Expert Poker Concepts

Let's analyze each of these factors in depth.

First, let's examine factor #1: your actual hand against your opponent's actual hand.

You

Opponent

Flop

Your chance of winning in this situation is about 87%. Thus, if you bet $50 and your opponent called $50, you would expect to receive $87 on average. Basically, making that bet earns you a profit of about $37 in the long run.

Of course, you do not always know what cards your opponents have. Because of this, you must take into the account the fact that you are putting your opponents on a spectrum of hands. Here is an example of where you are uncertain of what your opponent holds. The reality of poker is that you will never know for sure what your opponent holds. Nevertheless, your reads of your opponent will enable you to make an educated guess of what your opponent's hole cards may be. Consider this example, where you have made certain reads on your opponent:

Your Hand

Board

You have deduced that there are only two possible hands your opponent could have: A K or Q J. There is an equal chance that he holds each of these hands.

In this situation, the EV = 0.5 * (chance of winning against A K) + 0.5 * (chance of winning against Q J).

Your chance of winning against A K is about 13%, and your chance of winning against Q J is about 53%. Thus, in this example, your overall chance of winning is about 33%. As you can see, your overall chance of winning is highly dependent on the percentage chance you think your opponent has Q J versus A K.

Being able to reasonably estimate the chance than an opponent has a certain hand takes a lot of experience and skill. The important thing to remember is that when evaluating a mistake in retrospect, you need to factor in the chance that the opponent may have had a different hand. Poker is a game of limited information, which needs to be accounted for when evaluating the severity of a mistake.

The final component of analyzing the size of your mistake is the size of the pot. This is because the size of the pot affects the expected value of decisions you make at the table. For example, suppose your opponent bets $20, and you know through your infinite poker prowess that you have exactly a 25% chance to win. Should you call?

Well, it depends on the size of the pot. If the pot is $0, clearly not. That would be a mistake of roughly $10 ($40 * 25% - $20). However, let's say the pot is $100 before your opponent makes the bet. In this case, you would only be putting in $20 into a final pot of $140 ($100 plus the two bets of $20). Since you have a 25% chance to win, your expected value from the pot is $35. Since $35 is much more than $20, you should call. If you had folded, you would have made a $15 mistake.

The most obvious real-life example of how the size of the pot affects decision-making at the table is pot odds. Essentially, pot odds is a shortcut for evaluating factors #1 and #2. You assume that your hand probably is not good enough to win as is, and you assume that you will win the pot if you are able to hit your draw. Hence, the chance of hitting the draw is equal to the chance of winning. What the other players have does not matter since you are assuming they are able to beat you unless you improve your hand.

Thus, with pot odds, you simply calculate the expected value of staying in the pot. If the EV is higher than the amount of the bet, you go ahead and call.

There are many situations where people may make large mistakes and do not realize it. Consider this example:

Your Hand

Board

You flopped top pair. There is $10 in the pot. A fairly tight player goes all-in in front of you for $25. What do you do? Let's analyze this situation given the three factors mentioned previously.

Factors #1 and #2: Your opponent could have many different hands. He could have top pair with a better kicker (AK), a set (88), two pair (A8), or a draw (K Q). He could also have some other hands like a pocket pair (QQ) or top pair with lower kicker (A9), though these hands are much less likely.

Factor #3. The size of the pot. In this case, the size of the pot is small in comparison to the size of the bet. The pot is only $10, so your opponent's bet is 2.5 times the size of the pot. You need a fairly high chance of winning to justify the call.

Now, let's analyze the different hands your opponent could have, and the EV of calling if he actually had it.

As you can see, you should probably fold. You are at best a moderate advantage and very likely at a huge disadvantage. If the pot is so big that you are only putting in 10% of the pot, you should probably call. However, in this case, the bet is large in relation to the pot and you expect this player to probably have a good hand. Because of this, you should fold.

The above example illustrated a player making a huge mistake because he made a loose call. However, many poker players have the opposite problem: they make huge mistakes at the poker table and do not realize it because they think they are making "great laydowns." The decision to fold in a pot can be even more disastrous than the decision to call a bet.

For example, suppose you are playing a $5-$10 fixed-limit hold'em game. There is $80 in the pot.

Your Hand

Board

A player bets $10, two players call. You decide to fold.

