Effect of Serve Rules/Scoring in Ping Pong/Table Tennis

In ping pong, there seem to be three primary rule variations around switching which player serves:

• Switch every 2 cumulative points (points can be scored regardless of serving)
• Switch every 5 cumulative points (points can be scored regardless of serving)
• Switch upon loss of a point, but don't record a point (points can only be scored when serving)

How do these rule variations benefit/hurt the better player?

In all cases, I'm going to assume a best-of-5 series (first player to win 3 games wins), and each game is to 11, win by 2.

For calibration of the simulator, I'm assigning a "favorite" and an "underdog", where the "favorite" has to have a slightly higher chance of winning a point when they serve vs when they return:

• Favorite wins point on serve: 55%
• Favorite wins point on return: 50%

I then ran each rule set described above 10,000 times, with the "favorite" serving first. The probability the "favorite" wins a best-of-5 match:

• Switch every 2 points: 73.83%
• Switch every 5 points: 73.93%
• Switch serve on lost point, no point recorded: 76.69%

So switching serve when the returning player wins the point (and not recording a point) is a huge advantage to the better player - because it effectively lengthens the game, because points are only recorded while serving. It's a well known phenomenon that the shorter the game, the more randomness is exhibited, and the better the chances are for the underdog.

However, this flips when the "underdog" serves first, but only for the third rule set:

• Switch every 2 points: 73.58%
• Switch every 5 points: 73.77%
• Switch serve on lost point, no point recorded: 74.39%

The first two serving patterns (switching every 2 or 5 points) are more fair, since switching serve is independent of who scored points, and results in virtually the same win probability regardless of who serves first.