OK, imagine a game called pong, which is played between two players as a sequence of points. The first player to win three points wins the match, so a match can have three, four or five points in total.

Suppose the same two players, A and B, play ten pong matches. We intend to model pong on the basis that every point is independent of every other point, with the probability of player A winning any particular point being p (so that the probability of player B winning any point is 1-p). We don’t know the value of p, but we would like to estimate it based on the results of the ten matches that these players have contested. Suppose we have the complete point-by-point results of these ten matches, showing which player won each point in sequence. For example, the data set might be as follows (where each row is a match and the winners of the points are read left to right):

ABBAA

BAAA

AABBA

BBB

BABB

AAA

AABA

BAAA

AABBB

AAA

Someone suggests that for each match we calculate the proportion of points won by player A in that match, and then we take the mean of these values across the ten matches as our estimate of p (let’s call our estimator p’). With the example data set, the proportion of points won by player A in the first match is 0.6, in the second is 0.75, in the third is 0.6 etc., and p’ would then be calculated as the mean of these ten values, in this case 0.61.

Someone else suggests defining p’ as the total number of points won by player A in the ten matches divided by the total number of points played in the ten matches. With the example data set, this would give a result of 24/40 = 0.6 as our estimate of p.

I am wondering about these two suggestions for estimating p, and in particular to determine whether either estimate of p is unbiased. (We say that p’ is an unbiased estimator of p if and only if, for every value of p with 0<=p<=1, the expected value of p’ is p.) Are there any alternative (and perhaps better) suggestions for estimating p based on the data? I would ideally prefer unbiased estimators to biased ones, but why would any unbiased estimator be any better than any other one? Might a biased estimator actually be better in practical terms than an unbiased one?

Thanks!