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.

Tuesday, December 29, 2009

Key Concepts in High-Low Poker: Part II

This is a continuation of a post in October on concepts for high-low poker. If you have only played variants of poker in which the high-hand wins the entire pot, and are trying to learn some of the split-pot games, here are some key concepts for high-low play that apply to Seven-Card Stud High-Low Eight-or Better Poker (Stud-Eight), as well as Omaha High-Low Eight-or-Better.

Split-pots generate action:
Many players complain about split-pot games, because it is possible for a powerful (monster) hand, such as quads, to split a heads-up pot resulting in a slight loss after the rake is subtracted. It seems unfair that you can hit a hand as rare as quads, and have to share the winnings. But, while winning the hand isn't a problem when you hit a monster, getting action on that hand often is a problem.

Consider holding pocket Aces in Hold'em or Omaha and flopping quads. Pairs on the board are scary enough, and since no one else has Aces you are unlikely to generate much action. Even someone with a full boat might be cautious. The same problem exists in Seven-Card Stud. It is possible to hold quads, or a boat, without showing a pair, but unlikely. If you have an exposed pair of Aces your action will dry up.

But, in split-pot games, action is possible in these circumstances because potential lows might try to draw. Sometimes they will hit the low, and you will have to share. But a certain fraction of the time, the low will not qualify, and you keep it all. Just remember, in each case you are receiving action that might not have occurred in a high-only game. If more than one low-draw joins the fray, you are assured of winning money, and can raise without risk of losing or driving out paying customers.

It might be frustrating to share large pots with a nut-high, but remember the pot might not have had any value had competition for a low-pot not been possible.


Starting card requirements should be tighter:
Split-pot games have different starting card requirements than high-only games, because the goal is to scoop. Starting hands must have the potential to win the high-pot or low-pot, and ideally both. Because high-low games offer so many different kinds of drawing opportunities, many players think that starting hand requirements can be looser than for high-only games, such as Seven-Card Stud or Hold'em. Actually, the reverse is true. Starting hand requirements need to be much tighter, because coordinated starting cards that have the potential to attack both sides of the pot are rare holdings. In Hold'em, each player receives just two cards, and that makes just about any pair, or any two high-cards a potential winner. But, in Stud-Eight, three starting cards are less likely to coordinate than two. Starting hand requirements are discussed at great length at StudHighLow.com. However, there are some general principles.

-- Aces, either singly or paired, are the most valuable starting cards because of their potential to complete both a high-hand, and a low-hand.

-- Connected low-cards, suited low-cards, and connected-suited low-cards have the greatest potential to scoop large multi-way pots.

-- Mid-range cards, especially 9s, 10s, Js, play poorly, and should be avoided. However, even 8s, Qs, and Ks can lead to problems.

Strong starting hands in Stud-Eight also need to be "live," meaning that the cards needed to improve must not exposed in other hands. An A, 2, 3, all in Clubs is a great looking starting hand in Stud-Eight, unless you are looking out at a board with the 5, 6, 9 in Clubs, and two of the remaining Aces. It is frequent to receive starting cards that look great, but need to be mucked, because too many of the cards needed to improve are exposed in other hands.

Fold equity is reduced:
There are two ways to win a pot in poker-have the best cards at showdown, or induce the other players to fold before a showdown happens. The act of betting is intended to put more money in the pot to win when you have the best cards. But, betting also forces opponents to make decisions, and folding is one alternative when confronted with a bet. The chance that a bet will win the pot uncontested is known as "fold equity." Bets in high-low games have much less fold equity than bets in high-only games, especially late in the hand if the pot is large. Players that failed to improve for the high-pot, might still have many outs available for the low-pot, so it is not in their long-term interests to fold to hands that will clearly win the high-pot. The reduced fold equity means that aggression, and bluffing, must be combined with cards that work together to attack both sides of the pot. Aggression will not win by itself, because players call much more frequently in high-low games.

Most fold equity in Stud-Eight is available early in the hand, when players with marginal cards will fold to bets and raises, because they do not want to be sucked too deeply into a hand with limited opportunities. However, players, who stay until Sixth Street, are usually in it until the end. Part of the problem, is that by Sixth Street, many pots become so large, that drawing to half the pot becomes mathematically correct. This is especially true for players who missed their high chances, but still have ten or more outs for a low-hand. Contesting a small pot on Third Street, with a hand that can only compete for the low-pot makes no sense mathematically. But, a large multi-way pot on Sixth Street is a different case, particularly if a draw to a low-hand is still live. My own experience, playing Stud-Eight, is that fold equity on Sixth Street is close to zero. Opponents willing to call a large bet on Fifth Street, almost never fold to a Sixth Street bet. An opponent willing to fold on Seventh Street is slightly more common, because of missed low-draws, but it is still an uncommon event. The fact is that you will show down more hands in Stud-Eight compared to most other variations of poker.

