Thursday, March 29, 2007
By Bluff magazine
Before I get going on this article, I want to stress that the following concept is one of the most important in poker. You will not meet a great player who doesn't understand this idea, whether he has put it into words himself or not. I haven't seen this concept explained well enough for the average player to understand, so I'm going to do my best here. It involves some boring math, but it will be worth it for your game if you can get through it. So pay attention.
In one or more of his many acclaimed books, David Sklansky introduced the term "Sklansky Dollars." If you don't know what these are, you should pick up "The Theory of Poker" and read it today, but I'll touch on it briefly for the purposes of this article. Sklansky Dollars allow you to assess how you're doing without your results being as affected by luck. To keep track of how many Sklansky Dollars you win, you look at the percent chance you had to win the hand when money went into the pot, and multiply that by the money that goes in.
So, if you get all in preflop with A-A versus J-J for $10,000 and you lose, you lost $10,000 in real dollars; however you won about $8,000 in Sklansky Dollars because you should win that hand 80 percent of the time. As Sklansky explains, as long as you are making Sklansky Dollars, you will make money in the long run because luck evens out. In the ultimate long run, your Sklansky Dollars earned and real dollars earned will be the same. It's a good way for you to keep calm and make the right decisions when facing the swings that inevitably come with poker. It's a great concept.
I, however, am improving on it. And because I'm as egotistical as Sklansky, I'll be naming my idea after me. Introducing "Galfond Dollars" (G-Bucks for short):
First, I want to make sure you understand hand ranges. Let's say, for instance, you raise under the gun in a nine-handed game with Ah-Kh. Your hand is Ah-Kh, but your range is so much more than that. Your range is every hand for which you would take the same action. So your range for raising under the gun might be A-K off-suit, A-K suited, A-Q suited, and all pairs 9-9 and above. Let's say the button calls, and the flop is Q-6-5 rainbow with the queen of hearts. Now you bet two-thirds of the pot. Your hand is still Ah-Kh, but let's say that you always (you probably don't always do anything, but roll with it for the sake of simplification) check the flop with your smaller pairs, 9-9 to J-J, and you check your Q-Q half the time for deception. Now your range is A-K off-suit, A-K suited, A-Q suited, A-A, K-K, and half of your pocket queens.
Do you follow? Unfortunately, I can't hear your answer, so I'll just keep going. Now, let's say the button calls and the turn is the 2h, putting two hearts on the board. You bet three-fourths of the pot. Your hand is still Ah-Kh, but your range has changed once again. Let's say that you give up with your non-heart A-K hands and you check A-Q suited for pot control. Now your range is Ah-Kh, A-A, K-K, and half of your pocket queens. You following me? OK.
The button calls and the river is the 2d. Tempted to bluff? Hold that thought. We'll come back to this hand in a little while.
Now that you understand a range, let's talk about Galfond Dollars. The way that Galfond Dollars work is similar to the way Sklansky Dollars work. However, instead of taking your hand and seeing how it does against your opponent's hand, you take the entire range of your hand and see how it does against his hand. (The next level would be taking range versus range, but that gets very complicated mathematically.) So, let's go with a simple example:
You're playing $50/$100 no-limit heads-up. Your opponent has only $1,000 on the table and you have him covered. You're in the small blind. You decide before the hand that you will shove all-in with K-Q, J-J, Q-J suited, and 7-6 suited, and not push other holdings. (This shoving range is a bit far-fetched, but just go with it for this explanation.) You're dealt Qs-Js and, just as you planned, you go all-in. Your opponent thinks for a bit and calls with Kc-9d. The board comes Qh-5s-6d-Kh-2h and you lose the $2,000 pot. Let's see how you did in real dollars, Sklansky Dollars, and G-Bucks:
In real money, you lost $1,000.
In Sklansky Dollars, you lost $80 (Q-J suited is about 46 percent to win versus K-9 off-suit times $2,000 in the pot equals $920; $1,000-$920 = $80).
