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Let's say I've developed a rating for a series of horses in a race and I want to convert those ratings to a win probability. What techniques/approaches can be used to do so?

Horse 1 - 67.6
Horse 2 - 56.3
Horse 3 - 78.3
etc.

I have sufficient data to test the system on unseen data... but just looking for a little kick-start to help me approach this the right way.

Thanks!

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Conversion from rating to win probability is strictly connected to rating algorithm that produces ranking score. For example, in most known Elo system (that works for chess or other one-on-one games/sports), if Player A has a rating of $R_A$ and Player B a rating of $R_B$, the formula for the expected score (i.e. probability of winning) of Player A is

$E_A = \frac 1 {1 + 10^{(R_B - R_A)/400}}$.

Similarly the expected score for Player B is

$E_B = \frac 1 {1 + 10^{(R_A - R_B)/400}}$.

This could also be expressed by

$E_A = \frac{Q_A}{Q_A + Q_B}$ and $E_B = \frac{Q_B}{Q_A + Q_B}$, where $Q_A = 10^{R_A/400}$ and $Q_B = 10^{R_B/400}$.

For a match played by players A and B, where $R_A=1613$ and $R_B=1477$, for example, it leads to $E_A = 0.686$ (and so $E_B = 0.314$).

Obviously, that formula doesn't work in a different rating structure for the same kind of game, saying for example that you get two point after a win, and lose one point after a loss, but even a minor change in same Elo structure (e.g. denominator) drives to different winning probability calculation.

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