Suppose it turns out that other the three players had the following hands:

Opponent #1

Opponent #2

Opponent #3

It turns out that you had a 12.5% chance of winning this hand. In this situation, you would have put in $10 into a final pot of $120 on the turn. If you hit your straight on the river, you probably would have won an additional $10 to $20. Thus, in reality, you would only be putting in $10 to win about $135. If the 9 came out, you probably would not end up paying off on the river because the other players would start raising each other with four clubs on the board.

Thus, your mistake in this situation was worth $6.87. The EV of staying in the pot was about $16.87 and you would need to pay $10 to stay in that pot. In a $5-$10 game, EV of $6.87 is a pretty big mistake, since that's about 0.7 big bets. Most people average a fluctuation of about one big bet per hour at each limit hold'em table they play, so this is a pretty big mistake.

River Betting

The river is a unique round. During all previous rounds of betting, each person had a chance of improving his hand. On the river, bets are pure value bets or bluffs. There is no need to knock someone out, and it is impossible to semi-bluff. These changes in gameplay require an alteration in strategy. While each river is different, I will highlight three common situations where an advanced player can gain an edge.

Multi-way pot with semi-strong hand

Your Hand

Board

You are in middle position. You wait for the check-raise. The button bets, small blind calls, you check-raise, button and small blind call your raise.

The turn comes another jack, so now you hold trips. You bet. The button and small blind call.

The river comes a ten, but there is no flush draw on the board.

Your Hand

Board

Besides you, two other players are still in the pot. The small blind bets. It is your turn and the button is after you. What do you do?

Clearly, you do not fold. There is simply too much money in the pot to lay down trips with top kicker. So your two options are to raise or call. In this situation, I would definitely call. The reason is is that if you call, there is a high likelihood that the player on the button will call. However, if you raise, that player will probably fold. If you raise, the small blind will likely reraise you if he has you beat. However, he will only call you if you have him beat. Therefore, if you have the best hand, you will likely win the same amount whether you call or raise. However, if you hold the second-best hand, you will likely lose two more big bets if you raise instead of call.

So let's say there is a fifty percent chance you hold the winning hand. Let's also assume there is a fifty percent chance the button will call if you call but will always fold if you raise.

Now, we should analyze the proper play based on two scenarios: when you call, and when you raise.

If you call: If you call and lose, you will lose one bet. However, if you call and win, you will probably win 1.5 bets (50% chance button will call and lose to you).

If you raise: If you raise and have the losing hand, you will lose three bets. This is because the small blind will reraise you. If you have the winning hand, you will win two bets. The button will fold and the small blind will call your raise.

Obviously, this is an imperfect example. These probabilities are arbitrary, but they do prove a point. If you call, you expect to win .5 bets (1.5 when you win an 1 if you lose). If you raise, you expect to lose 1 bet (2 if you win, 3 if you lose). Thus, calling will earn you an expected value of 1.5 bets!

On the river, there is no need for you to worry about being drawn out. In the type of situations described above, you have a hand that can beat most hands. The few that can beat you are certainly not going to fold if you raise (they will much more likely reraise you). Thus, you are only focused on winning as many bets as possible or minimizing your losses if you hold the losing hand. Getting an extra bet by having someone call after you has just as much value as a call from someone you raised. When you have a strong, but potenitally beatable holding on the river, it is often better to allow people to make crying calls after you then to try to extract one more bet from one guy by raising him.

Heads up when you have been the aggressor

When you hold top pair in a heads-up pot, you are more than likely betting it through the flop and turn. Now, suppose the river forms a scary hand. What do you do?

It really depends on your position. Let's say you have position on him. For example, he is in mid-position and you are on the button. Your opponent checks. You are contemplating making a value bet, though you are wary of a check-raise. Let's say he will always raise if he has you beat and will always call if you have him beat. This means you need a 2/3 chance of winning to bet. This is because you will win one bet if you have the better hand and lose two if he has the better hand.

Of course there are situations where he will still just call with a better hand and not raise you. There are other times when he will raise you even when you have the better hand.

You will need to use your own judgment on whether to bet in these situations. There is very little I can offer as advice on this subject because this decision is so situational. Just bear in mind that you should not be scared if only one type of holding can beat you. Provided there are many second-best hands that will probably make a crying call, you should bet. However, if the only thing that he could be calling with is a draw that missed and any other hand that he could possibly hold will now beat you, you should probably check after him.

