Consider the Perfect Diplomacy Player. This gent, by merely suggesting an idea, and by the force of his very likable character, successfully manages to get everyone to move as per his wishes throughout the game. When they are in danger, he manages to talk them into looking the other way, and wins. PDP is so highly skilled that he is perfectly in control, and requires no luck whatsoever. Similarly for the Perfect Chess Player. Such levels of skill may be extremely rare, but that is beside the point.
Consider, on the other hand, the Perfect Coin Toss Player. Try as hard as he may, he, on an average, beats his opponent no more or no less than half of the time, which in a two-player game, is what you'd expect, purely by luck. So, while luck plays a big role in the coin toss game, if it plays a comparable role in my diplomacy games, it is only because I'm not good enough, and I need to improve!