# cbloom rants

## 3/23/2006

### 03-24-06 [poker]

If you have a 3:1 chip lead in heads up, should you race with the worst of it? Assume the ICM model, so before the hand you are 75% to win. (the ICM model is the assumption that your chance of winning is proportional to your chip count). If you lose an all in, it will be 50/50, an even match. What odds do you need to get all in ?

0.75 = P * 1 + (1-P) * 0.5
0.75 = 0.5 * P + 0.5
0.25/0.5 = P
P = 0.5

So, you should take any race where you're >= 50% to win. If you take any worse race, you're giving up EV. Actually, this is not really interesting, it's built into the ICM model. The ICM model assumes that you're running your chips against each other at a 50% win rate. It's an open question how accurate the ICM model is; there are some good data mining projects which suggest ICM is not quite right.

Now, that only applies to calling an all in. If you're putting him all in you have the bonus of fold equity, and if you have to take a race as a dog it's balanced by all the times he folds. For example, say you push something like Q9 or K2. If called, it's probably a 60/40 race with him in the lead. If the blinds are tiny, pushing is a bad idea. What size do the blinds need to be for pushing to be right?

His stack is 1, yours is 3, and he's in the big blind of size B, you're in the small blind. You push. He calls with the top 20% of hands (roughly any ace, any pair, better kings and queens). 80% of the time you win +B and the stacks become {3+B,1-B}, so your chance of winning becomes (3+B)/4. 20% of the time he calls and you race as a 60/40 dog, in which case your chance of winning is 0.4 * 1 + 0.6 * 0.5 = 0.7

0.75 = 0.8 * (3+B)/4 + 0.2 * 0.7
0.75 = 0.6 + 0.2 * B + 0.14
0.01 = 0.2 * B
B = 0.01/0.2 = 0.05

So any time the big blind is >= 5% of his stack, this push is goot. (that's M = 13.33 for him). This is actually sooner than I thought from intiution. I don't usually start making this push until his stack is more like 8 big blinds or less, which means I'm actually waiting too long. Good Sit-N-Go pros know all this and are super-aggressive pushbots once they have a chip lead.