Adjusting for the third player (part 1)

When you type a hand into Perfect Preflop Play, two kinds of information are given you: on the top left corner, you see the hand’s Q-scale class and other related things; and on the rest of the screen you see the pot odds required by your hand against each range, as well as their translation into numbers of big blinds.

Normal mode (non-heads-up) screen for Deuces in a game with antes. Results for normal mode are split in two screens. This one shows up right after you type in the hand. If you tap the right-hand third of the screen you get to the no-antes screen, and from there you repeat to select your next hand. Tapping the left third instead will always take you back one step (you may want to type in 2-A-s instead of 2-2), and tapping the top from either spreadsheet will take you to the beginning of hand selection. PPP is designed to work as fast as you can possibly move your fingers (and succeeds at it); for that we use a lot of invisible, tap-once-and-done buttons, instead of tap-wait-confirm or scroll-swipe-release commands. We really don't wanna waste your time in the heat of battle, whether you have your phone sitting next to your keyboard or sneakily hiding on your lap in a live tournament.

The first type of information helps you decide whether to go all-in with your hand. For instance, with 10 BB it makes sense (but feel free to disagree) to move in under the gun with any class-A hand, in middle position you can use classes A and B, on the button maybe you want to include class C, and in the small blind you may be comfortable shoving even class-D hands. Note that this method helps you be more coherent in your decisions, as now, with just a glance, you may stop yourself from shoving T2s in the small blind, for it’s a class-E hand which performs horribly against any reasonable ranges the big blind may use to call your bet. You may vary your play, be more aggressive to explore opponents who seem intimidated – but with PPP you’re always swimming in shallow water and can plant your feet any time to make sure you’re not doing anything stupid.

The second type of information helps you decide whether to call someone else’s all-in bet. Each hand has some equity against each of the ranges, and from that percentage we get the minimum pot odds required for a call, and from these pot odds we get the maximum numbers of callable big blinds for each position at the table. Because you have already paid the forced bet before the hand began, you are getting more generous pot odds in the big blind than on the button, and you can frequently call a player’s all-in in a given situation where it would be wrong to call the same all-in bet from the same player who had the same stack, if the only difference were your position changing from BB to button.

Of course, you are almost always going to need only one of the two types of information provided, because the situations are mutually exclusive. But there are exceptions. Let us explain them using the simplest possible examples, where only one player enters the hand before you by going all-in, and you must decide whether to call in the small blind – but the big blind is still live and has enough to cover you both. The examples will not feature antes.


                                 

1.      You are all-in when you call

It’s important to study these situations because, any time you are not the player closing the hand, there is the chance the hand will be played by more than two players, which means there are more variables. Nevertheless, as we’re gonna see, PPP’s results are still very precise – in fact they may be scarily precise because, even though it may seem risky to call a big bet with Fives against a button you have a great read on, but with the BB still possibly lurking with Aces, we see that the fact that two players have already gone all in for a big amount makes the big blind fold their hand the vast majority of the time. So, it’s obvious that the eventual BB-with-Aces scenario is unpleasant, but it is so rare that it has a very small weight on the whole equation.

Let’s jump right into an extreme example. You hold As8c in the SB, and for some reason are certain that your opponent on the button is going all-in for 150 big blinds with the “loose” range of 31% of hands. Well, if you are correct, then PPP tells you it’s right to call up to 165.7 BB. You have exactly the 149.5 BB necessary to cover the button’s shove, so the call is correct here, and folding would be a mistake. However, as if it weren’t enough, the player in the big blind is sitting on 200 BB of his own. I wonder if this piece of information will suffice to make us fold this hand that we would otherwise play.

Let’s see. We’ll give the button QsJs, a hand belonging to the range we know he has. In the direct confrontation between As8c and QsJs, As8c has 54.2% equity. So, if the big blind folds his hand, the final pot will be 301 BB and you will walk out with an average 163.14 BB. But what if the BB plays?

The first thing to note is that the button played very poorly, like a lunatic. Even so, he was a lunatic opening the hand on the button, which is much less risky than being a lunatic by calling 150 BB from two players. That is, even if the big blind is on tilt and generally inclided to taking big risks, it will be extremely rare to see a player call here with a marginal (say a class-B) holding. So we are only going to look at two realistic options: that he calls the 149 BB with the “ultratight” range, or even tighter with just {KK+}.

So we have two possible triple all-ins: As8c versus QsJs versus {JJ+, AKs, AKo} and As8c versus QsJs versus {KK+}. In the first one, our As8c has 19.27% equity in the 450 BB pot, resulting in an expectation to walk out with 86.72 BB. In the second, our As8c has 19.25% equity, for 86.63 BB.

So when the big blind plays, you tend to finish the hand with less than 87 BB, instead of the safe 149.5 you would retain by folding. But all that means is that you should fold your As8c if the BB had a very strong hand 100% of the time (he’d need dozens of Aces up his sleeve). What is going to happen much more frequently is an all-in showdown between you and the button, which is profitable. All that’s left is finding out the frequency at which the big blind joins the party, and we’ll know how good the call is.

