# Modelling Races

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?

• It seems difficult to rank competitors based on a couple of biological features and a single trial or race. You suggest that there is a 'different race' which implies multiple trials or races. Perhaps those multiple trials should be integrated into the model? Feb 5, 2018 at 12:55
• One classic model for rankings is the Plackett-Luce model. In general ranking problems can be viewed as permutation/matching problems, so it would be worthwhile to look into models for permutations. Feb 5, 2018 at 17:59
• Ironically I have a paper on this sort of problem due in 3 weeks, so will hopefully comment here with that, then. Feb 5, 2018 at 22:46
• @N. McA. : is the paper finnished? Nov 25, 2018 at 15:03
• I think I was referring to this. Not really sure how relevant it is. (future readers, sorry if I break this, I may do) n-mca.github.io/work/… Nov 25, 2018 at 16:38

Google "learning to rank", "Optimizing for the AUC" or similar. One idea is to predict probability that person $i$ wins over person $j$ as
$NN(x_{ir},x_{jr})$ with $x_{ir}$ being features of subject $i$ at race $r$, NN some neural network or model. Ground truth is binary. Predicting who wins a race is then pairwise comparisons among all contestants (=ranking)