For big money invest in low-straights and low-flushes: In high-only games, when you think of monster hands, what come to mind are full houses and quads. Ironically, in high-low games filling up can often be detrimental to your financial outcome. For example, in Stud-Eight, if you hold A, A, 3, 3, 4, 6 going to the river, a final card of 2 might scoop the pot with Aces-up for a high-hand, and a 6-high for a low-hand. But, if an A, or 3, hits the full house disqualifies your hand from competing for the low-pot. Because two-pair, and trips are vulnerable hands in games with many competitors going to the river, the hands that have the best change of holding up for the high-pot, and winning the low-pot are the small straights, and Ace-high flushes with a complement of small cards. A hand such as 6, 5, 4, 3, 2, is simultaneously a 6-high low-hand, and a 6-high straight. The hand A, 2, 4, 5, 7, all is Spades is simultaneously a 7-high low-hand, and an Ace-high flush. In both of these examples, the hand is simultaneously a powerful low-hand and a powerful high-hand. In the case of the straight, it beats anyone holding two pair or trips, and in the case of the flush it beats anyone holding a straight.

Low-straights and low-flushes are also rare hands that can never be completely exposed, because five cards are required. That means if the pot is large, any player with two pair, or trips, will have to call you down at the end to make sure that your hand is not just a low-hand. Draws to these hands also offer excellent freeroll opportunities. Suppose after Fifth Street, you have A-Hearts, 2-Hearts, 4-Hearts, 5-Hearts, 6-clubs. Your hand is a 6-high low-hand, that might have a lock on the low-pot. At the same time you have two cards to come, and any Heart gives you an Ace-high flush, a 3-Diamonds, 3-Spades, or 3-Clubs, gives you a 5-high straight, and a 3-Hearts gives you a 5-high straight flush. The draws to all of these possibilities are risk-free.

Monday, December 7, 2009

Catching Bad versus Catching Good on Fourth Street

Fourth Street is a point in the hand when it is easy to get out before becoming too committed to the pot. Many players who catch a bad card on Fourth Street will make quick exits, and many times, a raise will hasten the exit of marginal hands. Before betting or raising into multi-way pots, it is important to think ahead about how potential raises will effect the action.

Consider these scenarios:

High-pair with no low:


You (J, K) K, 10
Alice (X, X) 2, 3
Bob (X, X) 4, 8

You have a big pair with no hope of a low-hand, against two players who have each caught low cards. However, neither Alice nor Bob has a low-hand yet, and your pair would stand a better chance of scooping in heads-up play. You should lead with a bet. If Alice has four little cards, she will raise to force Bob, her other low-hand competitor, off the hand. Then you can re-raise Alice, to make her pay to draw out on you for the low-pot. If you instead check to the lows in this hand, Alice will bet, Bob will call, and you won’t have much better to do but call and hope for the best. A check-raise after Alice bets and Bob calls, will probably not force anyone off the hand. Bob will stay at that point, even if Alice raises, because he is into the hand too deep.


A low-draw:

You (4, 5) 6, A
Alice (X, X) K, 9
Bob (X, X) Q, 10

You have caught a decent card, although, it does not connect with your initial three and that lessens the chance of scooping with a low straight. The Ace does act first on this board. It would be tempting to bet, but then Alice might raise Bob off the hand. If that happened, you would be drawing for just one-half the pot, heads-up against Alice. It would be better to check to Alice, let her bet and Bob call. Then you could call and keep both players in. Most likely, Alice and Bob each have a big pair and no chance for a low-hand. If you pickup a low hand later on, then you can drive the action make it difficult for both Bob and Alice.

Small pair:

You (4, 5) 4, J
Alice (X, X) K, Q
Bob (X, X) 6, 5

In this situation, you should check-fold your hand no matter how Alice or Bob play. Your hand is unlikely to make low, which means that all you really have is a pair of 4s for high against a probable pair of Kings. You will be playing catch up for one-half the pot, a terrible situation to be in. But, I’ve seen many players continue in the hand until the end with any kind of pair, a pattern that is sure to lose money over the long-run.


