Mucking Aces Preflop at the World Series of Poker - Part II

You are at the final table of the World Series of Poker championship and hold pocket aces in the big blind. There are three other players left. Each of you has the same amount of chips. The player under the gun goes all in … the button goes all in … the small blind goes all in. First place pays $1.5 million; second place, $1 million; third place, $750,000; and fourth place, $500,000. What would you do and why?

Since the publishing of "Q's Question" above in the Aug. 4, 2000 issue of Card Player, literally hundreds of players have approached me to discuss their points of view regarding what they would do in the above scenario. Additionally, players from all over the country have been debating the issue, and Hiroshi Shimamura, head of Japan's largest poker Internet site, www.straddle.net, translated "Q's Question" into Japanese, and it has been discussed on the other side of the world! Hundreds of others chose to E-mail me their thoughts and other related questions. When it was a thread on the Internet poker newsgroup rec.gambling.poker (RGP), it produced reactions from more than a hundred online readers.

That being said, the purpose of this column is to pose a question that does not have a "right" or "wrong" answer, that does not have an absolute mathematical solution, and that brings in dimensions to consider other than the play of the hand (such as tournament payout structure and what your personal priorities are). This brings me to "Q's Question, Part II." I will begin with the salient points of my last article. Then, I'll consider what winners are and what they would (and should) think about in relation to answering this question. Following that, I'll examine the necessary mathematical and psychological applications that make up the balance of the objective and subjective thinking necessary to answer this question intelligently and completely. Finally, I will come to the bifurcation point inherent to this and all questions, the point where the objective and subjective aspects of the question, after being weighed and balanced in your mind, finally separate, and the path to the final conclusion is taken. Here is where either your objectivity or subjectivity on the subject will prevail. Here is where objective thoughts (What is the payout structure for the top four places? What are the risk vs. reward considerations? What are the expected theoretical values of the various hands under differing scenarios in this situation? What are the odds of my pocket aces winning?) and subjective thoughts (What is my "read" on the other three players? What is it that I really want to win, money or the title?) must all come together, and a conclusion formulated. Here is where the fork in the road comes and you must ask yourself, "Now that I am cognizant of all of these stats and imponderables, and have considered my personal financial situation, should I muck or call?"

In the first "Q's Question" in August 2000, 77 astute, consistent winning players were surveyed; 41 of them said they would muck the hand and 36 said they would call.

If you wanted opinions on Q's Question, who would you ask? Winners! They are the ones whose opinions you would respect. What is a winner? A winner is a focused individual with a positive mental attitude who optimizes his time, thus achieving successful results. What do winners possess that you admire so much that you would want to know what their thinking is on this and related subjects? They possess a mental edge that is utilized to gain advantage during play. They exude a strong positive "table image" that is associated with a formidable adversary. They continually perform well by having confidence in their abilities, knowledge of their game, discipline in their play, and patience, and continually approach all of their endeavors in life with intelligence and perseverance. Most anyone would wish to possess all of these qualities.

Are there theories that winning players have in common that would apply here? One is the notion that sound risk management is the key to financial success. Let's examine the risk vs. reward in the above scenario from the standpoint of chip equity and the expected value of mucking or calling with the pocket aces. Each of the four players started the hand with 25 percent of the chips. If one were to muck the pocket aces and only one other player was left after the hand was played, the mucker's equity would be the guaranteed $1 million second-place money plus 25 percent of the difference between the first- and second-place payouts: .25 x (1.5 million - $1 million) = $125,000, totaling $1,125,000. Now, if the player decided to call instead and happened to lose the hand, his equity would most likely be: .33 x ($500,000, fourth-place money + $750,000, third-place money + $1 million, second-place money), totaling $750,000. Therefore, the question you have to resolve is whether or not calling would be sound risk management, considering the fact that you are risking $375,000 ($1,125,000 - $750,000) to make $375,000 ($1.5 million, first-place money - $1,125,000). Furthermore, if you called and tied with one other player, each of you would now have 50 percent of the chips and equity of $1,250,000: $1 million second-place money + .50 x ($1.5 million, first place - $1 million, second place). Taking all of the above into consideration can put even the best of players into a real quandary. Do I want to put $375,000 at risk to possibly make an additional $125,000 (caller's equity if tied of $1,250,000 - mucker's equity of $1,125,000)? Do I really want to lay 3-1 by calling in this position? Has the positive expectation that I usually have with pocket aces been considerably diminished in this situation?

