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Multinomial logit models(MNL) are generalization of binary logit models. Your DVs are still 1/0 variables. They are grouped together by race. For example, in a 5 horse race, your data would be made up of 5 observiations one for each horse. Each observation would have an indicator variable (1/0) if the horse won the race and post performance data associated with that horse. Unless there is a DH, one of the 5 observatations will have 1 and other 4 will be zero.
You would have to have a stat package that estimates MNL models. I use SAS. It is expensive, but I use it for my regular work. There is a very good free open source stat package called R that I know has the ability to estimate these type of models.
I have a copy of the original 1986 Bolton paper and could send you a scanned copy next week. To show you how far we have come in 25 years, the paper only uses 200 races to estimate its models because of the data entry time require to enter past performance data by hand.
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