One More Thought on A-Q: Phil Gordon Making Me Say Huh?

In Phil's Little Green Book, he summarizes a hand he played online (Phil Gordon, Little Green Book, p. 115) and I think it's worth examining because the logic behind Phil's play is not immediately apparent to me. Phil was in middle position with A-Q suited diamonds. He was first in and raised to $150, three times the big blind. The button called.

Ok, so there's $375 in the pot before the flop ($50 big blind, $25 small blind, Phil's $150 raise, the button's $150 call). The flop comes Q (spades), 9 (clubs), 2 (clubs). Phil bets $150, which makes some sense. He's got an excellent chance of having the best hand and he's offering the button 3.5 to 1 odds to call ($150 call to win $525 pot). The button called. So far so good.

The turn is K (diamonds). Worried about the straight draw, Phil checks. So does his opponent. At this point, Phil put the button on a flush draw or A-9. Let's take a break here and ask if Phil's check makes a lot of sense. Granted, if he bets and his opponent has a straight or a straight draw, then Phil will likely be called or raised. One might think that it is worth it to bet something here to find out what the button has. The problem is that there is now $675 in the pot. Phil has got to bet enough to: 1) make calling and playing a draw a bad bet; or 2) show that his opponent's hand is good enough now to justify a call. Figure the button is holding any one of a number of hands which will likely win if they get the right draw (such as J-J, 10-10, 10-9, A-J, A-9, or A-Q), or he's holding J-10 or A-k in which case Phil's likely already beat. The chance of holding any of these hands is roughly 25% (each of these hands has about a 1/11 X 1/11 chance of popping up, call this 1% x 8 of these hands = 8%). We'll raise the 8% up a bit since the button has shown some strength and let's call it 20%. This is 4 to 1, so all Phil has to do from a mathematical expectation standpoint is make a bet which makes this a losing proposition. With $675 in the pot, a $300 bet will do it ($300 to possibly win a $975 pot = roughly 3.3 to 1 pot odds). If the button calls this bet, he's either made a bad play or he's got Phil beat. Instead, Phil checks, the button checks, and the guy gets a free card. I don't think this makes sense. With a so-so hand, don't you want to know where you stand, force your opponent to take the worst of it, or, preferably, win the pot right there.

But Phil checked and the river is the 9 (diamonds). Phil checks and his opponent bets $300. Phil calls and wins the showdown when the button turns J-8 (both clubs). This was the right play for Phil because he was facing pot odds of 3.3 to 1 (see preceding paragraph) and he had to figure that he had a better chance than that to win the pot based on probabilities. Phil says that by checking, he made more money because the button would have folded if he'd bet.

Something Bothering Me About Pot Odds

Sklansky's Fundamental Theorem of Poker is widely accepted and it makes sense: make fewer bad decisions than your opponents and you win. How do you judge decisions? Well, there are a number of ways but one of the most prominent is pot odds. Pot odds tell you whether you're getting sufficient return on your money given the risks you are taking. Pot odds of course are based on mathematical expectation, in other words: (the chance that you will win X the amount you will win) - (your chance of losing X the amount you lose). If the difference is positive, you are making a good play, according to the standard wisdom.

As you've gathered, however, one of the goals of this blog is to question the standard wisdom and I'm going to question it here. First of all, I've got no problem in general with judging your plays based on pot odds. However, let's dig a little deeper. I want to make two points. First, there are differing degrees of positive expectation: your expectation could be slightly positive or it could be highly positive. Second, the accuracy of probability calculations depends to some degree on the number of trials. What do I mean by this: I mean that when you have an Ace-Six and you're hoping to draw another Ace, the probabilities look like this on the flop (3/50 + 3/49 3/48). This adds up to roughly a little better than a 1 in 6 chance overall of getting the Ace on the flop. Now you might expect that if you look for this draw 6 times, you might reasonably hope to get one Ace but not so. Probabilities are most accurate over large numbers of trials (see law of large numbers). You might hit 3 Aces or you might well hit none at all. Flip a coin ten times and it's highly unlikely you come up with 5 heads and 5 tails.

So what does all this mean? To me, it means this: before applying pot odds robotically, ask yourself how many hands of NLH you're planning on playing a year. If its tens of thousands, then play by the pot odds book. If, however, you're going to play in 3-4 tournaments a year, which the Hold 'Em Scholar is probably looking at right now, then you've got to be careful about playing tournament hands where the odds against your winning are 2 to 1 and the pot odds are 3 to 1. You just don't have that much of an edge in this situation, given the risk. In other words, you probably want to choose situations where the pot odds are more like 4 to 1 when the odds against you are 2 to 1 and you know you aren't going to be playing an NLH tournament every weekend.

