Hi guys,
In a thread started by "Power" below, the correct play of KK on a A-rag-rag flop in a large multi-way pot was discussed.
The usual (and sound) advice of "fold unless the pot is very large" was given in many forms by various posters. I posted outlining some scenarios where it would be correct to play on, and things to consider if one chose to do so.
In part of my post, I gave an example of a situation where it might be even be correct to raise with a second best hand. This was a point heavily contested by other posters, and a long sub-thread has developed, complete with an attempt at an EV calculation by Mike Haven.
Since Mike's attempt was a valiant one, but not without flaws, I just finished a detailed calculation of my own, and came up with a few results which, frankly, surprised me.
I'll post the full calculation a little later, but I thought I'd post the scenario I considered, and see where you guys thought the dividing lines between call, raise, and fold might be. (Basically a test of your "playing instincts," as it were ... it turned out my instincts were a little off)
So here's the scenario. You have KK on the button. There are several limpers, and you raise. Everybody calls, and you see a flop of Ah 7d 5s. There is a bet and two callers to you.
We'll say that you are reasonably certain that two guys are on weak aces, and one guy is on the straight draw. Your opponents are all passive calling stations who will not be aggressive without a very good hand, but will chase and pay off on a regular basis.
How big would the pot have to be to make a call marginally profitable? (e.g., 15 small bets? 19 sb? 23 sb?)
Does raising ever become a better play than calling?
If yes to above, how big do you think the pot has to be to make a raise a better play if: a) you are guaranteed to get a free card on the turn by raising, b) you only get a free card if they don't hit their second pair or the straight on the turn? (Reality probably lies somewhere in between, of course.)
At any rate, please drop me a quick note indicating where you think the dividing lines are. I'll post the "answers" and my calculation either later this evening or early tomorrow morning.
Dave
"How big would the pot have to be to make a call marginally profitable? (e.g., 15 small bets? 19 sb? 23 sb?)"
If your opponents will not become more aggressive if the pot gets bigger, then you would only continue with large enough pot odds for your two outer. But if opponents can get tricky in big pots and will bet draws, and opponents are capable of raising with less then an ace (to chase out medium to big pocket pairs) then it surely WOULD become a function of potsize, related to the number of opponents (who percentagewise can hold an ace). Definite numbers on this very much dependent on player tendencies IMO.
"Does raising ever become a better play than calling"
If the pot odds are big enough to chase, and an ace is not likely to 3-bet, and you're quite certain it will get you a free turn card, then yes. But this are a lot "if"s.
"If yes to above, how big do you think the pot has to be to make a raise a better play if: a) you are guaranteed to get a free card on the turn by raising,"
I have calculated this myself against any number of opponents and any reasonable potsize, but I dont have the sheet here atm (not at my own compu right now), so I might get back to that later.
"b) you only get a free card if they don't hit their second pair or the straight on the turn?"
That would be a more difficult situation, since you have to know how much collective outs they have against you. So that's either only to be exactly calculated afterwards, or you have to base it on your handreading. The second option would be alot of work and not very accurate, since you have to assign percentages to your amount of certainty your read is accurate.
But summarizing (without my own calculated stats);
Against weak/passive opponents, who have the tendancy to check/call draws and hands weaker than a pair of aces, I would fold in pots without the drawing odds (with the implied odds calculated), and will raise if opponents are not 3-betting aces here, and opponents will most likely passively check the turn, and I have slightly above the drawing odds (to call).
So that would mean against these opponents, I will fold if the pot is less than about 20 SB, and raise if the pot is >25SB. Between 20-25 it doesnt matter much either way IMO. But I would guess if your opponents are somewhat observant, and can deduce your raise could very well mean a free card raise, then raising becomes very quickly more incorrect. I would guess (again without the numbers) that if they would bet the turn >10% of the times you raised the flop, it becomes incorrect for you to raise the flop. But also consider that it will become quickly more correct to get somehow a showdown if your opponents could have a worse hand than you (due to big pot size).
