quinta-feira, 30 de agosto de 2012

'Relevant profit' mode coming soon for free!

Okay, so it's an eight-handed table, you hold Deuces in the big blind and the button, who has been bullying everyone for the last three orbits because we're close to reaching the money, again shoves around 25 BB from the button. You're entirely certain he has a range not an inch tighter than the 31% range, and you must call you last 16.7 BB to go to showdown. Should you do it? 

Yes! Am I right?

Well, yes, but also...

What kind of player are you? How much is this bubble? Do you care about this particular mincash? Are you in a fighting mood, or do you really just want to guarantee a winning session?

Granted, in a perfect world where you always have thousands of buy-ins and no feelings regarding what happens, but instead you only focus on optimum play an the long run, the answer should be: fuck the bubble, I've read the situation well, I know what this guy is doing, calling is absolutely a positive-expectation play, not only immediately, but it may later stop people from messing with my blind, so I call.

But if those questions indeed bug your mind, it's no fault of yours, and you're not wrong to consider them - you are what is known in laymen's terms as a human

A human (by Flickr user vagueonthehow).

The number 17.1 BB is the exact point where calling the bet against the 31% range is break-even, meaning calling 17.1 BB has the exact same expectation of folding - that is, to have 17.1 BB by the end of the hand when you fold, and to end the hand with a total 171,000 BB over 10,000 hands when you call (averaging 17.1 BB). Any number greater than that is unprofitable to call (your expectation is lower than folding), and any number smaller than that is profitable. And 16.7 is smaller, so this is a profitable call. But...

What Perfect Preflop Play (get it here) is neglecting to tell you is, by how much? That is, if by calling my non-desperate stack of 16.7 BB I expect to end the hand with 16.8 BB but also expect to bust out before the bubble more than half the time, do I want to do it? I'm guessing most of you will answer "no."

By Flickr user lindenbaum

And that is why we're coming up with a free update that is going to tell you what you really want to know: When does calling become so profitable that it's a shame not to do it? 

It's really simple. We're not going to erase the information you currently find on PPP, but by messing around with the original formula just a little bit, we're going to bring you a whole new mode that you can choose to show you at what point your call starts yielding an extra 5% to your stack, or an extra 1.5 BB for stacks bigger than 30 BB. 

So now that we demand to increase our stack by at least 5% to warrant making the call, what is the cutoff? For this precise situation, it's 10.87 BB. That's because, when you make the all-in call with 10.87 BB, the total effective pot becomes 25.04 BB, and you have 45.6% equity in that, which amounts to 11.42 BB, which is bigger than 10.87 BB by 0.55 BB, which is 5% of 10.87 BB.

So if you have 10.87 BB or less against this exact player in this spot, then by calling you are going to increase your stack by at least 5%, effectively turning the call into too good to miss. So from a practical standpoint, this update (which also includes 6-max mode) is going to make PPP a whole lot better, and for free. 

The "relevant profit" concept will also be applied to Heads-up Monster, to make sure you're only shoving the small blind (especially with a decent stack) when there's enough to gain from it. 

terça-feira, 28 de agosto de 2012

PPP Heads-up Only coming for just $1.99!!

So we have figured out how to make what is commonly known as a "light" version of our app (buy full version here). We weren't going to give you a tool that would stop working after a few sessions, nor one that would only give you results for say, suited hands. So this is our approach:

It's, of course, Perfect Preflop Play's heads-up mode, which we are soon launching as a standalone app. It will not contain 9-max or the upcoming 6-max mode, or the PPP book, but anyway it's pretty easy to use if you read the free articles and videos on this blog, and for those of you who specialize in heads-up tournaments it may just be all you ever need, as it covers everything you need to know when facing a push heads-up (and of course the Q scale and class are going to guide you when it's your turn to shove). 

This will also cover situations in non-heads-up tournaments when you're in the small blind against a shove from any player, or when you're in the big blind against a shove from the small blind, but as long as there are no antes. That's because the with-antes column for heads-up assumes only 2 antes posted, which is, of course, not the case for a full table. And the Q scale and class are going to guide you in playing  or folding your hand from any position. 

So it is a little crippled for non-heads-up, but it's still a perfect utility whether you play heads-up tournaments (where there are never antes), or you arrive at heads-up play after a series of eliminations at a bigger table (where tipically there will be antes, then you use the rightmost column). 

But if you are not a heads-up specialist and are considering buying the full version, this is still the way to go, simply because there is no waste. You get PPP Heads-up Only for $1.99, you see what it's about, and then you upgrade to the full version for $4.99 if you ever feel like it - and no, there will be no pop-up asking you to do it. No restrictions - you get the exact full version, book and all, and you actually save one cent!

