Expert Heads Up No Limit Hold’em: Excerpt 2

This is the second excerpt from Will Tipton’s book Expert Heads Up No Limit Hold’em Vol 1: Optimal and Exploitative Strategies. The book is on sale at www.dandbpoker.com from 14th of December 2012.

Excerpt 2: Expected Value

Expert Heads Up No Limit Hold’em
by Will Tipton

Our use of the term ‘EV’ is subtly different than that in many other poker texts. We feel that our convention is simpler and more intuitive, especially when studying complicated situations. However, the break from tradition raises the possibility of confusion. That said, the material in this section is extremely fundamental to the rest of the book.

We will often encounter and want to talk about numbers whose values are unknown. The expected value of some quantity which depends on random events is the average of all the possible values of that quantity weighted by the likelihood of each value. This is also known as the expectation of the quantity or its EV for short. A simple example is found in the case of a single roll of a fair, six-sided die. The die can land with any number one through six face up. Each of these outcomes is equally likely and happens with probability 1/6. To find the average value of the outcome of a single roll, we sum the value of each possible result times the chance of that outcome. Thus, the expected value of the result of one roll is

Of course, in no single trial can you roll a 3.5. However, if you rolled the same die a very large number of times, then the average result would be about 3.5. Now, technically we can talk about the expected value of many different quantities whose values are uncertain. However, in poker, our stack size is by far the most important quantity. So, we will use the term EV to refer specifically to the expectation of stack sizes. In particular, we will want to focus on expected stack sizes after we take a certain action. For example, we will refer simply to “the EV of calling” when what we mean is “the expectation value of the size of the player’s chip stack at the end of the hand if the player makes the decision to call”. When faced with a poker decision, we will generally seek to find the expected value of the size of our chip stack after taking each of the options available to us, averaging over things we can not control (the cards which will come) or we do not know (our opponent’s holding). We can then make the most money on average by going with the choice which has the largest expectation.

A simple example will help to make this clear. Suppose Hero finds himself facing an all-in bet on the river in a hand where both players started with 75 BB. There are 50 BB in the pot, and Villain’s river bet size is also 50 BB. Thus, Hero must risk 50 BB to call the bet and have a chance to win 100 BB. Hero knows that he has the best hand 40% of the time. What is the expected value of his stack size after calling and after folding, and which should he choose?

Firstly, if Hero folds, he knows exactly how many chips he will have at the end of the hand: 50 BB. If he calls, however, there are two possibilities. He will double up for a stack of 150 BB 40% of the time, and he will go broke 60% of the time. Thus, the expected size of his stack if he calls is

150BB x 0.4 + 0BB x 0.6 = 60BB.

Since 60>50, the correct play is for Hero to call. Calling is worth 10 BB more than folding, on average.

So what happened here? We considered a situation where Hero had two options, and then we found how many chips he would have at the end of the hand (on average) as a result of each choice. We chose the more profitable action. This is the basic idea behind the solutions to almost all of the strategic decisions we will face. We will choose between our options by calculating the EV of each one and picking the largest.

Now, the sophisticated reader may have noticed that the convention used here is actually different than that found in most poker literature. In this book, we will work in terms of expected stack sizes at the end of the hand. A more common choice is to deal with the expected change in stack size relative to the current stacks. Notice first that this is not a strategically-significant difference. The option which leads to the largest expected stack size at the end of the hand is the same as the one which maximizes expected change in stack size from the current decision point onward. Let us take an example which relates to the discussion in the next section to see exactly how these different choices of convention can play out. Suppose both players have 10 BB stacks. It is the beginning of the hand, Hero is in the SB, and he is choosing between folding and raising. First, what is the EV of folding? Second, what is the EV of raising if we know that the BB will fold to our raise?

Using our convention, this is easy.

EV(folding) = 9.5 BB

EV(raising given that the BB folds) = 11 BB

If the SB folds, his total stack size at the end of the hand is 9.5 BB. If he raises and the BB folds, then he ends up with 11 BB.

Now, what if we are working in terms of the expected change in stack size relative to that at the point of Hero’s decision? In this case, we have

EV(folding) = 0 BB

EV(raising given that the BB folds) = 1.5 BB

since if we fold, our stack size at the end of the hand is the same as at the decision point. That is, there is 0 change. If we raise and the BB folds, our stack size is 1.5 BB more than it was at Hero’s decision point.

Thirdly, especially in the case of preflop decisions, you may sometimes see people work in terms of changes in stack size relative to the stacks at the beginning of the hand. What would our result be if we took this approach? We would have the following.

EV(folding) = −0.5 BB

EV(raising given that the BB folds) = 1.0 BB

By folding, we end up with 0.5 BB less than we started the hand with. By raising, we earn 1 BB more than we started the hand with, assuming the BB folds.