Second-best low:

Alice (X, X) K, Q
You (A, 3) 4, 8
Bob (X, X) 6, 5

If Alice leads with a bet in this situation, you should fold because of your position. You never want to be in the middle between the best high-hand and best low-hand because it will cost you four bets to see every card. If you call Alice, Bob will raise and Alice will re-raise. If you cold-call that two-bet raise, Bob will cap it. Your best hope after that is that Bob catches bad on 5th street and does not complete the low-hand, but you catch good and do. But, even if that happens—for example Bob gets a Jack while you get a 7—you are still in a tough spot because you have the worse possible low-hand—an 8-7—while Bob has a draw to a better low. If you both catch good cards—for example both you and Bob get 7s—you are in a terrible position on Fifth Street because you are not sure where your low-hand ranks and you have almost no chance of scooping. It is best to get out cheaply on Fourth Street before being faced with all these difficult and expensive decisions.

Potential monster with a brick:

Alice (X, X) Q, 7
You (3d, 4d) 5d, Ks
Bob (X, X) 6, 5

Alice leads with a bet and you are faced with a situation similar to the previous one, in which a call could lead to capped action. The difference is that you’ve caught a bad card for one of the best three-card starting hands. Your hand could develop into a monster, which means that you do not want to be raised out of it. At the same time, the bad card means that you do not want to put a lot of money in the pot. One possible solution is for you to make the raise. That might cause both Alice and Bob to pause. They would have to consider the possibility that you have wired Kings and just completed trips. If Bob raised, Alice might leave the hand and you might even have the highest hand at that point. If Bob called and Alice called, you would make it to Fifth Street for the cost of two small bets, instead of four, and could decide what to do from there. You might even get free cards on Fifth Street if Alice and Bob don’t improve.

Friday, November 27, 2009

Overplaying the Eight

One of the most common mistakes I see in Seven-Card Stud High-Low, is overplaying 8-high low-hands. Many players raise the moment they make any kind of low-hand, even if it's on the river, and cold-call raises with a draw to any kind of low-hand, even if its an 8-high.

The problem is that the 8 is actually one of the deadliest cards you can hold, because an 8-high is the worse possible low-hand. Remember, that a low of 8, 4, 3, 2, A, loses the low-pot to 7, 6, 5, 4, 2, because low-hands are ranked by the highest card, not the lowest.

I once saw a player river an 8-high low-hand and fire an immediate raise. He was promptly called and scooped by an opponent with a 7-high low-hand and an exposed pair of 3s for high. The raiser could not beat the board for high, and yet raised on the basis a weak low-hand.

A river raise will hardly ever move a bettor off a hand in Seven-Card Stud High-Low. In fact, you will show more hands at the end then just about any other form of poker. Most players who bet until the end have something going for at least one-half of the pot. Remember, it is a fixed-limit game with pots that can grow very large by the end. With only one bet to call at the end, players with any kind of chance for any half of the pot usually pay to see the raiser's holdings. Many players with a small pair such as 3s, will call if they can beat the board for high, even if they miss a low-hand. The call often works because if multiple players are competing for low-hands only, it is not uncommon for a small pair, or even A-K, to take the high-pot at the end.

Because hands with an 8 qualify for low, many players automatically play them aggressively and often set themselves up for disaster. In one hand, I bet out a 7-5 low-hand on Fifth Street and was promptly raised by an opponent with an 8, 7, 5, showing on the board. Knowing it would be difficult for her to better my low-hand, I re-raised and ended up scooping the entire pot when at the end, I completed a 7-high straight.

In another hand my opponent was luckier. I had completed a 9-high straight on Fifth Street, and my opponent, who had an 8, 6, 10 exposed, cold-called my two-bet raise to draw to an 8-high low. On Sixth Street I picked up the 4 to make it 6 sequential cards, while my opponent hit a King and kept calling. He saved half the pot on the end when I caught a Queen that failed to improve my 8-7 low-hand and he picked up a 2 to complete an 8-6 low-hand. He got his money back, but calling all those raises in order to draw to second-worse low-hand that you can have, (the 8-high straight I held is the worse) will not win over the long-run.

The pitfalls of 8s, means that many of the so-called "Razz hands," such as starting cards of 8, 2, 3, or 8, 7, 3, or even 8, 2, A, should not be played. These hands are playable in Razz because the lowest hand takes the entire pot with no qualifier necessary. But, in Seven-Card Stud High-Low these cards are competing for one-half the pot, and are often second best for that half of the pot. If you play low starting cards, the cards should have some possibility of "connecting," that is complete a low straight, and rank better than an 8-high for the low-hand.

