Predictability without considering sequence order

We summarize a setter’s overall distribution by calculating the proportion of sets they made in each of our four categories. So, for example, a setter who sets left- and right-side sets (in equal numbers) would have a distribution of [0.5 0.5 0.0 0.0] whereas a setter who sets equal numbers of sets in all four categories would have a distribution of [0.25 0.25 0.25 0.25].

We can quantify the predictability in this distribution using a quantity called information entropy. Information entropy (also known as Shannon entropy) has its roots in information theory, and is a measure of the amount of information in a message. This might sound like a strange thing for us to be interested in, but it is useful because it tells us about the predictability of the setter. The “message” in our case comprises the set choices that a setter makes. A setter who only ever sets one type of set would have an entropy of 0, because there is no information in this message (their set choice never changes). This setter is also completely predictable: we will always know where they are going to set. A setter who sets left- and right-side sets in equal numbers has some information in their distribution (the set choice does change back and forth between the two possibilities) and has an entropy of 1. A setter who sets equal numbers of sets in all four categories has the maximum entropy of 2. This last setter is also the least predictable, in the sense that any given set is just as likely to go to any of the four options.

Considering sequence order

So far we have just looked at a setter’s overall distribution (their proportion of sets in each of the four categories), but ignored the sequence in which those sets were made. We can add a temporal aspect to this process by looking at pairs of sets in sequence. For example, if a setter made the sequence of sets L, L, L, 1T, L, R, this can be broken into the pairs L/L, L/1T, 1T/L, and L/R (order matters, so L/1T and 1T/L are not the same as each other). We can then use these pairs as the basis of our distribution, and tabulate it as before — so in this example, 40% (2 of 5) pairs were L/L, and the remainder (L/1T, 1T/L, and L/R) each occurred once (20%).

From this we can calculate entropy (let’s call this “sequence entropy” to distinguish it from entropy calculated without considering sequence order). The sequence entropy gives us some insight into the predictability of a setter’s choice given the last set that they made.

Some examples:

These three examples each contain five L’s and four R’s, and so all have the same entropy value of 0.99 (ignoring sequence order). However, if we take sequence order into account, the sequence entropy values differ. The first example sequence contains only two pairs (L/R and R/L) and has the lowest sequence entropy of the three examples. A set to L is always followed by a set to R, and vice-versa. The second contains contains four pairs (L/L, L/R, R/R, R/L), and so its sequence entropy is higher than the first example — but note that L/L and R/R appear more often than L/R and R/L. This makes it more predictable (because if we just guessed that L always follows L and R always follows R, we’d be right 6 out of 8 times). The third example also contains four pairs, and in this case they occur equally often, and so its sequence entropy is the highest of the three.

We can extend this approach to higher-order patterns, by looking at sub-sequences of three sets (second-order patterns), four sets (third-order), and so on. Here we use sub-sequences from length two to length five, and average these entropies to get an overall measure of sequence entropy. (Note: what we are calling “sequence entropy” is known in the literature as “permutation entropy”¹).