Monday, December 27, 2010

Fifth Street Tactics: Keeping opponents in versus driving them out

If you have a made low on Fifth Street and no one else does, it is often advantageous to keep players in the hand contributing to the pot. If the high hand is willing to bet, calling might be a more profitable option because you do not want to just get your money back if it becomes a heads-up contest with the high hand. However, there are situations where raising with the intent of driving others out of hand is preferable. Some examples:

-- Your low is vulnerable to another player with a low draw. Suppose you have (2, 4) A, 7, 8 and another player has an exposed 6, 3, 3. The pair means a low has not been made, but she could have a draw to a better low and she holds three cards that would improve your low. Your low hand is very vulnerable to becoming second best later in the hand. If the high bets you should raise and attempt to force her out.

-- You can freeroll your low for a powerful high hand and do not want other hands to compete against it for high. Suppose you have (2, 5) 3, 4, 7 in a multi-way pot in which one of the players appears to be on a flush draw. Another player with an exposed pair of 8s bets. You should raise to force out the flush draw. If you make your straight, you want it to hold up for high.

-- The high hand appears weak and possibly the result of a busted low. Suppose you have (6, 3) A, 5, 7 against a player with an exposed 2, 2, J. The pair of 2s might be the only thing he has going for him and pairing any one of your cards could lead to a better high. Aggressive raising might win the pot outright. He could judge that it is too risky to pursue the hand for high.

-- You have a lock on low in a multi-way pot in which raising will not drive the others out. There could be several powerful high hands that have developed or are developing in such a way that your opponents will pay any price to stay in the hand. Obviously in that situation, raising will maximize the amount in the pot and comes with no risk, even if you are drawing dead to high.

Monday, November 29, 2010

Missed Opportunities on the River

Extracting maximum value from winning hands is as important as limiting financial damage from losing hands. Here are some examples of mistakes I've made on the end.

Missing an extra bet

Opponent: (X, X) A-Clubs, A-Spades, 10-Hearts, 10-Spades, (X)
Me: (3-Clubs, 5-Hearts) 4-Clubs, 6-Spades, 5-Diamonds, 5-Clubs, (J-Clubs)

Action: Obviously I was unhappy to see the brick on the end that denied me a qualifying low-hand. My opponent had led all the way in this hand with the exposed pair of Aces, and I had serious doubts that my trip 5s would hold up for high. But, on the river my opponent checked and after I thoughtlessly checked back, I won the entire pot.

Analysis: I missed picking up an extra bet on the end because it is not possible for my opponent to have trip Aces or trip 10s. If he has three of either rank his hand would be a full house, which is a holding that he would certainly bet. In fact he would bet quads, a full house, a flush, or an Ace-high straight. The only reason for a check is that he has none of these holdings, and fears losing to a possible small straight. Therefore my trip 5s has to be the nuts and I should bet. With Aces-up, he has to call because the pot was large and I could be betting with only a low-hand.

Missing a chance at half the pot

Opponent: (X, X) 10-Clubs, 9-Hearts, 8-Spades, 7-Clubs, (X)
Me: (2-Spades, Q-Clubs) Q-Spades, 5-Spades, 6-Spades, A-Clubs, (7-Diamonds)

Action: This was a heads-up hand, that because of the high door-cards, I bet out thinking that no qualified low-hand would result. I checked on Sixth Street and when my opponent responded by betting into my Queens, I read him for a straight and stayed because of my flush draw. I missed the flush-draw on the river but backed into a 7-high nut-low. My opponent bet on the end and I made the mistake of calling. He won the high-pot with two pair 10s and 7s.

Analysis: There was no reason for me not to raise in this situation. I have no risk of being scooped by a straight and a raise would force him to make a difficult decision if he missed his draw, which in this case he did. Do you call someone raising on the end with two overcards, when all you have is two small pair? He's not expecting a low-hand on my side anymore than I did. Most likely he would fold because his river bet amounts to a semi-bluff.

Part of the reason for my errors in both these cases was backing into a different kind of hand than what I had sought. In the latter case I had too much mental focus on playing a high-hand without thinking about the backdoor low possibilities. The call was an afterthought because I had not been looking for a low-hand. In the former case I was looking for a low straight because I believed trip 5s and even 5s-full would lose. But, backdoor low-hands and backdoor high-hands occur frequently in Stud-Eight. You need to quickly switch your thought processes when they occur and think about the new tactical possibilities that they present.