Regards
I feel, even though their is a mathematical approach to this, you are better off mucking those Kings, right then and there on the flop, you can't look at this situation mathematically, but rather psychologically..I was playing, in a fairly tight 5-10 game,where 2-3 people are seeing the flop yesterday at the Taj, not one single person, gave up on that Ace, later on during the night, not one single person would give up on their Ace, unless 3 flush cards appeared, so unless you have outs, such as a straight or a flush, just muck those kings, especially in small stakes, some people just don't know how bad their Ace is and will continue to play it.
Hi guys,
Sorry for the delay in responding, but I had to help some friends move, and got delayed in typing this up. (I also underestimated the amount of time it would take to convert my rough notes and calculation into a readable form for consumption on the forum.)
The short answers are as follows (to within 5% or so):
There needs to be around 18 small bets in the pot to make calling +EV. Raising does indeed become more profitable than calling as the pot becomes larger ... albeit strongly dependent on your ability to get a free card on the turn. If you always get a free card on the turn, the threshold is around 21 small bets, while if you only get a free card when they miss their hands, the pot needs to have 28 small bets.
Since the maximum pot size on the flop when we make our decision in the scenario outlined above is 23 small bets (everybody limps, you raise preflop, and all opponents call for a total of 20 sb preflop, then 3 sb on the flop to you), we can also express this condition as the absolute minimum percentage of the time that we have to get a free card on the turn when our opponents make their hands in order to consider this play. This minimum percentage works out to be approximately 67%.
I did the calculation as a "proof of concept" demonstration that a situation could exist with KK on a A-rag-rag flop in which it was both a) +EV to Raise and b) Raising was a greater EV play than just calling.
Some quick observations before I get to the nitty gritty of the calculation itself...
Note that the fact that there is a straight draw possible significantly impacts these thresholds. You lose a surprising amount of equity when you consider that even when you do spike, the straight draw has 8 redraw outs to beat you. On a rainbow flop with no straight draw possible, these thresholds will shift lower by 3 or 4 small bets. With a flush draw on board, and you holding the king of the appropriate suit, they shift downwards slightly ... roughly one small bet only. Your additional value from your runner runner draw is offset by the fact that you will likely be forced to pay a couple of bets on the turn while still drawing.
If more players were to have called (in the scenario described, I only have three players who call a bet on the flop), the thresholds also shift downwards, although not a dramatically as one might think. Each new player improves one's implied odds, but some players might also bring additional redraw outs against you.
THE CALCULATION
For those of you who want to scrutinize the calculation which led me to these numbers, and the assumptions I have made along the way, I give it below:
ASSUMPTIONS
The Scenario
1. You hold KK on the button.
2. The flop is A 7 5, of three different suits.
3. You face a bet and two calls on the flop, for a total of p small bets in the pot.
4. The game is a $5/$10 with a 5% rake to a max of $5, with an additional $1 drop for a jackpot taken at $100.
Your Playing Style
5. You will not pay off a bet on the turn or river without improving.
6. You will always bet the river if you improve to a set; likewise, you will raise if there is no straight potential and you are bet into, but just call if a straight is possible.
Your Opponents
7. Your opponents are loose passive calling stations.
8. Given the size of the field, and the fact that the pot has not been raised, it is extremely unlikely that anyone has anything stronger than one pair. (This assertion is sure to raise some objections; however, I'm aiming this analysis at a particular type of game which is not altogether uncommon in the local rooms where I play. I will comment at the end on the likely effects of relaxing this assumption somewhat.)
9. For the sake of the argument, we'll say that only the three opponents who have put money in on the flop will play further. (If there are more opponents, you do much better on the implied odds.)
10. We'll give two of your opponents weak aces without runner runner straight possibilities (for simplicity sake ... this has a net impact of overestimating the EV by around 1% to 2%), and the third an open-ended straight draw.
11. If you raise on the flop, the three opponents will just call and not three bet.
12. All opponents will call on the turn for one bet. If the turn goes to two bets, only the straight draw and one of the aces will call.
13. If one of your opponents should improve on the turn, you will only be able to get a free card q% of the time. (A free card could still happen if your opponent is very passive, or loves to check-raise big hands.)
