Is There a Mathematical Advantage in Winning a Game If You Serve First?
No.
I guess I should explain. Some believe that serving first is a mathematical advantage in winning a game, since it means you will sometimes serve more than an opponent in a game. For example, suppose you serve first, and win 11-7. At that point you will have served ten times, your opponent eight. So you won because you served first? No!!! Even if your opponent had the two missing serves, and even if he happened to win both points (the odds are against it), you still would have won 11-9. Mathematically, serving first means you have a better chance of winning by a larger margin (or keeping it closer when you lose), but it makes zero difference mathematically in who wins.
Here's another way of looking at it. A game to 11 is really a best of 20, where we go to deuce if it reaches 10-all. In the case of deuce, you alternate serves, and both players have served ten times before that, so there's no mathematical advantage there to serving first. If it doesn't go to deuce, it means that one of the players scored 11 points within the first 20 points - it just so happens that in our scoring system, we stop the game as soon as someone reaches 11, and so don't play out the entire 20 points. If we did, to use the example above, then both players would serve ten times, and it might change the final score, but no matter how you work it, the player who scored 11 points first is going to win that game, even if the other player were given his missing serves.
In the example above, if the loser were to win both points on his missing two serves, he'd still lose 11-9. If he split the two points, he'd lose 12-8. He might even lose both of them, and lose 13-7.