Thursday, October 14, 2010

The problem with one-way low-hands

Many players who have four low-cards after Fourth Street in Seven-Card Stud High-Low will stay until the end, and even raise before completing a low-hand. However, a hand with four low-cards and nothing else going for it has more problems than you might imagine. Consider a common scenario in which four unconnected low-cards are heads-up against a high pair. Consider this example:

You: (8, 7) 3, 2
Alice: (Q, K) K, 9

If your hand is completely live there are 16 outs to complete low-hand. That means that by the end you should complete a low-hand 73% of the time. The majority of hands, in which you split the pot, will return your money plus half the money already present from the antes and bring-ins. However, 27% of the time you lose all the money you invested on the later streets. Clearly this is a negative expectation contest because you win no additional money from your later bets the 73% of the time that you succeed, but 27% of the time you will lose all the money you invested. The money that already exists in the pot from antes and Third Street betting is rarely so large that half will offset this negative expectation.

In a three-way pot, your expectation is positive, but not as high as you might think.

You: (8, 7) 3, 2
Alice: (Q, K) K, 9
Bob: (J, J) 10, Q

If you, Alice, and Bob, each contribute $50 to see the final three cards, there will be $150 at stake. If this situation is played 100 times, you will have spent a total of $5000 to win half of $150, or $75 for the 73 low-pots that you will win on average. Your total return is $5475, which is less than a 10% return on your investment, barely enough to cover the rake. However, if you are in a hand such as this against two one-way high hands, there is the possibility of making your low-hand early and being able to freeroll on later streets.

However, a dangerous situation arises when you have a one-way low-draw against a high hand and another low-draw. In this case the probability of making a low-hand decreases because your draw is usually not completely live. The reduction in outs can be exacerbated by mucked low-cards after the deal. Consider a deal in which a 5 and 6 are mucked on Third Street and the following three-way hand develops:

You: (8, 7) 3, 2
Alice: (7, 5) 4, A
Bob: (J, J) 10, Q

This is a terrible situation to be in. Alice has three of your outs and two other outs are dead. There are only 11 cards available to complete your low-hand, which means the probability has decreased to 59%. While this is still a better than even chance it shifted your expectation to negative. If you, Alice, and Bob, each contribute $50 to see the final three cards, there will be $150 at stake. If this situation is played 100 times, you will have spent a total of $5000 to win $75 for the 59 low-pots that you will win on average. Your total return is $4425, a loss of $575 or 11.5%. That figure optimistically assumes that you win the low-pot each time that you make a qualifying low-hand. In fact, Alice is drawing to a better low-hand than yours, and a significant fraction of the time she will win the low-pot even if you qualify. That means that your expected losses will be much worse than 11.5%.

However, if your hand has scoop potential the expectation shifts to your favor. Consider having connected low-cards:

You: (3, 4) 5, 6
Alice: (Q, K) K, 9

Alice will still scoop the 27% of the time that you fail to make a low-hand. But 44% of the time you will complete a straight that most likely will scoop, and 29% of the time you will win the low pot. If you and Alice each contribute $50, there will be $100 at stake. Consider 100 trials of this scenario. At $50 for each trial it will cost you $5000 total. On average, you will win $100 the 44 times you hit the straight, and $50 the 29 times you make a low-hand only. Your total winnings over 100 trials will average $5850,which is a return of 17%

In a three-way pot against two high hands your positive expectation is even greater if both high hands stay until the end and a straight holds up for high. Consider this example:

You: (3, 4) 5, 6
Alice: (Q, K) K, 9
Bob: (J, J) 10, Q

If you, Alice and Bob each contribute $50, it will cost you $5000 to play this scenario 100 times. On average, you will win $150 the 44 times you hit the straight, and $75 the 29 times you make a low-hand only. Your total winnings over 100 trials will average $8775,which is a return of 75%. In practice this large positive expectation will be offset by the times when the high-hands improve to better than a 7-high straight which will still result in a split-pot.

These examples show how important the possibility of a scoop is to determining expectation. The challenge when you play the high side of these scenarios is to judge if your opponent has scoop potential so that you can avoid playing a hand in which you have a negative expectation. In the examples discussed, I specified the hole cards so that I could present a precise calculation of expectation. In practice you don't see your opponent's hole cards and must infer the values. In you fold a high pair any time that your opponent has two exposed low-cards, you are giving up in a situation in which you have positive expectation. However, anytime you are playing into a sequence of four connected low-cards, or four suited low-cards, you have a negative expectation.