14. When unimproved, at least one, and occasionally two, of the aces will pay off on the river, but the draw will certainly fold.
15. If any of your opponents has improved to two pair or a straight by the river, you will be bet into.
16. If you check behind on the turn, the aces will bet into you on the river around 50% of the time.
It seems like quite a lot, but all I'm doing is setting out a plausible course which the hand might take as various things happen. Once again, when I talk about making a particular play, I have a certain lineup in mind ... one based on the types of players and games that come up locally from time to time.
The only assumption which I feel is a little dodgy is #8. When I first started the calculation I had included a third free parameter for the % of the time I would get 3-bet on the flop, but the analysis started to get very complicated. So I put #8 in place to give me a more easily solvable problem. Therefore, the play recommendations and thresholds encountered here are for use in situations where the 3-bet is very unlikely (passive opponents, chronic slowplayers, tells from people that they are folding, etc.) Even so, the risk of a 3-bet will reduce the EV of raising by at least 5% in the long term.
Note that I have left two free parameters: p, the size of the pot when you make your flop decision, and q, the probability of getting a free card even when your opponents hit their hand.
With all that said and done, we are ready to undertake the calculation. You have three choices: raise, call, or fold. Let's look at each in turn:
FOLDING
If you fold here, you neither win any further bets, nor lose any further bets, so the long term EV of this choice is 0 BB. This is our baseline.
CALLING
Ok, here goes. Bear with me, things are a little complicated, but not all that bad.
The most likely outcome is that you miss on the turn. This will happen 45 of 47 times, or 95.745% of the time. In this case, you fold, and you only lose 0.5 big bets.
When you hit, since you didn't raise on the flop, it is likely that the turn action will again be a bet and two calls to you. You raise, and get on average two calls (the draw, and one of the aces). You have now put 2.5 big bets in total at stake, and stand to win 4.4 big bets (accounting for the rake) above and beyond the money in the pot prior to the your flop action.
However, things aren't that simple, since the straight can still outdraw you. We now have to consider the 4.255% of the time when you play the river
The one ace has 4 outs to trips and two pair (since the other ace has folded), which, although they won't give him a win, they will cause him to bet into you. The straight draw has 8 outs to beat you. Since we assumed that the aces did not have runner runner straight potential, the aces and the straight cannot both hit at the same time. (Which makes things somewhat simpler.)
4 times of 46 = 8.696% of the time, the ace bets his two pair, the straight draw folds, and you raise, and get paid off. You have a net win of whatever was in the pot on the flop plus an additional 6.4 big bets.
8 times of 46 = 17.391% of the time, the straight draw gets there, you get bet into, and since a straight is possible, you just call. You have a net loss in this case of 3.5 big bets.
The remainder of the time, you extract one bet from the lone remaining ace, giving a gain of 5.4 big bets since the flop.
Therefore, the net EV for calling in this case is:
EV(call) = 0.95745*(-0.5)+0.04255*(0.08696*(p/2+6.4)+0.17391*(-3.5)+0.73913*(p/2+5.4))
This can be rewritten as
EV(call) = -0.31111 + 0.01758 * p
which can be set equal to zero and solved for p. Accordingly, we find that if there were 17.7020 small bets in the pot when you made your flop decision, the EV for a call would be zero, by this calculation.
Therefore, if the pot contained 18 small bets or more, calling should be at least marginally profitable. Obviously, if the pot contains 17 or fewer bets, KK should be folded.
RAISING
Does raising ever become a better choice than both folding and calling? Let's find out ...
Say I raise the flop, and, as per my assumptions, all my opponents just call. I have put 1 big bet at stake, and I would net 1.5 big bets above and beyond what is already in the pot if there were no further action.
On the turn, there are 2 cards that make my hand, 7 that make two pair or trips for my opponents, 8 that make them a straight, and 30 blanks.
This is a little complicated, as there are 11 cases we need to consider. We'll start off simple, and work towards the more complicated ones.
