Pair under pair
Fuel55 recently pointed out an article by Phil Gordon, which gives a formula for the chance of a pocket pair being dominated by a higher pair in a hand still to act. Specifically:
Please read the original article for examples, explanation and motivation.
I'm about to criticise this principle, so let me say at the outset that Phil makes it clear that this is only an approximation, a useful rule of thumb. It is an excellent rule of thumb for short handed situations acting UTG. However, as we depart from these situations, it can become a rather poor approximation. The good news is, that it consistently overestimates the chances of being dominated by a higher pair, and if you keep this in mind, it may still be a useful tool in decision making.
Let's begin with an understanding of how we might arrive at Phil's rule. Consider the next player following you. In each of the R ranks dominating your pair, there are 6 pairs he can hold of that rank (the six different combinations of two suits). So there are 6×R dominating pairs altogether. Since you already have two cards, there are only (50)×(49)/2 hands he might hold, and so the percentage chance that he has a higher pair than you is
(100×6×R×2)/(50 × 49) = 1200 × R/ 2450 ≈ R/2.
Phil's formula is a straightforward extrapolation from this -- if this is the chance that a single player might hold a pocket pair higher than yours, then multiply it by N to get an approximation that at least one of N players will. What's wrong with that?
Quite a lot actually. The main problem is that when the player next to you does not hold a higher pocket pair, his cards will generally interfere with the pairs that any further players might hold. This is most marked when your pair is low, say for example 22. If the next player does not have a higher pocket pair, then he will usually have two overcards and the following player usually has only 66 possible dominating pairs instead of 72. Obviously this effect is less marked when your pair is higher, for example 99, since then quite often the next player will have only one, or no higher cards.
You want the data? You can't handle the data!
Sorry, seemed to be channeling Jack Nicholson there. Ok, I'll give you data. Suppose that you are UTG (I'll explain this restriction later), with a pocket pair, and N players remaining to act. Then the chance that your pair is dominated is shown below:
This table is a result of simulation (ten million independent trials for each possible starting pair) as exact computations of these numbers past the first column or two would be truly messy indeed. Note that in the lower left half of the table (i.e. below the diagonal from upper left to lower right), it agrees quite closely with Phil's numbers. But six handed with 44, Phil's approximation would give you a 30% chance of being beaten, whereas you're actually beaten only 26% of the time, and at a full table with 22 Phil's formula makes you a favourite to be dominated by a higher pair, while the actual chance is only 42%.
Why did I specify that these numbers assume that you are UTG? What's the difference between being UTG at a six handed table, and fourth to act at a nine handed table after three folds? Well, the latter situation (particularly if blinds are high relative to stack sizes) indicates that the three hands which folded were unlikely to contain pairs. Again, if you have a lowish pair, this reduces considerably the number of higher pairs that are probably available to the following players.
Just one example here, you have 33 in the small blind at a four handed table. All stacks are relatively short compared to the blinds, and there are two folds to you. The table above would assume that the next player can hold 66 pairs dominating yours. But, up to 12 of these 66 may have been eliminated as we may assume that neither player who folded held a pair.
Edit Preliminary simulations suggest that the latter effect is not significant. That is, the table above can be used without assuming that you are UTG, only that there is no action in front of you and there are N players still remaining to act.
The Gordon Pair Principle
Let C = percent chance someone left to act has a bigger pocket pair, N = number of players left to act, R = number of higher ranks than your pocket pair. Then, C = (N × R) / 2.
Please read the original article for examples, explanation and motivation.
I'm about to criticise this principle, so let me say at the outset that Phil makes it clear that this is only an approximation, a useful rule of thumb. It is an excellent rule of thumb for short handed situations acting UTG. However, as we depart from these situations, it can become a rather poor approximation. The good news is, that it consistently overestimates the chances of being dominated by a higher pair, and if you keep this in mind, it may still be a useful tool in decision making.