Here are some guidelines for making that judgment.
  • Count your opponent's outs for a low-hand. If many of the low-cards needed are dead the probability that your opponent will qualify for the low-pot by the end drops considerably.
  • Note possible implied outs. A hand with a low door-card that limped in on Third Street and mucked after catching a high card on Fourth Street, probably removed two additional low-cards from play, not just the one exposed.
  • Pay attention to the blockers and take special note of the 4s and 5s. As explained in the previous section if the either rank-4s or 5s-are dead, low straights cannot occur.
  • Note gaps in exposed low-cards. An 8, 2 showing is much less of a threat than an exposed 3, 2.
  • Most importantly, track your opponent's tendencies. A tough, aggressive opponent who always plays to scoop is much more likely to have connected low-cards than an opponent who consistently limps in with any random set of low-cards.
The common Fourth Street confrontation between a high-hand and a draw to a low-hand has a precarious balance. The player with the low-hand has more information because it is difficult to hide a high-hand, while the player with the high hand must guess at the quality of the low-hand. However, no qualified low-hand can exist on Fourth Street which means that high hand is poised to scoop unless drawn out against.

Wednesday, September 15, 2010

Counting Outs in Seven-Card Stud High-Low

Having identified the different kinds of drawing hands last month, we can now count the outs available in order to determine likelihood of improvement. However, Stud games, in contrast to flop games, usually have many more exposed cards to account for when adding up outs. As a result, counting outs is much more situational in Seven-Card Stud High-Low than in Hold'em.

Consider the following extreme example. For the coordinated high example of 2-clubs, 3- clubs, 4-clubs, 5- clubs, 5-diamonds, 5-spades there are a total of 31 cards that will improve its high ranking, qualify the hand for the low-pot, or do both. But the 31 total is an upper limit on the number of outs available to improve the hand. Suppose the 9- clubs, 7- clubs, 4-diamonds, and 6-spades are exposed other hands. All these cards are no longer available as outs, so the count needs to be reduced by four to a total of 27


Let us also suppose that another player has all four Aces exposed in her hand. Now not only must the Aces be removed from the total outs, but most of the high draws for this hand are now dead. There is only one out available for this hand to take the high-pot-the 6-clubs-and for the low pot all that remain are the two other 6s, three other 7s, and the four 8s. Instead of 31 outs available, there are actually only 10 cards that matter, because if someone already has quad Aces hitting quad 5s or any of the full houses, or high flushes, will not win anything.

The consideration of this extremely unlikely scenario is useful because it illustrates four important points about counting outs. You need to subtract from the total for the following reasons.

Dead cards: You must subtract from the total available outs all the helpful cards that you see in other hands because they are no longer available. This is different than in Hold'em, in which every card that you see is potentially playable.


Dead draws: You must subtract from the total available outs all the cards that you see that improve your hand but do not win. Again in contrast to Hold'em, you can often see in a Stud game when you are drawing dead, because certain improvements will not beat your opponent's exposed cards ("beat the board").


Weighted outs: Not all outs are of equal value. In high-low games some outs win the entire pot while other outs win just half. In the example of the straight flush draw versus quads, the 6-clubs wins it all while the nine available low cards win just half the pot. To account for this difference, in this book I introduce the concept of counting "weighted outs." To count weighted outs, add 1 for each out that scoops, and 1/2 for each out that only wins half the pot. For this example there is one out that scoops and nine outs that take the low-pot, so the total number of weighted outs is 1 + 9 x (1/2), or 5.5. I will show in a future post how a count of weighted outs can be used to determine pot equity, which is a measure of how much you should invest when betting on a hand.

Implied outs: You can often subtract from the total number of outs, cards that you cannot see because your opponent's actions imply the contents of their hole cards. Suppose three players in the hand after Fourth Street show 2-3, 5-7, and A-2, and all act as if they are on draws to low-hands. If you are looking for a low-card, it is clear that six of them are dead, but in this situation you can imply that 12 are dead, because the six unseen hole cards are most likely low.

Once an accurate count of outs is determined, the probability for improvement can be found by dividing the total outs by the number of unseen cards.

