# Quantifying + comparing the effects of conditioning on Shannon entropy

I'm working on a project that compares how predictable/unpredictable individuals' actions are in terms of how they transition between actions. We consider their actions to part of a first-order Markov chain, where $$X$$ is a random variable describing their present action and $$Y$$ describes their next action. We calculate $$P(Y|X)$$ based on observed data and then compute $$H(Y|X)$$ for each person.

Ideally, we'd like to be able to compare the transitional/behavioral unpredictability between individuals. Our concern is: we have $$n$$ human-defined "types" of actions, so some individuals engage in the full set of actions while some only perform a subset of them. Thus, the "maximum" possible conditional entropy differs between people: $$|Y| \neq n$$ for everyone. This makes me think that $$H(Y|X)$$ can't be compared between individuals as-is.

My first idea was to use Mutual Information instead because it accounts for how much uncertainty is contained in $$Y$$ in the first place: $$I(X;Y)=H(Y)-H(Y|X)$$. However, the maximum value of $$H(Y)$$ is also still affected the number of action categories. Instead, I'm thinking of proposing that we compare $$\frac{H(Y|X)}{H(Y)}$$ between participants: the percentage of uncertainty left in $$Y$$ after viewing $$X$$. Thus, we can compare how informative a person's present action is when predicting their subsequent action; less predictable people will have a greater percentage of the original entropy $$H(Y)$$ remaining than more predictable people.

However, I've been unable to find any mention/usage of the quantity $$\frac{H(Y|X)}{H(Y)}$$ in other papers. Is there a term for this value, or is there a reason why I can't find it anywhere? The closest comparable quantity I could find divides by maximum possible entropy instead ($$\log(n)$$), but this makes me uncomfortable because it requires us to assume that each person is only capable of the actions seen in the data and not all $$n$$ action categories.

EDIT: $$\frac{H(Y|X)}{H(Y)}$$ is just $$1-U(Y|X)$$ where $$U$$ is the uncertainty coefficient: https://en.wikipedia.org/wiki/Uncertainty_coefficient.
The actual formula is $$\frac{I(X;Y)}{H(Y)} = \frac{H(Y)-H(Y|X)}{H(Y)}$$ and describes the fraction of information/bits you can predict from seeing $$Y$$. It's a form of normalized mutual information.