Playing Small Pocket Pairs in Limit Poker

Playing Small Pocket Pairs in Limit Poker

I am often asked whether you should play these hands and, if so, under what circumstances. To consider this question we first need to determine if small pocket pairs can be played profitably at all (show positive expected value (EV)).

I describe small pairs as anything from a pair of fives down to a pair of deuces.

Can small pockets show positive expected value?

According to the theory, small pocket pairs should be profitable, in the long-run, if they are played against five or more opponents and you fold if you don't hit a set. This might seem a bit counterintuitive to some. Surely you are a better chance of winning if you are up against fewer opponents? That philosophy assumes that you are willing to go to showdown every time, but with small pocket pairs if you don't flop trips you should be done with the hand. If you went to showdown every time you would loose a pile of money. This is a major mistake that many amateurs make. You are playing to flop trips and that is it. So the decision to play small pocket pairs is easy. You compare your likely payoff to the probability of flopping trips. If your likely payoff is high enough to justify the risk than you should play. This occurs if and only if there are five players or more. So it appears that small pockets can be played profitably.

Why do you need five players or more?

The chance of flopping a set, or better, starting with a small pocket pair is 7.5/1 or (11.8 %). Let's assume we are playing a $5/$10 game and it is $5 to see the flop. You will make your hand 11.8 % of the time and miss your hand 88.2 % of the time. So to make your initial investment worth it you need to recoup $37.5 when you make your hand (5 x 7.5/1). To show this numerically, assume you play 100 times with small pocket pairs, each time paying $5 for a total outlay of $500. If you hit your hand your return is $37.5 and if you miss your hand you lose $5.

88.2 % of the time you will lose $5 (100* 0.882*5 = - 441)

11.8 % of the time you will get $37.5*.118*100 = 442)

After allowing for rounding, you can see that this is zero expectancy fair game. So if your payoff is going to be higher than 7.5 times your initial outlay (for a $5/10 game $37.5) then it is a positive expectancy bet. Using this simple analysis you would need 8 callers, including yourself, to justify making the call.

Playing Small Pocket Pairs in Limit Poker

Our analysis is not yet done. The astute reader will be asking, what about the next three betting round including the turn and river? This is where the analysis becomes a bit tricky. After you have made your hand you are a huge favourite to win. You will occasionally lose but any bet you make on the next three betting rounds will return massive positive expectancy in the long run. This reduces the overall initial return you need. Calculating the exact reduction is not straightforward, because there are countless combinations of possible hands. However, it is generally accepted that you need about 5/1 pre-flop (5 callers including yourself) to make the overall bet positive expectancy. Indeed this is the figure that David Sklansky and Mason Mulmuth arrive at in their book "Hold'em Poker for Advanced Players".

Pre-flop raise

Lets consider the mathematics involved if there is a pre flop raise before you act. Now instead of calling $5 you need to call $10 pre flop. The total return you need has increased to $75 ($10*7.5). However, the analysis has changed a bit since the next betting round, assuming no raises, is half what was put into the first round ($5) and the next two rounds are the same as your initial outlay and not the usual double. So you lose some expectancy on future rounds compared to your initial outlay when the pot has been raised. If you were to consider this a bit more deeply you might gain some of this EV back when you take into account that, in the circumstance where there has been a raise, your opponents are likely to be more pot committed, on average, given the increased size of the pot.

To summarise, you lose some EV because subsequent betting rounds are not as large relative to your initial outlay, but gain some back because players are more likely to be pot committed in subsequent rounds. The net effect of all of this is ambiguous. However, I still think that 5 players is a good rule of thumb to cold call a raise.

cop a raise behind

Before moving on, let's consider the situation where you call and someone behind you raises. It is almost always worth calling this raise. Lets consider the worst possible situation where everyone else folds and you are heads up. If you invest $5 more then you will win your $10 + your opponents $10 + the $7 blinds at a minium if you hit your hand. You are getting 54/10 which is better than 5/1, even if you are heads up. Sure your expected value after the flop decreases, but you are extremely unlikely to end up heads up so your overall payoff on most occasions will be better than this for the reasons I describe in the next section.

How can you be certain five or more players are going to be in?

The answer to this question depends on position, position, position. If you are one of the last to act you have the advantage of seeing how many callers are likely to eventuate. let'ss say you have pocket deuces and you are one position to the right of the button. Someone from an early position calls and two people from middle position also call for a total of three callers with you next to act. If you make the call, that makes 4 and, assuming no raises, the big blind makes 5. So a good rule of thumb is, if there are 3 callers before you then you can almost always call with small pocket pairs and be guaranteed of having at least 5 callers pre-flop. What if you call and someone else raises behind you? This is fine, because the raiser makes a 5th player and it is unlikely that any of the original callers will fold without seeing the flop. You might even pickup the big blind which will make six. So the golden rule is - if there are 3 or more callers before you act, then you can call with any pocket pair but fold if you don't hit your set.

Have small pocket pairs historically shown a profit?

Playing Small Pocket Pairs in Limit Poker

A well known poker site, PokerRoom.com, has expected value stats for each hand from each position taken from actual playing hands. Now we know from theory that small pocket pairs should show positive expectancy, from a late position, if every player is following the golden rule that I have laid out above. The table below shows the expected value for each table limit from playing pocket deuces from the button where 10 players have been dealt-in. The expected value is measured as average profit in big bets. You can see that pocket twos show negative expectancy for limits from $1/$2 to $5/$10 and positive expectancy from $10/$20 and above. Obviously, low limit players at PokerRoom.com have not read this article. They are either seeing too many flops where there are less than five players and or not folding if they don't flop a set.

Playing 2,2 in a 10 handed game from the button.

Table Limit

EV

$1/2

-0.13

$2/4

-0.12

$3/6

-0.25

$5/10

-0.06

$10/20

0.13

$25/50

1.15

It is impossible to deduce how a player plays after the flop based on these statistics, but we can test how the expected value changes if you play the same hand from an early position, where you have no idea how many callers you might get. We can see that playing the hand from second position produces negative expectancy for all limits (which is what we expect). Interestingly the expected value does not change that much for the lower limit games. This indicates to me that low limit players play 2,2 extremely badly from all positions or PokerRoom.com is an extremely loose site. I suspected the former.

Playing 2,2 in a 10 handed game from second position.

Table Limit

EV

$1/2

-0.08

$2/4

-0.14

$3/6

-0.16

$5/10

-0.1

$10/20

-0.08

$25/50

-1.06

Are there ever circumstances where you can break the Golden Rule?

Predicting the number of players

From time to time you might have to make some judgement calls when there are only two callers before you. Once again position is important here. If there are still several people to act, who have proven to be loose in the past, you might make the call by assuming that on average there will be 5 or more callers. This is particularly true if the table has proven to be very loose in the past. It is harder to call a raise with only two people acting before you because you can't rely on one of the blinds calling and never call with only one caller before you.

Playing on after the flop

Lets say you hold pocket fives and the flop comes 2,2,2. In rare circumstances like this it might pay to play on after the flop sticking in a post-flop bet. If a blank hits on 4th street try your luck again with another bet. However, it is likely that over cards will fall and it is very hard to withstand any heat. If in doubt fold, or check and call if the over card comes on the river.

Conclusion

Theory tells us that small pocket pairs will produce positive expectancy if you follow the golden rule - if there are 3 or more callers before you act then you can call with any pocket pair but fold if you don't hit your set. However, empirics have shown that pocket deuces do not show positive expectancy from a late position in low limit poker. We can deduce from this that many low limit poker players are not following the golden rule

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