MSE-optimal K-factor of the Elo rating system for round-robin tournament

Result 1

(Approximate Asymptotic Limit of the Updated Rating Vector r* under the MSE-Optimal K-factor as m → ∞). Let r*, r, and r ^t be the n-dimensional vectors of post-tournament ratings, pre-tournament ratings, and the true ratings during the tournament, respectively — as defined in Section 3.2 . Define r ̄ t and r ̄ as the arithmetic mean of the elements in r ^t and r, respectively. Assume that the true ratings obey the conservation of Elo ratings so that the sum of the elements in r ^t equals the sum of the elements in r, i.e., n r ̄ t = n r ̄ .

Consider an m-replicate round-robin tournament, i.e., every player meets every other player m times, in which the MSE-optimal K-factor k_o, given in Eq. (11), is used in the Elo Rating System to update the ratings. Then r* converges to r ^t approximately (in the linear sense) as m → ∞.

Proof

By Eq. (5) and the definition of z given in Section 3.2,

where X is the vector of the scores X _ij . Both X _ij and p are defined in Section 3.2. With k = k_o,

as m → ∞, since X/m → p ^t almost surely by the Law of Large Numbers.

Because treating the model for win probability in greater generality facilitates the proof, we shall consider the general paired-comparison model H, rather than the specific Bradley–Terry model given in Eq. (2). The function H is continuous and strictly increasing within its domain and hence its inverse H⁻¹ exists over the interval {p : 0 < p < 1}. This inverse can be well approximated within a large portion of the interval by a line centered at p = 0.5. A linear approximation of R _i − R _j is therefore given by

for i > j, where α and β are constants that depend on the model H. Likewise, R i t − R j t ≈ α + β π i j .

Note that, for i > j,

where e i j ′ is a (1 × n) row vector in which the ith element is 1, the jth element is −1, and all other elements are zeroes.

Define the n 2 × n matrix W as follows:

where the indices in the subscript of e′ follow the standard order, as described in Section 3.2. By Eqs. (22) and (23), it follows that αj + βp ≈ Wr and αj + βp ^t ≈ Wr ^t , where j is a vector of 1’s and in this instance of dimension n 2 . Thus,

Before the proof is finally demonstrated, three straightforward short results are needed:

where I is the identity matrix of dimension n and J is the (n × n) matrix of 1’s. This result, which can be shown fairly easily, leads to the following one.

1. U = W′ where U is the design matrix of the round robin with n players.

With the aid of all the results above, the limit in Eq. (21) becomes

and thus r* → r ^t approximately, as claimed.

Note that the limit will be exact if a uniform distribution is used for the paired-comparison model, i.e., H ( x ) = c x + 1 2 for − 1 2 c < x < 1 2 c and c > 0.

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