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I'm working on a problem that involves ranking based on pairwise comparisons (it's for a scientific problem, not actually for games). My comparisons return a numerical score (in practice roughly normally distributed around zero), not a binary outcome (win/lose or draw) like in Elo. For an A vs B comparison score>0 means A wins and score<0 means B wins, but the magnitude of the score is related to how "strong" the win was. Scores in repeat comparisons are likely to be roughly normally distributed around the average of previous scores, so a score with large magnitude also gives high probability that future scores would have the same sign.

Is there an extension to Elo or an Elo-like model which can use numerical scores?

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    $\begingroup$ This paper may be helpful: Extension of the Elo rating system to margin of victory $\endgroup$
    – Scene
    Commented Nov 22, 2023 at 18:56
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    $\begingroup$ What you do is use the difference in ELO scores to predict the comparison difference (symmetrically so $0$ ELO difference predicts $0$ comparison difference), then adjust the scores by some (possibly linear) function of the difference between the actual comparison difference and the predicted comparison difference (one up and the other down by the same amount). $\endgroup$
    – Henry
    Commented Nov 22, 2023 at 19:35
  • $\begingroup$ @Henry Sure, that would work for predicting the difference. I was also hoping that using the magnitude of the difference when ranking can result in more stable rankings with (much) fewer comparisons - any ideas along those lines? $\endgroup$
    – Alex I
    Commented Nov 22, 2023 at 19:38
  • $\begingroup$ Volatility is affected issues like parameters chosen ($k$ in ELO), randomness in comparisons, and transitivity between comparisons of multiple items. $\endgroup$
    – Henry
    Commented Nov 22, 2023 at 19:43

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