Question: Is it appropriate to use a multinomial logit if there is only a single outcome for all but one of the three possible outcomes? How could I reframe to incorporate the possible outcomes.

Background: I am modeling a three-horse(or elephant) race(Trump, Kasich, Cruz) for the state of New York(The Republican Party Primary), and I have a dataset with 62 records that looks like this:

County      |Winner |Trump Votes|Cruz Votes | Kasich Votes | Demographic Variables...
Cattaraugus |1      |1000       |       500 | 700          | .57
Erie        |1      |39589      |      7964 | 13136        | .38

I was planning on using a multinomial logit (multinomial logistic regression) to determine significant independent variables, however...

it just so happens that one horse / elephant (Trump) wins all but one of the data points(counties) in my dependent variable(Candidate). One of the three candidates does not show up in my dependent variable as a victor of any county, while the third wins only one county.

Would this be a poor sample/dataset to use a multinomial logit on?

If so.... I also have individual votes for each candidate so that they could be ranked as 1st, 2nd, 3rd or by some percentage margin etc. Is there be a better way to code these in this instance?

Note: Here is a little more background on my analysis via a question asked earlier


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