So let's assume a new entity, gamer-economus. Gamer-economus plays board games hoping to defeat their opponents by demonstrating their skill differentials across a variety of games. They prefer to minimize the time spent on games where achieving victory is beyond their control. Luck Time is a new metric designed for the gamer-economus in many of us.
Luck time = L*T
L = Luck component. If a game is 50% luck, the completely dominant player will win 75% of the time. This is a number generated by a lot of hand waving, because as we know from chess even high skill games have lower skilled players being higher skilled players occasionally.
T = Average time to completion.
Combined, the Luck Time measure gives us a sense of how much time Gamer-economus "wasted" on a game that was in the end decided by a dice roll.
Examples:
In "flip the fair coin" - the game is dominated by luck. L=1 It's also a quick game, so T approaches zero.
In chess or go - the game is dominated by skill. Lack of skill may cause unpredicted lucky situations to arise, but a computer with enough time foresee these situations. The better player may not win every game if the skill gap is close enough of if the losing player learns from their mistakes, but any time the game is played skill generally determines the outcome. L approaches 0. T varies.
In the above games, the LT is close to zero. In "flip a coin" it is because the time played is close to zero (a match of "flip a coin" to 100 would be pure luck time), while in chess or go LT approaches zero because L is close to zero.
Applying this analytical method to games in-between is where it becomes interesting. If backgammon and Settlers of Catan both have similar amounts of lucky events that even out over the game to the same extent, then the two games can still be quite different in the luck time measure. This is because the time variable between the two games is vastly different. Settlers can last over an hour, while one game of backgammon is often over in 10 minutes. Given these assumptions, settlers has six times as much Luck Time as backgammon.
If a single game of backgammon was expanded into a match, the time would go up while the luck would go down significantly. This is because there are additional mechanics in the match, such as "doubling" that make the longer match more skill intensive. The LT might actually decrease and the average luck time per point of the match definitely decreases.
But despite their similar reliance on dice rolling, backgammon and Settlers have very different mechanics. Luck Time is more relevant in games where the core mechanic is similar. A shorter amount of Luck Time is a core reason for why many gamers prefer The Resistance over Battlestar Galactica. They are both "spot the traitors" games, but The Resistance finishes each game in between ten and thirty minutes, while Battlestar can take multiple hours. There are reasons to play Battlestar, but it would be aimed at the gamer who enjoys themes and different mechanics for their own sake.
Some games take a while to learn but once the optimal strategy is learned there isn't much variation in how it is applied. The first games played would have low levels of Luck Time, but once everyone figures out the game's strategies the LT would be much higher if there aren't enough subtleties in the game or if the subtleties have a small impact relative to the luck of the game. This is a problem with many board games in general, and it is one of the reasons board gamers switch between many different games instead of repeatedly playing just one or two.
What about Kingmaking?
One slight modification to Luck Time would be Luck Time without Kingmaking and Luck Time plus Kingmaking (Total Wasted Time). Kingmaking occurs when a player who is not going to win gets to use the mechanics of the game to essentially pick the player who ends up winning. Sometimes its occurrence emerges as part of the strategy (the player who was mean earlier will be punished), but sometimes it is pure social dynamics unrelated to the game itself. These social dynamics are part of why Settlers of Catan is a decent tournament game but a subpar general social game.
Luck Time = L*T
Total Wasted Time = L*T + Km*T
Km = Percent of time the winner is picked capriciously by another player
Munchkins is an example of a game that has both a low skill ceiling as well as significant king making. Once everyone knows how to play, the question becomes if players will let someone get to level 10 before them. It has high Luck Time, but even higher Total Wasted Time - especially when particularly aggressive play at lower levels can make the game drag on for hours.
When thinking about games, there are many factors to consider. The style of the game, how fun the mechanics are, the level of the skill ceiling, how fun the theme is, how easy the game is to learn and how long the game takes should all be taken into account. But even if gamers don't think of themselves as a gamer-economus, the potential for high amounts of Luck Time and Total Wasted Time is a metric that ought to be explicitly considered.