Case 1: We All Miss on Turn, I Get Free Card, Miss the River
We all miss on the turn 30 times out of 47 = 63.83% of the time. In this case, they check to me, and I take a free card. Now on the river, 44 out of 46 = 95.65% of the time, I miss my hand, and either check it down, or fold if bet into. In this scenario, I still lose only 1 big bet.
This will contribute a term of the form
EV(1) = 30/47 * 44/46 * -1
which becomes
EV(1) = -0.61055
Case 2: We All Miss on Turn, I Get Free Card, Hit the River
The other 2 times out of 46 = 4.35% of the time on the river, I hit my set. By assumption 16, around 50% of the time, the aces will bet into you on the river, which gives me a chance to raise. In this scenario, I will collect somewhere between two and three big bets, depending on whether the second ace get trapped for a bet somewhere along the way. The other 50% of the time, I will get between one and two big bets, again, depending on whether the second pair of aces calls on the river.
Combining these factors, it seems to be a fair estimate that I will make on average 1.6 BB on the river. After rake, then, I make 2.5 big bets above and beyond what was in the pot on the flop.
The EV term for this case is therefore
EV(2) = 30/47 * 2/46 * (p/2 + 2.5)
which simplifies to
EV(2) = 0.06938 + 0.01388 * p
Case Three: I Hit Turn, Opponents Miss River
Let us now consider the case where I hit my hand on the turn, which we know from above will happen 2/47 = 4.26% of the time. Given my passive, calling station opponents, they will check to me, I will bet, and all 3 will call me. I now stand to lose 2 big bets, or to win 4.5 big bets beyond what was in the pot on the flop.
Now, there are 31 blanks that can fall on the river. In this case, I bet, and get called by one ace for sure, and also by the other ace at least 10% of the time. In the long run, then, I collect at least 1.1 big bets on the river. Therefore, after rake, I net 5 BB above and beyond what was in the pot on the flop.
Combining all this, we have a term of the form
EV(3) = 2/47 * 31/46 * (p/2 + 5)
which simplifies to
EV(3) = 0.14339 + 0.01434 * p
Case Four: I Hit Turn, Opponents Hit Two Pair/Trips on River
My opponents have 7 cards to hit for two pair or trips. They bet into me, I raise, and they call. I make a minimum of 2 big bets on the river, and could infrequently make 4 or more big bets if the case ace falls. (Both aces won't lay down ...) Something like 2.85 BB of action on the river seems a reasonable estimate. Therefore, I stand to win 6.75 BB above and beyond what was in the pot on the flop after rake is deducted.
EV(4) = 2/47 * 7/46 * (p/2 + 6.75)
which gives
EV(4) = 0.04371 + 0.00324 * p
Case Five: I Hit Turn, But Opponent Hit Straight on River
There are 8 cards which will complete the straight draw. In this scenario, you get bet into on the river, and, since a straight is possible, you just call. You lose 3 big bets in this scenario.
EV(5) = 2/47 * 8/46 * -3
which gives
EV(5) = -0.02220
Case Six: Opponent hits Two Pair/Trips on Turn, Bets, and I Fold
My opponents will hit two/pair or trips 7 times of 47. We assumed above that even if my opponents hit, I will still get a free card q% of the time. Therefore, they will bet (1-q)% of the time, I lose 1 big bet, and the EV term in this case is:
EV(6) = 7/47 * (1-q) * -1
which can be rewritten as
EV(6) = -0.14894 + 0.14894 * q
Case Seven: Opponent hits Pair/Trips on Turn, Checks, and I Miss River
I will miss the river 44 times of 46. In this case, I still lose only 1 big bet.