There's also a no-antes screen, and 6-max mode for more accurate results for a shorter table. And of course you get to keep heads-up mode  

segunda-feira, 27 de agosto de 2012

Heads-up Monster coming soon as a paid update

We have decided that our Q scale for classifying hands in their preflop shoving strength, albeit a powerful tool for keeping your game consistent and stopping you from going stray and making any horrible mistakes, was still not enabling you to take advantage of every single blind versus blind opportunity (mind you, if you play heads-up poker, by blind versus blind I mean every single hand played; if you play non-heads-up poker, I mean every time it is folded around to you in the small blind, which is still thousands of hands a year, and of course, also every time you get heads-up from a bigger SNG or MTT).

Well, enough of that. Heads-up push/fold poker is just about as finite as it gets for no-limit hold'em, so as it turns out, there is a formula, and we found it. There is some heavy lifting involved, so this PPP (app store link) update may take a while to come out, but we are confident it will revolutionize how you play the game. A lot of the results provided are incredible, as in you would have never, in a million years, guessed you can play that hand that way. We were in disbilief too, but the formulas have been triple-checked and there isn't a single hole. 

By Flickr user JMR_Photography

This will be like nothing you have ever seen. It's not your old "shove this for 7BB" shtick - no, that's far too simple. We are way more ambitious. We are talking how much you can shove with any hand, against any of the six ranges your opponent may call you with, for any number of antes posted. 

And as you know, Perfect Preflop Play is not about creating idiot human bots. Your intelligence comes into play in that your notes and feel for the game are going to help you gauge what range your foe may call you with. We're talking about abusing an opponent who just can't bring himself to call with less than AT, while showing (proving, actually) that maybe you have to let the same hand go if you have a more perceptive adversary who can adjust and look you up more often. We're talking playing T4s in the small blind against Phil Ivey in the big blind and not losing money. We're talking never again making a negative expectation play heads-up preflop. 

Heads-up Monster will integrate seamlessly with PPP for just an extra $1.99, and you will have options as to how to see its results only when useful, so as to not slow down the already awesome PPP experience. For that value you will also get another exclusive Perfect Preflop Play book, that will explain to you all of the math behind Heads-up Monster and is sure to make you a believer. We're also going to tackle other important issues, to build on the foundation laid by the original PPP book, which comes free with the main app. If you know some of our work, you know we're onto something. We want to evolve with you, and we're just getting started. 

6-max mode coming soon as a free update

Perfect Preflop Play (app store link) is getting even better soon, and for free! 6-max mode is getting implemented in version 1.1 for free in just a few weeks. 6-max mode is going to give you more accurate results for short-handed games when there's an ante, as compared to what is now normal mode (soon renamed to 9-max). 

So if you start out playing only 6-max tables, you want to just use 6-max mode from the start until heads-up. But if you play 9-player SNG's, then you want to use 9-max at first, then switch as the table shrinks. But don't worry - if you forget to switch, 9-max is still going to guide your game very well, but using 6-max at the right time will enable you to make some razor-thin value calls. 

quarta-feira, 22 de agosto de 2012

So what's a close call worth?

I won't try to fool you here: tournament structures bend the values of chips. You should be well aware of it, but chances are you are, because, well, every player is. The clearest example of this is satellites. In satellites you often have from a few to hundreds of prizes that are worth the same - you have many people tied for first. Say a satellite awards seven seats to the Sunday Million. There's no three-handed or heads-up play. There will be no champion. The tournament ends the exact moment someone busts out in eighth, and at that point the player who has 10 million chips gets the same as the one who was holding tight with one ante left - although the first guy was probably a lot less stressed toward the end. 

By Flickr user imagesbyferg

The type of close call you should always pass on, as there is just too much to lose. Poker is never like this, however (unless you're a complete bankroll maniac, in which case please get help). Playing within a comfortable bankroll, you should take a lot of showdowns where you think you're gonna win a few chips on average. If your reasoning is based on solid foundations - and PPP will make sure it is! - then you should let it ride!

It's common for people with some seven big blinds to fold AK, QQ or even AA in this sort of position, which would normally be preposterous. Of course, with Queens you have the expectation to win chips if you open-shove 7 BB, so why would you open-fold them? In such a satellite bubble, the answer is almost always this: you think someone else is going to bust first, maybe this very hand, and you are not the chip leader, so by folding you make sure you are not all-in this hand, and you may win your ticket by just waiting around. Since you are not going to keep playing down to a single winner, there is no more need for chip accumulation. If there is one player all-in in the big blind, then automatically the goal of every other player is to knock him out, simply because it would be the quickest, safest way for everybody (but him) to get what they want.