Thursday, November 19, 2009

Probabilities for Simultaneous High and Low-Draws: Weighted Outs

The big-money hands in Seven-Card Stud High-Low are the low straights and low flushes. These hands often scoop and better yet, usually get paid. Any player with at least two-pair will pay if the pot is large, because it is not possible to know for certain that the hand qualifies for low, let alone know that it is a straight or flush.

However, if you start with connected or suited cards, and then pick up a helpful card on Fourth Street, you still do not have a winning hand. Here is a summary of the outs available to complete the low-hand, high-hand, or both for some common Fifth Street-draws. For these tabulations I will introduce the concept of "weighted outs". In many cases the total number of outs to make a high-hand, low-hand, or both, are the same. But some circumstances are more favorable than others, because the numbers of outs that can make both a high-hand and a low-hand are greater than the number that make high-hand only or low-hand only.

To reflect this discrepancy, I define a "weighted out" as equal to 1 if the out completes a high-hand and a low-hand, and one-half (0.5) if the out completes a high-hand or a low-hand. Totaling "weighted outs" is a better measure of your actual equity in the pot. For example, if you have seen 16 cards after Fourth Street, there are 36 unseen cards. If your total weighted outs are 12, you have 33% equity in the pot. Some of the outs might result in winning only half the pot, and some the entire pot, but on average you can expect to pick up 33% of the money on the table if the scenario is repeated many times.

Here are the tabulations:

Open-ended straight-flush with low-draw

Example: 3, 4, 5, 6 all in Spades

Improve to:
Outs
Weighted Outs
Straight-Flush
2
2
Straight
6
6
Low-flush
2
2
High-flush
5
2.5
Low-hand
6
3
Totals
21
15.5

Four-flush with inside straight and low-draw

Example: A, 3, 4, 5 all in Hearts

Improve to:
Outs
Weighted Outs
Straight-Flush
1

1

Straight
3

3

Low-flush
3

3

High-flush
5
2.5
Low-hand
9

4.5

Totals
21
14

Four-flush with low-draw

Example: A, 3, 5, 6 in Hearts

Improve to:
Outs
Weighted Outs
Low-flush
4
4
High-flush
5
2.5
Low-hand
12
6
Totals
21
12.5

Open-ended straight with low-draw

Example: 3, 4, 5, 6 rainbow

Improve to:
Outs
Weighted Outs
Straight
8
8
Low-hand
8
4
Totals
16
12


Inside straight with low-draw

Example: 3, 4, 6, 7 rainbow
Improve to:
Outs
Weighted Outs
Straight
4

4

Low-hand
12
6
Totals
16
10

All of these tabulations count the maximum number of outs possible, which only occurs when the draw is completely live. It is rare that a Fifth Street draw, in which 10 to 15 cards are already exposed, would be completely live. That is why counting weighted outs on the later streets is so important. Consider the most favorable draw-an open-ended straight-flush with low-draw. If the 6 outs to a low-hand are missing, the weighted outs drop by 3 to a total of 12.5. But, if the 6 outs for the straight are missing, the total weighted outs drops by 6 to 9.5.

The tabulations above are still valid on later streets. In all likelihood there would be fewer unseen cards on later streets, but more missing outs. As is usually the case in Seven-Card Stud High-Low, pot equity can change drastically from street-to-street as cards are exposed.

Tuesday, November 10, 2009

Cold-Calling a Blocking-Raise with a Low-Draw: Scenario 2

Last week, I analyzed a tactic in Seven-Card Stud High-Low Poker I call a "blocking-raise," in which a player with a made low-hand on Fifth or Sixth Street, raises a bet from a high-hand to force out a player on a draw to a better low. The issue is whether it is correct for the player on a draw to cold-call the raise. Analysis showed that if a scoop is possible for the player on a draw, cold-calling does gain equity. The amount of equity depends on the number of outs that are still live. But what happens if no scoop is possible?

Consider a second scenario:

After Fifth Street you are in a three-way pot with the following (hole cards are in parenthesis):

You (A, 2) 6, 7, J
Bob (x, x) 3, 4, 8
Alice (x, x) K, 9, J

Action: Alice leads with a $2 bet; Bob raises to $4. It is your turn to act.

Again for computational purposes we will assume a $1-2 game with eight players dealt into the hand.

Total seen cards = 16 (Eleven that you are looking at, plus five mucked door cards).
Total unseen cards = 36.

A scoop is less likely in this case because you cannot complete a straight. Your only hope to scoop is to pair the Ace and hope it holds up against the high-hand. If your hand is completely live, your outs to make low-hand are as follows:

Outs to make a 7-high low=10 (three 3s, plus three 4s, plus four 5s)
Outs to make an 8-high low = 3 (three 8s)

Again we can divide the number of outs by 36 to arrive at Sixth Street probabilities. Your chances on Sixth Street of having a:

7-high low = 28%
8-high low = 8.3%

Again we can approximate your equity in the pot by making the following assumptions:

A 7-high will win the low-pot.
An 8-high will win the low-pot about half of the time.