Sunday, August 15, 2010

Kinds of Drawing Hands in Seven-Card Stud High-Low (Stud-Eight)

In high only poker the concept of drawing to improve the rank of your hand is simple--the hand can only increase in rank. Therefore, cards drawn either raise the rank of your hand or they do not. But, in high-low forms of poker, defining improvement is more complicated because there are three kinds of improvement--increasing the rank of your high-hand, qualifying for a low-hand, and improving your low-hand. All hands, no matter how low, have the potential to win the high-pot, but not all hands qualify for the low-pot. To complicate matters further, some drawn cards can simultaneously improve your high and low hands, while some drawn cards can improve your high hand at the expense of disqualifying you from holding a low-hand. To account for these complexities we will define two categories of hands--half-made hands and drawing hands, and within each of these categories define three kinds of hands. That makes a total of six different kinds of drawing hands.

Half-made hands are holdings that can win half the pot, but need improvement to win the other half. The three kinds of half-made hands in order of desirability are:

Coordinated highs are hands that still need to qualify for the low-pot, but have outs for a low-hand that will improve the rank of the high hand. For example consider the best possible coordinated high that you could hold on Sixth Street--2-clubs, 3-clubs, 4-clubs, 5-clubs, 5-diamonds, 5-spades. This hand is already trip 5s, but it has four sequential suited low-cards. It can improve its high ranking to quads, a full house, or a 9-high flush or better without qualifying for the low-pot. But, it can also improve it high ranking to a 5-high or 6-high straight flush, 7-high or 8-high flush, 5-high or 6-high straight and at the same time qualify for a low-hand. It can also qualify for low without improving its high ranking. For example it can become a 7-high or 8-high low-hand, and remain trip 5s.

Uncoordinated highs still need to qualify for the low-pot, but in doing so cannot improve their high ranking. While coordinated highs might have some outs to qualify for the low-pot that do not improve the high ranking, for uncoordinated highs none of the outs for a low-hand improve the high hand. Consider the best uncoordinated high-hand that you can hold--A-spades, 2-spades, 3-spades, 5-diamonds, Q-spades, K-spades. This hand is an A-K-Q-high flush with four low-cards. There are 16 outs to qualify this hand for the low-pot--any of the 4s, 6s, 7s, or 8s--but none of these outs will change the fact that the high hand is a flush.

Lows are hands that qualify for the low-pot, but rank poorly as high hand. For these hands you are drawing for the high half of the pot. For example the holding--2-hearts, 3-hearts, 4-hearts, 5-hearts, 7-diamonds, 8-spades is a qualified 7-high low-hand, but only an 8-high high hand. However, it has many outs available to improve its high ranking. An A-hears or 6-hearts would result in a straight flush, a K-hearts, Q-hearts, J-hearts, 10-hearts, 9-hearts, 8-hearts, or 7-hearts, would make it a flush, while an A-clubs, A-diamonds, A-spades, 6-clubs, 6-diamonds, or 6-spades, would make it a straight. Any of the remaining 2s, 3s, 4s, 5s, or unsuited 8s or 7s would make it one pair.

Drawing hands are holdings that need improvement to win any part of the pot. In order of desirability the kinds or drawing hands are:

Scoop draws have outs available that can simultaneously win the high and low pots. Consider a Sixth Street holding of 2-clubs, 3-clubs, 4-clubs, 5-clubs, 9-hearts, 10-spades. This open-ended straight flush-draw has outs that will improve to a high-only flush (K-clubs, Q-clubs, J-clubs, 9-clubs, 10-clubs), outs that will qualify for the low-pot (7-hearts, 7-spades, 7-diamonds, 8-hearts, 8-spades, 8-diamonds) along with outs that qualify for the low-pot and improve the high hand. The 7-clubs or 8-clubs completes a flush and qualifies for low; the A-clubs or 6-clubs completes a straight flush and qualifies for low; the A-diamonds, A-hearts, A-spades, 6-diamonds, 6-hearts, 6-spades, would all complete a straight and qualify for low.

One-way high-draws can never qualify for the low-pot. The only outs available improve the high-ranking. Consider K-spades, K-diamonds, K-hearts, Q-hearts, J-hearts, 10-hearts, played against a 3-hearts, 4-diamonds, 5-spades, 6-clubs, 7-diamonds, 8-hearts, 8-spades. The trips Kings will lose to the 8-high straight, but the hand has many outs for a better high hand. It can improve to a straight flush, quads, full house, flush, or straight, and all of these hands would beat a low straight for the high-pot. However, the 8-high straight has a lock on the low-pot because a hand with trip Kings can never qualify.

