## Structured Hand Analysis SHAL

Most poker opinions are formulated from a combination of experience and intelligent guesswork. Consider problems like the following:

1. "I'm at an eight-handed table, in third position with ace-jack suited. My M is down to 5. The players behind me have been tight. There's one stack even smaller than mine. He's getting desperate. Am I strong enough to push all-in here?"
2. "I'm in the big blind at a six-handed table. I'm holding king-ten offsuit. The guy in first position went all-in. His M was 4. I think he was desperate, but I'm not sure. I've got plenty of chips. Is my hand good enough to call?"

These are tough questions. If you went to a poker tournament and posed these questions to a collection of good players, you'd get a bunch of reasonably informed opinions, based loosely on their experiences over the years. (Most would answer "no" to both questions.) But when you were done, would you really know anything?

Actually, you wouldn't. Guesswork, even informed guesswork from strong players, isn't the same as real knowledge. In poker, as in every other area of human endeavor, the considered opinions of a collection of the world's best practitioners might be right, but it might also be quite wrong. The consensus of what is considered true seems obvious and inevitable until some brave soul comes along and says "No, the truth is really like this."

In this section, I'll show you a way to get beyond opinion and guesswork and figure out some real answers to tough problems-

It's a method i call Structured Hand Analysis (SHAL for short). SHAL gives us a way of actually calculating answers to a variety of endgame problems. It's especially useful in resolving all-in problems of the sort posed at the beginning of this section, but it's useful for others as well. To give you the idea, I'm going to pose a question and work through the answer in a step-by-step fashion.

Before we get started, however, there's a caveat. SHAL isn't a method that you can use at the poker table. It's a technique for posing and solving problems at home, when you're out of the action. You'll need time, some facility with using a spreadsheet like Excel, and one of the programs that lets you input a couple of starting hands and gives you the probability that each hand triumphs in a showdown. (I like PokerWiz, but there are others.) If you're willing to put in the time and effort, and you enjoy noodling over numbers for an hour or two, you'll be able to figure out things that no one else knows. If that kind of work doesn't appeal to you, no harm done. Just skip this section and move on to the problems at the end of the chapter. If the thought of exploring the dark corners of poker and discovering hidden treasures is appealing, then let's get to work.

We'll start by considering a sample problem.

Example 9. It's the final table of a major tournament. Nine players remain. The blinds are \$3,000/\$6,000, with \$300 antes, so the pot is \$11,700 to start. You're fifth to act, with a stack of \$90,000 after posting the ante. The first four players all fold.

### You pick up

The four players yet to act behind you are a mixture of tight and loose. All have stacks slightly larger than yours, although none are very much larger. You briefly contemplate moving all-in with your ten-eight offsuit since you've been playing tight for awhile and no one would have any reason to suspect you'd be making a play. But your better judgment prevails and you muck your hand.

The cut-off seat, the button, and the small blind now fold, and the big blind takes the pot. Again you wonder, should I have made a play with that hand? And if not, how good a hand did I need to make a play?

Answer: We've really posed two questions here, one specific (what happens if I go all-in with T484?) and one more general (what's the minimum hand I need to make going all-in a profitable move?). We'll start by trying to answer the specific question. It's easier, and it's also possible the answer to the specific question will contain a big clue to the answer to the general question.

Our first step is to build a profile of the four players who act behind us. In our first statement of the problem, we just described them as a mixture of "tight and loose." Now we need to get more detailed, which will also necessarily involve a little guesswork and speculation. Let's say that the cut-off seat player is rather tight, as is the player in the big blind. Let's say that the button is a considerably looser player, and the small blind is the loosest of all. From now on we'll refer to these players as A, B, C, and D, in order from the cut- off seat to the big blind.

To finish our profile, we're going to write down the specific hands that each player would use to call an all-in bet. Here are my guesses:

• Player A (tight): Will call all-in with AA through QQ> and AK (suited or unsuited).
• Player B (looser): Will call all-in with AA through 99, AK and AQ (suited or unsuited).
• Player C (loosest): Will call all-in with any pair, AK, AQ, AJ, and KQ (suited or unsuited).
• Player D (tight): Same requirements as Player A.

Do these estimates make sense? I think so, although a reasonable person could certainly argue about the exact distribution of calling hands. Remember we specified that each player had a stack slightly larger than ours, although not enormously larger. Since our stack is \$90,000, our M is about 8. These four players will have bigger Ms, let's say in the range of 8 to 11. None of them should feel particularly desperate, and since we specified that we'd been playing tight, no one has any reason to think that we're moving in with other than a good hand.