Then if (P) represents the pot-size, and (E) represents the equity in the pot, then under these assumptions:

E = (0.28)P/2 + (0.083)P/4 = (0.16)P

Because you are on a draw for only one-half the pot, your equity is only one-sixth of its value. To break-even on this play you need a pot-size of about $24. Unless there was a great deal of prior action on the early streets in the hand, it is unlikely that the pot is that large. It is also unlikely that your low-draw is completely live if Bob already has a low-hand. In this scenario, without a possibility of a scoop, it is best to fold to the raise.

Monday, November 2, 2009

Cold-Calling a Blocking-Raise with a Low-Draw: Scenario 1

A tactic, that Seven-Card Stud High-Low (Stud-Eight) players frequently encounter, is what I call a “blocking-raise.” This is a raise intended to force out a player on a low-draw. Usually, a player who has just made a low-hand, and wants to force out a player drawing to a better low, executes this kind of raise. Blocking-raises occur most often on Fifth or Sixth Streets, when a player has completed a weak low—such as an 8-high—and wants to force out a player showing two undercards, but no other low-cards. Any player, with only two exposed low-cards, cannot have a low-hand until the river. In this post, I will analyze a scenario to determine if cold-calling a blocking-raise is a correct response. As is usually the case in high-low poker, whether or not a scoop is possible, has large effect on the expected value for a play.

For these computations, I will assume a $1-2 game with eight players dealt into the hand. Hole cards are in parentheses.

After Fifth Street you are in a three-way pot with the following:

You (2, 3) 4, 5, J
Bob (x, x) 3, 4, 8
Alice (x, x) K, 9, J

Action: Alice leads with a $2 bet; Bob raises to $4. It is your turn to act.
Total seen cards = 16 (Eleven that you are looking at, plus five mucked door cards).
Total unseen cards = 36.

If your hand is completely live, your outs are as follows:

Outs to make a straight=8 (four As plus four 6s)
Outs to make a better low = 4 (four 7s)
Outs to make comparable low = 3 (three 8s)

That means we can divide the number of outs by 36 to arrive at Sixth Street probabilities. Your chances on Sixth Street of having a:

Straight = 22%
7-high low = 11%
8-high low = 8.3%

We can approximate your equity in the pot by making the following assumptions:

A straight will scoop.
A 7-high will win the low-pot.
An 8-high will win the low-pot about half of the time.

Then if (P) represents the pot-size, and (E) represents the equity in the pot, then under these assumptions:

E = (0.22)P + (0.11)P/2 + (0.083)P/4 = (0.296)P

You expect about 30% equity in the total pot after the action is completed. If you cold-call the $4-raise and Alice closes the action with a call, the total pot-size needs to be at least $13 for your equity to equal the $4 call. We know that the antes and Fifth Street action alone is equal to $13. If there was any prior action on Third and Fourth Streets, the pot is larger than $13. Cold-calling the raise is a profitable response. Not only is the current pot-size large enough, but also the implied pot-size from action on Sixth and Seventh Streets is greater still. And you get a second chance to make the hand if you miss on Sixth Street. Depending on the exposed Sixth Street cards, a cold-call might still be favorable.

However, it is rare that your hand is completely live. The kinds of missing outs make a big difference in your pot-equity. If the 7s and 8s are dead, but the straight draw is live, your pot-equity falls to 22%. That means you need to take down an $18 pot to make a $4 investment worth it. But, if your straight-draw has missing outs, your pot-equity is considerably reduced. Suppose you only have four outs left for your straight, but the 7s and 8s are live. Now your pot-equity is 18.5%. You need a pot in excess of $21 to break-even. That could be a stretch. You might only get that kind of action if the high-hand can beat a low straight, in which case you have only half the equity that you thought you did.

Also keep in mind, that if Alice jams you, that is not close the action by calling, she might be unafraid of a low straight because she can already beat one. The kind of player Alice is has a big effect on your equity. Some players with high-hands are timid in response to a possible low-hand because they are afraid of being freerolled. These players will call the low-hand down, unless they can beat a low straight, in which case they will raise. Other players are more afraid of giving free cards than being freerolled, and will jam anyone raising with a low-cards, even if all they have is one or two pair. In that case your scoop possibility is still live.

Next week we’ll look at a scenario in which scooping is not a possibility.