= Solution
No intercept should be included unless the ordering of the two teams has an external meaning such as home advantage. If
$$
\eta_{ij,k\ell}
=\beta_i+\beta_j+\beta_{\{i,j\}}
-\beta_k-\beta_\ell-\beta_{\{k,\ell\}},
$$
then reversing the teams gives $\eta_{k\ell,ij}=-\eta_{ij,k\ell}$, so the <logistic model> automatically gives
$$
\mathbb P(\{k,\ell\}\text{ wins against }\{i,j\})
=1-\mathbb P(\{i,j\}\text{ wins against }\{k,\ell\}).
$$
An intercept would not change sign when the teams are swapped and would violate this required symmetry. Thus \b[an intercept does not make sense in the stated model].
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