Calculate advantage of Serves in Table Tennis In table tennis there's an ongoing controversy about "illegal" serves. Often the question arises how strong the influence of serves on winning probability is. I stumbled upon a recent paper, but I find it not convincing.
The problem to me is how to calculate the influence of serves while the two opponents have a difference in playing strength and a different number of serves. (In table tennis serve changes every two points and every game another player is serving first, so depending on the results of the games there may be a small difference in number of serves.)
Here's an example:
Player A vs. Player B: 11:7 9:11 11:5 11:9 11:7
Points played: 92
Points won: A = 53, B = 39
Serves: A = 48, B = 44 (A started serving in games 1, 3 and 5)

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|          | Won on own serve | Lost on own serve |
| Player A |               32 |                16 |
| Player B |               23 |                21 |
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Player A won 57.6% of all points. While serving he won 66.7%.
Player B won 42.4% of all points. While serving he won 52.3%.
It' obvious from these numbers that serving helps to win the point.
Given these numbers how can I calculate the advantage of having the right to serve in table tennis?
 A: With this information, you could set-up the following for every match you have data for as follows:
$$\text{logit} P( \text{player } i \text{ wins point} ) = \theta_i - \theta_j + \beta \times [ 0.5 - 1\{\text{Player }i \text{ does not serve}\} ],$$
where $i$ indexes the first player and $j$ indexes his/her opponent. It does not really matter, which of the two players you call $i$ and which one $j$.
For any particular match, this can take two values: $\text{logit} \pi_{ij1} = \theta_i - \theta_j + 0.5 \times \beta$ when player $i$ serves and $\text{logit} \pi_{ij0} = \theta_i - \theta_j - 0.5 \times \beta$ when player $j$ serves. The parameters $\theta_i$ indicate the strength of players (higher = more likely that that player wins a point).
If you then assume that that winning a point follows a binomial distribution with the probabilities $\pi_{ijk}$. It seems reasonable to ignore the effect of getting slightly more or fewer serves depending on how the match goes.
You could also try to account for how the strength of the players may vary across matches by making the $\theta$s random effects or estimating separate $\theta$s for the same player for different matches.
