In a simply setting, we have 2 players, player $A$ and player $B$, playing a game of chess repeatedly. Player 1 wins with probability $p_1$, player 2 with probability $p_2$ and a draw with probability $1-p_1-p_2$.
If $n$ games are played, the score is the $n_1$ - $n_2$ where $n_1$ is the number of times player 1 wins, and $n_2$ the number of times player 2 wins. Draws are disregarded for drawing purposes.
I would like to study $p_1 - p_2$, ie, the difference in probability of winning. To that end, I know an unbiased estimator is simply $(n_1-n_2)/n$. Can anyone give me an estimator for the variance of this estimator for $p_1 - p_2$? An estimator that does not need knowledge of $p_1$ and $p_2$ is ideal, but one that uses them works too since I can just plug in their estimators.
Thanks.
Let $N_3$ denote the number of games which ended in a tie. Note the random vector $(N_1,N_2,N_3)$ has a multinomial distribution with success probabilities $p_1,p_2,1-p_1-p_2$ respectively. Therefore
$$\begin{eqnarray*}V\Bigg(\frac{N_1-N_2}{n}\Bigg)&=&\frac{1}{n^2}V(N_1-N_2) \\ &=&\frac{1}{n^2}\Bigg[V(N_1)+V(N_2)-2\text{cov}(N_1,N_2)\Bigg] \\ &=&\frac{1}{n^2}\Bigg[np_1(1-p_1)+np_2(1-p_2)+2np_1p_2\Bigg] \\ &=& \frac{p_1+p_2-(p_1-p_2)^2}{n}\end{eqnarray*}$$