The fact that your data is grouped into matches really has no effect on the value of p. p will have that value no matter how you arrange the rules of how a match is won via winning individual points. The minimum error estimate of p is thus going to use all of the data as a big pool and ignore the fact that it is grouped into matches.
The estimate is obviously
and this is also an unbiased estimate.
The problem is much more interesting when you don't have access to individual point results but only have data on the results of matches. Then you will probably run into a biased estimator.