Do I have numerical data or ordinal data? I have collected data and I am analyzing data from a game with 20 contestants over 20 games.
I am running a regression to try to model the final position of a player in the game.
The position of a player is determined by when he/she gets voted out by the other players.
His position is 20th if he is voted out first and his position will be 1st (Winner) if he is voted out last (or equivalently when the 1st runner up is voted out).
I have first assumed the data to be numerical and hence I have ran a linear regression. But do I really have numerical data or do I have ordinal data with 20 levels?
 A: Technically you have ordinal outcome data. Then again, continuous data are also ordinal.
On one hand, as @David says in his answer, sometimes a linear regression will work OK if there are many levels.
On the other hand, sometimes continuous outcomes can be hard to model with regression if there isn't a simple linear relation between the outcome and the predictor values. In that case, a type of ordinal regression called a continuous probability model provides a flexible way to proceed. With that type of ordinal regression approach you don't need to model one intercept for each level of your outcome variable.
So if a linear regression seems to work OK in your case, that will be OK. But ordinal regression can potentially provide more flexibility while maintaining efficiency.
A: As per your explanation the target variable isn't continuous in nature and regression is also not a good fit for it. I would rather treat it as a classification problem with discrete levels for each rank. So, your models output layer should have 20 units followed by a softmax operation to convert it into class/rank probabilities. With this setting you can use cross-entropy loss for optimizing the network.
A: Since there are really a lot of levels, it's not a big mistake to consider your data as numerical (we do something similar with age all the time)
Ordinal regression would be a great idea if we had a ton of data, but, since you will have to estimate 20 intercepts rather than one, I don't think it is really worth it.
My approach would be to try fitting the regression model first, then validate it. If it is good enough, there is no reason to overcomplicate things
