The Elo rating system in chess

Nicholas R. Moloney (with Mariia Koroliuk)

SaclayOctober 3, 2017

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Outline

Elo ratings

Remarks on rating/rankings

ELO rating update formula

AR(1) modeling

Stationarity

Intrinsic strength

AR(1) analysis

Autoregressive parameter

Noise amplitude

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Outline

Elo ratings

Remarks on rating/rankings

ELO rating update formula

AR(1) modeling

Stationarity

Intrinsic strength

AR(1) analysis

Autoregressive parameter

Noise amplitude

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Examples of ratings/rankings

Women’s Volleyball

1. China 330

2. USA 298

3. Serbia 272

4. Brazil 230

5. Russia 206

Chess

1. Carlsen 2827

2. Vachier-Lagrave 2804

3. Kramnik 2803

4. Aronian 2802

5. Caruana 2799

Men’s Tennis

1. Nadal 9465

2. Federer 7505

3. Murray 6790

4. Zverev 4470

5. Cilic 4155

Coding

1. Petr 3676

2. rng_58 3670

3. tourist 3427

4. Um_nik 3274

5. xudyh 3271

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Rating update equation

Xn+ℓ = Xn + K(Wa − We)

Xn rating after n games

K scale factor

Wa actual score

We expected score

We(D) =1

1 + 10−D/400

0

0.2

0.4

0.6

0.8

1

-300 -200 -100 0 100 200 300

expe

cted

sco

re W

e

rating difference D

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Rating update: example calculation

opponent’s rating result

2515 1⁄2

2472 1

2430 1

2608 0

Xn = 2500

K = 15

Wa = 2.5

We = 1.97

New rating:

Xn+4 = Xn + K(Wa − We)

= 2500 + 7.95

= 2508 (after rounding)

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Incorrect formula in The Social Network

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Incorrect chessboard in watercolor

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

Elo rating

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

Elo rating

gain

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

Elo rating

gain loss

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

movingb. c.

Elo rating

gain loss

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

movingb.c.

Elo rating

gain loss

deflation b

gain

Elo rating

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

movingb.c.

Elo rating

gain loss

deflation b

gain

Elo rating

b b

loss

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

movingb.c.

Elo rating

gain loss

deflation b

gain

Elo rating

b b

loss

b

gain

Elo ratinginflation

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Complicated non-stationary dynamics

b b b bb b b b b bb b b b b b b

movingb.c.

Elo rating

gain loss

deflation b

gain

Elo rating

b b

loss

b

gain

Elo ratinginflation

loss

b

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Outline

Elo ratings

Remarks on rating/rankings

ELO rating update formula

AR(1) modeling

Stationarity

Intrinsic strength

AR(1) analysis

Autoregressive parameter

Noise amplitude

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Lifetime evolution of a player’s rating

1800

2000

2200

2400

2600

2800

0 10 20 30 40 50 60 70 80

ELO

rat

ing

age

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Lifetime evolution of a player’s rating

1800

2000

2200

2400

2600

2800

0 10 20 30 40 50 60 70 80

ELO

rat

ing

age

stationary

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AR(1) modelingXn+ℓ = Xn + K(Wa − We)

Key assumptions:

◮ player’s rating fluctuates around an intrinsic value X⋆.

◮ average rating of opponents X is close to X⋆.

As a consequence:

Wa ≈ We(X⋆ − X, ℓ)

≈ ℓWe(0)− ℓXW′

e(0) + W

′

e(0)

ℓ∑

i=1

X⋆i

≈ ℓWe(0)− ℓXW′

e(0) + ℓW ′

e(0)X⋆ +

√ℓW ′

e(0)σZ,

where approximately

Z ∼ N(0, 1).

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AR(1) modelingXn+ℓ = Xn + K(Wa − We)

Xn+ℓ = Xn + K[Wa(X⋆ − X, ℓ)− We(Xn − X, ℓ)]

AR(1) modelingXn+ℓ = Xn + K(Wa − We)

Xn+ℓ = Xn + K[Wa(X⋆ − X, ℓ)− We(Xn − X, ℓ)]

≈ [1 − ℓKW′

e(0)]Xn + ℓKW

′

e(0)X⋆ +

√ℓKW

′

e(0)σZn+ℓ

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AR(1) modelingXn+ℓ = Xn + K(Wa − We)

Xn+ℓ = Xn + K[Wa(X⋆ − X, ℓ)− We(Xn − X, ℓ)]

≈ [1 − ℓKW′

e(0)]Xn + ℓKW

′

e(0)X⋆ +

√ℓKW

′

e(0)σZn+ℓ

This is an AR(1) process with

φ = 1 − ℓKW′

e(0)

EXn = X⋆.

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AR(1) modelingXn+ℓ = Xn + K(Wa − We)

Xn+ℓ = Xn + K[Wa(X⋆ − X, ℓ)− We(Xn − X, ℓ)]

≈ [1 − ℓKW′

e(0)]Xn + ℓKW

′

e(0)X⋆ +

√ℓKW

′

e(0)σZn+ℓ

This is an AR(1) process with

φ = 1 − ℓKW′

e(0)

EXn = X⋆.

Putting in some realistic numbers:

W′

e(0) ≈ 1/700

K ≈ 10

ℓ ≈ 10-20

=⇒ φ ≈ 0.7-0.85

Only X⋆ and σ to fit!

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Outline

Elo ratings

Remarks on rating/rankings

ELO rating update formula

AR(1) modeling

Stationarity

Intrinsic strength

AR(1) analysis

Autoregressive parameter

Noise amplitude

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AR(1) analysis

2000

2200

2400

2600

2800

0 10 20 30 40 50

ELO

rat

ing

i

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AR(1) analysis

2000

2200

2400

2600

2800

0 10 20 30 40 50

ELO

rat

ing

i

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AR(1) analysis

2000

2200

2400

2600

2800

0 10 20 30 40 50

ELO

rat

ing

i

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AR(1) analysis

2000

2200

2400

2600

2800

0 10 20 30 40 50

ELO

rat

ing

i

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AR(1) analysis: autoregressive parameter φYn+ℓ = φYn + σZn+ℓ

0

0.2

0.4

0.6

0.8

1

2300 2400 2500 2600 2700 2800

φ

ELO rating

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AR(1) analysis: autoregressive parameter φYn+ℓ = φYn + σZn+ℓ

0

0.2

0.4

0.6

0.8

1

2300 2400 2500 2600 2700 2800

φ

ELO rating

ℓ = 15

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AR(1) analysis: autoregressive parameter φYn+ℓ = φYn + σZn+ℓ

0

0.2

0.4

0.6

0.8

1

2300 2400 2500 2600 2700 2800

φ

ELO rating

ℓ = 50

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AR(1) analysis: noise amplitude σYn+ℓ = φYn + σZn+ℓ

5

10

15

20

25

30

2300 2400 2500 2600 2700 2800

σ

ELO rating

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AR(1) analysis: noise amplitude σYn+ℓ = φYn + σZn+ℓ

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Summary

◮ ELO ratings are intended to be predictive.

◮ AR(1) approximates stationary rating trajectories.

◮ Computers may provide ‘objective’ ratings in future.

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Summary

◮ ELO ratings are intended to be predictive.

◮ AR(1) approximates stationary rating trajectories.

◮ Computers may provide ‘objective’ ratings in future.

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