Why are we taking the logarithm for risk-averse decision-makers? I'm a mathematics student learning a bit of Game theory. Many examples are given within a very economic setting and up to know I could follow most of it because they were very basic and I learnt some basic economics in high school.

Now I've come across some examples where a player is risk-neutral whereas the other is risk-averse (e.g. insurance company vs. customer). Say we have outcomes $(A_i)_{i \in I}$ which occur with probabilities $(p_i)_{i \in I}$, and they have payoffs $(w_i)_{i \in I}$. It seems the expected payoff for risk-neutral players is simply

$$\sum_{i \in I} p_i w_i$$

whereas for risk-averse player we apply the natural logarithm and end up with

$$\sum_{i \in I} p_i \log w_i.$$

In the lecture notes I am reading, I have found no justification for this and I assume it is covered in other lectures in the standard curriculum of any economics student.

Question 1: Why are we taking the logarithm? (mathematically and economically)

Question 2: Are there more "measures" of risk-attitude? If so, how does the expected payoff change?
Let's step back. In general, utility functions are (monotone increasing) functions $f(w)$ that map wealth into utility. That is, the input $w$ is money and the output $f(w)$ is utility. If we have a distribution $\vec{p}$ over possible wealth outcomes $w$, then my expected utility is $\mathbb{E}f(w) = \sum_i p_i f(w_i)$.

Now, what is risk aversion? Intuitively, it means I prefer a certain outcome to an uncertain one even though they have the same expectation.

Thus, a risk-averse utility function is defined to be a concave utility function $f$. This is natural because of Jensen's inequality: if $f$ is concave, then for any distribution over outcomes of $w$, we have $\mathbb{E}f(w) \leq f(\mathbb{E} w)$. In other words, I prefer to get $\mathbb{E}w$ for certain rather than some uncertain amount of money with that expectation.

To connect back to your question (1), you are comparing the expected utility from two different utility functions: $f(w) = w$ and $f(w) = \log w$. The latter is just a very nice and handy concave (hence risk-averse) utility function. But it is only one example (albeit an important one) of a risk-averse utility functions.

And to connect back to your question (2), to get risk aversion, any concave function will do. But the "more" concave, the "more" risk averse. Meanwhile, risk-neutral functions should satisfy $\mathbb{E}f(w) = f(\mathbb{E} w)$, so that's any "affine" function of the form $f(w) = a\cdot w + b$. Finally, risk-seeking utility functions are convex (and less-often studied as they seem less natural).