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!} …