This is the second of a two-part article on schooling. The first part can be found here.

Schooling is a controversial poker concept describing the behavior of groups of weak players, typically in low-limit games. The theory suggests that a large number of loose players in a limit hold’em game help each other out by pooling outs and giving each other better pot odds. In fact, some say that games with too many players are actually harder to beat than games with only a few loose players.

In last month’s article, we discussed the concept of schooling from a theoretical standpoint. We illustrated how it is possible that a large number of players calling with weak draws can serve to hinder a player with a vulnerable made hand. But then we proceeded to question whether this genuinely made the games harder to beat in practice. After all, when you have a big hand or a big draw, all these loose callers should give your hand more value than what they strip from it when you have a weaker hand.

But all the arguments on both sides were theoretical. In this article, we will try and add some cold, hard numbers to the debate. Using Poker Stove and the odds calculator on ITH, and starting with a basic model, we will attempt to empirically prove what effect schooling will have on your expectation and results in various poker situations.

A Simple Model

Let’s say you are playing in the strangest poker game ever. Each of your opponents fall into one of two categories:

1. Rocks: They don’t actually want to play poker, they are just passing the time. As a result, they fold every hand pre-flop so they can lose their money as slow as possible.

2. Maniacs: They want to gamble as much money as possible every hand. No matter what cards they hold, they will raise all the way to the river.

Obviously this is a completely unrealistic scenario, but it can be used to explore the concept of schooling quite effectively because it eliminates a lot of the variables that make poker complicated. You are unable to protect your hand, bluff, or value bet at all, because every street will be capped regardless of what you do (unless all nine opponents are rocks, in which case you can steal the blinds on every hand). You will also be completely unable to put anybody on a hand, because each opponent behaves the same regardless of what hand he holds.

Let’s say you are playing in this game and you are dealt aces in middle position (in fact, position is irrelevant in this game). Obviously you are going to play this hand, but how many opponents would you like to play with you? In other words, how many rocks and how many maniacs would you like in the game?

If all nine opponents are rocks, it is straightforward to see that you will win 0.75 big bets if we assume a standard blind structure. If n players are maniacs then your expectation will be (((n+1) x 12) x Y) – 12 big bets. This looks like a complicated equation, but it is fairly simple. (n+1) x 12 represents the money in the pot at the end of the hand. If every street is capped, then each player including you will put 12 big bets in the pot (two pre-flop, two on the flop, four on the turn, four on the river). Y is a variable representing your pot equity against n random hands. The 12 at the end is obviously your own investment that you need to subtract from the pot to calculate your expected net profit. Note, that if you have exactly one maniac opponent, then you can prevent every street being capped. However, we will assume that you will be happy to cap each street in this case. It takes a pretty horrific board for aces not to be favored over one random hand.

Here is a table showing your expectation against various numbers of maniacs:

A number of interesting things can be seen here. Despite the fact that aces are unlikely to make a big hand (trips or better), you can see that you are better off with many opponents as opposed to few. Each random hand decreases your pot equity, but increases your overall expectation due to the increased size of the pot.

However, you can observe evidence of schooling! When we add the ninth maniac opponent, your overall expectation goes down. A player capping every street with a random hand is actually decreasing your overall expectation, even though individually you figure to be a big favorite against him. It would be interesting to see what the effect of adding a 10th and 11th opponent would be, but unfortunately I don’t have the software to simulate this.

In order to try to observe this schooling effect better, I decided to try the same experiment with a different hand. Aces are always a fun hand to simulate, but intuition dictates that schooling will be more prevalent with a hand that has less monster potential. The following are the results when this experiment is repeated using AJo as the hero’s hand. This hand was chosen as a quintessential top pair hand that is traditionally thought to struggle in multi-way pots.

With this weaker hand, the effects of schooling are even more pronounced. Upwards of six opponents, you would welcome the additional maniacs, but beyond that point they are decreasing your expectation. However, while schooling is clearly a factor here, there are two things worth noting:

1. One difference you have holding AJo rather than aces is that you can afford to fold. If you don’t flop a pair or a draw with AJo against nine opponents, then your pot equity will drop to around six percent or below and you can safely fold. This makes situations with a large number of opponents more profitable than they initially appear.

2. Although schooling does decrease your expectation with more than six opponents, it does so only by a small amount. You are far better off having nine opponents rather than two, for example, which would certainly surprise the most devoted disciples of schooling.

