Tennis Game as a Markov Chain

Model a tennis game as a Markov process and compute win and duration probabilities

A tennis game can be viewed as a point-by-point Markov process on score states. Given a per-point win probability p, we can quantify the chance of winning the game and how long it tends to last. An explanation of tennis scoring system can be found here.

In a game of tennis between two players, suppose the probability of the server winning a point is

p=0.6

. There are 17 possible states:

In[1]:=

states={"GameA","GameB","Deux","AdvA","AdvB","40-15","15-40","40-0","30-15","15-30","0-40","30-0","15-15","0-30","15-0","0-15","0-0"};

Use a transition probability matrix to define the tennis Markov chain:

In[2]:=

proc=With[{p=0.6,q=0.4},DiscreteMarkovProcess[17,SparseArray[{{1,1}1,{2,2}1,{3,4}p,{3,5}q,{4,1}p,{4,3}q,{5,2}q,{5,3}p,{6,1}p,{6,4}q,{7,2}q,{7,5}p,{8,1}p,{8,6}q,{9,3}q,{9,6}p,{10,3}p,{10,7}q,{11,2}q,{11,7}p,{12,8}p,{12,9}q,{13,9}p,{13,10}q,{14,10}p,{14,11}q,{15,12}p,{15,13}q,{16,13}p,{16,14}q,{17,15}p,{17,16}q},{17,17}]]];

Visualize the Markov chain as a directed graph of state transitions:

In[3]:=

Graph[proc,VertexSizeLarge,VertexLabelsThread[Range[17]Map[Placed[Style[#,{Bold}],Center]&,states]],ImageSizeLarge,ImagePadding25]

Out[3]=

Find the probability of the server winning the game:

In[4]:=

PDF[proc[Infinity],1]

Out[4]=

0.735729

Repeat the calculation via the concept of reachability by computing the probability of ever reaching the "server wins" state from the starting state:

In[5]:=

MarkovProcessProperties[proc,"ReachabilityProbability"]〚1〛

Out[5]=

0.735729

Compute the expected number of points until the game ends:

In[6]:=

Mean[FirstPassageTimeDistribution[proc,{1,2}]]

Out[6]=

6.48418

Get the same value via transient visit counts. Sum the expected visits to all such states, which equals the expected points played before absorption:

In[7]:=

Total[MarkovProcessProperties[proc,"TransientVisitMean"]]

Out[7]=

6.48418

Find the mean number of states visited:

In[8]:=

MarkovProcessProperties[proc,"TransientTotalVisitMean"]

Out[8]=

5.71329

Find the average number of times the score will be tied at deuce. Deuce is the first transient state:

In[9]:=

MarkovProcessProperties[proc,"TransientVisitMean"]〚1〛

Out[9]=

0.877292

Simulate 10,000 games, stop when the path hits a win/loss state (1 or 2) or reaches 1000 points. For each path, count how many times state 3 (deuce) appears, then take the mean to estimate the expected number of deuce ties:

In[10]:=

Mean[Count[#,3]&/@RandomFunction[proc,{0,10^3},10^4,Method{Automatic,"StoppingFunction"Function[{len,pos},len<1001&&pos≠1&&pos≠2]}]["ValueList"]]//N

Out[10]=

0.8766

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Tennis Game as a Markov Chain

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Tennis Game as a Markov Chain

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