Normal, single-table sit-and-gos have that too, but to a smaller degree. If three are paid and there are four left and one guy is supershort, then the other two non-chip leaders often want to refrain from risky moves before they see if he doubles up or busts - thus guaranteeing the other players a cash. 

But look! There is a major different between this SNG and the satellite, and that difference is that there will be an actual winner, meaning you should still take the occasional very profitable chance if it's not entirely obvious the shortest guy is gonna bust out in fourth. I mean, if he has a half big blind left and he's the big blind next hand, that is a clear situation where he has no choice but to be all-in in what is looking like a pretty terrible spot, but things are not so clear if you have 8 BB and he has 6 BB. 

Many aggressive players (and even some passive ones, in a complete 180) will go absolutely crazy here. You should not. I mean, they have the chip lead with 4 left and one guy shorter than the other two, and they start shoving 100% of their hands. My friend, you should be aware that this is just almost never good. You either don't have such a big lead, and getting called by someone with your embarrassing 94o would cripple you terribly, or you do have a big lead, you're completely comfortable, you are gonna cash almost surely, and there is no need to spew chips like a retard! Even if the two medium stacks are cowards, remember that at least the shortest one is probably looking for a good spot to triple up, and you are handing it to him, just like that, and maybe turning a hopeless situation into a game where everyone has chances. 

With Perfect Preflop Play you will be very well equipped to deal with the idiot push-every-hand chip leader. After five straight pushes you can easily put him on the "psycho" range (and that's if you wanna be cautious, because he may have the all-hands range), and by taking one second to type in your hand, you will be surprised at how many good calls you can make with stuff like 98s and K2s. 

But there is one money bubble where chip values are not distorted, and that's the last one, when you're heads-up. Let's take the actual numbers from a 9-player, $15 knockout SNG on Pokerstars. Second place gets $30.45 and first gets $50.77, but since the winner will necessarily be knocking out the runner-up and not be knocked out himself, he also wins two bounties for his win, so we're gonna count first place as $56.41, so this is a $25.96 bubble between second and first. 

A Perfect Preflop Play spreadsheet for heads-up. Usually in a tournament you're going to be playing with antes, so you'd want the rightmost column for the most accurate results. But, since the numbers are very close (as there are only two antes in play, so there isn't that much money added to the starting pot), you can use the other one if you want to make sure you're not being too optimistic. Note that heads-up mode shows all results in one screen, as there is effectively only one position to consider. But if you're multitabling and don't want to shift between normal and heads-up modes all the time, you can simply stick to normal mode and use the no-antes SB column there, which will give you the same results as the no-antes heads-up column. You cannot use the with-antes SB column for heads-up play, however, as that one assumes 8 antes, which would terribly distort the results for a heads-up game. And, of course, disregard all of this if you're so quick at math that you can translate pot odds into a number of big blinds in a second. In that case you only ever need the gold column, from which the red ones are calculated.   

There is no more waiting around, of course, because there isn't a third player to knock out the other guy for you. We can, in fact, interpret the $30.45 as a thing of the past. You are both fighting just for that $25.96. This is now the same as a fixed-buy-in heads-up cash game that only ends when one player loses their buy-in. 13,500 chips is worth exactly (no relativity) $25.96; 0 chips is worth $0.

So before you two post for this hand you both have 6,750 chips, and since there is no distortion your expectation (disregarding skill disparity) is $12.98. The blinds are 300 / 600 with a 50 ante and you're the big blind. Your opponent is definitely capable of stealing, and he's been very active, and there is just no better spot to steal than heads-up with short stacks, so you're not gonna give him much respect for his raises - unless he starts folding the button several times. You pick up QJs and he goes all-in. From the description you should put your opponent on the "loose" or "psycho" range (you should seldom assume the all-hands range unless a player is ultrashort or in maniac mode; remember you are never being cautious if you assume 100% of hands - you're either right or you're optimistic).


Well, against the "loose" range PPP will tell you that you need 1.11 to 1 or greater to call, and that you can call up to around 20 big blinds. There are no more players that can bust if you fold, so this is just a plain positive-expectation call, so you call. Needing 1.11 to 1 means you are gonna win one time in 2.11, or 47.43% of the time. So your equity in this 13,500 pot is 6,403 chips, which, of course, is better than retaining your 6,100 by folding. 

