Analysis of when to call down to a turn raise

[jtr]

Hi, all.

I've done some basic maths on when you have the odds to call down to a turn raise in limit hold'em. Would appreciate comments / criticisms.

(This was originally posted over in small stakes, as a response to an old thread, but hasn't got a lot of attention so far. I'm hoping that's just because the forum is busy, but if you think what I've written is crap, please feel free to say so.)

First, a few caveats:

1) By default I'm talking about Party Poker 2/4 games. The assumptions I'm making about my opponents' postflop passivity will need to be altered for bigger games, I'm sure.

2) I do realize that these decisions are contextual / hand-specific. I intend the following as a rough guideline to get started.

So, here are the two cases I've thought about. Case one: you've got a big overpair, let's say AA. You raise, you bet the flop, you bet the turn, and get raised (possibly check-raised) by a single opponent. Case two is the same except you've got AK and have hit top pair, top kicker.

Hopefully everyone will agree that a decision on whether to fold or continue must be dependent on the size of the pot. In a truly huge pot, you have to continue; in a small one, you'd lean towards folding.

To simplify things, let's look at the EV for folding now (zero, obviously) versus calling the turn raise and calling the assumed-to-be-inevitable river bet. I know real life is more complicated than this, but given that in these situations you won't really know which cards are your outs (especially if the river card pairs the board) I think this is a decent starting point.

Again, keeping it simple: let's say there are three categories of hand that the opponent may have.

1) A set or monster made hand better than a set, against which we have zero outs, or 2 outs at best in the case of AA versus a set. Call the probability of this case M (for monster hand).

2) A typical raggedy two-pair situation; the opponent having hit on the turn or hit on the flop and slowplayed a little. Against this we have 8 outs in the AA case and 9 outs in the AK case, although we won't know which cards are our outs. Call the probability of this holding T (for two pair).

3) The hand we really hope he has: a complete bluff, top pair with a worse kicker, some crappy semi-bluff draw. Call the probability of this holding B (for BS hand). M+T+B=1, obviously.

Now, after we bet the turn and get raised we assume we're going to pay 2 big bets to see a showdown. Call the number of big bets in the pot at the start of the turn N. So your turn bet makes N+1, the raise makes N+3, and you stand to win N+4 bets for your 2 bets to call down.

Here are two formulas suggesting a minimum size for N to make calling down the right policy.

In the AA case, N(crit) = ( 92 / ( 2M + 8T + 41B )) - 6. In the AK case, there are no outs against the monster, so it's simpler: N(crit) = ( 92 / ( 9T + 43B)) - 6.

I know there are some simplifications in there, but I think those numbers are pretty close, so bear with me.

Now to the real point of my post. At Party 2/4 (or for whatever game you play) what do you think are reasonable values of M, T and B for the typical opponent? What about for a known passive fish? What about for an aggressive, thinking opponent?

I've made a pessimistic start at describing the average passive 2/4 player with M = 0.6, T = 0.35, and B = 0.05. I didn't just pull these out of the air. The two pair number is low-ish because of the number of times the really passive guys just call down with their raggedy two pair. The B number is very low because they just don't raise on the turn much without decent hands. These assumptions lead to cutoff points for N of 9.2 BB in the AA case and 11.3 BB in the AK case. Note that N is the number of big bets in the pot at the beginning of the turn (I did it that way so I can look at the pot size before I even bet the turn and think about whether it's big enough for me to call down if raised).

9.2BB or 11.3BB represent pretty big pots when heads-up on the turn, and imply that most times on 2/4 you'll be folding to a turn raise from a typical opponent. However, it's all very sensitive to the values you choose for M, T, and B -- especially B. If we imagine an aggressive opponent who is equally likely to have a set, two pair, or a semi-bluff draw in this case, we would conclude that N is actually a negative number, meaning that you absolutely call the guy down every time.

Thoughts on the logic very welcome, but also anyone's estimates for M, T, and B at different games. (Speaking selfishly, especially interested in hearing from 2/4 and 3/6 players.)

Cheers,
--JTR.