If you are in early position, you should be much more willing to bet. Why? Because even if he has a strong hand. he is going to bet or raise you. Thus, you lose only one more bet by initially betting. However, most of the time, he will probably not have hit that strong of a hand. He will still call you because the pot will be so large that he will need to make a crying call.

Even if he has a semi-strong hand, he may not raise you. When you bet to him, you show strength. He will be reluctant to raise becasue he will fear the reraise. However, if you check, he may sense weakness and will go for the bet. Therefore, whether you bet or check, you lose one bet. Your initial bet doesn't matter in this scenario.

Of course, there are some situations where you should not bet your made hand if you are first to bet. If you hold AJ and the board on the board is AJ5QT with a flush draw on the board, you should probably check. But the important point is that you should be more willing to bet when you are out of position on the river in Limit Hold'em.

Heads up when you have been drawing

If you hit a big draw, what do you do? If you have position on him, it's obvious. Bet and raise. You more than likely have the best hand, get your value out of it!

However, if you are out of position, it is much trickier. Do you go for the check-raise or do you go ahead and bet? It depends on your opponent.

If your opponent is very aggressive, I would recommend the check-raise. First, he may have nothing and has been bluffing into you the whole time. He may try to represent the flush, so go ahead and let him! Also, maniacs almost always go for value bets when the other player shows weakness, so a check-raise is in order.

However, if your player is tight, you should bet. These players get scared easily, especially by flush draws. If you hit some weird straight or trip, you can probably check-raise. But if you hit an obvious draw, you should bet.

If a tight player tends to fold to these river bets, then you should consider bluffing on the river in the future. In general, folding on the river is bad policy. See When to Fold for the reasons why. The tight player may have folded because he was bluffing or missed a draw. However, if it seems that a tight player is really willing to fold on the river, you should consider betting on the river in the future if you miss your draw but another scary draw comes. Players this tight are rare to come by, but you should take advantage of them when you find them.

Expected Utility

Utility of Money

Earlier we discussed the concept of expected value, or the amount of chips you expect to gain from making a certain move. In almost all ring-game situations, taking into account expected value and implied odds will give you enough information to make the right decision. But, underlying the concept of expected value, there is an important assumption: The assumption that every dollar has the same value.

This assumption is not always true. Let's say that that your bankroll is a million dollars. Someone offers to bet you a million dollars on a coin flip. Would you take that bet? Some people would, but most wouldn't. The reason for that is that, if you have a million dollars in the bank, then earning a second million won't bring you much utility (happiness). After all, if you have that extra million, what could you do with it? The utility (happiness) gained from that second BMW and house isn't quite equal to the loss you would experience if you were suddenly made penniless.

The reason for this is that your Utility of Money changes depending on how much money you have. Each level of worth or income has associated with it a certain level of utility. That utility is not necessarily going to increase uniformly. For most people, the second million is not worth as much as the first. This is called Diminishing Marginal Utility. People who fall into this category are called risk-averse:

Other people would take the bet, because the second million is worth exactly as much to them as the first million. These people are called risk-neutral:

Now, there are many people who wouldn't take the million dollar bet at even-money odds, but if they had a 55% chance of winning instead of 50%, they might take it. Of course, many people still wouldn't take it, but some would.

Now, what if the tables were turned? What if you were offered the million-dollar bet with 45% chance of winning? Believe it or not, there are some people who would take the bet. This is because the marginal utility they get from the second million dollars exceeds the utility they get from the first million. Perhaps they need exactly $2 million to save their business, and anything less would result in bankruptcy.

This condition (increasing marginal utility) also partially explains why people buy lottery tickets. In some states, you will get around 30 cents of EV on every dollar you play on the lottery. These are terrible odds! And yet, some people who realize this still play, because the utility they get out of becoming a multi-millionaire (or even just thinking about it) is worth that 70 cents to them. People who fall into this category are called risk-loving:

Risk-loving people are the most susceptible to becoming compulsive gamblers. This is because they enjoy taking bets with bad expected value.

