Simple model

You could model the score from each of the players as some function (in the example image below a beta-binomial, but you may choose whatever is appropriate to you) that is independent of the others.

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Based on those curves you can compute the probability of the ranking of a player. (The order statistic could be computed, but with multiple players, it may be better to simulate it)

For instance when we simulate the above 10 000 times then the outcome frequencies are:

          3rd     2nd    1st
Alice     7150    2078   772
Bob       1909    5518   2573
Carol      941    2404   6655

You could actually also compute that table simply from the observed frequencies of places, but with this method you can handle situations when the group of competitors are not always of the same composition.

Estimating the simple model

  • If you have insight into the scores then you will be able to use those data to estimate the curves for each player.

  • Otherwise you will need to estimate the curves based on the outcomes. This is not easy. The likelihood function is not easy to express and optimize. One way that you could tackle this is by using an approximate method for instance Approximate Bayesian Computation

More complex model

The above method represents the scores from each player as independent from the opponents. But there might be correlations. Especially when interactions in the game occur.

For instance, in a game of soccer, the probability of the score is dependent not only on the player quality but also on the opponent's quality. Even in a relatively individual game like darts you get influences, not just psychologically, but also because of the layered scoring system with legs and sets.

In these circumstances, you might still think of some underlying player score function, and match it against others, but it will not be so easy to match the scores to an outcome. Possibly you may also think of more complex functions (e.g. a score for defense and a score for offense).

Now the player score function will be an input for another model. You can see this as a latent variable model the reasoning of the above player score functions is very similar to the probit model which models a probability that determines some binary outcome as the probability that a latent variable that is Gaussian distributed, surpasses some level.

You could base a model on something like that (a categorical probit model), but at this point, you might need to explain the mechanics of your game better.