= Solution
Whenever players 1 and 2 occur, the <linear predictor> contains their parameters only through
$$
\gamma_{12}=\beta_1+\beta_2+\beta_{\{1,2\}}.
$$
For any constants $a,b$, replacing
$$
(\beta_1,\beta_2,\beta_{\{1,2\}})
\longmapsto
(\beta_1+a,\beta_2+b,\beta_{\{1,2\}}-a-b)
$$
leaves every fitted probability and hence the <likelihood function> unchanged. The parameterization is therefore <identifiability>[nonidentifiable]: at most $\gamma_{12}$ can be estimated uniquely. Hence \b[there is no unique maximum-likelihood estimate of the three separate parameters].
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