Warning: This post is my first in a series of multi-street, game theoretic analyses of common post-flop Omaha situations. All such posts will have three things in common: they will be very mathematical, very revealing and very long.

PRE-FLOP

$400 PLO, 100BB Stacks
We hold K♥Q♣J♦9♠ UTG
We raise to pot and are called by the BB.

FLOP A♠K♣T♦ ($30)

We shall model our UTG pre-flop range as the top 20% of all starting hands. Let’s assume we Cbet half-pot* ($15) with 100% of our range. We might be concerned that this is exploitable by an observant opponent who adjusts by check-raise bluffing the flop frequently. If he check-raises to $45 he needs us to fold the flop 50%+ of the time. Since we will flop {AK or better} 50% of the time we have an easy, unexploitable, flop defend. However he bet-sizes the turn and river we are going to be able to just call the nuts 3 streets if we so choose since 28% of our flop calling range contains QJ. We don’t care about most board changes since the rest of our range is sets/top two. Incidentally, if our opponent is savvy enough to use blockers (QQ) then our flopped-nuts drops to 8% total or 16% of our flop calling range. If he is willing to fire three this will be profitable for him. However:

1) He won’t have QQ or JJ with sufficient frequency compared to QJ for us to care. So if he check-raises {QJ,QQ,JJ} he is just playing better than simply check-raising QJ.

2) If he chooses to check-raise many combinations of QJ he is going to severely weaken his check-calling range as we shall see below

Now we are happy to C-bet 100% on this flop (I haven’t shown it’s optimal, just that it’s not exploitable), let us look at our opponent’s possible continuing hands. Against our entire range (convenient with 100%, huh?) his flop equity runs as follows:

QJxx: 73%
AAxx: 77%
KKxx: 65%
TTxx: 56%
AKxx: 56%
ATxx: 45%
AJ98ddss: 44%
AJ98: 40%
AQ98: 40%
A987: 29%

I have been more specific with the side cards with the weaker hands simply to illustrate the difference (or lack of) between AQ/AJ based hands and their counterparts with multiple backdoor flush draws. A useful question here is, how much equity does our opponent need to call this flop profitably?

*I have to make an aside now on flop structure. It is obvious that usually with position we have more opportunity to outplay our opponent. This amounts to being able to call with less equity in position and make up for it by bluffing later streets. More dynamic boards lend themselves to this, and we can outplay weaker/unbalanced opponents even OOP. However the AKTr flop OOP against a tight range is not one that lends itself to light flop calls. There are only 7 board changes on the turn and the nuts on the flop will often be the nuts on the river. The reason I have focused on this board is that it is similar in structure to a number of high lockdown boards, consider {AKQ, AKJ, AKT, AQJ, AQT, AJT, KQT, KJT, KJ9, QJT, QT9}r where other than board-pairs or running flushcards, the flopped str8 is always the nuts on the river. KT9r is not the same since a Q or J can make the A high straight beat QJ.*

Our opponent would need 25% to break-even on his call if the action ended here. However, since the weakest hands in the range cannot continue to a Pot-sized turn bet, he will need much more equity than this. As a crude assumption, let’s say he expects us to only pot a brick turn with 50% of our range {AK or better}. Let’s say he also expects us to never take bet/check/bet as a line. Now he needs 40%** equity to call the flop bet since when he is risking $15 to win 0.5*45=22.5, 15/37.5=0.4.

This fits in-line with our expectations, A loose bad player would probably call all his Axxx here, along with any gutshot to the nuts. Since these have around 29% equity he is clearly making a losing call. A LAG regular would probably call {AJ+}, especially with a backdoor nut flush draw or some non-nut straight options. A tight player would probably call {AK+}.

TURN Q♦ $60

On this turn 40% of our range can stack-off even against the pure nuts if we make a PSB and get shoved on. 36% is the nut straight and another 4% is set+FD (They all have the 37% equity needed to get it in once we PSB).

