# Rating Systems for Human Abilities: The Case of Rating Chess Skill (UMAP)

### Author: William H. Batchelder and Robert S. Simpson

This module illustrates how to construct and use a mathematical model for rating chess ability. The model is probabilistic in character, and it is based on Arpad Elo's chess rating system that is used both nationally and internationally to rate chess players. The key assumption is that the probabilities of a win, loss, or draw between two chess players each depend only on the difference between the two players' ratings. When this assumption is suitably augmented with other assumptions, a system is developed that rates newcomers as well as changes the ratings of established tournament players. Some empirical validation of the system is discussed.** Table of Contents:INTRODUCTIONTHE NATURE OF CHESS RATING SYSTEMSEXAMPLES OF MONOTONE RATING SYSTEMS** The Thurstone System

The Uniform System

**Obtaining Actual Ratings**

The Nature of the Problem

Applying Estimation Theory to Obtain Ratings

**ESTIMATION THEORY FOR THE ELO SYSTEM**

Initial Group Estimation for the Elo System

Newcomer Estimation for the Elo System

Dynamic Estimation for the Elo System

**EMPIRICAL VALIDATION OF THE ELO SYSTEM**

SOLUTIONS TO EXERCISES

REFERENCES

ABOUT THE AUTHORS

SOLUTIONS TO EXERCISES

REFERENCES

ABOUT THE AUTHORS

#### Mathematics Topics:

#### Application Areas:

#### Prerequisites:

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