There are 1,140 ways of ordering a pizza with 3-toppings, if the pizza offers 20 choices

First order take rates tell us about the relative popularity of different options. For example, consider a small set of possible pizza toppings.

Customer buying patterns really start with second order take rates, which tell us about pairs of options, or toppings. Second order take rates tell us about relative popularity, but they also reveal something deeper: dependence. If you know that a pizza has pineapple on it, there is a very good chance that it also has Canadian bacon. This is dependence. In this case the reason is that there is a widely known “Hawaiian Pizza” that has both of these toppings. In general, customers don’t flip coins or roll dice. They select options that “hang together” in some way. The patterns can be seen in the combined take rates. Let me illustrate with three examples that are contrived to illustrate some important points. First consider pepperoni and mushrooms together.

In this table you can see that pepperoni has a 40% take rate, since 32% of pizzas have pepperoni without mushrooms, and 8% have pepperoni with mushrooms. In the same way, 20% have mushrooms, because 12% have mushrooms without pepperoni and 8% have mushrooms with pepperoni. This illustrates the first law of second order take rates: the first order take rates must be preserved. But this table contains no new information. Customers are apparently ordering pepperoni and mushrooms independently. This is revealed by the fact that 8% is exactly 20% of 40%. Knowing that a pizza has pepperoni does not give us any clue about whether or not it has mushrooms. Similarly, knowing that it has mushrooms is useless in guessing if it has pepperoni.

As a second example, consider the two toppings that are on every Hawaiian pizza: pineapple and Canadian bacon.

For simplicity, I have made this an example of complete dependence: a pizza has pineapple if and only if it has Canadian bacon. Notice that the first order take rates (3% for each) are preserved.

The third example is the really important one: partial dependence. This is illustrated here by pineapple and green peppers.

In this case, the first order take rates are also preserved: 3% for pineapple and 10% for green peppers. But these two choices are not independent. The take rate for pineapple and green peppers together is 2%, which is much greater than 10% of 3%, which would be only 0.3%.

Exercise for the reader: show that if we know that the pizza has green peppers, then there is a 20% chance that it has pineapple. (Much greater than its 3% first order take rate.) And if we know that it has pineapple, then there is a whopping 67% chance that it has green peppers!

So second order take rates capture information about how customers are combining toppings, and we can use that information to make predictions.