofitable

  • Players who bluff with approximately the correct frequency are dangerous opponents because they often force you into the position of making an incorrect play. Therefore, it is important to try to stop or induce bluffs to lead opponents away from correct bluffing strategy. You should normally induce a bluff against players who already bluff too much and stop bluffs against players who already bluff too little.In the first case, you are in a situation where you would have to call if your opponent bets. By inducing a bluff, you increase your chances of winning that last bet since your opponent will bet more hands - including his bluffs - that you can beat than he otherwise would.In the second case, against someone who bluffs too little, you feel you would have to fold if that opponent bets, even though there is some chance he might be bluffing. By stopping his bluffs, you reduce the opponent’s chances of winning since he will bet only when he has made his hand, and you can comfortably fold.Besides artificial means, you try to induce a bluff by showing weakness on an earlier round; you stop a bluff by showing strength on an earlier round. Thus, inducing a bluff is something akin to slowplaying, and stopping a bluff is something akin to semi-bluffing.When you induce a bluff, you plan to call if your opponent bets since you have increased the chances he is bluffing. When you stop a bluff, you plan to fold if your opponent bets since you have reduced or even completely eliminated the chances he is bluffing

  • An obvious ploy to stop a bluff is to reach for your chips as though you’re anxious to call. If your opponent still comes out betting, fully expecting you to call, you throw away your hand. Of course, you have to use this play against the right player. An experienced player who sees you reaching for chips and suspects what you are up to is all the more likely to come out bluffing, fully expecting you to fold

  • A ploy to induce a bluff is to give the impression you intend to fold your hand. Now if your opponent bets, you call. But once again an experienced player who sees through the ploy might not bet without a good hand; realizing a bluff won’t work, that player saves money when he or she has nothing

How to play heads-up poker

  • Last Position Play
    • If you are second to act when all the cards are out and your opponent bets
      • Call if your hand is not worth a raise but has a better chance of winning than the pot odds you are getting. Your chances of winning are the sum of the chances that your opponent is bluffing, plus the chances that your hand can beat his legitimate hand
      • Raise if your opponent will still be the underdog after calling your raise. Raise also as a bluff if you think it will work often enough to have positive expectation Also consider raising with what appears to be a calling hand if your opponent is capable of throwing away a better hand than yours for one more bet
    • If you are second to act when all the cards are out and your opponent checks
      • Bluff if you think it will work often enough, remembering that a bluff does not tend to work as often in second position as it might in first position
      • Bet your hand for value if you are a favorite to have the best hand, even when your opponent calls your bet
      • Don’t bet in close situations to avoid a check-raise
  • First Position Play
    • If you are first to act when all the cards are out and have a very strong hand
      • Try to check-raise if your opponent will bet and call your raise more than half as often as he will call you when you bet
      • Come out betting if you don’t think a check-raise will work often enough to be profitable or if you think you can win three bets when your opponent raises and you reraise
    • If you are first to act and have a bad hand
      • Bluff if you can get away with it often enough for the play to have positive expectation
      • Otherwise check and fold if your opponent bets
    • If you are first to act and have a hand that is a favorite to win if called but not strong enough for you to try a check-raise
      • Bet if your opponent will call with more hands than he will bet with if you check
      • Check and call if your opponent will bet with more hands than he will call with
      • Never check and fold
    • If you are first to act and have a hand that is a small underdog to win when your bet is called
      • Bet if your opponent will call with more hands than he will bet, as long as some of the hands he would have bet, had you checked, would be worse than yours. Check and call if you think your opponent will check behind you with a significant number of hands better than yours but might still bluff with some hands you can beat
      • Check and call if your opponent will bet with more hands than he will call with, as long as your pot odds make it worth calling when he does bet
      • Check and fold if your opponent will almost never bet a hand worse than yours

Reading hands

  • There are two universally applicable techniques for reading hands. Most commonly you analyze the meaning of an opponent’s check, bet, or raise, and in open-handed games you look at his exposed cards and try to judge from them what his entire hand might be. You then combine the plays he has made throughout the hand with his exposed cards and come to a determination about his most likely hand

  • Reading hands well is a powerful poker weapon because it allows you to play correctly more often, according to the Fundamental Theorem of Poker. The better you read your opponents’ hands, the less likely you are to play your hand differently from the way you would play it if you could actually see what your opponents had. Weak players are difficult to read because there is little pattern to their play. Good players are easier because there is logic to their play. However, very tough players are more difficult to read because of their ability to disguise their hands.One way to read hands is to put opponents on a variety of possible hands and eliminate some of them on the basis of their play and the cards they catch from one round to the next, keeping track of the order in which they catch their cards. A second, complementary way is to work backward, looking at an opponent’s later plays in terms of how he played his hand in earlier rounds

  • You can also read hands by using mathematics, by comparing possible hands on the basis of Bayes’ Theorem. If you know an opponent will bet only certain hands, you form a ratio based on the probability of that opponent being dealt each of those hands. To simplify, you can divide his possible hands between those you can beat and those you can’t beat. The ratio tells you which of the hands he is favored to have.Finally, when reading hands you must consider the number of people in the pot. When there is a caller ahead of you, the caller and the original bettor cannot both be bluffing, so you must play on the assumption that you are up against at least one legitimate hand. When there is a raiser ahead of you with the same standards as yours, you should have more than your minimum raising hand to call that raiser because you have to figure your minimal raising hand is beat

Psychology of Poker

  • The psychology of poker is an important aspect of the game. You should think not only about what your opponents have, but about what they think you have and about what they think you think they have. You must go through such thought processes against good players in particular, but the better they are, the more difficult it is to figure them out. When you get to the expert level, the process sometimes becomes so complex and tenuous that you have to fall back on game theory.On the other hand, these thought processes can be costly against weak players because your opponents are not thinking on such an advanced level. Against weak players the best strategy is to play your cards in a basic, straightforward way

  • Thinking about what your opponent is thinking will improve your calling and betting strategy. If an opponent is sure you will call his bet, he is not bluffing; if he thinks you will fold, he may be bluffing. By the same token, if an opponent thinks you are strong, you may be able to bluff, but you should not bet a fair hand for value. If an opponent thinks you are weak, you can’t bluff, but you can bet your fair hands for value.Ordinarily you evaluate a poker play solely on its own merits, but you can occasionally make a bad play for psychological effect
  • to create an impression for the future.The psychology of poker is an extension of reading hands and using deception in the play of your own hands, and thus it is an extension of the Fundamental Theorem of Poker  
  • When deciding whether to call, your risk is a bet, and your reward is the pot. Any poker decision can be put into these terms. What do you have to gain (including future benefits on subsequent hands) by making a particular play? What do you have to lose? The ability to evaluate properly the risk-reward ratio for any poker decision is the ultimate test on the road to becoming a champion poker player

  • Accurately and quickly analyzing risk-reward decisions at the poker table in the heat of a hand comes only with experience. Some top players do it intuitively. Most of the time, when the choice of plays is problematic, your best play is the one likely to be correct more than 50 percent of the time. However, when the favored play has very bad consequences when it is wrong, and the less-favored play has only slightly bad consequences when it is wrong, it may be correct to choose the less favored play

  • There are endless kinds of mistakes you can detect in your opponents’ play, and when you detect them, there is always a way to take advantage of them. Following is a list of the most common mistakes poker players make, accompanied by the best strategies to use to take advantage of the mistakes:

Poker Mistakes 1 Poker Mistakes 2 Poker Mistakes 3

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