Additionally, what if the caller got knocked out of the tournament when the three other players tied with the best hand (for example, three players tie with a straight: Two players have K-Q and the third player has K-K, and the board comes Q-J-10-9-X) and received only $500,000 for fourth place and lost his chance to make an additional $1 million for first place? Is this an acceptable risk to take? These and other related questions would have to be answered during the ephemeral period of time allotted for the big blind's turn. By maintaining your focus on the issues at hand and utilizing what time is given you judiciously, you will always have the satisfaction that comes when you know you have done the best job you possibly could under the given circumstances. Although there were no "correct" answers to these questions, the knowledgeable player can at least live peacefully with the decisions he made and whatever results occurred.

Formulating a logical solution to this and other poker situations entails a lot of effort. First of all, one has to become a rational, competitive poker player to be able to determine the proper plays to make - from the simplest to the more complex, such as the situation being dissected here. To do this, one has to possess at least a conceptual understanding of probability plus the ability to "put players on hands." These are the minimum requirements. A good working knowledge of poker probabilities can be obtained in any number of ways. One can simply memorize stats from various publications, think through game theory and risk formulas, acquire information from knowledgeable people, or discover reliable resources that have answers to the concepts you need to input into your cerebellum in order to have all of your "poker synapses" firing properly. Improving your " hand reading" skills consists of practicing your powers of deductive reasoning. Players who are unwilling to develop these attributes and continually hone their skills are giving up a huge edge to the ones who are constantly working to improve their game. One needs to have a "feel" for the game being played, an understanding of what events are the most likely to occur and what are the most likely hands that your opponents are holding. Here's a case in point with regard to "Q's Question": What would your read be on the small blind's hand? Do you think it likely that it would be pocket aces or kings? If your answer is no, think again. What would you put yourself on for a minimum hold'em hand to overcall two all-in players at the final table of the World Series of Poker, given an equal chip count amongst all players and a payout structure that could swing your bankroll an additional $1 million? If your answer is still not pocket aces or kings, you are definitely on the shortlist of poker players I have been in contact with. If, however, you did put the small blind on pocket aces or kings, what minimum hand would you put the button on? Consider what the hands of the under-the-gun player could be - any pair, perhaps? Ace-king? Connectors? Ace-X suited? You cannot have any feel for what to put your opponents on until you take into account several factors, such as the type of players you are up against, position, chip counts, and the stakes you are playing for. Moreover, it is rather difficult to have any clear idea of "where you are at" in a hand if, in addition to the above, you do not at least have a general idea of the different events that may occur and their respective probabilities of occurring. Take the question at hand, for instance, and ask yourself this: In percentage terms, what are the chances of pocket aces standing up against pocket kings, queens, and jacks? Against three random pairs? Against three suited (or unsuited) connectors? Most importantly, overall, against a wide range of combinations, say, 80-100 different three-hand combinations? What is the statistical probability of pocket aces winning, losing, or tying against three other players when one of them also holds pocket aces? These are the types of questions that should be flowing through your mind when deciding to muck or call under these conditions. When you have at least a rough estimate of the range into which the above answers fall before you sit down to play, you will be more able to conceptualize your final solution when it is your turn to act - since, I think you would agree, there would not be adequate time allowed for you to figure out all of the answers (assuming that you knew what questions you needed answered). That being said, the following table shows the percent win of pocket aces (and percent tie when one of the three players also has pocket aces) vs. 85 other three-hand combinations. This table offers you the overview and subtle comparative differences needed to put this whole question into the proper prospective.

