Monday, October 26, 2009

Key Concepts for High-Low Poker, Part I

If you have only played high-only variants of poker, and are trying to learn some of high-low games, here are some key concepts you need to know. These concepts apply to both Omaha High-Low and Seven-Card Stud High-Low Eight-or-Better (referred to as "Stud-Eight" in the post).

Scooping: Routinely playing for high-hand only, or low-hand only, leads to trouble. Splitting the pot should be a saving out, not the reason for playing the hand. As a general rule, you do not want to be on a draw for half the pot. The goal should be to have a hand that is on track to win the entire pot. Ideally, opponents drawing against your hand should only win half the pot when their hands hit.

For example in Stud-Eight, many players routinely play uncoordinated low cards with the intention of winning low-only. But consider going the river with A, A, 5, 5, K against an opponent with 2, 3, 5, 7, J. The King on Sixth Street caused you to miss a chance at low, but you have a lock on high. Your opponent is hoping to pick up either an A, 4, 6, or 8 to hit low. Depending on the number cards exposed, that could be as many as 14 outs. But your opponent will only get half the pot if he or she hits, you get the entire pot if he or she misses. Repeat this scenario enough times and you will accrue much more money.

The value of Aces: Compared to high-only poker, Aces have even greater value in high-low games, although not for the usual reasons. Understanding the role of Aces is critical to success. In Hold'em a pair of "bare" Aces is a powerhouse, because it can often win on its own without improvement. In high-low games, bare Aces rarely win without improvement. Many players transitioning from high-only games over value pairs of Aces, and loose money by overplaying them. But Aces are the most important cards in high-low games because they are simultaneously the lowest and highest card in your hand. Having an Ace is the equivalent of having an extra card.

In Omaha High-Low, a starting hand such as A, 2, 3, J, can be part of a nut-low such as A, 2, 3, 4, 5 or a nut-straight A, K, Q, J, 10, depending on the board. In Stud-Eight, an Ace can be part of an Ace-high flush such as A, 3, 4, 5, 6 all in Spades, and these same cards also qualify for a 6-high low. Correct strategy that emphasizes playing coordinated cards that can scoop, relies on the dual capabilities of Aces.

Qualifying for Low: Most high-low forms of poker require that the low hand "qualify." In Stud-Eight and Omaha High-Low, the qualification is that a low-hand must contain five cards with none higher than an eight. If no hand qualifies for low, the high-hand takes the entire pot. It is sometimes possible to know that no hand qualifies for low. In Omaha High-Low, players must use two of their cards to form hands. Low hands are impossible if three cards appear on the board higher than an 8, and in that case, competition is only for the high hand. In Stud-Eight, any player with three exposed cards higher than an 8, cannot have a qualifying low. If all players have three exposed cards higher than an 8, the game reverts to ordinary Seven-Card Stud. But even if a low is possible, that does not guarantee that someone has a qualifying low. Judging when an opponent has a low is an important skill to develop.

Driving players out is not always correct: In high-only poker when you have the best hand, you want to make it expensive for the others to stay and draw out on you. Betting and raising are the usual actions. That is usually the case in high-low games, but not always. If you have a nut-low and are up against a high-hand that you cannot beat or move off the pot, the only way to profit is to keep others in the game. If the other players will cold-call raises, then by all means raise. But, if raising will drive them away, calling is the best play.

Freerolling: This term refers to situations where you can make risk-free bets that will payoff if your hand improves. Freerolls occur in all variants of poker, but the number of freeroll opportunities is much greater in Stud-Eight.

Consider the following situation in Pot-Limit Omaha. You hold A-Spades, Q-Hearts, J-Diamonds, 10-Spades. Your opponent holds A-Diamonds, 3- Diamonds, J-Hearts, 10-Spades. The flop is 9-Spades, 8-Spades, 7-Hearts. Both of you flopped the nuts. But your opponent's hand cannot improve. Yours can improve to the nut-flush, if any Spade hits later on, or a Queen-high straight, if a 10 hits. If a Queen hits you both have Queen-high straights but if it's the Queen of Spades you have made the nut-flush. Your opponent's nut-flush possibility in Diamonds is dead. That means if you raise all-in against each other, the worse that can happen to you is a split-pot where you recover your investment. But, you get two additional cards that might improve your hand so that you can take the entire pot. Your opponent risks his entire stack for half the pot. You risk nothing for a significant chance of taking the entire pot.