One-way low-draws are drawing dead for the high-pot, but can still qualify for the low-pot. For example the hand A-clubs, 2-spades, 4-diamonds, 7-hearts, J-clubs, K-spades can never make a high hand better than one pair. If it plays against an opponent showing two pair--10s and 9s--on the board the hand is drawing dead for the high-pot, but drawing any of the remaining 3s, 5s, 6s, or 8s will win the low-pot.

Wednesday, July 14, 2010

Drawing Hands in Seven-Card Stud High-Low

In high only poker the concept of drawing to improve the rank of your hand is simple. The hand can only increase in rank; cards drawn either raise the rank of your hand or they do not. But, in high-low forms of poker, defining improvement is more complicated because there are three kinds of improvement—increasing the rank of your high-hand, qualifying for a low-hand, and improving your low-hand. All hands, no matter how low, have the potential to win the high-pot, but not all hands qualify for the low-pot. To complicate matters further, some drawn cards can simultaneously improve your high and low hands, while some drawn cards can improve your high hand at the expense of disqualifying you from holding a low-hand. To account for these complexities we will define two categories of hands—half-made hands and drawing hands, and within each of these categories define three kinds of hands.


Half-made hands are holdings that can win half the pot, but need improvement to win the other half. The three kinds of half-made hands in order of desirability are:

Coordinated highs are hands that still need to qualify for the low-pot, but have outs for a low-hand that will improve the rank of the high hand. For example consider the best possible coordinated high that you could hold on Sixth Street—2-clubs, 3-clubs, 4-clubs, 5-clubs, 5-diamonds, 5-spades. This hand is already trip 5s, but it has four sequential suited low-cards. It can improve its high ranking to quads, a full house, or a 9-high flush or better without qualifying for the low-pot. But, it can also improve it high ranking to a 5-high or 6-high straight flush, 7-high or 8-high flush, 5-high or 6-high straight and at the same time qualify for a low-hand. It can also qualify for low without improving its high ranking. For example it can become a 7-high or 8-high low-hand, and remain trip 5s.

Uncoordinated highs still need to qualify for the low-pot, but in doing so cannot improve their high ranking. While coordinated highs might have some outs to qualify for the low-pot that do not improve the high ranking, for uncoordinated highs none of the outs for low improve the high hand. Consider the best uncoordinated high-hand that you can hold—A-spades, 2-spades, 3-spades, 5d, Q-spades, K-spades. This hand is an A-K-Q-high flush with four low-cards. There are 16 outs to qualify this hand for the low-pot—any of the 4s, 6s, 7s, or 8s—but none of these outs will change the fact that the high hand is a flush.

Lows are hands that qualify for the low-pot, but rank poorly as high hand. For these hands you are drawing for the high half of the pot. For example the holding—2-hearts, 3-hearts, 4-hearts, 5-hearts, 7-diamonds, 8-spades is a qualified 7-high low-hand, but only an 8-high high hand. However, it has many outs available to improve its high ranking. An A-hearts or 6-hearts would result in a straight flush, a K-hearts, Q-hearts, J-hearts, 10-hearts, 9-hearts, 8-hearts, or 7-hearts, would make it a flush, while an A-clubs, A-diamonds, A-spades, 6-clubs, 6-diamonds, or 6-spades, would make it a straight. Any of the remaining 2s, 3s, 4s, 5s, or unsuited 8s or 7s would make it one pair.

Drawing hands are holdings that need improvement to win any part of the pot. In order of desirability the kinds or drawing hands are:

Scoop draws have outs available that can simultaneously win the high and low pots. Consider a Sixth Street holding of 2-clubs, 3-clubs, 4-clubs, 5-clubs, 9-hearts, 10-spades. This open-ended straight flush-draw has outs that will improve to a high-only flush (K-clubs, Q-clubs, J-clubs, 9-clubs, 10-clubs), outs that will qualify for the low-pot (7-hearts, 7-spades, 7-diamonds, 8-hearts, 8-spades, 8-diamonds) along with outs that qualify for the low-pot and improve the high hand. The 7-clubs or 8-clubs completes a flush and qualifies for low; the A-clubs or 6-clubs completes a straight flush and qualifies for low; the A-diamonds, A-hearts, A-spades, 6-diamonds, 6-hearts, 6-spades, would all complete a straight and qualify for low.