Now that we have our player profiles, the next step is to figure out how often we'll be called, and by whom. This is pretty easy. We know there are 1,326 possible poker hands (52 times 51 divided by 2). Once two cards are removed, however, (in this case our ten and eight) the remaining fifty cards form only 1,225 hands. For every pair, there are six possible ways of dealing the pair. For every non-pair, there are 16 possible ways of dealing the hand, 12 unsuited and four suited. Let's start with player A and figure out how often he will call.

 AA 6 KK 6 QQ 6 AKs 4 Alto 12 Total 34

Of the 1,225 possible hands, only 34 are calling hands for Player A. So the probability that Player A calls our bet is 2.8 percent.

1,225

### We can do the same calculation for Players B, C, and D. I won't list them separately, but here are the answers:

• Player B: 65 calling hands or 5.3 percent
• Player C: 136 calling hands or 11.1 percent
• Player D: Same as A or 2.8 percent

Now we'll make an assumption which is just slightly inaccurate, but which simplifies the calculations enormously. We'll assume that only one person will call us. With that assumption in place, here's how the hand will turn out:

• Player A calls: 2.8 percent
• Player B calls: 5.3 percent
• Player C calls: 11.1 percent
• Player D calls: 2.8 percent
• No one calls: 78.0 percent

So almost 80 percent of the time we pull down the pot uncontested. The rest of the time we get called by somebody.

Now we're ready for the next step of the problem, figuring out how often we win against each opponent, assuming that we're called. Let's start with Player A (the easiest ease).

Our first job is to figure out how often our ten-eight offsuit actually wins in a showdown against the five possible different hands that Player A might hold. (This is where wc need a program that calculates the results for two hands matched against each other in a showdown.) Again, we'll make a simplifying assumption that is slightly inaccurate, but makes our work much easier. It affects the calculation a little bit if the suits of our cards match the suits in his hand, or if just one suit matches, or none. For simplicity's sake, just assume the suits don't match. Now it turns out that our T48* wins 18 percent of the time against his aces, 17 percent against his kings, 16 percent against his queens, 34 percent against ace-king suited, and 36 percent against ace-king offsuit.

Now we construct a chart that looks like this:

 A's Hand Probability of Hands Hand Count Winning Won AA 6 0.18 1.08 KK 6 0.17 1.02 QQ 6 0.16 0.98 AKs 4 0.34 1.36 AKo 12 0.36 4.32 Total 34 8.74

The winning percentage is 25.7.

The first line of this table, for instance, shows that Player A has six ways of being dealt a pair of aces, that we are 18 percent to win if he holds aces, and the average number of hands we win out of six is just 1.08 hands. Adding up the hands won in each category gives us, on average, 8.74 hands won out of 34 total hands, for a winning percentage of 25.7 percent. On average, if Player A does call us, we win only about one hand in four.

Player D's chart looks exactly like Player A's, of course. The charts for Players B and C are more extensive, since they call with more hands. Against Player B, we end up winning 26.7 percent, and against Player C's loose calling, we actually win 34.3 percent.

Now we're ready to put together a final chart, combining the probability that each player calls us, the probability that we win if they call, and the size of our stack if they call and we win. (If they call and we lose, our stack unfortunately goes to zero since everyone has us covered.) I'll lay out the final chart, then explain what the various entries mean.

 Event Probability After Play Expectation No call 78.0% \$101,700 \$79,326 A calls, I win A calls, I lose 0.7% 2.1% \$191,700 \$0 \$1,342 B calls, I win B calls, I lose 1.4% 3.9% \$191,700 \$0 \$2,684 C calls, I win C calls, I lose 3.8% 7.3% \$188,700 \$0 \$7,171 D calls, I win D calls, I lose 0.7% 2.1% \$185,700 \$0 \$1,342

Expectation if all-in: \$91,865 Expectation if fold: \$90,000

The first column, "Event," just lists the various possible outcomes of the hand. The second column, "Probability," shows the probability of the events in the first column. The most likely outcome, as we have seen, is that all players fold and you take the pot (first line). The second and third lines show what happens when Player A calls. Note that the sum of the probability that A calls and wins (0.7 percent) and the probability that A calls and loses (2.1 percent) add to 2.8 percent, which we already calculated was the probability that A called, given his mix of calling hands. The same is true for Players B, C, and D.