Of course, this very simple model has its limitations. No matter how bad you think your opponents are, you would be hard pressed to find even one player who plays like the maniacs described above, let alone a whole table full of them. Typically, you will be up against opponents who make many mistakes, but will play in a way that at least makes sense. Whether this will have a positive or negative effect on schooling is difficult to prove empirically. Obviously, you would much rather be up against opponents with completely random hands than ones with legitimate (albeit weak) draws. On the other hand, you won’t play many hands where you have six or more opponents who all have a legitimate draw on the turn, because there are only so many draws available on any given board.

A More Realistic Model

To explore the effects of schooling further, we will need to use a more realistic model, one where both you and your opponents make decisions as they might in a real game (this doesn’t necessarily mean correct decisions, of course). Unfortunately, by doing this we are going to lose a lot of the objectivity of the previous model, as whatever hand we choose is going to be somewhat contrived. However, it should give us a good indication of how schooling affects your expectation in real life hands.

In this hand, you hold KhQh. On the turn, the board is: KsJs7d2c

There are seven big bets in the pot. Your opponents happen to hold the following hands (you won’t know this of course).

MP: 8c9h

Button: 3h3d

SB: Ac6d

BB: Kd4d

This has the potential to be a classic schooling situation. All of your opponents have weak draws, but they are all chasing different outs and are, collectively, quite likely to beat you. Although you have the best hand with only one card to come, your pot equity is only around 68 percent. If all four opponents call on the turn, then you are going to lose almost one-third of the time, even though none of your opponents have more than four outs.

First of all, let’s consider the possibility that all of your opponents play correctly (of course, they have probably already made errors in the hand, but we will wipe that slate clean). MP actually has a reasonable draw. His gutshot straight draw gives him four outs, but this doesn’t give him pot odds to draw (although with implied odds it would be close). The button and SB should also fold. They have two and three outs, respectively, which have to be discounted in such a large field (heavily in the case of SB). BB only has three outs, because you have him dominated. Obviously he won’t necessarily know this, so it is difficult to say whether he should fold his top pair from a good poker standpoint. However, mathematically he should obviously fold. Hence, if this flop was played out with all players playing correctly, then you would bet and everybody would fold. You would win the seven big bets in the pot.

The question is: How much better would you do if one or more opponents called you down? Let’s assume that if you have the best hand on the river, your opponents will call, and if you get outdrawn the winner will raise and you will pay him off. Obviously, neither of these scenarios is totally realistic. However, the former is favorable to you and the latter is favorable to your opponents, so hopefully they will go some way to canceling each other out.

So to simplify, if you win the pot you will win seven big bets (the existing pot) plus two times the number of callers (one bet for each on both the turn and the river). If you lose the pot you will lose three bets (one on the turn and two on the river). The following table shows your expectation with different numbers of callers. I have tried to make it realistic in regards to which players will call your bet (BB most likely, then MP, then SB then Button).

As you can see, your expectation rises with each of these bad calls that your opponent makes. Although each caller is giving the others better pot odds, it doesn’t compensate for the additional money that is going into the pot when you are a heavy favorite. In fact, it doesn’t even come close. Even the fourth caller gives you an extra 0.31 big bets in expectation, which in limit hold’em is a big chunk of pie.

As illustrated earlier, it is quite possible that when you get up to seven, eight, or nine callers, schooling might start to hurt you. However, even in the loosest games it is very rare to get seven players calling a bet on the turn. And if you did, it is likely that some of them would be drawing dead and you really don’t mind players that are drawing dead coming along for the ride. For example, if you add a sixth and seventh player in the hand above with KT and A5, respectively, they don’t remove anything at all from your pot equity. Even on scary boards, there are usually only a finite number of feasible draws.

Hopefully this article has illustrated two things about schooling:

1. It is a theoretical possibility that, in a big pot with a lot of opponents, additional players making bad calls might harm your expectation.

2. In reality it is very unlikely to happen and if it does it will be an oddity of that particular hand that will not make the game significantly less profitable overall.

As we stated last month, schooling is one of those phenomenons that players seem to want to believe in, because it helps them to explain why they sometimes experience horrific beats in what appear to be very good games. But schooling, as these players see it, is a myth. You want as many bad players as possible at your table. Even though occasionally they might school together to reduce your expectation, this will be far outweighed by the times that their bad calls add huge amounts to the value of your flopped draws and flopped monsters.

Schooling is an interesting idea. Interesting, but just not that relevant.