Arpad Elo — The Rating of Chessplayers, Past& Present

Summary

Arpad Elo's 1978 book "The Rating of Chessplayers, Past & Present" is the original authoritative source on the Elo rating system, written by its inventor. The book provides the complete mathematical derivation of the system, including the normal distribution function underlying the rating formula, the performance rating formula, and the rationale for the 400-point scale.

The book covers: (1) the theoretical foundation of rating as measurement of chess performance, (2) the normal distribution assumption for game outcomes, (3) the rating update formula, (4) the performance rating formula, and (5) historical ratings computed retrospectively for famous chess players from the 1800s onward.

Key Concepts from the Book

  • Performance rating formula: RP = RD + 400 × (W - 0.5), where W = points scored / games played, RD = opponent rating average. This computes the rating a player "performed at" over a set of games.
  • Normal distribution assumption: Elo assumed game outcomes follow a normal distribution with mean equal to the player's true rating and standard deviation related to the 400-point scale.
  • The 400-point scale: The scale was chosen so that a player with a 200-point higher rating would be expected to score 76% against the lower-rated player — a "合理" advantage level.
  • K-factor rationale: Elo discussed how K-factor should vary by player experience: higher K for new players (more volatile), lower K for established players.
  • FIDE adoption: The book explains the adoption by US Chess Federation in 1960 and FIDE in 1970.

The Original Elo Formula

Expected score for player A:
$$E_A = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{(R_A - R_B)/200} e^{-t^2/2} dt$$

Elo originally used the normal distribution integral. The logistic approximation (1 / (1 + 10^(-dr/400))) is mathematically simpler and works equally well in practice.

Performance rating:
$$RP = R_D + 400 \times (W - 0.5)$$

Where R_D = average rating of opponents, W = points scored / games played.

Notes

  • This is the original source for the Elo rating system — the existing elo-rating-system.md note covers the modern logistic form; this source provides the original normal distribution derivation
  • Key historical note: Elo originally used the normal distribution, not the logistic. The logistic approximation was adopted because it's mathematically simpler and produces virtually identical results.
  • The book confirms that the 400-point scale was a deliberate design choice: it gives "reasonable" win probabilities (200-point difference → 76% win rate)
  • Elo's K-factor discussion: he recommended K=32 for new players (<30 games), K=16 for intermediate, K=10 for veterans. Modern sports adaptations use different K-factors.
  • The retrospective rating computation for historical players (Bobby Fischer, Emanuel Lasker, etc.) is a fascinating application — the system can compute ratings for players who never played under the system
  • For the World Cup model: the book's K-factor discussion is directly applicable — teams with few international games should have higher K (faster adaptation), established teams should have lower K (more stable ratings)