The question that I wish to ask is what is an appropriate likelihood model to use for races.
For example, suppose that we have 4 competitors in a 100m race. We observe the competitors weight and age as input variables ($X$). The rank of the racer is observed as the output variable, $y$.
The first method that I chose to tackle the problem was as a classification problem. eg. map to a latent variable $f_{ir}=w^TX_{ir}$, and then use cross-entropy loss to get a MLE estimage of the weights. Here $w$ is a weight vector, $r$ is the $r$-th race and $i$ is the $i$-th competitor in that race.
However, the problem is that a competitor with the same weight and age can come second in a different race, if there are better competitors. This is what convinced me that this is not ordinal regression (which is what is used in rating systems).
An option to go around this is to let $f_{ir} = f_{ir} - \underset{{i\in r}}{\text{argmin}} \,\,f_{ir}$ so that the latent function is reset with respect to the weakest competitor. This brings us to a second problem. Since ordinal regression gives a threshold, $\theta_k$ on where the $k-th$ rank starts, this would expect gap between the weakest competitor and the strongest to be atleast $\theta_1$. Meaning that if the gap is small two players can be inferred to have rank two, since it did not surpass the thereshold.
Firstly am I looking at the problem wrong? Second if not what would be an appropriate likelihood to represent ranks within a race?
