An objective measure for Luck doesn't exist, so no such comparison can be made.
What is luck, in game terms? One definition is, the chance that you can win or lose a game, due to random chance and not due to decisions that you or your opponent made. A game of 100% pure luck, cannot have any decisions that you or your opponent make have any impact on who wins (i.e. You should be just as likely to win/lose by making random moves). A game of 100% pure skill, cannot have the result of who wins or loses determined by anything other than decisions that you or your opponent make, (i.e. Your decisions should cause you to win/lose more often than random moves).
Luck can be equated with randomness. An objective measure of luck would be:
The difference between how often you win a game compared to how often a random player would win a game.
While this is a completely objective, it isn't very useful. Some examples are in order:
A very simple game
This game consists of a shuffled deck of 3 cards labeled 1, 2, and 3. The game is played over two rounds, with the deck shuffled between rounds. Each player chooses a number between 1-3, then the top card of the deck is revealed.
- If the card is >= to the highest guesses: the player with the highest guess scores the other players guess (if both are the highest, both score)
- If the card is < one player's guess: the other player scores the high guess.
- If the card is < both player's guesses: neither player scores.
Random Play: There are 27 possible outcomes in this game each round.
R(1)¦ 1,1 ¦ 2,0 ¦ 3,0 ¦ 0,2 ¦ 0,0 ¦ 0,0 ¦ 0,3 ¦ 0,0 ¦ 0,0
R(2)¦ 1,1 ¦ 0,1 ¦ 3,0 ¦ 1,0 ¦ 2,2 ¦ 3,0 ¦ 0,3 ¦ 0,3 ¦ 0,0
R(3)¦ 1,1 ¦ 0,1 ¦ 0,1 ¦ 1,0 ¦ 2,2 ¦ 0,2 ¦ 1,0 ¦ 2,0 ¦ 3,3
An expert player will never choose 3 for the first round. Doing so would result in their opponent scoring 3 points 33.3% of the time (ensuring a loss), drawing 4/9ths of the time, and only gaining points in 2/9 of the outcomes. An expert will never choose 3. If we compare a random player to an expert:
- Rnd1: E1,R1 = 3/9 Win, 2/9 Lose, 4/9 Draw
- Rnd1: E1,R2 = 5/9 Win, 4/9 Lose
Rnd1: E1,R3 = 5/9 Win, 4/9 Lose
Rnd1: E2,R1 = 14/27 Win + 6/27 Lose, 7/27 Draw
- Rnd1: E2,R2 = 3/9 Win, 2/9 Lose, 4/9 Draw
- Rnd1: E2,R3 = 14/27 Win, 8/27 Lose, 5/27 Draw
The expert player will open with Choose 2, and will win nearly twice as often as the random player. So, we will give this game a score of 60% skill (didn't work out the exact math for how often a random player draws, something close to 16%). So let us change the game. Instead of 3 cards, we will have 10 cards from 1-10. Now the random player has more bad choices. Instead of just 1/3rd of the choices being bad, now 4/10ths are bad. Is the game anymore skill based? An expert player will improve their win percentage against a random player, but what does that really mean?
We can make other rather trivial changes to the game like, not shuffling between rounds or removing several cards face-up from the shuffled deck (10-card version) that would increase an expert players win percentage versus a random player. These objective measures just don't have any meaningful value when comparing these trivial guessing games. If you take the fact that when trying to objectively measure the luck in these simple games, you can see the trouble of trying to apply these measurements to games with wildly variant forms of randomization.