EV(7) = 7/47 * q * 44/46 * -1
or
EV(7) = -0.14246 * q
Case Eight: Opponent hits Pair/Trips on Turn, Checks, and I Hit River
Ah, the case where my opponent really regrets giving me a free card! I hit the river 2 times of 46. This scenario is similar to Case Four in terms of the expected river action, so we'll also use the estimate of 2.85 big bets. In this case, we have a net gain of 3.75 big bets since the flop after rake is deducted. Hence,
EV(8) = 7/47 * q * 2/46 * (p/2 + 3.75)
which becomes
EV(8) = 0.02428 * q + 0.00324 * p * q
Case Nine: Opponent hits Straight on Turn, Bets, and I Fold
By analogy with case six above, we have
EV(9) = 8/47 * (1-q) * -1
which reduces to
EV(9) = -0.17021 + 0.17021 * q
Case Ten: Opponent hits Straight on Turn, Checks, and I Miss River
Once again, by analogy with case seven, we have:
EV(10) = 8/47 * q * 44/46 * -1
or
EV(10) = -0.16281 * q
Case Eleven: Opponent hits Straight on Turn, Checks, and I Hit River
Ah, the case where I really regret getting a free card! I hit the river 2 times of 46, and will pay off a bet given my simplistic river strategy. Hence,
EV(11) = 8/47 * q * 2/46 * -2
which becomes
EV(11) = -0.01480 * q
The Final EV Expression, and Discussion
Now, of course, we also need to factor in the indicated wind speed in Bangkok ... just kidding.
We now combine all the individual case EV terms to get the overall EV expression for raising in this circumstance. When the dust settles, we have
EV(raise) = -0.69542 + 0.03145 * p + 0.02336 * q + 0.00324 * p * q
Investigation with this relation is a little more complicated, since we're dealing with an equation in two parameters. Since we want to compare with the EV of calling, we take the difference of this result with our result for EV(call).
EV_diff = EV(raise) - EV(call)
which gives us
EV_diff(p,q) = -0.38431 + 0.01388 * p + 0.02336 * q + 0.00324 * p * q
Recall that q denoted the percentage of the time that I will still receive a free card when my opponents actually hit their card on the turn. The exact percentage is very player dependent, so let's first get an idea of the range of possible values.
First off, let's say my opponents are the sort who will always bet into me on the turn when they improve, such that q = 0. Then we have
EV_diff(p,0) = -0.38431 + 0.01388 * p
which can be solved to yield a flop pot size p = 27.69 small bets for which the raise exceeds the call as a greater EV play. Therefore, for 28 small bets (or more) on the flop, a raise is even higher EV than calling. This lies outside the range of possible pot sizes in the scenario under consideration.
To get the other bound, we take the other extreme case, q = 1. In this case, we obtain
EV_diff(p,1) = -0.36095 + 0.01712 * p
which shows that a flop pot size of approximately 21.09 small bets is where the raise becomes a greater EV play than raising. So the threshold is right around 21 small bets. This is inside the set of possible flop pot sizes for the scenario given, albeit very near the top of the range.
It is also of interest to determine the minimum percentage which will allow a raise to be a greater EV play than calling within the allowed range of flop pot sizes. We accomplish this by solving the equation
EV_diff(23,q) = -0.06513 + 0.09783 * q
which yields q = 66.577% as the minimum percentage of the time I need to get a free card when my opponents make their hand in order to even consider making this non-standard raise play.
This percentage is not wholly unreasonable ... it could be attained with some combination of passive opponents or opponents who would attempt the checkraise on the turn. (After all, you have indicated some strength by raising both preflop and on the flop ... it would not be unreasonable to assume that you would bet again on the turn.)
If any of you are still reading at this point, I'm very impressed. This was one of the first EV calculations I've done in fully general form ... i.e., leaving free parameters to allow for investigation of where exactly the various actions became + EV, superior to one another, etc. I found the whole process rather informative and instructive ... while I knew that, in principle, at some point raising would become superior to calling, my instinct as to where the dividing line lay proved to be a little off. I sealed a tiny leak in my game as a result of this calculation.
I'll gladly answer any questions you might have regarding my calculations, assumptions, or conclusions, etc. I could probably go on for another couple of pages on the effects of relaxing various of my assumptions on the location of the thresholds, but in the interest of length, I'll quit here. (Besides, who's reading this anyways? The math nerds will be fussing over the fifth decimal places, and the "poker is a psychological game" wonks will have stopped reading anyways. But I'm not bitter ... /images/tongue.gif )
Any comments/feedback appreciated.
Dave Shaw
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