Let's look at it in G-Bucks now
Remember, we match our range up against his hand. So let's first see how likely we are to be dealt each hand in our range:
There are 16 combinations of K-Q (Ks-Qs, Kh-Qc, Kd-Qh, etc.), six combos of J-J, four combos of Q-J suited, and four of 7-6 suited. In total, there are 30 hand combinations we can have. Now we will see how each hand stacks up versus his K-9 off-suit:
K-Q versus K-9 off-suit: 74.0 percent
J-J versus K-9 off-suit: 72.0 percent
Q-J suited versus K-9 off-suit: 45.5 percent
7-6 suited versus K-9 off-suit: 41.0 percent
Next, you multiply each win percentage by how likely the hand is to be dealt. In other words, how many hand combinations make up that hand compared to how many hand combos you have in your entire range. The best way to do this is to multiply each winning percentage by the number of hand combos and then divide by the total number of hand combos.
K-Q (.74 x 16) + J-J (.72 x 6) + Q-J suited ( .455 x 4) + 7-6 suited (.41 x 4) / Total hand combos (30)
11.84 + 4.32 + 1.82 + 1.6 = 19.58
19.85/30 = .653
So, your range is 65.3 percent against K-9 off-suit, meaning that, on average, you win about $1,305 from the $2,000 pot when he calls your shove with K-9 off-suit. That makes your average profit $305. So, when K-9 off-suit called your Q-J suited shove, you made $305 G-Bucks!
Real Dollars -$1,000
Sklansky Dollars: -$80
Galfond Dollars: +$305
The example was not very important to your poker game, but I want to make sure the concept of G-bucks is entirely clear. Let's move on to more interesting hands.
How about the hand that we left off with up top? You have Ah-Kh and have already fired two barrels after raising under the gun and getting one caller. You missed your flush. Remember, the board read Qh-6s-5c-2h-2d, and we found that your range for raising preflop and then betting both the flop and turn was Ah-Kh, A-A, K-K and half of your pocket queens. We were deciding whether or not to bluff the river. Let's say in this spot your opponent has something like top pair or J-J or 10-10: a hand that is moderately strong but can only beat a bluff on this river. Let's say you bet the full pot with your Ah-Kh, as well as with your entire range here once again. How much does your opponent make or lose when he calls?
We never gave the pot dollar values, so let's just say there's $5,000 in the pot. If you bet $5,000 and he calls, he makes $10,000 in real dollars and in Sklansky Dollars, since there are no more cards to be dealt. How does he do in G-Bucks?
Well, you have 13.5 hand combinations (three combos of Q-Q, so betting Q-Q half the time makes 1.5 hand combos), of which he beats only one. He loses $3,889 by calling your river bet versus your range. So, he would be making a terrible call if he had any idea of your range. An observant player would figure this out and would not pay you off here. Against weak players who call too much, almost always having the goods when you bet is a smart way to play. However, against higher-level players, you've made your game super exploitable by not bluffing enough. You have to think about manipulating your range so that you become more unpredictable, and you put your opponents to tougher decisions. If, for instance, you could have 8-7 suited, Ah-Jh, Ah-Th, and you followed through with half of your A-K off-suit hands, your opponent's decision would be closer on this river. In the example we gave, against half-decent players or better, you have to bet the river with your missed Ah-Kh, otherwise you become even more predictable, since you actually are bluffing zero percent of the time.
This is why, in tough games, you can't raise only 10-10 and above and A-K from under the gun. Your smart opponents will be able to put you on a hand too easily when the flop hits. If the flop comes 7-6-5 rainbow, they know that you can't like it. Even with an overpair, you might have to fold if they play back hard at you, and because of your preflop raising range under the gun, you could never have a set or a straight. In tough games, you can't only bet strong hands, and you can't give up every time you miss. You also have to learn to value-bet thinner. Here's an example of how value betting too tight can get you in trouble:
You are a solid, aggressive player. You put a lot of pressure on your opponents, which is a good thing. When you fire big bets on all streets, you can show up with a missed draw bluff sometimes, but you can also show up with the nuts. You aren't big on slow playing. However, you are careful not to get stacked with one-pair type hands. On to the hand
Everyone has $20,000 to start the hand. You raise 9h-8h to $700 on the button, and a smart, optimistic player calls in the big blind (I use the term optimistic to describe a poker player who is quick to put you on a hand he can beat if it's reasonable).
