For example, if you play a round of cribbage mathematically perfectly, someone will win because they got better cards on average. But if that difference in scores comes down to 10 points, that could come down to imperfect play or even gambling on a card.
Is it possible to say that cribbage is only for example, 50% skill?
I can think of two mathematical measures that might be useful, both based around Elo ratings.
Is the correct Elo distribution function. Let's suppose Player B beats Player A 64% of the time, and Player C beats Player B 64% of the time. How often does Player C beat Player A? Generally, speaking, the higher the number, the less of a role luck plays, though you should be suspicious of answers much above 83%. (83% is what you would expect if all player's performance in a game (either because of luck or variance in performance) is normally distributed, all with the same variance)
Is the Elo difference, usually the difference between the top player and an average player. Filtered through the answer to the previous paragraph, this basically means how often the top player beats an average player, except this is frequently hard to directly observe (either because top players never play average ones or because the answer is "always"), so you may need to measure this indirectly through Elo.
In many games, you want to think of a "game" as actually a sequence of games for this to make any sense (just like playing cribbage to 120, as usual, is in some sense a sequence of games).
The higher the luck factor in the game, the higher S.D(X) - E(X).
For example, lets look at a game of Poker with high and low amount of luck: Assume a player that have a small edge over a Poker table, meaning that her expected value is positive E[X]>0, now
S.D(X) > E(X).E(1MX) > S.D(1MX)
It would very difficult to come up with a metric that depends solely on the game itself, and not the population of players. It's much easier to answer the question "How large a role does luck play, in comparison to the skill difference of players?" than to answer the question "What is an absolute measure of how much of a role does luck play?"
If it's defined in terms of the players, then there are a variety of metrics. One would be to look at the variance of win rates among players; the higher the variance, the less important luck is. However, you'd have to adjust for the fact that players tend to play players of similar skill, which decreases the variance; if two players of exactly equal skill play, then the game will come down solely to luck. We could also look at how much Elo spread there is (different games use different parameters for their Elo assignments, but if you construct Elo scores using the same parameters across games, then comparing the Elo spreads should give some sense of the size of luck).
