Poisson Games and Sudden-Death Overtime

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 $$l = -{\lambda_1} + n \log{\lambda_1} - {\lambda_2} + n \log{\lambda_2} - 2 \log{n!} + \log{\lambda_1}-\log({\lambda_1 + \lambda_2}).$$ This gives $$\begin{equation} \eqalign{ \frac{\partial l}{\partial \lambda_1} &= -1+\frac{n}{\lambda_1}+\frac{1}{\lambda_1}+\frac{1}{\lambda_1 + \lambda_2}\\ \frac{\partial l}{\partial \lambda_2} &= -1+\frac{n}{\lambda_2}+\frac{1}{\lambda_1 + \lambda_2}. } \end{equation}$$ Setting both partials equal to $0$ and solving, we get $$\begin{equation} \eqalign{ (n-\hat{\lambda_1})(\hat{\lambda_1}+\hat{\lambda_2})+\hat{\lambda_2} &= 0\\ (n-\hat{\lambda_2})(\hat{\lambda_1}+\hat{\lambda_2})-\hat{\lambda_2} &= 0, } \end{equation}$$ and so $$\begin{equation} \eqalign{ \hat{\lambda_1} &= (n+1) \frac{2n}{2n+1}\\ \hat{\lambda_2} &= n \frac{2n}{2n+1}. } \end{equation}$$ For example, if both teams score $3$ goals in regulation and team $1$ wins in sudden-death overtime, our MLE estimates are $\hat{\lambda_1} = 3\frac{3}{7}, \hat{\lambda_2} = 2\frac{4}{7}$.

Intuitively this makes sense, because $2n$ goals were scored in regulation time, hence we "expect" that the overtime goal occurred around a fraction $\frac{1}{2n}$ of regulation, so team $1$ scored $n+1$ goals in about $\frac{2n+1}{2n}$ regulation periods and team $2$ scored $n$ goals in about $\frac{2n+1}{2n}$ regulation periods. The standard Poisson process MLE estimates here coincide with the estimates we derived above.

Does this work in practice? Yes! I tested it on my NCAA men's lacrosse model, and it increased the out-of-sample testing accuracy by 0.5%. Surprisingly large for such a small change!