A-A vs. % Win % Tie

A-A (J-10)s (6-5)s 2.1 50.4

A-A (J-10)s (4-3)s 2.1 52.5

A-A Q-Q 6-6 1.9 54.8

A-A J-J 6-6 1.8 54.9

A-A 10-10 6-6 1.8 54.9

A-A K-K 6-6 1.9 55.2

A-A J-J 5-5 1.9 55.2

A-A 10-10 5-5 1.8 55.2

A-A K-K 7-7 1.9 55.3

A-A J-J 7-7 1.8 55.3

A-A Q-Q 7-7 1.8 55.5

A-A Q-Q 5-5 1.8 55.7

A-A 10-10 7-7 1.8 55.9

A-A K-K 5-5 1.8 56.2

A-A 10-10 2-2 1.8 56.4

A-A K-K 10-10 1.9 56.8

A-A J-J 2-2 1.8 56.8

A-A K-K J-J 1.9 57.3

A-A Q-Q 10-10 1.9 57.3

A-A Q-Q 2-2 1.9 57.3

A-A 10-10 9-9 1.9 57.3

A-A J-J 9-9 1.8 57.6

A-A K-K 2-2 1.9 57.8

A-A K-K Q-Q 1.9 57.8

A-A (J-10)o (6-5)o 2.1 57.8

A-A Q-Q J-J 1.9 57.9

A-A J-J 10-10 1.9 57.9

A-A (J-10)o (4-3)o 2.1 60.2

A-A vs. % Win

K-K (J-10)s (6-5)s 46.1

K-K (J-10)s (4-3)s 47.0

K-K 10-10 5-5 52.4

K-K (J-10)o (6-5)o 52.5

K-K Q-Q 5-5 52.6

K-K Q-Q 10-10 53.4

K-K (J-10)o (4-3)o 53.4

K-K J-J 10-10 54.3

K-K Q-Q J-J 54.6

K-K 10-10 (A-Q)s 55.8

K-K J-J (A-Q)s 56.3

K-K 10-10 (A-Q)o 58.8

K-K J-J (A-Q)o 59.9

K-K 10-10 (A-K)s 63.7

K-K Q-Q (A-Q)s 63.8

K-K Q-Q (A-K)s 63.9

K-K J-J (A-K)s 64.2

K-K 10-10 (A-K)o 68.1

K-K (A-K)s (A-Q)s 68.1

K-K J-J (A-K)o 68.6

K-K Q-Q (A-K)o 68.7

K-K Q-Q (A-Q)o 69.0

K-K (A-K)o (A-Q)o 79.7

A-A vs. % Win

Q-Q (J-10)s (6-5)s 47.8

Q-Q (J-10)s (4-3)s 48.7

Q-Q (A-K)s (8-7)s 49.5

Q-Q (A-K)s (9-8)s 51.3

Q-Q (J-10)o (6-5)o 54.6

Q-Q J-J 10-10 54.7

Q-Q (J-10)o (4-3)o 55.5

Q-Q J-J (A-K)s 56.3

Q-Q 10-10 (A-K)s 56.3

Q-Q (A-K)o (8-7)o 57.1

Q-Q (A-K)o (9-8)o 59.0

Q-Q 10-10 (A-K)o 59.2

Q-Q J-J (A-K)o 59.3

Q-Q J-J (A-Q)s 63.9

Q-Q 10-10 (A-Q)s 64.3

Q-Q (A-K)s (A-Q)s 67.8

Q-Q J-J (A-Q)o 68.5

Q-Q 10-10 (A-Q)o 68.6

Q-Q (A-K)o (A-Q)o 80.0

A-A vs. % Win

(J-10)s (7-6)s (4-3)s 44.1

(K-Q)s (J-10)s (6-5)s 45.5

(A-K)s (J-10)s (6-5)s 46.6

(A-K)s J-J (6-5)s 49.2

(A-K)s 10-10 (6-5)s 49.3

(J-10)s 5-5 3-3 49.4

J-J 7-7 4-4 53.0

10-10 5-5 3-3 53.1

(J-10)o (7-6)o (4-3)o 54.1

(A-K)s (A-Q)s 7-7 55.1

(K-Q)o (J-10)o (6-5)o 56.2

(A-K)o J-J (6-5)o 56.7

(A-K)o 10-10 (6-5)o 56.8

(A-K)o (J-10)o (6-5)o 58.6

(A-K)o (A-Q)o 7-7 64.3

Since results vary slightly with different suited cards, nonconflicting suited card combinations were used whenever possible. The statistics above were generated by a computer simulation program run one million times each.

