Does the first-serving team have a structural advantage in pickleball?

In pickleball doubles with conventional side-out scoring, points are scored only by the serving team. The serve alternates during a game, with each team serving until it has faulted twice, except at the beginning of the game, in which case the first-serving team serves until it has faulted once. A game to $n$ can be modeled by a Markov chain in a state space with $4n^2+10$ states. Typically, $n=11$ or $n=15$. The authors, both pickleball players, were motivated by the question in the title. Surprisingly, the answer to that question depends on the number of points needed to win. In a game to 11, the first-serving team has a very slight disadvantage, whereas, in a game to 15, the first-serving team has a very slight advantage. It should be noted that these advantages and disadvantages are so small that they cannot be detected by simulation and are revealed only by an analytical solution. The practical implication is that a team that is offered the choice of side or serve should probably choose side. We investigate the probability of winning a game to 11, as well as the mean and standard deviation of the duration (or the number of rallies) of a game to 11. We compare these results with the corresponding ones when modified rally scoring is used in a game to 21. We also investigate the title question for a hybrid form of rally scoring that combines modified rally scoring and traditional doubles server rotation.
Comment: 25 pages, 10 figures