Freerolls occur more frequently in Stud-Eight. In high-only Seven-Card Stud, it is difficult to know if you have the nuts. After all, in Seven-Card Stud it is possible for an opponent to have a full house or quads without showing an open pair on the board. But in Stud-Eight it is possible to know if you have the nut-low. That often puts the player with a made nut-low in the drivers seat, with a re-draw to a possible high. Consider having A, 4, 5, 6, 7 against K, 4, 10, 10, K. If this were high-only Seven-Card Stud the player with the open-ended straight would have to call bets from the player with the two-pair in hopes of improving. But, in Stud-Eight the situation is reversed. The player with the two-pair can never make a low-hand and the player with the made low-hand knows that. With the low already made, that player can bet and raise with no risk, because half the pot is assured. If the straight hits and the two-pair do not improve, the entire pot goes to the straight.

The situation in Stud-Eight can get more interesting with multi-way pots. Consider having A, 2, 5, 6, 8 against two opponents with, A, A, J, 10, 10 and Q, 9, 9, Q, 9. Your opponents with well-hidden high hands, might start a raising war that you can encourage with additional re-raises, at no risk to yourself. You know that you get half the pot no matter which high-hand holds up. Put in all the raises you can, because the hand is guaranteed to make you money.

Monday, October 19, 2009

Reasons to Limp-in on Third Street

In the previous post, I concluded that you gain an edge from attacking players who limp-in on Third Street with three low-cards, and fold immediately if hit with a high-card on Fourth Street. That raises the question: Are there situations where it is correct to limp-in on Third Street? As a general rule, open-limping is a weak play. If you intend to play the hand, and have the first opportunity to complete the bet, you should complete and force the players acting later to define their hands. It is also risky to allow the bring-in a free card. However, there are situations when limping has advantages. The situations I will analyze are when you act after multiple limpers, and you (1) hold a live-pair other than Aces (2) hold the lowest hand.

These situations occur frequently in low-limit Seven-Card Stud High-Low (Stud-Eight) games, when it is common for several players to limp-in holding just about anything. These players are hoping for a cheap card on Fourth Street that will somehow improve their mediocre starting cards. As a general rule, completing the bet after these players have limped-in, will not drive them out. Even though they would have folded, if a bet were placed before they had a chance to act, they now feel committed to the hand, and will call, if you bet.

(1) Pairs other than Aces. In Stud-Eight, pairs can present problems because they are a poor start for a low-hand, and in many cases, a poor start for high-hand. Mid-pairs, such as 9s and 10s are almost unplayable against a large field. The hand is unlikely to make low, and stands a good chance of being dominated by for high. A low-pair with a low-kicker, has a better chance of making a low-hand, but a greater chance of being dominated for high-hand. However, rolled-trips are powerful holdings, and in cases when the trips are low, an extremely deceptive holding. Rolled low-trips often look like the best low-hand, when in reality it might be the best high-hand.

Consider a $1-2 Stud-Eight game with a $0.20 ante and a $0.25 bring-in. Eight hands are dealt, and three players limp after the bring-in. You hold a low-pair with a low-kicker and your cards are completely live. Therefore, when it is your turn, the pot has $2.60. If by limping, you end the action, the pot is offering 10 to 1 on your $0.25 payment. What are the odds of making trips? You have seen 10 cards--the three in your hand, plus the other seven door cards. Only two of the remaining 42 unseen cards make trips, so the odds against trips are 20 to 1. However, the implied pots odds are much greater than the current 10 to 1 the pot is paying. If you make trips, a strong possibility exists for you to scoop what could be a $20 to $30 pot. You will have to contribute a minimum of $7 to reach a showdown, but with trips after Fourth Street, you are a heavy favorite.

However, if you complete the bet, instead of limping, your initial pot odds are greatly reduced. If you complete, and the other three players call, you have contributed $1, while the others contributed $4.85. The pot is now paying about 5 to 1, while the possibility of making trips on Fourth Street remains 20 to 1. You need much greater implied pot odds to have an edge compared to the case of limping. And remember, rolled-trips is not a lock, the hand will be outdrawn at times.

This analysis assumes that you will fold on Fourth Street, if your hand does not improve to trips. Against a large field this will often be the case. Limping, followed by a quick fold in the case of no improvement, is a viable strategy in circumstances when aggression will not drive players out. You will almost always need to have strong cards against a large field of limpers, because, in most cases, at least one person will improve, and stay until the end.

