Raising AK in big blind vs 3 limpers

[elindauer]

In this post, I'll be discussing, again, the old question of whether or not to raise AK from the blinds. Since this is a very complicated topic, I'm going to try to limit the discussion in the following ways:

1. You are the big blind facing 3 limpers. the small blind folds.

2. All limpers have exactly the following hand range:
any ace. any suited connector. any pair below TT. any two cards totalling 20 or more.

3. the framework for discussion: we will calculate 4 numbers
- EV of AK in unraised pot when it improves
- EV of AK in raised pot when it improves
- EV of AK in unraised pot when it misses
- EV of AK in raised pot when it misses

4. the EV of our hand when we hit will be determined by calculating the EV of the draws against us, subtracting this from the pot. See this post for an explanation of why I think this is best.


So, for the purposes of this thread, statements like "AK has an equity edge preflop" and "AK makes more money on every call" are simply inadmissable as they are in violation of rule 4. If this is the way you prefer to think about profit, I respectfully request that you choose not to post in this thread.



Now, I'll get us started with some rough calculations of the 4 key numbers. Please reply with suggested improvements for any of these calculations.

- EV of AK in unraised pot when it improves
It's not uncommon to find draws out against AK that will lose money in this situation. I'll say that players calling down with weaker aces have -EV draws and balance out the EV paid out to hands like flush draws. Factor in the occassional flopped set and maybe we have 90% equity in the pot.

EV = 4.95 SB

- EV of AK in raised pot when it improves

In this spot, the draws out against us have all gotten more valuable, so we can't possibly give the whole pot to AK like we did above. Implied odds have gone up some as there is no way we will almost never get away from this hand. More draws can call profitably. I'll take a stab and say the draws suck out 70% of the 8.5 SB pot on average.

EV: 5.95 SB

- EV of AK in unraised pot when it misses

tough one. AK often can't call a bet, but he sometimes finds ways to show his hand down unimproved to win. I'll call it 10% pot equity in this 5.5 SB pot for it's draw.

EV = .55 SB

- EV of AK in raised pot when it misses

The bigger pot makes the draw more valuable. He'll have profitable opportunities to show down unimproved more often, not to mention more opportunities to draw to the overcards. Maybe 15% equity for the draw.

EV: .825 SB



I have made a good faith effort to stab at these numbers. In fact, I don't even know, as I type this, whether they suggest a raise or a check is correct preflop. I expect all of these numbers to become more clearly defined during this thread, but for starters, here are the results:

EV (check) = 1/3 (.55) + 2/3 (4.95) - .5 ~= 3 SB
EV (raise) = 1/3 (.825) + 2/3 (5.95) - 1 ~= 3.25 SB

These numbers are pretty close, clearly the margin of error in my guesses is greater.

Thanks.
Eric


PS. If anyone can calculate the odds of AK improving given the hand ranges described, that would be helpful.