The first concern with making a PSB is being exploited by our opponent check-raising us off of our hand. Let us assume he check min-raises (giving him the best possible price on his bluff/semi-bluff. With pure air he needs us to fold 50% of the time (he risks $120 to win $120). This allows us to C-bet as much as 80% of our entire range on the turn! If he instead plans to check-raise-call by semi-bluffing with say a set of Kings(!) he needs us to fold 60% of the time (Standard FE calc w 24% equity, maths not included here).

Now that we know our opponent can only check-raise the nuts (or set+FD) profitably, let’s look at his possible lines with his range:

  1. CR turn with nuts, CF everything else
  2. CR turn with nuts, cc sets and 2p+FD, CF else
  3. CC turn with nuts, CF everything else
  4. CC turn with nuts, sets and 2p+FD, CF else

I have disregarded mixed strategies in an attempt for (some) brevity. Against the first and third strategies, you want to make the smallest turn bet you can without starting to price in sets etc.

Against the fourth strategy our opponent will arrive on the river with a QJ hand around 65% of the time if he uses the tighter flop range. However, 44% of the time the river will pair or flush out. When this happens and we bet 2/3 pot on the river our opponent is going to need to call 60% of his range to stop us bluffing ATC. If he doesn’t start calling lots of his QJ combos we can print money with our bluffs by triple-barreling river changes. In this instance we want to maximize value by potting the turn in order to bloat the pot which our opponent surrenders on 44% of rivers Remember, because our opponent isn’t check-raising the turn with the nuts, we can bet the turn with our sets and flush-draws. 56% of the time we will lose the $60 but 44% of the time we will win the $60+the other $60 he puts in on the turn and folds out on the river=$120. If we choose only to bet stuff with equity on the turn, then {FD,sets,nuts} comprises 74% of our total range.

Against the second strategy our line is different. Since our opponent will check-raise us off our hand when we have a set or FD, we want to check these back. We now want to polarize our range so that we bet {nuts, air} on the turn. This time we are barreling board non-changes. Again the best sizing on the turn is pot. 65% of the time we get check-raised off of our bluff, losing $60. A prosaic analysis would assume since our PSB gets wasted 65% of the time we are burning money. 0.35*0.56=20% of the time the board does not change and we can bet the river to get folds: 0.65*-60+0.2*+120= -$15. However if our opponent ever check-raises the flop with the nuts our bluff will quickly become +EV (it is hard for him to do this <1/3 of the time if he ever does it since most human beings find it tough to randomize; if he does it 1/3 of the time the 0.65 changes to close to 0.4 and we start breaking even, any more and we profit.) Furthermore, if his strategy remains static we simply bet for value and reduce our turn bluffing frequency. Again the best turn sizing is POT.

In conclusion, the most interesting thing about this turn is that our turn/river play should be reversed depending on our opponents’ tendencies. Note that a nit will probably take strategy 1 or 3 (and you should bet smaller), but a decent reg will take 2 or 4 and you should PSB the turn then bluff board changes or non-changes depending on whether or not he check-raises the nuts on the turn.***

I appreciate this was REAAAAAALLY long, so thanks to (anyone) who read to the end. A quick reminder that this will be useful on {AKQ, AKJ, AKT, AQJ, AQT, AJT, KQT, KJT, KJ9, QJT, QT9}r boards also and in other spots where your range is super-nutted on the turn.

I’m going to go have a glass of water.

Quad

*We can use a different sizing, but flop sizing is something most good regulars get right pretty easily. Needless to say, if he is a massive nit on the flop bet small; if he is a clown who calls any Axxx just pot it.
**Since the hands we double-barrel will be the stronger parts of our range he will actually have more equity against our turn check back range so might be able to call slightly lighter. Since the next class of hands drops to 29% equity I didn’t bother pursuing this.
***You may think, but what if he donks the river with the nuts when he binks/the board doesn’t change and he still has the straight? This only increases the EV of our bluffs when he checks the river, since he can check/call us less frequently and we don’t have to waste a river bet.