Flop is Qh-10h-5d ($1,500 in pot). He checks and you bet $1,500. He calls.
Turn 4c ($4,500). He checks and you bet $4,500. He calls.
River 5s ($13,500). He checks and you go all-in for your last $13,300. He insta-calls with Ac-Jc. You look down at your 9h-8h, think how bad of a call he just made, and that you would play Q-Q the exact same way, and muck. He gets the $40,000 pot.
But what the smart player knows about you is that you don't bet hard enough with top-pair type hands, and you always bet hard with your draws. He knows that with K-Q or A-A, you would check behind on the turn to control the pot. So the only hands you bet for value on the turn are two pair and sets. You also bet with any open-ended straight draw on the turn and with any flush draw. Let's look at your range on the river and see how bad a call it was.
Your range for raising preflop and betting all three streets:
Q-Q, 10-10, Q-10, 5-5, 4-4, Ah-5h, 7-6, K-J, J-9, Ah-2h to Ah-Kh, Kh-9h, Jh-8h, 9h-8h, 8h-7h, 9h-7h.
I'm trying to get you to recognize how many hand combos make up a certain hand. For instance, when you think someone has a set, there are only three possible combos of each set, whereas there are 12 hand combos of top pair-top kicker. So if someone takes a line in which he has to have a set or a bluff, realize how unlikely it is that he has a set. Similarly, suited hands are much less likely than unsuited hands.
Anyways, let's see how bad his river call was in G-Bucks. Since all the cards are out, G-Bucks analysis is simply what percent of your hands he can beat now. If it were earlier in the hand, we would also factor in percent to improve to the best hand by the river.
I won't go through the whole hand counting analysis again because it's as boring for me as it is for you. There are programs available online where you can input a range of hands and see how your hand does against that range, and it accounts for hand combinations. If you are so inclined, try this one by hand to see how often your opponent made a good call versus the range I gave above. Remember to account for his Ac-Jc in your range, because he knows you cannot have the Ac or the Jc.
I'm going to put the range into the program and see how often he has the best hand. If he's right, he wins $26,800, and if he's wrong he loses $13,300; so he has to be right only about 33 percent of the time to make a call to break even. Looking at the math, he has the best hand 70.5 percent of the time! That's way more than enough to call on the river. His call made him almost $15,000 Galfond Dollars ($26,800 real dollars) and is clearly the right play against you.
Let's talk about a couple of other aspects of the hand. First, his call on the turn. Second, how you can manipulate your range to make this a more difficult decision for him.
While your opponent's river call was very standard versus your range, his turn call was much shakier. River decisions are very simple in that they can be solved completely with numbers. Pre-river decisions are much more complicated. Let's look at his call on the turn from a Galfond Dollars perspective. When you bet the turn with that same range, and he calls with Ac-Jc, he is 54 percent against your range of hands. Since he has to call $4,500 to try and win the other $9,000 in the pot, it might seem like he's making a good play. If your $4,500 bet put you all-in, and your range was the same, his call would be making him $2,790 G-Bucks (see if you can get that number on your own). However, with a draw-heavy board, and being out of position with money still to go in, his call isn't as good. I don't have a clear-cut figure for you, but you should be folding in spots that result in positive G-Bucks when certain situations arise.
Here are some examples of times that you should fold when G-Bucks calculations are telling you otherwise:
• You're out of position and there is some money left behind.
• The board is draw-heavy, and you don't know which cards help your opponents.
• Your opponent is a strong, aggressive player.
• Your hand has little chance to improve.
On the other side of things, there are some spots in which you can call when G-Bucks calculations make a call seem slightly wrong:
• You are in position and there is some money left.
• The board is draw-heavy, and you have a disguised draw (especially in position).
• Your opponent is very predictable -- too loose or too tight -- and you are a strong, aggressive player.
• Your hand has outs to become very strong.
(These factors all increase in effect the deeper the stacks are.)