The third column, "Stack After Play," shows the size of your stack if the event in the first column happens. In the case where no one calls, for instance, your new stack becomcs your old stack (\$90,000) plus the existing pot (\$ 11,700), which equals \$ 101,700. In the other cases where you are called and win, your stack more than doubles, although you win a little less from Players C and D since they were in the blinds and had already contributed some money to the pot.

The last column, "Expectation," is the key. To find the expectation for a given event, multiply the probability in column 2 by the stack size in column 3, giving the expectation in column 4. By adding all the expectations for all possible events, you get the expectation for the play itself (going all-in). At the bottom of the table, you can sec that the cxpected value of your stack after the play is \$91,865. Now compare that to your expectation if you fold instead, which is just the size of your current stack, or \$90,000. The comparison shows that going all-in is a positive expectation play (meaning the play earns money, rather than loses money), with an average earn of \$1,865.

That' s a pretty startling result, so let's take a step back and See just what we think we have learned.

We began by trying to solve two problems: Was moving ail-to with a ten-eight offsuit in fifth position a good play, and what Was the minimum hand we needed to go all-in from that position'.' °ur assumptions were correct, we answered the first question and came close to answering the second question. The ten-eight offsuit showed a small profit on average, so moving all-in was a reasonable play, and it only showed a small profit, so it's probably close to the theoretical worst hand needed.

Still, however, this is a pretty startling result, so before we jump to conclusions, let's review our assumptions and see if we really trust them.

Our key assumption is our estimate of the hands that each player would require to call an all-in bet at this point in the tournament. Once we settled on a hand distribution for each player, the rest was just straight mathematics. So let's look at those hand distributions again.

• The tight players. For Players A and D we specified that they would need one of the three high pairs or ace-king (suited or not) to call the all-in. Reasonable? 1 think so. After all, their stacks were stipulated to be eight to ten times the pot, so they weren't really desperate, and we were supposed to be a tight player, so they have no reason to believe we're making a move with other than a pretty good hand. Would you put your tournament up for grabs with a pair of tens or ace-jack after an all-in move from a solid player? I know plenty of players who would, but they're not players I'd describe as tight. In fact, Players A and D are following the strategy recommended inmostpoker books: don't call an all-in unless you have a really strong hand.
• The loose player. We characterized Player B as loose, and said that he would call an all-in with any pair down to nines, plus ace-king and ace-queen. Again, this seems reasonable for a not-so-tight player with an M of 8 to 10.
• The loosest player. For Player C, we specified he would call the all-in with any pair, plus the ace-jack and king-queen combinations. That's pretty loose. Would you call an all-111

bet with a pair of deuces? I wouldn't unless 1 was way down in the Red Zone and just looking for a chance to get into an even-money situation for all my chips. With an M of eight or ten I wouldn't consider this play. Player C qualifies as loose in anybody's book.

All in all, I'm comfortable with our description of the playing styles of our four imaginary opponents. But to get a better handle on the problem, let's look at three additional cases and see what happens. All three of these cases can be solved just as our initial problem was solved. (In fact, if you do the calculations in a spreadsheet, it becomes a trivial matter to just plug in new assumptions and get a new answer.) I'll spare you the details and just give you the final answers:

• Case 1: Four tight players is the best possible case. Your expectation rises to \$98,524, for an expected profit of S8,524. Going all-in is now a clear play.
• Case 2: Four loose players is not as good as our original distribution of profiles. Your expectation is now \$90,863, for a profit of just \$957. All-in is still a positive expectation play, but just barely.
• Case 3: Four of the loosest players behind you is bad news. Your expectation dips to \$85,229, for an expected loss of \$4,771. Now going all-in is a bad play indeed.

Of course, these results are just what we might expect. Going aU-in against loose players with a weak hand just plays into their Style, since they call you with hands that aren't very strong, but are still good enough to beat you.

In short, our speculative play was able to show a profit against a mix of loose and tight players, and a big profit against tight players only. It was just the case where we faced loose players only that we were better off laying the hand down.

Conclusions. If this play is theoretically profitable, should we use it? The answer, as in many other plays in poker, is that we can employ the play as part of a balanced strategy, but any abuse of the move will quickly render it useless. Don't forget that we postulated in the original problem that we were both a tight player and seen as tight by the other players. The first time we use this play, they'll give us credit for making a value bet and only call with hands they regard as appropriate, given their individual styles. As we use this play more and more, their calling requirements will start to dip, and they won't need to dip very much before the play is unprofitable. However, we learned a valuable lesson and a good tactical play: From middle position with four players left to act, all with Orange Zone-type stacks, and with mixed styles, you can make a profitable first all-in move with hands down to something like ten-eight offsuit.