One-way high-draws can never qualify for the low-pot. The only outs available improve the high-ranking. Consider K-spades, K-diamonds, K-hearts, Q-hearts, J-hearts, 10-hearts, played against a 3-hearts, 4-diamonds, 5-spades 6-clubs, 7-diamonds, 8-hearts, 8-spades. The trips Kings will lose to the 8-high straight, but the hand has many outs for a better high hand. It can improve to a straight flush, quads, full house, flush, or straight, and all of these hands would beat a low straight for the high-pot. However, the 8-high straight has a lock on the low-pot because a hand with trip Kings can never qualify.

One-way low-draws are drawing dead for the high-pot, but can still qualify for the low-pot. For example the hand A-clubs, 2-spades, 4-diamonds, 7-hearts, J-clubs, K-spades can never make a high hand better than one pair. If it plays against an opponent showing two pair—10s and 9s—on the board the hand is drawing dead for the high-pot, but drawing any of the remaining 3s, 5s, 6s, or 8s will win the low-pot.

Sunday, February 28, 2010

Betting on the end into a heads-up split-pot

I frequently see players berate other players for betting, or raising, a heads-up hand on the end in situations in which the pot is split. The complaint is that such an action costs both players money by contributing unnecessarily to the rake. The thinking goes that if a player knows that the pot will be split, it is foolish to throw additional money on the table for the house to rake before giving it back to the players. I've witnessed some very nasty comments directed at players who have made bets on the end with full knowledge that calling the bet would result in a split-pot. I've also witnessed players thank an opponent for not betting on the end and accepting the split-pot without further action.

Actually the players making these kinds of comments frequently misunderstand how the rake is calculated. In many of these situations the additional action is not generating rake for the house. The rake structure at most cardrooms usually includes a cap on the total rake taken from a single pot. Once a pot becomes large enough that the rake is capped, further bets and raises do not cost the players additional money. However, the cap depends on the number of players dealt into the hand (not the number seated at the table) which makes knowing when the cap is in effect difficult at times.

For example, consider the rake structure of a $2-4 Stud-Eight game at Full Tilt Poker . The rake is 5%, or $0.50 per $10 in the pot. But, if only two players are dealt into the hand the rake is capped at $0.50. In heads-up play, with the two players betting on every street, the cap will be reached by Fifth Street. That means action before a showdown will not cost the players or benefit the house. But, if it's the same $2-4 limits with seven players dealt into the hand, the rake is still 5%, but the cap rises to $3.00. It now takes a $60 pot to cap the rake. In a $2-4 game $60 is a large pot, even with seven players dealt in because most will fold early in the hand. In most cases, additional action on the end will generate more rake for the house.

The reason for the discrepancy is that in a two-player hand, it is common to see each player make $2 Fourth Street bets followed by $4 Fifth Street bets and a $10 pot that caps the rake is reached. But is very rare in a seven-player hand to see every single player contribute $2 on Fourth Street and $4 on Fifth Street. Even if that happened, the $42 contributed is still less than the $60 needed to cap the rake.

The situation is a little different in the $2-4 games at Poker Stars. In a two-player game the rake cap is $1, however, only $0.50 is taken from the first $20 in the pot. In a seven-player game the rake is capped at $3.00, but $1.00 is taken from a $20 pot, which is the same as at Full Tilt.

What is important to realize is that the $3 cap at Full Tilt and Poker Stars applies to all games at $2-4 limits and higher. In $5-10 games, in which $60 pots would be common, the same 5% of the pot up to $3.00 applies. Also, both cardrooms reduce the rake cap for limits less than $2-4. For $1-2 games the maximum rake at each site is $1.00, and for $0.50-1 games is it $0.50.

This means that the rake structure for the $2-4 game is the least favorable for the players because at that limit, the $3.00 cap on the rake will rarely be reached. Additional bets on the end will generate more rake for the house. You are better off playing at either a higher or lower limit than $2-4.

Both sites, by the way, have policy of not raking a hand unless it gets to Fourth Street. No money is raked if no one calls the bring-in, or if everyone folds to a raise on Third Street.

It is clear from examining the rake structures, that the complaints that players are wasting money with bets on the end are not always true. For low-limit games-$1-2 and below-a pot size of 10 large bets caps the rake. For medium to high-limit games-$3-6 and above-a pot size of 10 large bets or less caps the rake. It is only at the $2-4 limit that extra action at the end costs the player money.

Of course, even if the action on the end is generating more rake for the house, players need to be careful about automatically assuming that their opponent will not fold to a bet. Even if the bettor can't beat the board, that doesn't mean that the opponent will have enough confidence in a small pair showing along with a missed low-hand to call. That situation occurs frequently in Stud-Eight. Inducing a single fold of a better high hand could pay for all of the additional rake spent on hands in which the other player did not fold.