The reason your opponent should fold the Ac-Jc if you are a competent player, in my opinion, is that you will make his life very difficult on the river. If you don't have hearts and a heart falls, you can bluff him off the best hand. Or you might hit a straight and he'll pay you off thinking you missed a flush draw. The main thing is that you have another street to act on where you have everything going for you. You should earn money on the river on average, if you are as good as your opponent or better, because of all of the factors above. So he should give up a little bit of money in value on the turn to make up for the value that you should gain on the river.
A good example that I like to give: same game, $100/200, $20,000 stacks. The button, a good, aggressive player, raises to $600 and you call in the big blind with 5-5. Flop is Jd-10d-2h. You check and the button bets $1,000, which he does with almost every hand he raised with. You usually have the best hand. But a fold is still correct. Think about that and make sure you understand. You likely have the best hand, definitely over 65 percent of the time, and you have over 2-to-1 pot odds, but a fold is still clearly correct. First of all, you are an underdog to finish the hand ahead. You're about 44 percent against a reasonable button-raising range on that flop. Even with the pot odds, which would make the call appear to net you some Galfond Dollars, you have to factor in your opponent's advantage on later streets because of the examples I gave above.
Let's go back now to the Ac-Jc hand and talk about how you can make your turn and river play better. Remember the action looked like this:
Flop is Qh-10h-5d ($1,500 in pot). He checks and you bet $1,500. He calls.
Turn 4c ($4,500). He checks and you bet $4,500. He calls.
River 5s ($13,500). He checks and you go all-in for your last $13,300. He insta-calls with Ac-Jc.
And we said that you would take this action with the following hands: Q-Q, 10-10, Q-10, 5-5, 4-4, Ah-5h, 7-6, K-J, J-9, Ah-2h to Ah-Kh, Kh-9h, Jh-8h, 9h-8h, 8h-7h, 9h-7h.
Now, his turn call is very close, as it would be with weak one-pair hands, so your turn play doesn't need too much work, except for the fact that we need to tweak it a little to help your river range. So, let's try checking behind on the turn with your ace-high flush draws besides Ah-Kh and Ah-Jh. I like checking these a little bit better because we know where we stand when a heart hits, while we don't if we check behind 9h-8h and a flush comes, and also because our ace outs might be good, and I would hate to get check-raised off of a hand with that many outs on the turn.
We also need to add some more one-pair hands to our range. Remember our opponent is smart and optimistic, so on a board this draw heavy, he is going to call down quite often. In addition, it's unlikely that he would only call your flop bet with a hand like a set of fives or Q-10 when so many scary cards could come on the turn. There's no reason for you to think that a hand like A-Q is beat here. So, let's also pot the turn with K-Q, A-Q, J-Q, K-K, and A-A. We'll stop there and not bet Q-9 because he can have K-Q or Q-J often, and he could check-shove the turn with a draw and you might have to fold the best hand.
Let's look at your new range: Q-Q, 10-10, Q-10, 5-5, 4-4, A-A, K-K, A-Q, K-Q, Q-J, 7-6, K-J, J-9, Ah-Kh, Ah-Jh, Kh-9h, Jh-8h, 9h-8h, 8h-7h, 9h-7h.
I'm certain now that his turn call is bad against this range.
Let's see how good his Ac-Jc river call is against your new range, assuming you bet all of these on the river. Running the numbers
he still has the best hand 43.3 percent of the time, making his call correct and netting him some G-Bucks. Since he's kind of a calling station, we want to make his call incorrect. At the very least, we want to make it a tougher decision. So let's try to get his hand down to 30 percent against you. This means we have to check behind with some of our bluffs on this river. Well, for starters we should check behind A-K and A-J of hearts, even though they actually aren't bluffs versus his hand. We should check them behind if we're going to check any more hands behind because they can sometimes win the pot without betting. How about we check behind with open-ended straight draws? So K-J and J-9. We give up with those hands, so now we are betting: Q-Q, 10-10, Q-10, 5-5, 4-4, A-A, K-K, A-Q, K-Q, Q-J, 7-6, Kh-9h, Jh-8h, 9h-8h, 8h-7h, 9h-7h.
