What do the notations $\sim$ and $\Delta (A) $ mean in the paper "Fairness Through Awareness"?

Question

In this paper Fairness Through Awareness, the notation $\mathbb{E}_{x \sim V} \mathbb{E}_{a \sim \mu_x} L(x,a)$ is being used (page 5 top line), where $V$ denotes the set of individuals (so I guess set of feature vectors?) and the meaning of the other variables can be found in the paragraph above the mentioned notation. What does the $\sim$ in the expectation stand for?

Another notation that I do not know is $\Delta (A) $, where $A$ is the set of outcomes, for instance, $A = \{ 0,1\}$. What does it stand for?

Answer 1

The $\sim$ symbol means that a random variable is drawn from the given distribution, i.e. if I were to say $X$ has a Standard Normal distribution I would write $X \sim \text{Normal}(0,1)$. They write two explicit expectations here because $a$ is a random variable with distribution $\mu_x$ but $X$ is also a random variable with distribution $V$. I believe you are right that $V$ would be analogous to a set of features. So we are saying that $X$ is a random variable over the distribution over the features, or individuals in this context.

As for the $\Delta(A)$, I have never seen this notation used before -- I am not sure if it is standard notation. I Googled to see if it was something that I just hadn't seen before, but there was no such answer. However, from the context of the paper they define $M: V \rightarrow \Delta (A)$ to be mappings from individuals to probability distributions over $A$, so I guess that $\Delta (A) $ is probability distributions over $A$.