Let's begin with an understanding of how we might arrive at Phil's rule. Consider the next player following you. In each of the R ranks dominating your pair, there are 6 pairs he can hold of that rank (the six different combinations of two suits). So there are 6×R dominating pairs altogether. Since you already have two cards, there are only (50)×(49)/2 hands he might hold, and so the percentage chance that he has a higher pair than you is
(100×6×R×2)/(50 × 49) = 1200 × R/ 2450 ≈ R/2.
Phil's formula is a straightforward extrapolation from this -- if this is the chance that a single player might hold a pocket pair higher than yours, then multiply it by N to get an approximation that at least one of N players will. What's wrong with that?
Quite a lot actually. The main problem is that when the player next to you does not hold a higher pocket pair, his cards will generally interfere with the pairs that any further players might hold. This is most marked when your pair is low, say for example 22. If the next player does not have a higher pocket pair, then he will usually have two overcards and the following player usually has only 66 possible dominating pairs instead of 72. Obviously this effect is less marked when your pair is higher, for example 99, since then quite often the next player will have only one, or no higher cards.
You want the data? You can't handle the data!
Sorry, seemed to be channeling Jack Nicholson there. Ok, I'll give you data. Suppose that you are UTG (I'll explain this restriction later), with a pocket pair, and N players remaining to act. Then the chance that your pair is dominated is shown below:
Players Remaining
Pair(R) 1 2 3 4 5 6 7 8 9
22 (12) 5.9 11.4 16.6 21.5 26.1 30.4 34.5 38.3 41.9
33 (11) 5.4 10.5 15.3 19.9 24.2 28.3 32.1 35.8 39.2
44 (10) 4.9 9.6 14.0 18.2 22.2 26.0 29.7 33.1 36.4
55 (9) 4.4 8.6 12.6 16.5 20.2 23.7 27.1 30.3 33.4
66 (8) 3.9 7.7 11.3 14.8 18.1 21.4 24.5 27.4 30.3
77 (7) 3.4 6.7 9.9 13.0 16.0 18.9 21.7 24.5 27.1
88 (6) 2.9 5.8 8.6 11.3 13.9 16.5 18.9 21.4 23.7
99 (5) 2.4 4.8 7.2 9.5 11.7 13.9 16.0 18.1 20.2
TT (4) 2.0 3.9 5.8 7.6 9.5 11.3 13.0 14.8 16.5
JJ (3) 1.5 2.9 4.4 5.8 7.2 8.6 9.9 11.3 12.6
QQ (2) 1.0 2.0 2.9 3.9 4.8 5.8 6.7 7.7 8.6
KK (1) 0.5 1.0 1.5 2.0 2.4 2.9 3.4 3.9 4.4
This table is a result of simulation (ten million independent trials for each possible starting pair) as exact computations of these numbers past the first column or two would be truly messy indeed. Note that in the lower left half of the table (i.e. below the diagonal from upper left to lower right), it agrees quite closely with Phil's numbers. But six handed with 44, Phil's approximation would give you a 30% chance of being beaten, whereas you're actually beaten only 26% of the time, and at a full table with 22 Phil's formula makes you a favourite to be dominated by a higher pair, while the actual chance is only 42%.
Why did I specify that these numbers assume that you are UTG? What's the difference between being UTG at a six handed table, and fourth to act at a nine handed table after three folds? Well, the latter situation (particularly if blinds are high relative to stack sizes) indicates that the three hands which folded were unlikely to contain pairs. Again, if you have a lowish pair, this reduces considerably the number of higher pairs that are probably available to the following players.
Just one example here, you have 33 in the small blind at a four handed table. All stacks are relatively short compared to the blinds, and there are two folds to you. The table above would assume that the next player can hold 66 pairs dominating yours. But, up to 12 of these 66 may have been eliminated as we may assume that neither player who folded held a pair.
Edit Preliminary simulations suggest that the latter effect is not significant. That is, the table above can be used without assuming that you are UTG, only that there is no action in front of you and there are N players still remaining to act.
Labels: math
posted by Michael Albert at 8:50 am
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