Consider a statistical model for a competition experiment in which each observational unit is an ordered pair \((i, j)\) of distinct competitors (chess players). The state space consists of three outcomes won, drawn, lost, ordered from the viewpoint of player \(i\), and coded for convenience as \(1,2,3\) so that \(Y_{i, j}+Y_{j, i}=4\). Consider the ordinal trinomial model

\[
\text { logit } \operatorname{pr}\left(Y_{i, j} \leq right)=\gamma_{r}+\alpha_{i}-\alpha_{j}
\]

depending on two threshold parameters \(\gamma_{1} \leq \gamma_{2}\) plus player strength parameters \(\alpha_{1}, \ldots, \alpha_{n}\).

In what sense is this model inconsistent? How do you modify it to resolve the inconsistency?