(2) Holding the lowest hand. Low-cards, that are unsuited and unconnected, are generally weak starting cards in Stud-Eight. These cards usually compete for just one half of the pot, so the pot odds are greatly reduced in comparison to a hand that can scoop. However, if you are the lowest hand on the board, against a number of players who have limped-in with high cards, it can be worth limping to pick up a cheap Fourth Street card. In this case you are looking to pick up a low-draw, not to make a low-pair. If your low-draw is completely live, there are 20 unknown cards that can help  (the 9 other low cards will pair your hand). In an eight-handed game with seven other door cards, 42 cards remain unseen. You are looking at "coin-flip" chances of picking up a low-draw on Fourth Street--20 out of 42, or 47.6%. However, a low-draw is not yet a qualified low, so you will be investing money on the later streets before you know if the hand has any value. But on later streets, a low-hand against multiple highs, can be a profitable holding because it can freeroll and jam the high hands.

However, limping with low-cards, after multiple players with low cards have also limped, is a questionable play. Even if your hand is the lowest of the lows on Third Street, the numbers of outs that you have to make a "viable" low-draw on Fourth Street is greatly reduced. By "viable," I mean a card that will allow your hand to remain the lowest on the board. Being second-best low is a dangerous position in Stud-Eight. Suppose you limp-in with a 6-high, such as 2, 3, 6. There are only 12 cards that can maintain a 6-high low-draw after Fourth Street (As, 4s, 5s). Even if none of those cards are on the board, chances are that some of the limpers with low door cards, have some of your outs for hole cards. The realistic chances of having a 6-high low-draw after Fourth Street are only about 25%.

In summary, I think there are some situations where late limping is a viable strategy. But, those situations are ones when a questionable holding after Third Street, has a small chance to improve to a powerful holding on Fourth Street. Because the improvement chance is small, the Fourth Street Card should be seen cheaply.

Monday, October 12, 2009

Attacking Third Street Limpers

Of the board games, Seven-Card Stud High-Low (Stud-Eight) attracts the most Third Street limpers. By "limpers," I refer to players who call the bring-in bet, rather than raise to the complete bet allowed by the limit. I am not referring to callers of a completed bet.

Limping is rarely seen in Razz because there is no reason not to show aggression with an exposed low-card. The hole-cards are almost immaterial in the early part of a Razz hand. In Stud-high, limping is seen, but it is generally regarded as a weak play. Even if a starting hand is marginal, aggression should be used early on to force the other players to define their hands.

But, in Stud-Eight, there are certain types of hands in which it is advantageous to entice a large number of competitors, rather than drive opponents out. The ideal situation is a low hand versus two or more high hands. In that situation, the low-hand can jam the others while being assured of half the pot.

As a result, it is common to see many players with three low-cards limping, in the hope that they can pick up a fourth low-card cheaply, and see if the hand develops into the only viable low. Many of these players will make a quick exit if their fourth cards are high.

Of course in poker, any player showing a predictable pattern should be a target. The question is what is the best way to get an edge? Should you attack or limp-in yourself? Here is a mathematical analysis of a typical scenario.


  •  $1-2 Stud-Eight game with a $0.20 ante and $0.25 bring-in. (These limit values are computationally convenient because they scale easily to higher and lower limit games.)
  • You act near the end, and after one player, who has limped-in.
  •  That player has demonstrated a pattern of limping with three low-cards, and only continuing in the event of making a low-pair or low-draw on Fourth Street.
  • The bring-in folds in response to a completion.

Scenario 1: Full table with eight players making antes.

Suppose you attack the limper with a complete bet while holding three low-cards; the bring-in folds and the limper calls. The pot size is the $1.60 in antes, plus the $0.25 bring-in, plus the $2 in bets, for a total of $3.85. You have bet $1 for a chance to win the $2.85 on the table uncontested if your opponent's next card is high. The pot is paying 2.85 to 1. What is your chance of succeeding?

The deck contains 32 low cards and 20 high cards. The minimum number of low cards in play is 7. There are three low-cards in your hand, three for the limper, and the bring-in must have had one low-card exposed to be the bring-in. (If not you or the limper would be the bring-in, because you each have three low-cards.)Your worse case scenario is that the five hands that mucked on Third Street, all had door-cards that were high. That means, that if we total the known and unknown cards, there are 25 low-cards that are unknown, and15 high-cards that are unknown. The chances that an unknown card will be high are 15 out of 40, or 37.5%. That means, the odds against this play succeeding are 1.67 to 1. Because the pot pays 2.85 to 1, the play has a positive expectation. It cost $8 to make this play 8 times, but it brings back $3.85 three times out of eight, for a total of $11.55. We expect to receive back about $1.44 for every $1 invested. The expected value (E. V.) of the bet is $0.44.The edge increases if more low cards are exposed in the five mucked hands.
No. Low-Cards Mucked
Fraction High-Cards Remaining (%)
Odds Against High-Card
E. V.
That means that in the ideal situation of five mucked low-cards, the locations of 12 of the 32 low-cards are known. Because no high-cards are visible, we are getting coin-flip chances on an outcome that pays 2.85 to 1. That is a significant edge.