Expected Utility and Poker

You might ask, what does all this have to do with poker? Well, there are a few things. First of all, the type of person your opponent is (risk-averse, risk-neutral, or risk-loving) will affect how he plays at the table. Many players, when they try to move up limits, play scared. This is because the increased amount of money they have at the table is high enough that they are no longer at a constantly sloping point in their utility curve. This helps to explain why you should not play at a limit which is over your head. If you are risk-averse at that limit, you will be conceding alot of small edges. The opposite situation is also true: you may do well at the $10-$20 because you like precisely that amount of risk. If you played $5-$10, or $2-4, you may now be risk-loving, and be playing too loose!

At first glance, one might think that risk-loving is the same as loose, and risk-averse is the same as tight. This is not true: a person may be playing tight simply because that is the best strategy given the type of opponents he is facing. A strong player in a No-Limit game will vary between playing tight and loose, but he is risk-neutral. He will neither turn down a bet with a slight edge, nor lay a slight edge to his opponent. The condition that allows him to vary between tight and loose is that the play of his opponents requires him to.

It's important to make sure that you yourself are playing risk-neutral, and also to read your opponent correctly. If your opponent is playing tightly, don't simply assume that he is risk-averse, and start bluffing like crazy at him. It is important to learn quickly whether he is truly risk-averse or simply playing tightly to take advantage of the other players.

Bankroll Risk vs. Stack Risk

So if you should be risk-neutral with your stack, what about your bankroll? We advise that you be risk-averse with your bankroll. The amount of money that you win or lose in any one session or even a few sessions shouldn't matter too much to your bankroll. Otherwise, your emotions will come into play too greatly, and playing scared is always a bad idea. But how does this make sense? How can you be risk-neutral with your stack and risk-averse with your bankroll at the same time? The answer is that the shape of your utility curve changes as the area you are looking at changes. If the area is small enough, the risk-averse curve will start to look more like a straight line. This is kind of like how the earth is round, but the area we can see with the naked eye is so small that it looks flat to us.

Utility of Tournament Chips

Expected Utility Theory also explains why tournament play is different from ring game play. You may have noticed that people are more risk-averse in tournaments than in ring games. This is because of expected utility: if you have 1,000 in chips early in a tournament, it is most likely a bad idea to take an all-in on a 50-50. This is because getting that second 1,000 in chips is not worth that much to you, but if you lose your first 1,000, you're out of the tournament.

On the other hand, in the middle of a tournament, it may become a good idea to take that very same bet! This is because having a big stack will give you some extra utility because you can now steal blinds. Also, if you are down to a short stack later in the tournament, you would probably gladly take that 50-50 all-in. This is because the utility of having that first 1,000 is much diminished because of the blinds.

Your utility curve in a tournament is dynamic, or changing over time. A winning tournament player is aware of what utility he gets from the chips he is betting, and how his utility curve changes throughout the tournament. He doesn't settle for a positive expected value in chips; if he talks of a positive expected value, it means a positive expected value in prize money. In other words, he demands a positive expected utility on all his bets.

Utility of Deals

Towards the end of tournaments, it is common for poker players to strike deals. Often, the prize structure is very top-heavy, giving first place nearly twice as much money as second. However, the blinds are so high at this point in the tournament that luck will be the primary factor determining the victor. Instead of battling it out, poker players often would rather strike a deal and give each player a proportion of the prize pool. These deals take into account the utility curves of the players. If a poker player is a billionaire and is playing in a $200 buy-in tournament, he probably does not care that much about the variance involved with winning. However, a player who barely scraped together that much money to enter the tournament is likely to be eager to strike a deal. When making a deal, take into account your opponents' and your utility curves. Do not let them take advantage of you because they suspect you are risk-averse. For example, it is speculated that there was no deal in the 2003 WSOP because Sammy Farha thought he could bully Chris Moneymaker at the final table. Farha, a multi-millionaire, knew that the money involved intimidated Moneymaker, an average player. Ultimately, Moneymaker kept his cool, played solid poker, and won the World Series.

Most of the time, people should play as risk-neutral, meaning that their expected utility is equal to expected value. However, there are times when people should be risk-averse, meaning their expected utility is lower than their expected value. For instance, most people would not bet their whole bankroll on a 51-49 shot. While this is a slightly +EV move, the utility a person gets from doubling their bankroll is likely much smaller than the utility they stand to lose from blowing their whole bankroll.

