There are many games that even though they include some random component (for example dice rolls or dealing of cards) they are skill games. In other words, there is definite skill in how one can play the game taking into account the random component. Think of backgammon or poker as good examples of such games.
Moreover, novice players or outsiders might fail to recognise the skill involved and attribute wins purely to luck. As someone gains experience, they usually start to appreciate the skill involved, and concede that there is more to the game than just luck. They might even concede that there is "more" skill than luck. How do we quantify this? How "much" luck vs skill? People can have very subjective feelings about this balance. Recently, I was reading someone arguing that backgammon is $9/10$ luck, while another one was saying it's $6/10$. These numbers mean very little other than expressing gut feelings.
Can we do better? Can we have an objective metric to give us a good sense of the skill vs luck component of a game.
I was thinking along these lines: Given a game and a lot of empirical data on matches between players with different skills, a metric could be:
How many games on the average do we need to have an overall win (positive win-loss balance) with probability $P$ (let's use $P=0.95$) between a top level player and a novice?
For the game of chess this metric would be $1$ (or very close to $1$). For the game of scissors-paper-rock it would be $\infty$.
This is an objective measure (we can calculate it based on empirical data) and it is intuitive. There is however an ambiguity in what top-level and novice players mean. Empirical data alone does not suffice to classify the players as novices or experts. For example, imagine that we have the results from 10,000 chess games between 20 chess grandmasters. Some will be better, some will be worse, but analysing the data with the criterion I defined, we will conclude that chess has a certain (significant) element of luck. Can we make this more robust? Also, given a set of empirical data (match outcomes) how do we know we have enough data?
What other properties do we want to include? Maybe a rating between $[0, 1]$, zero meaning no luck, and one meaning all luck, would be easier to talk about.
I am happy to hear completely different approaches too.