The most basic inflection point problems arise in the Red Zone. Do we go all-in or not? Problems 9-1 through 9-14 show examples of these situations.

Orange Zone play allows a little more discretion in how you play your hand. Problems 9-15 through 9-17 show a few examples.

Problems 9-18 and 9-19 show some more issues involving all-in play. Problem 9-20 discusses what to think when facing an all-in bet.

Playing small pairs and suited conncctors is always aproblem with small stacks. We'll look at a few examples in Problems 9-21 through 9-24. The last two problems, 9-25 and 9-26, cover some miscellaneous tactical ideas.

Hand 9-1

\$70,000

\$50,000

\$300,000

\$170,000

\$170,000

\$250,000

Situation: The final table of a major tournament. This is the firsi hand of the final table, and you have no information on the other players.

Action to you: You are first to act.

Question: Do you fold, call, raise \$30,000, raise \$50,000, or go all-in?

Answer: Under normal circumstances, a pair of nines under the gun wouldn't be considered a particularly strong hand. Early in a tournament, a call or a small raise might be in order here, hoping to see a flop cheaply.

But here, the time for small bets and cheap flops is long past. You've made it to the final table, but you're not in good shape at all. Your stack of \$90,000 is less than four times the size of the blinds and antes, so you'll be blinded away after only four more turns around the table. You need to make a move now, while your stack is still large enough to cause some pain even to the largest stacks at the table. I move all-in with this hand.

Am I hoping to steal the blinds, or am 1 rooting for a call? The quick answer is — I don't much care. I'll be satisfied if I pocket a quick \$24,000 with my nines. But if I get called, I'm only an underdog against one of the five larger pairs. Against every other hand, I'm a favorite to double ray stake, and get back in the running for one of the top positions. Given my smallish stack, this hand is a great situation for me, and I'm going to try to make the most of it.

Some players would play this hand quite differently. If you showed this hand to a bunch of reasonable players, you'd hear at least some advice that would go like this — "I'd make a small raise, to about \$30,000. That's enough to chase out some weak hands, but if I get reraised behind me, I can still get away from the hand because I've only invested a third of my stack." A plausible argument, but quite wrong.

Suppose you try this approach. What's likely to happen? Sometimes, you will win the blinds, just as you did when you went all-in (but not as often). Sometimes, you'll only be called, and the flop will mostly not contain a nine. Now, depending on the flop, you're throwing your hand away when someone bets at you. And if someone reraises you before the flop, you're throwing your hand away as well (sometimes in situations you would have won had the hand been played to a conclusion.)

When you bet and throw your hand away, your stack is now down to \$60,000. On the next two turns, you're in the big blind and small blind, which will most likely reduce your stack another \$24,000, down to \$36,000. That means that three hands from now, you've gone from \$90,000 to \$36,000. That's a disaster for you! Now any bet you make will certainly be all-in, and will be treated as an all-in bet from a very short stack — hence one that must be called. You'll have lost any ability to steal pots, which is a crucial part of your overall equity in any hand.

With that fate just three hands away, it's clear that a pair of nines in this situation is a fantastic hand for you — very likely the best hand at the table right now. You must take advantage of it, so shove in your chips.

Hand 9-2

Situation: Middle of the second day at a major tournament.

Action to You: Players A through F all fold.

Question: Do you play or fold?

Answer: Here's a basic, and easy, inflection point play. With the blinds and antes totaling \$1,700, your M is just under 3. Your \$4,900 stack is enough to last you just three rounds of the table. You're up against only three opponents, so your ace-7 offsuit isn't even objectively a weak hand any more. You're definitely playing the hand.

How much should you bet? The minimum bet now is twice the big blind, or \$1,600. That's one-third of your dwindling stack. If you made the minimum bet, then exited the hand later, you'd be even more crippled than you are now. So go all-in. Notice that the three active players behind you have Ms ranging from slightly more than 4 to slightly less than 6. Right now your stack is big enough to ruin any these players if they call and lose, so only a strong hand rates to call you, and the odds of not finding a strong hand in three random hands are quite good.

No need to look or feel nervous when making this play. Sure, if you make it, get called, and lose, you're out of the tournament. But you're almost out of the tournament anyway, and "conservative" play won't save you.

Resolution: You go all-in, and the three players behind you fold.

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