There are times when people might even be risk-averse with their chips at their table. Most of the time, this is an incorrect play, but there are times when this is warranted. For instance, many tournament players are risk-averse with their chips. This is because you are not guaranteed to win anything if you double up in a tournament. However, you are guaranteed to be out of the tournament if you lose your entire stack. (It should be noted that a lot of players are overly risk-averse in tournament situations to the point that it is heavily -EU).

Nevertheless, there are some instances when it might even be in a player's interest to be risk-loving with their chips at the table. This means that an -EV move might actually be a +EU move. There are not many instances where this is the case, but it can occur.

An -EV move that is +EU means that the action is definitely -EV for the individual hand, but it may turn out to be +EV in the long run assuming certain other factors come into play. Basically, the move is a mistake if you just consider the one pot alone. However, when factoring in later pots or later play in the tournament, the move may turn out to be +EV.

Before proceeding further into this article, it needs to be made clear that this type of move is for expert players only. Do not attempt to employ these theories if you are an average or even above-average player. Generally, only professional-level players are able to profitably apply these theories.

Opportunity Cost (Tournaments)

If you are playing an online poker tournament, you can likely multi-table a cash game or two on the side as well. Some players only like to play one game at a time, but most internet experts are comfortable with multi-tabling.

However, in a live casino game, multi-tabling is either extremely difficult or impossible. You will almost certainly miss out on hands in the tournament, which will certainly decrease your EV from the tournament. If you attempt to multi-table, most people get extremely angry with you because you end up slowing down the game considerably. Because of this, some poker rooms will simply not allow it.

Thus, when you are playing a tournament at a live casino, you are giving up the opportunity to play in a cash game. When deciding whether to play in a tournament, you would need to consider the hourly rate you expect to make from the tournament versus the hourly rate from the cash games at the casino.

If there happens to be very juicy cash games that night or your hourly rate from cash games are just really high in general, an extra aggressive style of play at a tournament might make sense. It may be profitable to be risk-loving in the tournament (hence certain -EV situations turn out to be +EU).This is because you will quickly be in position to win the tournament (hence the expected value of continuing to play in the tournament would be really high), or you would be able to play in the cash games.

Simply holding on to a puny stack might be the lowest EU of all situations. Mathematically speaking, it may make more sense to just give up on your stack on go play in a cash games because your expected hourly rate is higher in the cash game than holding onto that stack.

The reality of the situation is that people are not solely concerned with their hourly rate when playing poker. Many want to play in tournaments just for the enjoyment factor of the tournament. It is also very difficult to know for sure what one's expected hourly rate of a tournament is compared to the cash games at a casino.

This sort of opportunity cost situation does not apply as much to online players because playing a tournament just means one extra window on their computer screen. Sure, people generally max out on the number of tables they can play at once. Nevertheless, playing in a tournament is generally less intensive than playing in a cash game because you tend to fold more preflop. Thus, it really does not affect your ability to play cash games nearly as much as a live tournament does.

Bullying (Tournaments)

Some players only play well when they possess a lot of chips. This is because they frequently try to bluff people out of the pot and make other aggressive moves. They also want to be in a position to capture anyone's entire stack in one hand. These sorts of players most often aim to win the entire tournament. Their focus is not so much on survival as capturing first place.

Sometimes, this sort of play makes logical sense. In tournaments that are winner-take-all or heavily top-weighted, one should certainly go for the gold. For aggressive players in these sorts of tournaments, an ultra-aggressive strategy may pay off. These sorts of players would attempt to build their stack up as quickly as possible, even if it means they make some slightly -EV decisions for the chance of doubling their stack up.

By gambling it up, these players hope to build a powerful stack that will let them bully their opponents into submission. For these players, the -EV decisions they make at first turn out to be +EV over the long run because they play so much better with a larger stack.

Big Fish (Cash Games)

Sometimes, a poorly-skilled player will have a huge stack in a no-limit cash game. This person may have just bought in for a lot more money than most people, or this player may have just had a huge winning streak.

In a no-limit game, this player is apt to lose this entire stack in a hand or two, given the right conditions. When a fish with a huge stack is in a game with several good players, all of these players are gunning to get their stacks large enough to be able to take out the fish in one hand. Not only does the fish have a lot of money to lose, this type of player is not used to playing a large stack. Playing an extra-large stack requires different strategy than playing a small or mid-sized stack, and many casual players are ill-prepared to play such a large stack.