In a complete vacuum, where we don’t know the whereabouts of a single card, there are 1,326 hold’em hands (52 cards x 51 / 2), and the “ultratight” range contains 40 of them. In this instance, however, we know where the As, 8c, Qs and Js are, so they are not parts of the hands that can still be formed. Now the hands that can be formed are 1,128 (48 x 47 / 2) and the size of the “ultratight” range has also changed, since it can no longer use the As, Qs or Js to form hands. The range, which contained 16 AK and 6 of each pair Aces through Jacks, now contains 12 AK, 3 AA, 6 KK, 3 QQ and 3 JJ, totaling 27 hands. Twenty-seven is 2.39% of 1,128 possible hands, so the BB is going to crash the party only 2.39% of the time. Your expectation takes this into account and looks like this:

(2.39 x 86.72 + 97.61 x 163.14) / 100 = 161.31 BB

Meaning even a seemingly high-risk call of 149.5 BB with only A8o with the big blind possibly overcalling our whole stack still has a positive expectation if we are right about the first player’s range. Our expectation, which would be to end the hand with 163.14 BB if the big blind never called, is reduced only a tad to 161.31 BB, but is still much better than the 149.5 we would have if we folded. So listen: folding here against this lunatic button is a big mistake! Beware of similar (if less extreme) opportunities in your games.

Now we could do the same math to see what our expectation is if the BB played only {KK+}, but you must have noticed that our expectation for the triple all-in is about the same as before (86.63 BB). Except now the big blind is going to play the hand much less frequently, which means the weight of the (profitable) heads-up all-in will be even bigger than before, and our expectation will be better.

Now let’s give the button 5c5d and the big blind {KK+}. In the heads-up all-in we have a 44.73% share of 301 BB, amounting to 134.64 BB. (Calm down: it doesn’t matter that we lose money versus Fives, but rather how we perform against the whole range.) In the triple all-in we have 17.68% equity in 450 BB, giving us 79.56 BB. The universe of possible hands available is 1,128 and there are 3 combinations of Aces and 6 of Kings, totaling 0.8% frequency. So look at what happens. When an opponent decides they are only going to call your pushes with superstrong hands, it follows that they are also going to call with superlow frequency. We already know that, by entering the hand only one time in 125, it’s impossible for the BB to seriously affect your expectation. Make no mistake about it – in these precise examples the BB is correct to play few hands, given the stack sizes; but in less extreme situations, the widening of the calling range against an aggressive opponent is good for the defender and awful for the attacker. Use PPP to look for these spots and exercise no mercy. All right, so the equation here is:

(79.56 + 124 x 134.64) / 125 = 134.2 BB

Again the data shown by PPP allows for a good decision, notwithstanding the big blind occasionally catching us playing A8o for 150 BB! Remember these are extreme examples (huge bet by the button; big blind covering us). If the big blind had 100 or 70 BB, the situation would be even better for us, because we would be playing for 150 BB with an advantage, but only for a fraction of that when facing the infamous triple all-in.

P.S.: Just so it's clear, I'm adding a much more realistic example of the other side of things - with large blinds and a big-blind player willing to call a double push at a higher frequency.

This time we have antes in play. The initial pot is 2.3 BB and the button goes all-in for 8 BB with what you figure to be the "aggressive" range (which is far more common for a player with 8 BB left than with 150 BB!). You have exactly the 7.5 BB necessary to cover his bet in the small blind, and the big blind has you both covered. We're giving you Ks7s and the button QcJd. Seeing all the action unfold, the BB decides he needs a hand in the "tight" range in order to call. That is, he requires a pretty strong hand, but he is not stupid and has seen you both play aggressively, so he believes he will be doing all right with something as low as KQo.

This example is relevant because now the BB's range is 8.1% of hands, which means his presence is going to be felt more frequently in the expectation equation. On the other hand, his average hand is going to be weaker than in the previous examples, which means he is not going to diminish your expectation by that much in triple all-ins. As we are going to see, again you can just follow PPP's advice just as if he weren't there. 

Well, in the double all-in we have 58.44 equity in the 17.8 BB final pot, resulting in the expectation to end the hand with 10.4 BB.

In the triple all-in our expectation is to end the hand with 6.89 BB from the total pot of 24.8 BB. 

Again the big blind's range is part of a universe of 1,128 hands. Hands containing a King, Queen or Jack are going to be less frequent than in a vacuum, because there are only 3 of each of these cards left in the deck. In the example, the "tight" range is going to have six AA, three KK, three QQ, three JJ, six TT, six 99, twelve AK, twelve AQ, twelve AJ, four ATs, 9 KQ and two KJs, for 78 hands, or 6.91% of the universe. 

Our expectation equation is: 

(6.91 x 6.89 + 93.09 x 10.04)/ 100 = 10.15 BB

Again, very close to the result obtained if the BB never entered the hand. So, whether you're playing against a huge shove or in a more typical situation of shortstacked players in a sit-and-go, the presence of one (or a few) opponents that may still enter the hand after you're all-in is negligible. That is, if PPP tells you the hand if profitable, then you should still play it. But if it looks break-even, then the addition of more players could make it slightly losing. But it shouldn't be this very small difference that makes you fold a hand, but rather the fact that you must not be looking for a lot of marginal calls to make! Stick to the ones that seem to be profitable, and you'll do great.

RQ