the math of the small blind
The small blind is the serpent's apple of Texas Hold 'Em. It is almost deliberately designed to tempt the player to put money in the pot he shouldn't put in there. If there are no pre-flop raises, you get to play your hand for "half price". If there are raises, you still get a "discount". And if you fold, you are giving away money that you were forced to put into the pot and didn't want to give away.
But what are your odds when you complete a small blind with a marginal or bad hand? Nobody argues, of course, that you shouldn't play-- and raise-- from the small blind if you have a good hand such as a high pocket pair or Ace-King. I would also argue that the math is pretty clear that if you have enough limpers behind you to justify playing suited connectors, you should play them. What I am going to focus on is the hands that people play from the small blind that they wouldn't play from another position because they are getting a "discount"-- let's use a hand like King-4 offsuit.
So let's go to a table of $1/$2 limit and see how this plays out. Hero is in the small blind with K4 offsuit. There are 3 limpers and Hero now calls with his K4, and the big blind checks his option. The pot is $10 and $5 is raked, leaving $5 in the pot.
The flop comes and one player bets and there are 3 callers. The pot is now $13.
On the turn, the four players check around.
On the river, one player bets and there is one caller. The pot is $21.
In this typical hand, the small blind is getting 20-1 odds on his $1 pre-flop call. That sounds pretty good. But let's consider what you are getting for your 20-1.
A hand like K4 offsuit has a miniscule possibility of making a flush or straight, so those aren't the hands we are concerned with. Rather, you are likely to win the hand one of three ways:
1. Hitting trips (either Kings or 4's), about a 1.5 percent possibility.
2. Hitting 2 pair, Kings and 4's, a 2 percent possibility.
3. Pairing your King, a 17 percent possibility.
So, you have approximately 20.5 percent chance of flopping a potential winning hand. But while we can assume for this discussion that your two pair or trips are going to hold up, the problem is with point 3. First, you have to worry about an ace coming up and pairing another player's ace. There's actually a 25 percent chance of that happening by the river. Further, you also have to worry about another player flopping 2 pair or trips, which is about a 3.5 percent chance for each additional limper. In our example, with 4 other players, that's 14 percent. And you have to worry about another player having a king and outkicking you. The chances of each other player having a king is about 11 percent, so with 4 other players, that's 44 percent.
So your pair of kings will only hold up about 19 percent of the time. (That's what weak kickers do, folks.) And that's not even considering the possiblity of straights and flushes. But this means we can only count equity for 19 percent of the 17 percent chance of a pair of queens. That's about 4 percent.
Now, having done that, what are our real odds? About 7.5 percent. That does justify taking the 20-1 odds, but just barely. And this assumes perfect post-flop play, i.e., you are never bullied in to folding your pair of kings with a weak kicker by another player's raises. Good luck with that, especially since you are going to be out of position!
Bottom line: though one could justify making a call in limit in the small blind with a marginal hand, you would need to have pretty strong confidence in your abilities to play post-flop out of position with top pair and a weak kicker. Most players should fold it.
Now let's try no limit. The blinds are $1/$2. Again, Hero is dealt K4 offsuit in the small blind. There are 3 limpers, hero completes his small blind, and the big blind checks. The pot is $10 and $5 is raked, leaving $5.
Flop comes and there is one bet of $3 (half the pot) and 3 callers. The pot is now $17.
Check around on the turn.
On the river there is one bet of $9 (half the pot) and 1 caller. The pot is now $35.
So here's the good news. Now we are getting 34-1 on our money.
What's the bad news? Your post-flop play needs to be even better. Top-pair weak-kicker out of position is a tough enough hand to play in limit, but in no limit, it's horrible. You are basically going to have to fold it to any big enough bet from a tight player. So it is unrealistic to say that we are going to be around at the river at anything close to all of the 7.5 percent of the time that we have the best hand. (The alternative is worse-- if we call down aggressive plays with this hand, we will end up losing even more money because so often TPWK is beaten.) Let's say we are able to get to show down 50 percent of the time. That makes our real probability of winning the hand about 3.75 percent. That, again, is perilously close to the 34 to 1 odds that we are getting paid on our small blind.
Again, just like in limit, unless you really feel confident in your ability to play TPWK post-flop from early position, this is not something you want to be doing.
Finally, a word about blind structure. Not enough players pay attention to the differences in blind structure in different games. For instance, a common structure online is $0.10/$0.25. In a $0.10/$0.25 game, you are no longer getting 34-1. You are getting 22-1. And to make 22-1 a favorable play, you better be able to win 85 or 90 percent of your hands when you are ahead with TPWK in early position (while still folding every time you are behind). I doubt even a professional poker player can do that.
The blind structure $0.10 / $0.25 cuts your implied odds from the small blind by a third. A structure like $1 / $3 cuts them even more.
So the lesson is, if the blind structure requires MORE than half the big blind to complete the small blind, you need to tighten up the range you play in the small blind, A LOT.
Finally, I should note that I haven't concerned myself here with such topics as using the small blind to steal blinds or modifying your small blind play to project a table image. Those things can all be plausbile and profitable things to do in certain circumstances. All I am concerned about here are your implied odds when completing a small blind with a marginal hand.
So, to digest, here's what the implied odds tell us about playing the small blind:
1. Don't play marginal hands from the small blind in limit or no limit unless you are entirely confident in your ability to play TPWK in early position post-flop and to read opponents and determine whether you are ahead in the hand.
2. Whatever your range in the small blind is, tighten it up a lot if the blind structure requires a small blind player to pay more than 50 percent of the big blind to complete the bet.
[Addendum: One other piece of food for thought. When there's a raise on the board, yes you are still getting a "discount", but the discount is only about 1/2 the big blind. In a standard blind structure in limit, you are now putting in 3x the original small blind, i.e., you are cutting your implied odds by over 50 percent (if the pot size were the same it would be 66 percent, but since the pot size is likely to be bigger in a raised pot, I am shaving the figure a bit). In no limit, where the pre-flop raise is likely to be more than a min. raise, you are cutting your implied odds even more. That pretty much narrows you down to the range of "good hands" as well as hands that may play well post-flop. Further, of course, if the raiser is a tight player, you should narrow your range even further because of what he is representing his hand to be.]