Thursday, February 18, 2010

Low Hands in Seven-Card Stud High-Low Eight-or-Better

I often see a great deal of confusion about the ranking of low-hands in Seven-Card Stud High-Low (Stud-Eight). This post is meant to explain the rankings.

When ranking low-hands, an Ace is always ranked as the lowest card. In Stud-Eight, a low hand must "qualify" to win the low pot. In contrast, the game of Razz awards the entire pot to the lowest hand with no conditions attached. To qualify for the low-pot, a low-hand must contain five cards with none paired, and none ranked higher than an 8. For example: 8, 5 4, 2, A and 7, 6, 4, 3, 2 are qualifying low hands. A hand such as 9, 4, 3, 2, A does not qualify for low. When comparing low hands, the high cards are compared first. Therefore 8, 4, 3, 2, A would loose to 7, 6, 4, 3, 2. When high cards match the second highest cards are compared and so on until there is a discrepancy. For example 8, 7, 4, 3, 2 would loose to 8, 6, 4, 3, 2 and 8, 6, 4, 3, A would beat 8, 6, 4, 3, 2. If all cards in two or more qualifying low hands match, the players split the low pot.

Straights and flushes do not disqualify a hand from low. As a result, the best possible low hand is 5, 4, 3, 2, A, a hand that could also compete for the high pot as a 5-high straight. A hand such as A, 2, 4, 5, 7 all in spades, could compete as an Ace-high flush for the high pot, and 7-high low for the low pot. In high-only poker, the dream hand is the "royal flush" (Ace-high straight flush) because it out-ranks all other hands. In high-low games, the dream hand is the "steel wheel" which is an A, 2, 3, 4, 5, all in the same suit. Simultaneously it serves as the best possible low hand, and as a 5-high straight flush. Even thought it is the lowest-ranked straight flush, it would win high against any player with four-of-a-kind.

There are a total of 56 qualified low hands in Stud-Eight. Ranked from best (lowest), to worse (highest), the low hands can be categorized by the highest card in the group. Here is a listing of the low-hands in order of rank, with the total number of each kind of hand in parenthesis.
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5-high hands (1)

5, 4, 3, 2, A

6-high hands (5)

6, 4, 3, 2, A

6, 5, 3, 2, A
6, 5, 4, 2, A
6, 5, 4, 3, A
6, 5, 4, 3, 2

7-high hands (15)

7, 4, 3, 2, A

7, 5, 3, 2, A
7, 5, 4, 2, A
7, 5, 4, 3, A
7, 5, 4, 3, 2

7, 6, 3, 2, A
7, 6, 4, 2, A
7, 6, 4, 3, A
7, 6, 4, 3, 2
7, 6, 5, 2, A
7, 6, 5, 3, A
7, 6, 5, 3, 2
7, 6, 5, 4, A
7, 6, 5, 4, 2
7, 6, 5, 4, 3


8-high hands (35)

8, 4, 3, 2, A

8, 5, 3, 2, A
8, 5, 4, 2, A
8, 5, 4, 3, A
8, 5, 4, 3, 2

8, 6, 3, 2, A
8, 6, 4, 2, A
8, 6, 4, 3, A
8, 6, 4, 3, 2
8, 6, 5, 2, A
8, 6, 5, 3, A
8, 6, 5, 3, 2
8, 6, 5, 4, A
8, 6, 5, 4, 2
8, 6, 5, 4, 3

8, 7, 3, 2, A
8, 7, 4, 2, A
8, 7, 4, 3, A
8, 7, 4, 3, 2
8, 7, 5, 2, A
8, 7, 5, 3, A
8, 7, 5, 3, 2
8, 7, 5, 4, A
8, 7, 5, 4, 2
8, 7, 5, 4, 3

8, 7, 6, 2, A
8, 7, 6, 3, A
8, 7, 6, 3, 2
8, 7, 6, 4, A
8, 7, 6, 4, 2
8, 7, 6, 4, 3
8, 7, 6, 5, A
8, 7, 6, 5, 2
8, 7, 6, 5, 3
8, 7, 6, 5, 4

Notice that in each grouping, the low-hand that also competes for the high-pot as a straight is the worse low-hand that you can have. An 8-high straight loses the low pot to all other 8-high low-hands, a 7-high straight loses the low-pot to all other 7-high low-hands, and a 6-high straight loses the low-pot to all other 6-high low-hands. Also notice how common an 8-7 high low-hand is compared to the other low-hands. Of the 56 possible low-hands, 20 are 8-7 high low-hands, which is more than all the 7-high low-hands combined (15 total).

