WA/WB OOP Theory: Doing the math

All the discussion of bet-fold vs. check-call the last few days prompted me to do the math. Sorry for starting a new thread, but I wanted to focus the discussion a little more on the actual probabilities involved.

I was thinking about whether the way ahead/way behind (WA/WB) OOP hands are best played on the river by

1. bet-fold
2. check-call
3. check-raise.

Here is my analysis.

Preliminaries

Basically, the only difference between WA/WB hands and any others is that whoever is behind has very few outs to improve. That means you usually stay behind through the hand or stay ahead throughout. So we can ignore the probability of drawing out, to a first approximation.

There are a bunch of probabilities to look at:

P(IB): If you check on the end, the probability you can induce a bluff in an opponent with the worst hand.
P(FW): If you bet on the end, the probability your opponent will fold a winner.
P(CL): If you bet on the end, the probability your opponent will call with a loser.
P(CCR): If you check-raise on the end, the probability your opponent will call the raise.
P(CW): If you check on the end, the probability your opponent will check through with a winner.

And, of course,

P(A): probability you are ahead,
P(B): probability you are behind.

EV Calculation

If you are way ahead, your EVs for the three situations are:

EV(bet-fold | ahead) = P(CL)
EV(check-call | ahead) = P(IB)
EV(check-raise | ahead) = P(IB) P(CCR) + P(IB)

If you are way behind, your EVs are

EV(bet-fold | behind) = P(FW)*POT_SIZE - 1*[1 - P(FW)]
EV(check-call | behind) = -1*[1 - P(CW)]
EV(check-raise | behind) = -2

Note that, in the check-raise case, I'm assuming that an opponent with a good to great hand isn't going to fold to your bluff raise on the end and that you'll fold to a 3-bet, which isn't a bad approximation I think. The sum of the EVs in each case above is your net EV:

1. EV(bet-fold) = P(A) P(CL) + P(B) P(FW)*(POT_SIZE + 1) - P(B)
2. EV(check-call) = P(A) P(IB) + P(B) P(CW) - P(B)
3. EV(check-raise) = P(A) P(IB)[1 + P(CCR)] - 2 P(B)

Examples

This seems to be very complicated to work out in general, so let's look at some extreme player types:

1. Calling station (loose-passive): Let say he'll always call a bet/raise with a loser and never fold a winner, never bluffs, may or may not check through with a good hand (probably depends on how much aggression you've shown):

EV(bet-fold | CS) ~ P(A) + 0 - P(B) = P(A) - P(B)
EV(check-call | CS) ~ P(B) P(CW) - P(B) = P(B) [P(CW) - 1]
EV(check-raise | CS) = - 2 P(B)

Since P(CW) is always less than 1, the play with probably the best EV is bet-fold against a calling station, especially since a bet from them almost always indicates a good hand. Makes sense to bet into a calling station, so the math works out so far.

2. Maniac: Always calls a bet/raise with a loser and never folds a winner, always bluffs, never checks through a decent hand:

EV(bet-fold | LA) ~ P(A) - P(B) + bluff EV
EV(check-call | LA) ~ P(A) - P(B) + induced bluff EV
EV(check-raise | LA) ~ P(A) * 2 - 2 P(B) = 2 [P(A) - P(B)]

So the best play against a maniac if you are ahead it to check-raise. If this maniac has a good chance of raising you if you bet into him, it might be better to bet-raise, but let's ignore that for now to keep it simple. When you are behind more than half the time, you should check-call or check-fold. I think check-call is best because a maniac is much more likely t bluff than to fold a winner. So then your play is to either check-raise or check-call, depending on whether you are ahead most of the time against his likely range of holdings.

3. Rock: Almost never calls with a loser and sometimes folds a winner, rarely bluffs, may check a decent hand:

EV(bet-fold) | TP) ~ P(B) P(FW) (POT_SIZE + 1) - P(B)
EV(check-call | TP) ~ P(B) P(CW) - P(B)
EV(check-raise | TP) ~ - 2 P(B)

Since P(CW) < 1, checking and calling is -EV, so the only play that might yield a profit is to bet-fold, assuming your opponent will fold enough to make it profitable. Check-raising in the worst and check-calling is likely always -EV. So the best way to minimize losses or make a profit is to bet-fold against a rock. Note that if the rock folds his winners no more than the pot odds he's getting, the bet-fold line can be bad, but in this case he wouldn't be very rocky.

4. TAG: Calls with a loser if has pot odds, bluffs appropriately to game theory (P(IB) = 1/POT_SIZE), never folds a winner, never checks through a winner, never calls a check-raise.

EV(bet-fold | TA) ~ P(A) / POT_SIZE - P(B)
EV(check-call | TA) ~ P(A) / POT_SIZE - P(B)
EV(check-raise | TA) ~ P(A) / POT_SIZE - 2 P(B)

Play against the TAG is tougher. The best plays are check-call and bet-fold. To choose between the two, this is the classic: will he bet with more hands if checked to than he will call with, or will he call with more hands if bet into than he will bet with himself? This, I leave to you and PT.


So in summary, I think the best plays are:

1. Against a calling station: bet-fold
2. Against a maniac: check-call if behind most of the time, check-raise if ahead most of the time
3. Against a rock: bet-fold
4. Against a TAG: check-call or bet-fold, depending on bluff tendencies


Whew! Thoughts?

Let me try to amend these extreme player types to give more general advice:

1. Loose-passive: bet-fold if he will call with a wider range of hands than he will bet with (likely).

2. Loose-aggressive: Check-call if behind against most of his hands. If ahead, check-raise if he will bluff and call a raise more than half as often as he will call a bet; value bet if he won't. This depends on how passively or aggressively you played the hand, of course.

3. Tight-passive: bet-fold if he will bluff raise less often than the pot odds you are getting. If he will bluff raise more often (unlikely), call the raise.

4. Tight-aggressive: check-call if he will bluff with a wider range of hands than he will call a bet with; bet-fold if he won't.

[Abbaddabba]

Two of your assumptions about a TAG are self defeating.

He can't never fold a winner if he never calls a check/raise, unless you never check/raise the river without a winner. If he thinks that you never check/raise the river without an almost certain winner, you should check/raise the river with virtually anything. You get value from bluffs that wouldn't call, and you usually have better hands folding.

Ahhh, nice. Intelligent commentary. That's the kind of response I was hoping for. So do you think we should add a small probability of him folding a winner, or of him calling a check-raise? Or both?

Edit: I think adding a probability of folding a winner is necessary, in which case bet-fold would be better. I think check-raising a TAG in this spot is dangerous, as you are much more likely to get reraised when behind than you are to get a call when you're ahead. But that's just pure speculation.

Edit #2: Let me also add that if a TAG is following the same line as us, he will never check a winner through, since he will also be on the bet-fold plan.

Edit #3: The statement about the TAGs bluffing tendencies still stands, except now you have to account for the probability he will fold a winner. If he will call with more hands than he will bet with, then the obvious answer is bet-fold, but if he is *much* more likely to bet with more hands than he'll call with, we should still check-call.

Wouldn't a typical TAG that has position on us check behind us if unsure but call a bet with any decent hand if the pot is large enough? Sounds like bet-fold is best.