Since each chip has a fixed dollar value, it's obvious that 6,100 chips are worth $11.73 and that 6,403 chips are worth $12.09. So 36 cents is what you gain by calling here against the "loose" range. Doesn't sound like much, but this is meant as an example of a close call. Short-stack heads-up play can be very brutal in that every pot conceded increases your disadvantage by a lot, so you must be ready to take a stand against aggressive players soon, or not at all. 


quarta-feira, 15 de agosto de 2012

Play every coinflip you can

Okay, this is pretty basic stuff, so you may just skip it if you have any grounds to say you're at least an intermediate player, or if you've read Perfect Preflop Play's book (get it here). That's the real useful theory, as well as the app itself, which will give you concrete numbers against each preflop range.

However, this here is to address a wrong mindset, and while I'm at it I guess I'm gonna name it. Let's call it the Phil Hellmuth mindset. Now, this is not meant as an attack on Phil Hellmuth, but the name is appropriate. It's tough to argue against his 12 bracelets (sick, right?), but we ought not to be blinded by them either. A lot of great players say Phil plays horribly, and I agree. This is simply because, even though Phil is well aware of the math of the game, he feels it does not apply to him. 

That's just fucked up. I mean, he raise-folds 13 BB with Queens, he folds a straight flush draw on the flop if he thinks he's flippling against a set, etc. I won't even get into how dreadfully horrible such plays are. I'll just tell you that, at least with large blinds, you should always play every coinflip you can get your hands on!

Heads you win $1.15; tails you lose $1. And you're gonna fold that! (Photo by Flickr user redwood 1)

Now, wait a second. If you've read some articles in this blog or the PPP book, you know that isn't quite how it works, because you always have a defined hand (yours) against a range (opponent's), so you don't analyse hands like "I have KQo versus Fives," but rather "I need 1.18 to 1 with my KQo versus the 17% range," which is actually a number that PPP gives you in two seconds to help you decide and play accurately in real time. 

But you do hear some people say, "I avoid coinflips," and Hellmuth is the most high-profile of those players. So, if you somehow know that you're gonna play a coinflip, here's why you should always play it. 

First, in a coinflip you are usually a little under or a little over 50% to win. And because of the way that poker betting works, you should always take a bet where you're a little over 50%, and almost always take a bet where you're a little under that. 

First, let's take an actual coinflip, with a coin. You and your friend each bet 1 dollar on each side. Needless to say, this is a break-even bet. You have one chance in two (50%) to win, which breaks even for a bet of one to one ($1 versus $1). Now let's give you Fives in the big blind, and your opponent JTo on the button.  There are antes in play at an 8-player table so the starting pot is 2.3 BB, and he has 10 BB behind against the 9 BB you have after posting. This is the closest matchup I could find to an actual coinflip (each hand is almost exacly 50% to win). Now the button goes all-in for 10 BB and you know his exact hand and you must call 9 BB to stay in the hand. Should you do it?

This is the point, and I hope you get it. It really doesn't matter what player you are (sorry, Phil). There is a mathematical truth to the situation. You should definitely call, because, even though having 55 vs JTo is just like having heads versus tails, you are no longer getting 1 to 1. The way poker betting is designed, there are always forced bets (blinds, antes) before play begins. But when you call someone's bet, you only need to match their bet in order to be elligible to win the money left behind by others. In this case you would be calling 9 BB to try to win the current pot of 12.3 BB. Because the blinds are high relative to the stacks, the amount the pot is laying you is a big deal (unlike calling 60 BB to win 62.3 BB), and now you're getting a whopping 1.37 to 1! Meaning you would break even if you had as little as 42.2% equity. So calling with 50% should be a no-brainer. Your expectation is to end this hand with 50% of the resulting 21.3 BB pot, which is 10.65, instead of the 9BB you would retain by folding. Calling is a great play, and folding is an awful mistake. Hellmuth would rationalize this by saying he can wait for a better spot. Guess what, this is a great spot. Just by making a simple call you'd increase your stack by 18%, just like that. If your 9 BB have any value to you, then 1.65 BB should have some too - otherwise your logic is totally messed up, like Phil's. 

And that's it. You should definitely play a lot of coinflips with high blinds, because of the simple fact that the pot in poker is always giving you more than 1 to 1. You never have to be an actual favorite in order to play profitably!


quinta-feira, 9 de agosto de 2012

Perfect Preflop Play now for sale!

PPP is now for sale on the Apple app store, here. To find out more about it, read the articles on this blog or visit our Facebook page by clicking the tab on the top. Don't hesitate to ask us questions on Facebook - we'll be glad to tell you all about how you can use PPP to make money.

quinta-feira, 2 de agosto de 2012

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.