Let me give an example of a hand I wouldn't play, that Phil Gordon apparently would. This example comes from Phil's Little Green Book (Gordon, Little Green Book, p. 42). In discussing a sandwich play, Phil says that he'd push in all his chips where the pot odds are roughly 2.5 to 1 and the odds against him are about 1.5 to 1. Mathematically, this is the right move. Pot odds tell you to make it, Sklansky would presumably tell you to make, but I ain't making it in the early to middle rounds of a tournament. Why? Because I'm only playing a limited number of tournament hands a year and the odds are that I'm going home if I make this play. I don't know that I'm going to play enough hands over the course of the year to make this play pay off. You'll have to make your own decisions...

Hold 'Em Scholar Wrap-Up: Playing A-Q Pre-Flop

So this is where I take all the advice I've written about in previous posts regarding playing A-Q pre-flop and synthesize it. The majority on this issue says that you should generally play A-Q pre-flop from all seats at a 10 person table. The only exception is T.J. Cloutier, who says that you should not play A-Q in a tournament from seats 1 through 5. So, the real question is whether T.J.'s advice is sound.

So let's break down how good or bad a hand A-Q really is. Now, for starters, A-Q is an underdog to any pair ( ). What are the chances that someone else is holding a pocket pair at a 10 seat table? About 16.4% (math below if you're interested). So you're not likely to be facing a pair pre-flop. Even if you are facing one, you can beat most pairs by drawing an ace on the flop.

Now, in terms of overcards, only A-K is a better non-pair hand. At a 10 seat table, there is a roughly 1.3% chance that one of your opponents is holding A-K. So, combine the odds on pairs and this and there is about an 82% chance that you have the best hand when holding A-Q pre-flop. This would seem to support a bet from any position at the table, absent better information.

T.J. Cloutier's argument, however, is that what do you do when an A comes up on the flop? How do you know that one of your opponents isn't holding A-K? Obviously, you don't. The chances that someone is holding A-K, however, are quite small, as I just discussed.

Ok, so you generally want to play A-Q based on straight probabilities without knowing anything else about what your opponents are holding. But sometimes you do know more about what they are holding. What do you do with A-Q then? Let's say you're in middle or late position and there is an early position call -- a limper. Could be the limper is slowplaying a better hand than you but the odds are long against it. So the limper is probably playing a worse hand and hoping to see three more cards cheap. When you're holding A-Q, you want to raise the limper so he either folds or puts more money into the pot (at which point you can better assess whether he's slowplaying).

Now what if you're in the same position but there is an early position raise. Do you call, reraise, or fold? Obviously, the permutations here are endless (size of raise, what you know about player, etc.) but let's explore them a bit. A raise indicates some strength, so you now have to lower that 82% chance that you're holding the best hand. Indeed, it is somewhat likely that the early raiser is holding a pair, face cards, or A-x at least. Let's say you're now 50-50 to be holding the best hand and 50-50 to have the best chance of winning a showdown. Based on these odds alone, a conservative tournament player should probably fold here, particularly in the early rounds where blinds are low. The larger the raise, the worse your odds become, all else being equal, and calling such a raise is not something that a tournament survivor wants to do.

What about when you're in early position with A-Q. As I see it, it doesn't make much sense to limp in with this hand. If you do, you'll likely find yourself in a multi-way pot with a hand that could easily be second-best or worse. So, you want to put in a decent raise when you open the betting with A-Q, to discourage others from playing their hands.

So that' s the wrap. Whose advice does this sound like: a lot like Harrington's and Phil Gordon's to the extent we can tell. Now, the exception from Harrington is that he has you calling early position raises when you have A-Q suited and folding if you don't. Harrington has you raising early position limpers with A-Q suited, simply calling when you don't. So, I've essentially reasoned my way back to Harrington here.

Playing Ace-Queen Pre-Flop: Hellmuth

Phil Hellmuth's book, Play Poker Like the Pros, doesn't have a great deal of detail on NLH but it does have some interesting things to say. He talks about playing A-Q pre-flop in some detail (Hellmuth, Play Poker Like the Pros, 141-143). Specifically, Phil lays out 3 ways of playing this hand.

First, Phil's basic approach is that A-Q is a good enough hand to take a flop with. He says you can call a small raise or raise with it about the size of the pot. Phil doesn't advise calling a reraise or even a large raise, however. The basic principle is that you want to win the pot outright or see the flop fairly cheaply. Although the advice for post-flop play is not specific, Phil seems to be looking for a set on the flop to continue with this hand. He implies that flopping another ace alone can get you into trouble, for reasons I've discussed elsewhere (Q kicker may give you second best hand).

Phil also discusses Frank Henderson's approach to playing A-Q. According to Phil, Henderson likes to limp with A-Q in the hope of flopping a set. As Phil points out, Henderson won't win any pots pre-flop this way but does build the pot in the event he hits a set. Presumably, Henderson folds if he doesn't hit a set.

The last approach Phil discusses to A-Q pre-flop is what he calls the megalomaniac theory. Simply put, a megalomaniac will raise and reraise pre-flop with A-Q in the hope of winning the pot before the flop. Hellmuth says this approach is effective in tourneys because it steals a lot of antes. Of course, when you play it and someone else is holding A-K or A-A, you can lose a lot of money.