Some brick-and-mortar casinos allow a player to buy in for as much as he or she wants. So in these cases, a professional player could just buy in for as much as the big fish has, hoping to double up. However, all internet poker rooms and a lot of B&M casinos have limits on how much people buy in for. This means that in order to double up through the fish, one would have to increase his or her stack a lot by winning chips.

For instance, suppose you are in a $5-$10 no-limit hold'em game. The max initial buy-in is $1,000. You have $1,000 in chips, player A has $500, player B has $1,000, player C has $2,500, player D has $3,000, and Player E has $3,000. Player D and E are huge fish, while players B and C are about average.

In this case, even if you hold a monster hand against Player D or E, you can only get about 1/3 of their chips. In contrast, Player C could potentially get 5/6 of Player D or E's chips, a much heftier sum.

Against a big fish, one can potentially have huge edges. This is because inexperienced players will frequently pay off a player who clearly has the nuts. Thus, if you are able to build a huge stack, there is the chance that you will be in a situation where you are both all-in where you are a lock or at least a huge favorite.

It is likely once Player D or E loses their chips, they won't be back for awhile, at least not with that type of stack. Thus, there is a certain sense of urgency to get a larger stack in order to bust out the fish before others do.

However, getting that large stack is not always easy. One would have to win quite a few hands to get the type of stack needed. In this case, it may pay off in the long run to be fairly risky with your stack. Loosen up and see more flops. If you are making a decision for all your chips and are somewhat unsure of what move to make, it may be better to just go ahead and push because the rewards from having a huge stack in this game might be really high.

Of course, this does not mean you should be foolish with his stack and go all-in when you are a significant underdog. However, in cases where it's probably close to 50-50, it may be beneficial for one to lean towards gambling it up instead of avoiding a confrontation. Knowing when a situation is around a 50-50 is very difficult for poker players, which is why this move should only be done by expert players (expert players are also more likely to be able to bust a fish once they have a huge stack, too).

Inducing Tilt (Cash Games)

A lot of players in no-limit games can play very well, except when they go on tilt. When this occurs, these players will frequently drop a lot of money at the table. Many of the higher-stakes no-limit games are virtually unbeatable unless a player at the table goes on tilt. Most people should just avoid games this difficult, but many play in them anyway.

Nevertheless, once a player or two goes on tilt, profit opportunities may arise. This means that in these sorts of games, making plays to induce a player to tilt may pay off in the long run.

For example, certain types of bad beats may tend to destabilize a player. When on tilt, this player may tend to play "revenge poker" against the person who bad beat him or may just play bad in general. This is a heavy +EV result for the person who set the player on tilt.

So in these types of games, it may make sense for a person to make a -EV move if the person thinks there is a good chance that it may set another player on tilt. An example would be calling a large bet with a draw. Some players get extremely upset when others draw out on them, so calling and hitting a draw may set them on tilt.

Again, this sort of move should be used with extreme caution. For it to work, a player needs to get lucky enough to win the hand, the other player needs to go on tilt, the other player needs to not leave the table, and the other player needs to lose money to you while he or she is on tilt. These are a lot of "ifs," which means that even an expert should be wary of making even a modest sized -EV move in an attempt to set an opponent on tilt.

Advertising (Cash Games)

Many beginners' poker books talk about the "advertising" value of the bluff. This means you should show a bluff or try to get caught bluffing so your opponents will be more likely to pay you off later when you hold a hand.

Most books advise against this practice, with good reason. Most people play in poker games where players do not really pay attention to each other. This is especially the case with internet poker games. This means that people will not pay attention to your advertisement, so you just end up wasting money. A lot of players also have heard of this move and will not be fooled by your deceptive play.

However, there are a few cases when "advertising" may make sense. For example, suppose it turns out that someone new will be joining your buddies' weekly home game indefinitely. This player does not know how to play, and you happen to be a very tight, straightforward player.

It may pay off to try to bluff out this player at first, so his first impression of you is that you are a maniac. First impressions are very important in poker, just as they are in life. This player may continue to play you as a maniac in later sessions, even when it should be abundantly clear to him that you are a tight, straightforward player.