Bridge is a card game played by 2 teams of 2 players each. A bridge club records the outcomes of many games between teams formed by its $m$ members. The outcomes are modelled by

$$\mathbb{P}(\text { team }\{i, j\} \text { wins against team }\{k, \ell\})=\frac{\exp \left(\beta_{i}+\beta_{j}+\beta_{\{i, j\}}-\beta_{k}-\beta_{\ell}-\beta_{\{k, \ell\}}\right)}{1+\exp \left(\beta_{i}+\beta_{j}+\beta_{\{i, j\}}-\beta_{k}-\beta_{\ell}-\beta_{\{k, \ell\}}\right)},$$

where $\beta_{i} \in \mathbb{R}$ is a parameter representing the skill of player $i$, and $\beta_{\{i, j\}} \in \mathbb{R}$ is a parameter representing how well-matched the team formed by $i$ and $j$ is.

(a) Would it make sense to include an intercept in this logistic regression model? Explain your answer.

(b) Suppose that players 1 and 2 always play together as a team. Is there a unique maximum likelihood estimate for the parameters $\beta_{1}, \beta_{2}$ and $\beta_{\{1,2\}}$ ? Explain your answer.

(c) Under the model defined above, derive the asymptotic distribution (including the values of all relevant parameters) for the maximum likelihood estimate of the probability that team $\{i, j\}$ wins a game against team $\{k, \ell\}$. You can state it as a function of the true vector of parameters $\beta$, and the Fisher information matrix $i_{N}(\beta)$ with $N$ games. You may assume that $i_{N}(\beta) / N \rightarrow I(\beta)$ as $N \rightarrow \infty$, and that $\beta$ has a unique maximum likelihood estimate for $N$ large enough.