How to Measure Luck vs Skill in Games?
Think about the Elo Rating System used to measure the relative skill of Chess players. This is essentially an equation that takes a series of wins and losses and produces a number that can be used to predict the chances of one player winning a game against another player.

One of the inputs to the equation is a "distribution" that describes how much a difference in skill changes your odds of winning. This part is tweakable - they started with a normal distribution, but some groups have switched to a logistic distribution because they've found it to be more accurate.

If we had a fully deterministic game like "Which player weighs more?", the Elo rating would just be your current weight (or some deterministic mapping thereof), and the distribution parameter would be the Heaviside function (0 if less than 0, 1 if greater than 1).

If we modify that game a bit to make a game like "Roll a D6, add that to your weight, and see who has the highest number", the Elo ratings for everyone would still be the same, but the distribution function would have to change so that the behavior from -6 to +6 was a stepwise thing based on the distribution of die rolls.

If we had a zero-skill game like War, everyone would have the same Elo rating (and the distribution function wouldn't matter).

So if we have a two-player game that produces win/loss results, and a set of players, we can force the players to play games against each other and establish Elo ratings for every player, using some distribution function (maybe we start with a normal distribution like chess did). Eventually we'd be able to look at the results of all the games and determine that a logistic distribution (or a log-normal distribution, or some other distribution) did a better job of predicting the results than a normal distribution.

The shape of that distribution - how 'wide' it is, how it tapers near the ends, how steep it is in the center, and so on, is the answer to your question. Of course, this takes a huge amount of effort to measure, but it's theoretically possible to determine that curve.



Addendum: Here are some distribution curves (courtesy of Wolfram Alpha). Apologies that these don't have 100% consistent X/Y axes, but hopefully you get the point.

This is the 'normal' distribution (rather, its CDF). For example, using the yellow line, we'd expect a player with a skill of "+2" to win about 80% of the time.



The logistic distribution is very similar, but has heavier tails (this is really subtle!). The X-axis here is different (why, Alpha?); again using the yellow line we'd expect a player with skill "+50" to win about 90% of the time.



Note that with both the logistic and normal distributions, there is a parameter to tweak how 'wide' the spread is.

Heaviside:



Stepwise (using the "weight + d6" example, this should be transposed up and flatten out at the left/right extremes):