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.

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.

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.

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.

Assumptions:

•  $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.
0	37.5	1.67:1	$0.44
1	40	1.50:1	$0.54
2	42.5	1.35:1	$0.63
3	45	1.22:1	$0.73
4	47.5	1.10:1	$0.83
5	50	1:1	$0.92

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.