Now that you have had the opportunity to scrutinize the table, let us address the following questions and come up with some answers. What are the chances of pocket aces winning against a field of three all-in players who are collectively holding pocket kings, queens, and jacks? As you can see, the pocket aces will win 54.6 percent of the time. How about against three random pairs? It seems that the answer is in the same ballpark. Against three unsuited connectors? Once again, the same neighborhood! But notice that some of the combinations consisting of three suited connectors produced the least favored scenarios for the big blind's aces. In fact, in some unlikely cases, for example, (J-10)s (7-6)s (4-3)s, the field is expected to win approximately 56 percent of the time. What is the probability of the big blind winning if one of the three all-in players also has pocket aces? Obviously, not very good - there's only about a 2 percent chance of winning the tournament on that hand. However, there is about the same chance of tying with the other pair of aces as there was of winning the hand (excluding some triple combinations of suited connectors).

The focus of the overall picture here should now be getting sharper to you. Sometimes, we need to see many examples to be convinced what will and will not most likely happen under certain circumstances. Once a person understands the extremes of a given situation, all else in between can be easily understood. Are you now comfortable enough with the figures from the table and the above extrapolated information from it to be able to have a rough estimate of what the probabilities are of winning, losing, or tying with a pair of aces against the remaining three all-in stacks? You can readily see what an edge winning poker players enjoy by having a feel for what the most probable outcome of the hands they are playing will be. They do not have to know what the exact statistics are carried out to five significant digits, or what the standard deviation from the mean is in a given range. Just a general idea of what their chances are will suffice. Do you think it would have helped if you were to have known before playing the hand what a rough estimate of your chances were if you held a pair of aces and were up against three all-in players? If you had previously read this article and knew that the big blind's pocket aces would win (or tie) approximately 55 percent (+/- 10 percent) of the time, do you think that it might have enabled you to conceptualize this problem more efficaciously?

We cannot go over all of the subjective aspects of this question and the personal traits and characteristics that would have a bearing on what we would eventually end up doing at this final table. But, my discussion of the most pertinent objective and subjective aspects of this question should enable you to be armed with enough mental ammunition to arrive at your own intelligent conclusions.

In the beginning of this article, I made reference to the bifurcation point inherent in all questions, where the objective and subjective aspects must separate and decisions must be made. We are now at that point with respect to this question. So, what would you do? Muck or call?

Your mind just took that fork in the road, where the confluence of objective and subjective thoughts that were ebbing and flowing together went their separate ways. Now, only one of those streams of thought is carrying your final opinion on this matter. The very last thought that was in your mind before you made your decision was the force that impacted your thinking to make that decision. It appealed either to the emotional, creative, and subjective right side of your brain, or to the more logical, pragmatic left side of your brain. After all of your efforts to come to a decision, your final thought was something like: "The heck with it! If I can't call preflop with the very best starting hand, I shouldn't be playing this game at all!" And you called. Or, you thought, "It's better to error on the side of caution." And you mucked.

I would bet that it took you no more than five seconds to arrive at your answer. Did you surprise yourself? Are you comfortable with your decision? I would venture to guess that you are. Do you remember your last thought before telling yourself how you would play the hand? Was it based on objectivity or subjectivity? Probability or gambling? I am sure you know, and once you do, many other decisions you will make during poker games will wind up being based on the same thought path. If you thoroughly think through a problem and then check out the intangibles on the subject, the conclusion you will reach will always be one you can rely on.

How far are the concepts and questions presented in this article from reality? The first four places of the hypothetical payout structure I considered in "Q's Question" a year ago were amazingly close to the actual payouts of this year's WSOP championship. Q's Question (August 2000) payout structure: $1.5 million for first place, $1 million for second place, $750,000 for third place, and $500,000 for fourth place. WSOP (May 2001) payout structure: $1.5 million for first place, $1.1 million for second place, $700,000 for third place, and $400,000 for fourth place. And, pocket aces came into play on the last hand played at this year's championship with two players left vs. four players left in last year's Q's Question. One did have to give pause when Dewey Tomko's A A lost to Carlos Mortensen's K Q, which allowed Carlos to claim the 2001 World Series of Poker title. What do you think was going through Dewey's mind when he glanced down and saw the "eyes of Texas" staring back at him in that situation? Even though Dewey's chances of winning the hand were 82.1 percent preflop, 51.5 percent after the flop, and 75 percent after the turn, it goes to show that you just never know what the "card gods" may have in store for you.

I hope that you found this article interesting.

Editor's note: Q welcomes comments at: Q7654321@aol.com.