07-22-06 [poker] - 1

Consider a simple "pushbot" situation. I'll just assume a cash game or early in an MTT so payout structures don't matter, just chip EV.

Folded to you on the button. Only the blinds remain. You have C chips (in units of big blinds) and the blinds cover you (effective stacks all C). If you can only push or fold, should you push?

The correct way to do this is to guess some push range for you. Then assume the blinds know your push range and compute the correct call range for them. Then iteratively trying changing your push range, so that you find your push range that maximizes your EV over the whole range. (in each case assume the blinds know your range and make the perfect decision).

To see if you should push a hole H, you see if H is in the optimal range for you.

Okay, so as far as I know that's just review. In order to compute this you have to iterate many times over all holes for each person because you try a range for yourself, then try all calling ranges for both blinds to compute their optimal actions, then tweak your range & repeat until it stabilizes.

Here's an idea I had for a simple approximation :

It's folded to you on the button and you look down and see hand H. Assume your push range is all hands >= H. Now compute what the blinds should do given that range for you. Now consider pushing the actual hand H against the blinds' calling ranges. If it's +EV, then push, if not, don't.

Now, obviously this is an approximation and it doesn't get the actual EV right at all, but does it get the range right? I think it's very close and it's hard for me to see how/where it could fail.

Say for example in a given situation the correct push range is hands >= AQo. Folds to you and you have AQo, you'll make the right decision of course. What if you have AJo ? This should show up as -EV to push because if it was +EV it would've been added to the correct range, so we should get this right. What if you have QQ? We'll assume your range is only QQ+ so the blinds will fold much more, but it still should show up as a +EV push and tell you the right answer.

Anyone have an idea how this approximate procedure could yield a different result than the correct procedure?

BTW this is different than the Sklansky-Chubukov numbers because I'm assuming the blinds know your exact push range and make the perfect decision, but they don't know your actual hole cards.

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