Poker is a game of mathematical ranges. Novice players will look at their exact holdings and compare their chances of winning to their opponent's exact holdings. With each step in a player's evolution ranges will take place of exact holdings, not only their opponent's but their own range as well. This is where David Sklansky's "Sklansky Dollars" ends and Phil Galfond's "G-Bucks" begins.
In his book The Theory of Poker, David Sklansky explains a concept called Sklansky Dollars.
This formula shows how much money you should be winning if you take all the luck out of an all-in situation. This will give you an idea as to how well you are playing and not just how you are running.
You are playing in a $1/$2 No-Limit Holdem cash game on PokerStars and you are dealt A♠ A♣ and get all the money in against K♠ K♥. The pot is $500. You are an 82% favorite to win the hand, meaning you are entitled to 82% of the pot. Multiplying $500 by 82% gives you $410 in equity
You invested $250 and are getting back $410, which means you earned 160 Sklansky Dollars. Your opponent has put in $250 but is only getting back $90 in equity; this means he lost 160 Sklansky Dollars. As long as you are making more Sklansky Dollars than what you are investing, you will be a winning player in the long run.
Phil Galfond wrote an expansion of Sklansky Dollars called G-Bucks.
G-Bucks take into account your range and your opponent's range in the spot where you put all the money in. Let's look at an example and set up some ranges for both ourselves and our opponent.
You are playing the same hand in the same situation as above holding the A♠ A♣. Your opponent in middle position raises to $8. You re-raise on the button to (3-bet) to $23. Your opponent has 4-bet to $86. You decide to 5-bet all-in to $250 and he calls and shows K♠ K♥. How did you do in G-Bucks? This is where your poker math skills will be put to the test.
Phil Galfond plays as OMGClayAiken on Full Tilt Poker.
You need to figure out what hands YOU would be willing to put 5-bets in in this spot. Since you know this player is slightly tight, you decide that you would only be shoving J♠ J♣ or better.
Next you need to see how many combinations of hands you have in your range:
Two Jacks: six combinations
Two Queens: six combinations
Two Kings: one combination
Two Aces: six combinations
Total Combinations: 19
Note that there is only one combination of a pair of Kings you can have because our opponent is holding the K♠ K♥.
You would then take each hand and multiply the combinations by the winning percentage:
Two Jacks (6 * .18), Two Queens (6 * .18), Two Kings (1 * .5) Six Aces (6 * .81)
Adding those winning percentages together you get:
1.08 + 1.08 + 0.5 + 4.86 = 7.6085
Diving our total winning percentage will show you your G-Bucks equity:
7.52 / 16 hands = 47% odds of winning
When you have figured out both Sklansky and Galfond Dollars you can see some interesting numbers. Your hand has held up but let's see how you really did:
Actual Money: + $250
Sklansky Dollars: + $159
G-Bucks: - $15
Wow, what happened? Your range against exactly K♠ K♥ is too wide to be profitable. If you were to fold all pocket Jacks, you would still be a small loser in G-Bucks at 49.5% equity.
More often than not you will not know your opponent's exact holdings. They will be playing several hands in a similar fashion. We will take a look at an actual hand played in an online cash game.
This is a $1/$2 Full Ring No-Limit cash game on America's Cardroom. You are a solid tight-aggressive player and have raised to $7.50 with Q♦Q♣ in middle position. What hands would you do this with?
The player on your left has 3-bet to $26, everyone folds to you. You have him covered and effective stacks are $128.30. The pot is currently $36.50.
He has a very wide 3-bet range of 12.1%. For simplicity sake we will assume no bluffs ever make up his range.
His range may look like:
7♦7♥+, A♦9♦+, K♦T♦+. Q♣T♦+. J♦T♦, A♦T♣+, K♦J♣+
Calculate the Combinations
Our next step would be to see how many combinations of hands he can have:
Two sevens: six combinations
Two eights: six combinations
Two nines: six combinations
Two tens: six combinations
Two jacks: six combinations
Two queens: one combination
Two kings: six combinations
Two aces: six combinations
Ace-Nine suited: four combinations
Ace-Ten suited: four combinations
Ace-Jack suited: four combinations
Ace-Queen suited: two combinations
Ace-King suited: four combinations
King-Ten suited: four combinations
King-Jack suited: four combinations
King-Queen suited: two combinations
Queen-Ten suited: two combinations
Queen-Jack suited: two combinations
Jack-Ten suited: four combinations
Ace-Ten off-suit: 12 combinations
Ace-Jack off-suit: 12 combinations
Ace-Queen off-suit: six combinations
Ace-King off-suit: 12 combinations
King-Jack off-suit: 12 combinations
King-Queen off-suit: six combinations
A total of 139 combinations.
Combinations x Winning % = Equity
You would now take the combinations and multiply them by the winning percentages:
Two sevens: 6 * .19 = 1.14
Two eights: 6 * .19 = 1.14
Two tens: 6 * .18 = 1.08
Two jacks: 6 * .18 = 1.08
Two queens: 1 * .50 = .50
Two kings: 6 * .82 = 4.92
Two aces: 6 * .82 = 4.92
Ace-Nine suited: 4 * .32 = 1.28
Ace-Ten suited: 4 * .32 = 1.28
Ace-Jack suited: 4* .32 = 1.28
Ace-Queen suited: 2* .34 = .68
Ace-King suited: 4 * .46 = 1.84
King-Ten suited: 4 * .32 = 1.28
King-Jack suited: 4 * .32 = 1.28
King-Queen suited: 2 * .35 = .7
Queen-Ten suited: 2 * .16 = .32
Queen-Jack suited: 2 * .16 = .32
Jack-Ten suited: 4 * .18 = .72
Ace-Ten off-suit: 12 * .29 = 3.48
Ace-Jack off-suit: 12 * .28 = 3.36
Ace-Queen off-suit: 6 * .3 = 1.8
Ace-King off-suit: 12 * .43 = 5.16
King-Jack off-suit: 12 * .28 = 3.36
King-Queen off-suit: 6 * .31 = 1.86
Adding up all of the results you should get 44.78. Dividing that by 139 combinations you get should get .322 or a winning percentage of 32.2%. Against this player's range we would have 67.8% equity
You see in this players data he never folds to a 4-bet, so the correct play would be to re-raise all-in.
As with many poker calculations and statistics there is a time and a place to use them and not use them. G-Bucks calculations can become misleading in drawing situations. If you have an inconspicuous draw it can be correct to go against your G-Bucks equity, more so when the stacks are deep and the implied odds are great.
On the other side, if you are not sure about which draws your opponent can have and are deep stacked, you'll have terrible reverse implied odds, and should play cautiously.
Constructing a range and optimal play is vital at Bovada Poker's anonymous tables.