I have created a motorcycle race prediction model that is given pairs of racers and outputs the probability of each rider beating the other in each pairwise comparison. That info is then processed using the Bradley-Terry model to determine each rider's probability of winning.

With that information (pairwise probabilities and winning probabilities), how may I calculate the probability of each rider finishing in nth place?

So far, I've tinkered with the Harville method, which has been pointed out to have some unrealistic assumptions. I'm currently working with the method in https://github.com/microprediction/winning, but I'm not totally sure I understand the output can be used to determine the probability of each racer finishing in a specific position.

I'd like to know if there is anything obvious I can do with the info & models I'm currently using and/or if there are other models I should look into.

Much help from: Peter Cotton (https://math.stackexchange.com/users/92982/peter-cotton), Given every horse's probability of winning a race, what is the probability that a specific horse will finish 2nd and 3rd?, URL (version: 2021-09-10): https://math.stackexchange.com/q/3260483

  • $\begingroup$ Please share a minimum reproducible dataset and code to help others tinker with the problem $\endgroup$
    – Iyar Lin
    Sep 5, 2022 at 19:00

1 Answer 1


It is possible to use the Elo rating system to rank players in competitive sports. Elo assigns points to wins and losses based on the frequency of beating opponents, the value of the points is relative to the strength of the opponents. The ELO point values could be converted to ordinal / nth prediction.

  • $\begingroup$ I actually used the Elo Rating System before moving to the Bradley-Terry model, but the Elo system can be pretty influenced by the order of the pairwise data being fed into it until the dataset is large enough. I have witnessed more reliable finish order results when I switched to the Bradley-Terry model. What I'm actually looking for is the probability densities of each racer in each position, not just the likely finish order. $\endgroup$
    – bdwilson24
    Sep 7, 2022 at 15:07

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