Interesting spot at Red Rock Station yesterday between two villains.

BTN straddles for $5. Action for this type of straddle starts UTG and it folds around to MP, who has $55 behind. MP calls the $5. Folds to the CU who raises to $25. BTN and the blinds fold and it's back to the shortstacked MP with some dead money in the pot.

Two players, one left to act: (Pot $38) - Effective stack $50.

MP tanks for a few seconds and then CU--who may have thought he took down the pot preflop--tables QQ. The hand is still live. MP looks at CU's QQ and folds.

I think this was a big mistake. Let's look at it and maybe ignore the fact that most of us wouldn't be sitting at $1/$2 with $55, never mind open limping the straddle with it.

Now, MP's limping range equity is around 23% against QQ, and he would need around 42% equity to just ship in his remaining $50 preflop with essentially zero fold equity, so that would be a spew.

However, with perfect information on our opponent's hand, I believe we should be taking a flop for $20 and then either continuing when we're ahead of QQ, or bluffing our remaining $30 into a $58 pot if the flop comes A or K high.

Obviously, CU isn't going to fold very often in the latter case, but will he fold say...25% of the time?

I'm not entirely sure how to approach this question properly, but if you've looked at my blog you'll know that that's never stopped me before, so let's gooooo.



As I mentioned, Equilab gives MP's limping range 23% equity vs QQ if the hand goes to the river. Does that mean that MP would be ahead roughly 23% of the time if the hand was somehow stopped on the flop? I would think so. Certainly, I could be wrong, but let's run with it...say MP is beating QQ on the flop 23% of the time. Therefore he's folding 77% of the time and losing his $20 investment.

Not so fast, because MP is bluffing A or K high flops. How often do those show up? According to a Poker News article, the flop comes A or K high 41% of the time when someone's holding QQ. (For now, let's not count blockers if MP is holding an A or K, because then he's betting for value, and that instance will fall into another section further below.)

So, instead of folding 77% of the time, MP is only folding 45% of the time and then losing his $20 investment on the flop. The rest of the time, he's either bluffing or value betting. So when he folds it's 45% times a $20 loss which equals (-$9.00) in EV.

With 45% of instances covered by folding, that leaves us 55% of instances to go. If MP is ahead on the flop 23% of the time, that means he's bluffing 32% of the time, as these two figures add up to the remaining 55%.

Now, sometimes MP is going to suck out after bluffing on the flop and getting snapped off, and sometimes QQ is going to suck out when it's behind and MP was value betting on the flop. Let's just take those two instances and say that they roughly cancel each other out, and give them a 0 EV overall, as the percentage of instances when MP value bets and when MP bluffs and gets called are very close to each other.

Granted, QQ is much better than MPs range, but it's also harder for pairs to suck out when they're behind on the flop, as they often only have two outs to a set or some unlikely four straight or four flush outs (or counterfeiting board pairings with overpairs vs lower two pairs.) But we'll call it even and move on.

We'll break down the bluffing next. MP bluffs 32% of the time, during which he gets snapped off 3/4 of the time and gets a fold 1/4 of the time. That means he gets snapped off in 24% of total instances and loses his $50 for a (-$12.00) EV loss. He gets a fold 8% of total instances and wins $88 for +$7.04 EV gain.

Finally, MP actually wins the full $118 pot 23% of the time for a +$27.14 EV gain.

Eh, not so fast...because QQ is going to be either folding or checking down some of the time, particularly when an A or K flops. We'll give MP his full $118 win 3/4 of the time and only the $58 in the pot on the flop 1/4 of the time...well no, that's not right either, because even on bad runouts, MP might still be able to fish another $10 or $15 out of QQ on the turn, and then QQ will be almost obligated to call off another $15 or $20 into a roughly $100 pot on the river.

So, let's say out of the 23% of value spots, MP wins the full $118 in 20% of total instances for a +$23.60 EV gain, and wins $58 in 3% of total instances for a +$1.74 EV gain.

The final tally, then, is:

(-$9.00) + (-$12.00) + $7.04 + +$23.60 + +$1.74 =

+$11.38 in EV.

Now, some of you may have noticed that MP appears to have lost EV overall from bluffing, so what happens if MP never bluffs?

In that case, he folds 77% of the time and loses his $20 for a (-$15.40) in EV.

For the remaining 23% of value, it's the same as the top equation: +$23.60 + +$1.74.

This gives MP a total EV of +$9.94, less than if he had used a bluffing range. So what happened here? If my math is right, MP loses less money bluffing than he does folding. I know that folding is supposed to be 0 EV, but here we're accounting for the $20 MP put in on the flop...so it's the EV of call/fold, call/bluff, or call/value bet.

So does that mean MP should always bluff? Probably not, because QQ may never fold on a flop that doesn't have an A or K, so that brings the overall fold equity way below 25%.

I'll mention here that I left out MP semi-bluffing flopped draws on non-A or K high flops. If anyone can think of some math to work that out, then cheers to you, it would be great to see it. I'll venture again that for all types of bluffs we'll have to assume very low fold equity on non-A or K high flops.

What interests me most about this hand is thinking about how I would play this if we both had $300 stacks, instead of the dinky $55 effective.

^Oops, I forgot to take rake into account, which is 10%, $5 max including the promo, so keep all the red numbers the same, and reduce all the green numbers across the board.

This is definitely incorrect. MP's limping range's equity increases with # of cards. For the hands that include an A or K, the flop will contain one of them roughly 12% of the time. All of MP's other hands have significantly less equity than that with only 3 cards.

It is very, very hard to outflop an overpair without overs. On a flop like J72r QQ has 17% equity vs. that range, but QQ is the best hand a whopping 98% of the time. These hands need to see five cards to realize their equity, so QQ should bet any flop that's good for them for pure equity denial.

At first I was going to write a long post covering the theoretically optimal strategy for QQ to minimize profit vs. a "strong" preflop continuing range of Ax/Kx, but then I realized there's no point in doing that if a very simple, weak strategy already makes Ax/Kx unprofitable preflop calls.

So, consider if QQ always stacks off on the flop, either by calling off a jam, or jamming itself when checked to in order to deny equity.

The possible combinations of flops given 4 known cards is 48 choose 3 = 17296
The possible combinations of flops that do NOT contain our high card to beat QQ is 45 choose 3 = 14190. Therefore, the probability of flopping at least one of our high card is (1-14190 /17296)

I've ignored flops which have a Q because it's more work and it only makes OOP's odds worse.

When we flop our pair, we will GII with roughly 91% equity 100% of the time, for a $118 pot which we will have to pay $30 into. The EV of this is:

.91*118-30 = $77.38.

This event happens at frequency (1-14190/17296), about 18%, for an average postflop EV of (1-14190/17296)*77.38 = $13.90.

OOP paid $20 preflop, so he's not recovering his investment. The probability of flopping trips with our low card adds very little EV, about 50 cents.

It's unlikely considering the EV from flopping draws with suited cards makes up a $6 gap. Keep in mind I didn't consider that some of these flops contain a Q, or may offer QQ more than 9% equity.

Fold everything.

The deeper you get, the more valuable perfect information is. I believe first you'll start adding Ax/Kx, then pocket pairs, then suited connectors, and eventually trash very deep.

nice analysis browni