How to extend Elo: a Bayesian perspective

A Details of the Taylor series expansion

We already noted that the update is given by:

and that

The second derivative t ″ ( δ ) is given by:

And hence the update is:

B Derivation of margin of victory update

The likelihood and prior are given, respectively, by:

The log posterior is thus proportional to:

Hence

and

Evaluating these at the prior mean μ _δ leads to the update:

with k s h a r e d and s p r e d defined as in the main text.

C Details of correlated skills derivation

D Comparison between derived update and Glicko

In (Glickman 1999), Elo is derived as a special case of a more general approximate Bayesian rating system. Here, we compare the equivalent Elo update derived there with the one in this paper.

Equation (13) in (Glickman 1999) recovers Elo as a special case of Glicko under the assumption that the opponents’ skills are known exactly. Considering only a single contest in a period and matching the notation with that used in this paper, the equivalent k becomes:

Since σ 2 = σ δ 2 / 2 , this k-factor is:

This k-factor is almost exactly the same as the one given in Eq. (17), with the exception of the factor of 1/2 in the second term in the denominator. This factor may be due to the assumption that opponents’ skills are known exactly in the derivation.

E Derivation of approximate log marginal likelihood

The log marginal likelihood for a given match is:

The second integral is analytically tractable and reduces to the log density of s given a normal distribution with mean c 1 μ δ + c 2 and variance σ o b s 2 + c 1 2 σ δ 2 . The first integral is not analytically tractable but can be approximated well by γ ( b μ δ / α ) , where α = 1 + π σ δ 2 b 2 / 8 (Crooks 2009). The log marginal likelihood can thus be approximated by:

as stated in the main text.

F Quadrature estimate of the posterior mean

As stated in Eq. (13), the posterior distribution is proportional to:

Hence,

where c is an unknown normalising constant.

We seek the posterior mean of δ, that is:

We first compute c using Gauss-Hermite quadrature:

We then evaluate the integral in Eq. (63) using Gauss-Hermite quadrature a second time and obtain the final result by multiplying its value by c.

G Estimated Grand Slam additions

Table 8 shows the players with the largest and smallest estimated Grand Slam additions, as estimated at the end of the 2019 ATP season. Rafael Nadal has the largest estimated addition, adding around 52 rating points when playing at a Grand Slam. Stan Wawrinka also ranks highly which is consistent with his reputation as a “big match player”.^[7] On the other hand, Juan Monaco, Carlos Berlocq and Alexander Zverev are all estimated to perform worse at Grand Slams than at other events.

H Equivalence of EKF and Newton-Raphson means

In this section, we show that the approach proposed in Section 2.5 leads to the same posterior mean estimate as an extended Kalman filter using a particular choice of measurement error covariance and observation function.

We first state the equations used to update the mean estimate in extended Kalman filtering. Writing the state estimate before the update step as N ( μ θ , Σ θ ) , they can be written as (Särkkä 2013, p. 70):

where z is the vector of observations, g( x ) is a non-linear function from the state vector to the observation vector, R is the measurement error covariance, and G is the Jacobian matrix of g evaluated at x = μ _θ , the prior mean. Note that the equations above are typically written with subscripts, e.g. y ˜ k , to denote the update at time step k, but we drop these since we consider only a single updating step.

To show the equivalence with our derived approach, we now apply these general equations to the win-only update. We set

which implies

where we have defined the scalar quantity η = γ ( b a ⊺ μ θ ) ( 1 − γ ( b a ⊺ μ θ ) ) to simplify notation in the following. Please note that here, to be consistent with the Kalman filtering literature, the Jacobian is written as a row vector, while other Jacobians in this paper are written as column vectors.

The other equations thus become:

and

implying the following update to the mean:

where z has been set to 1, reflecting the winning outcome.

The approach derived in this paper in Section 2.5 has

which are used in a single Newton-Raphson step:

yielding

We thus see that for the two updates in Eqs. (65) and (66) to be equivalent, we must have

We rewrite the matrix inverse term on the left hand side using the Sherman-Morrison formula, yielding:

Multiplying this expression by a and making use of the associativity of matrix multiplication, the second and third terms in the numerator cancel, leaving

which is identical to the EKF update if

which is the variance of a Bernoulli random variable with success probability γ ( b a ⊺ μ θ ) and thus is a reasonable choice for the measurement error variance.