Ace in a Race

It is the World Series of Poker and you are in it! True, you won a satellite to get in, and you know you are not a superstar. But you're no slouch, either. You have survived the first six hours, built your $10,000 to $12,000, and are now playing with $200-$400 blinds.

Being a good, but not great, player, you know you have to gamble a bit. The very best players are trying not to commit lots of chips when they have only a small edge, but that is not your strategy. On the other hand, you have no desire to throw caution completely to the wind. You are willing to bet lots of chips with a small edge, but not otherwise. On the other side of the coin, part of your strategy is to avoid confrontations with the best players, if at all possible. There are plenty of players in the tournament who play worse than you. So, you know you will feel sick if you get played out of lots of chips by one of the champions, especially if you have a hand you didn't have to play.

As you are contemplating these thoughts, you look down to see the A 8 in the small $200 blind. This is not usually a playable no-limit hand, but when everyone folds to you, things change – because it is very likely you have a better starting hand than the player in the big blind. Thus, a raise, or at least a call, seems in order – except for one thing. It is Phil Ivey who's in the big blind. Now what do you do? If you were low on chips, it would obviously be right to move them in and take your chances. But you still have plenty of chips – $12,000. Against weak players, you might try a moderate raise, but you don't like that move against Phil. He will probably call. Then what do you do if an ace doesn't flop, or, for that matter, if it does?

Perhaps it would be better to just call the $200 – except that you know Phil will often raise. And even if he doesn't, you still wind up in a similar situation – out of position, with a tricky hand, against one of the best tournament players in the world. Thus, you remember your admonition to yourself about avoiding champions, and simply fold.

As smart as that seems, I can prove your play is wrong, because I can prove that moving in $12,000 is a better play than folding. Other plays may be better, but from a pure mathematical standpoint, folding can't be right because moving in has a higher expected value (EV).

At this point you may be thinking that it is impossible to calculate the mathematical expectation of moving in, because you don't know what hands Phil Ivey will call with. That is, of course, true – except that it turns out that it is right to move in even if Phil plays perfectly! In other words, he knows your hand and calls with all the hands he should and none that he shouldn't. Hopefully you see that it is correct to move in if Phil plays precisely A-8 to A-K, or a pair. It is even more correct if he folds some of those hands (or calls with others).

It is surprising, but true, that with an A-8, you would need more than 70 times the small blind in your stack for it to be right to fold your hand, even if the big blind sees your cards. (This statement does not take into account the "bunching factor," the idea that you can assume the big blind's cards are better than random when everyone else folds. That's true, but computers have shown that the effect is negligible in hold'em.)

Here's the math:

Assume $1-$2 blinds. Because the big blind is getting only very slightly above even money when you move in, it is correct for him to call only with pairs or a better or equal ace. Of the 1,225 two-card combinations he could have, only 141, or about one-ninth, will he play. So, there is about an eight-ninths chance you will steal the big blind by moving in.

Now let's see what happens when you are called. With 45 of the 141 combinations he could have (deuces through sevens, plus A-8), he is about even money. With eights through kings, he is an average of about 70 percent. That's 33 combinations. The three combinations of aces are more than 90 percent. And the 60 combinations of bigger aces are about 75 percent. If you average these out, you get the big blind winning about two out of three times he calls.

Suppose this situation came up 54 times. If so, you should steal the $2 blind eight-ninths of those situations, or 48 times. Of the six remaining times, you will win your all-in bet (plus $1) twice and lose that amount four times. So, with X dollars to move in, your net result, after 54 hands, would be (48)(2) + (2)(X + 1) – (4)(X+1)

That result is better than folding as long as it exceeds -54 (which would be your result if you folded every time). The break-even point occurs when (48)(2) + (2)(X+1) – (4)(X+1) = -54.

Simplifying: 2X = 148, or X = 74.

With 60 times the small blind and Phil Ivey almost certainly folding A-10 and A-9 when you move in, your decision should be easy.

Note: What you just read has some major implications, especially in a tournament. The fact is that any two cards can be "rated" according to their heads-up, faceup, all-in value. The rating for A-8 is about 70. A-2 is significantly less. Two kings are about 1,000. I expect to be writing a list of ratings for all two-card hands in the near future.