Scenario 2: Three-player game.

Interestingly, the edge does not go away if the game becomes shorthanded, even though the sum of the antes at stake is smaller. For example, a three-player game would have $1 less in antes in the pot. You would now be wagering $1 to win $1.85. The pot is now paying 1.8 to 1. We still know about the 7 low-cards, but have no information on the high cards. There are now 45 unknown cards. The chances of the play succeeding are 20/45 or 44.4% or 1.25 to 1 odds. In other words, 4 out of 9 times the limper will be hit with a high-card and fold immediately to a Fourth Street bet. You spend $9 making this Third Street-play 9 times, but it wins back 4 times on average, an amount of $2.85, or $11.44 total. You expect to receive back $1.27 for every $1 invested-an E. V. of $0.27.

Of course these calculations assume an idealized situation, in which the bring-in folds, and the limper is completely predictable. Often the bring-in will defend in these situations because he or she has observed the same pattern from the limper. Also, the limper might have a wider range of hands than three low cards. Split or wired-pairs might be included and a quick exit on Fourth Street not planned if he or she has a pair.

But those are all reasons to show aggression and not limp as well. Completing the bet forces the bring-in and the limper to define their hands. If they don't back down on Fourth Street, you will know something is up, and can proceed more cautiously.

In summary, if you see a player exhibiting this pattern, attacking on Third Street will give you an edge. Conversely, if you exhibit the pattern, alert players can gain an edge against your play.

Monday, October 5, 2009

The Possible and the Impossible in Seven-Card Stud High-Low Poker

Every variation of poker has conditions on the kinds of hands possible, given the exposed cards (the board) and rules of the game. For example, in Hold'em full houses and quads are possible only if the board is paired. Knowledge of the possible is vital throughout the hand, when deciding to place or call bets. Here is a summary of what is possible and impossible in Seven-Card Stud High-Low.

High Hands

It is possible for a player to have a full house or quads without an exposed pair. The three hidden cards can combine with any one or two of the four exposed cards to make these kinds of hands.

If just two of the four exposed cards have matching suits, a flush is possible.

It is impossible for a player to have a flush when all four exposed cards are in different suits.

It is always possible for a player to have a straight, because the four exposed cards can never be arranged in such a way, that the three hidden cards couldn't fill a gap.

It is impossible for two players to have trips of the same rank. As a result, kickers never play when trips are compared. (The same is true for quads.)

It is possible for two players to have two-pair of identical ranks. In that case, the kickers decide the hand.

Exposed Hands

The following kinds of hands require all five cards and can never be completely exposed on the board:

Full houses
Flushes (including straight-flushes)
Qualified lows (low hands that meet the eight-or-better condition)

The following kinds of hands require less than five cards and can be exposed on the board:


Low Hands

It is impossible for a player with three exposed high cards to have a qualified low.

It is impossible to have a qualified low and a full house or quads

It is impossible to have a qualified low with an exposed low-pair and two high cards. For example, a player with 5, 5, J, K, cannot make low.

For the same reason, it is impossible to have a qualified low with an exposed two-pair, when one of the pairs is high. For example, a player with J, J, 8, 8, cannot make low.

A player with an exposed high-pair and two low cards can have a qualified low-hand. However, the low-hand cannot be better than the highest exposed low card. For example, 9, 9, 7, 3 cannot be better than a 7-high low-hand.

A player with an exposed two-pair, with both pairs low, can have qualified low-hand, but it cannot be better than the rank of the highest pair. For example, 3, 3, 8, 8, cannot make a low-hand better than an 8-high.

It is always possible for a player with two exposed wheel cards to make a wheel. For example, a player with J, J, 2, 5, could have an A, 3, 4 hidden.

Similarly it is impossible to have a low hand better than the two lowest exposed low-cards. For example a player with 8, 7, 6, J exposed, can have at best a 7-6-high low.