Running the numbers once again, it looks like his A-J is good only 26.6 percent of the time now, so we made his call a bad one. So, now, when we bet this river and he calls with A-J (or Q-J or A-10 or 8-8), whether we have a full house or nine-high, he loses G-Bucks and we make G-Bucks. That means that in the long run we will earn money if he keeps making that call. To be exact, that river call cost him $2,633 Galfond Dollars and earned you the same amount.
Why, you ask, should we just not bluff this river? Then he'd be extremely wrong to call, right? Definitely a good question. If you are up against a very loose, very unintelligent player, you should probably bluff this river close to zero percent of the time. The problem with doing it against good players, even if they are loose, is that they're smart enough to catch on. They'll notice that you aren't bluffing enough and they'll not give you any action. Remember the Ah-Kh example up top where we weren't bluffing enough? Your goal is to make the most money on average, not necessarily on the present hand. You have to bluff sometimes against smart players in order to get paid off other times when you have a big hand. So, if you are only going to play five minutes against this player, and you think he will almost always call the river -- sure, don't bluff. But if you play against him often, you have to occasionally bluff so that he doesn't figure you out and start to play correctly against you.
Against a loose player, you want to mix up your play in a way that makes his calls as incorrect as possible (from a G-Bucks perspective) without making him realize and start folding. Against a tight player, you want to mix up your play so that he's as incorrect as possible when he folds, without making him loosen up enough to play correctly. So against a loose but observant player, you might want to bluff this river around 18 percent of the time, and against a tight but observant player, maybe around 45 percent. Against a very loose but unobservant player or a very tight but unobservant player, you would want to almost never bluff or almost always bluff, respectively.
Another tip I have for you is to see how cards that fall affect ranges. You raise preflop and the big blind calls, and the flop is 7-8-2 with two spades. He checks, you bet, and he calls. You think he raises most draws so you don't put him on spades or a straight draw. The turn is 10s. You should usually bet almost everything when he checks because his range can't like that turn card and probably can't stand the heat.
Or let's say you raise preflop under the gun six-handed with Q-J off-suit and only the small blind calls. Flop is 10-8-4 rainbow. He check-calls a bet from you. Turn is a 2 off-suit. Check, check. River is an ace. He checks. This card, while not improving your actual hand, improved your range a lot. Plenty of the hands that you play like this contain an ace. That makes this card a good card to bluff on. You shouldn't necessarily always bluff in this spot, but you should more than if the river were a 10 or a five. Just like before, you should balance your play so that it's a tough decision for your opponent based on your range.
One trap I've seen people fall into is the following:
You raise with 6-5 off-suit on the button and the BB calls. The board comes out Qd-Jd-4h-4s-7d.
Your opponent check-calls a pot-sized bet on every street and wins with Qs-10s. You think to yourself, "What a terrible call. My range is like, flush draws, sets, two pair and straight draws, and then, every once in a long while, just pure air. Every draw hit except for the straight draws. He's way behind my range."
So you almost never have complete air, but you just decided to bluff this time with it? That could be true, but it's probably likely that you would've fired all three streets with a lot of air hands here. Be honest with yourself about your range.
Don't use G-Bucks analysis as an excuse to make bad plays. Don't use it to console yourself after losing a hand by proving how bad your opponent's play was. Use Galfond Dollars to balance your play, to exploit your opponents' weaknesses and to keep level-headed when you make a play that happened to be incorrect this time but is still the right play. The worst form of running bad is when you make good calls or bluffs or value bets that happen to be wrong over and over again. It makes you feel like you're playing badly when you aren't.
I hope you got something out of this. If you don't understand it yet, or don't know yet how to apply it, save this article and reread it when you are ready. Reread it if you find yourself making what appear to be bad call-downs or bad bluffs. Go over your ranges and your opponents' ranges and see if they really were bad plays. Reread this if you find yourself unsure how to play against an opponent who plays too aggressively or too loose or too anything. Think about what his ranges are on every street, and the best way to play against him.
Good luck at the tables. I hope you all become Galfond Billionaires.
Phil Galfond regularly plays online under the alias OMGClayAIken. If you enjoyed this article, subscribe to the No. 1 poker magazine in the country, Bluff magazine.