Note for Omaha High-Low players:

Low-hands are ranked the same in Omaha High-Low and in Stud-Eight, but I've seen players become confused in determining the rank of their low-hands. In Omaha High-Low, you must use three community cards combined with two in your hand. That means that if the community cards include an 8, 7, 6, and you hold an A, 2, your hand is the best possible low-hand. It is still an 8-high low-hand, but no one can make a better low given the community cards. That rule makes an A, 2 a powerful holding in Omaha High-Low. But in Stud-Eight, if you have an 8, 7, 6, A, 2, you lose the low-pot to a player with a 7, 6, 5, 4, 3. You have an 8-high low-hand and your opponent has a 7-high low-hand. The 7-high bests the 8-high for the low-pot. The fact that you have A, 2, for your lowest cards, does not matter because it is the high card in the hand that counts.

It is also worth noting, that in contrast to Omaha High-Low, being "quartered" in Stud-Eight is a rare event. In Omaha High-Low, it is common for two players to each have the best possible low-hand and split the low-pot (each receive one-quarter of the total pot). For example if two players each have an A, 2, and the community cards included 8, 7, 6, each player has the same low-hand. But, in Stud-Eight there are no community cards, which means that to split a low-pot all five cards in the players' low-hands must match. A single un-matched card will decide the low-pot. For example, a player with 7, 6, 5, 3, 2 would lose the low-pot to a 7, 6, 5, 3, A, because the Ace beats the 2 for low.

Sunday, January 31, 2010

Low Straight Blockers

When betting on a high pair in Seven-Card Stud High-Low, against a player showing a bunch of scary-looking low-cards, the possibility of being scooped by a low straight must be considered. Conversely, if you have a bunch of low-cards, how likely is it that your hand will also fill out into a low straight? The chances of a bunch of low-cards turning into a low straight depend on the availability of the "blockers." That is the location of cards that are necessary for a low straight to form. If the needed cards are dead, the low straight is blocked, and the relative value of the high-hand goes way up in relation to the low-hand.

That means that when reading the board, you should look for exposed cards that block the completion of low straights. For example, if you started with 2, 3, 4, and picked up a 6 on Fourth Street, be on the lookout for the 5s. If three of the 5s are exposed on the board, your hand has gone way down in value. Your straight potential is down to just one out. You are essentially drawing for just one half the pot, which is a violation of the most fundamental principle of high-low poker, that you should always play to scoop.

Knowing the blockers is also important if you are going high. Suppose your hand is (4, 4,) 4, J and Bob raises with (X, X) 3, 5. You see the remaining 4 in Alice's hand. Your draw for quads might be dead, but more importantly; Bob's draw for any kind of low straight is completely dead. He will not hit a wheel, or even an 8-high straight. You can re-raise Bob all you want, because he will not beat you with a straight.

Here is a table of blockers for low straights (8-high or less):

Dead Rank straights blocked straights possible
A 5-high 6-high, 7-high, 8-high
2 5-high, 6-high 7-high, 8-high
3 5-high, 6-high, 7-high 8-high
4 5-high, 6-high, 7-high, 8-high none
5 5-high, 6-high, 7-high, 8-high none
6 6-high, 7-high, 8-high 5-high
7 7-high, 8-high 5-high, 6-high
8 8-high 5-high, 6-high, 7-high



It is important to note in the above table the critical role of the 4s and 5s. If either rank is dead, no low straights (8-high or less) are possible. If the 4s are dead the minimum allowed straight is a 9-high, and for the 5s the minimum is a 10-high. Players with exposed low cards (8s and less) are less likely to reach straights that high.

Also note from the table that these two-card combinations that will block all low straights.

A, 6
2, 6
3, 6
2, 7
3, 7
3, 8

In other words, if any single wheel card plus the 6s are dead, no low straights are possible. Dead 2s or dead 3s combined with dead 7s block all low straights. Dead 3s and 8s block all low straights.


Keeping track of blockers is especially important on later streets in determining hand values. Two 5s might be gone on by Fourth Street, but if all are dead by Sixth Street, that is useful information for deciding whether to value bet on the end.