Markov Chain Models in tennis

#1
If you already know tennis rules you can skip this wall of text and go directly to the end.

If you don't know them... well could you read please? It's a simple introduction on how it works.




In tennis, each game is won when one player achieves both of two goals: her score 1 must reach at least 4 points, and (2) must exceed that of her opponent by 2 points. Scores start with “Love” (0 points), then “15” (1 point), “30” (2 points) and “40” (3 points). If both players are tied with at least 2 points, the score is “Deuce”. If both players have a least 2 points, but one is ahead by 1, the score is “Advantage”.

For each point played, only one of two things can happen: one player (let’s call her “A”) can win the point, or the other (let’s call her “B”) can win. The model assigns the value p to the probability that Player A wins a single point, and a value of q to probability that Player B wins. For each point, there exists no other possibilities, so p + q = 1. Importantly, the model assumes that these probabilities never change, no matter how the game unfolds. In addition, the result of each point is independent of the results of the others.

I place each score of the game on a diagram (see below) and indicate, with arrows, which scores can lead, in the next point, to which other scores. In Markov chain terminology, each score represents a state of the game, from which zero or more transitions can occur to other states. As mentioned above, from most states there exist exactly two possible transitions (A wins the next point, or B wins). In tennis, there exist 15 such states, shown as circles. Markov chain terminology refers to them as transient states. From exactly two more states, the game does not continue and no further transitions will occur. I show these exceptional absorbing states as rectangles, labeled, respectively, “A Wins” and “B Wins.” Our modeling assumptions imply that with certainty, the game will eventually leave all of the transient states for one of the absorbing states (hence the terminology).


My question is (see the image below):

Let's say that the server had an initial probability of 30% of doing a 15-0 against an adversary when he's at the server.

Let's assume that he makes 15-0. Now what is his probability of making 30 - 0 ? And what is the probability of making 30 - 15 ?

And if he goes to 30-0... what is the probability of doing a 40-0 ?

1) Should I build a prior model on the probabilities of the tennis player at the service and then to fill the probalities into the Markov chain model?

Example: I make a regression and I find out that this tennis player has 30% chances of going from 30-0 to 40-0 when playing with a particolar kind of opponent (this is based taking in consideration a regression model built using the data about this tennis player)

2) or should I simply do a simple cumulative probability since the Markov Chain is not based on the past state?

Example: tennis serving has a 30% of making a 15-0. He makes it. From now on is the probability of going from 15-0 to 30-0 ?
 

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hlsmith

Less is more. Stay pure. Stay poor.
#2
I truly know next to nothing about Markov chains beyond independence and the Markov blanket (all you need to know is the parents and children of a node then the node is independent of everything else). Wouldn't your example break the independent state assumption I mentioned since every serve is conditional on the last game, set, etc., the full context of interactions between the opponents? But I don't know if this means you can't use the process, but just the conclusions may be a little biased.