The Angry Statistician

Let's say we have a game that can be reasonably modeled as two independent Poisson processes with team $i$ having parameter $\lambda_i$. If one team wins in regulation with team $i$ scoring $n_i$, then it's well-known we have the MLE estimate $\hat{\lambda_i}=n_i$. But what if the game ends in a tie in regulation with each team scoring $n$ goals and we have sudden-death overtime? How does this affect the MLE estimate for the winning and losing teams? Assuming without loss of generality that team $1$ is the winner in sudden-death overtime. As we have two independent Poisson processes, the probability of this occurring is $\frac{\lambda_1}{\lambda_1 + \lambda_2}$. Thus, the overall likelihood we'd like to maximize is $$L = e^{-\lambda_1} \frac{{\lambda_1}^n}{n!} e^{-\lambda_2} \frac{{\lambda_2}^n}{n!} \frac{\lambda_1}{\lambda_1 + \lambda_2}.$$ Letting $l = \log{L}$ we get \[l = -{\lambda_1} + n \log{\lambda_1} - {\lambda_2} + n \log{\lambda_2} - 2 \log{n!}...