## Sauceda Rating System Description

This rating system is based on the Elo rating formula for chess. I
developed it a few years ago for football, and recently expanded it for
other sports.
For each game, 1 "game point" is disputed. The winning team gets the
largest part, and the loser gets the rest. The formula is:

** gp = 1 - 0.4 ^ (1 + pd/11) **
(Note: the 11 constant is a typical value for basketball; for other sports it may be
smaller).

As an example, here are sample **gp** values for several margins of victory.

- Team wins by 1 point:
**gp = 0.632** (the opponent gets **0.368**)
- Team wins by 10 points:
**gp = 0.826**
- Team wins by 20 points:
**gp = 0.924**

Elo's formula states that a team's "winning expectancy" (**we**) can be determined by

**we = 1 /(1 + 10 ^(Rb - Ra)/400)**
where Ra is the team's rating and Rb is the opponent's rating. Basicly, **we** gives the percentage of game points that a team should win against a given opponent.

A team's total **we** and **gp** can be obtained by summing over each of its games.
The new rating is calculated by

**Rn = Ro + K (gp - we)**
where Rn is the new rating, Ro is the original rating, and K is an constant that determines the speed of convergence.

If a team's overall performance is better than expected (i.e. **gp > we**), then its rating will improve (notice that this does not necessarily require winning!). Conversely, the opponents' ratings will decline.
Each team begins with the same initial rating (1000). With each iteration these ratings will adjust themselves until equilibrium is reached.

The published ratings also include a homefield correction variable that likewise reaches an equilibrium.
The modified formula is:

**we = 1 /(1 + 10 ^(Rb - Ra +/- h)/400)**
**Making Predictions**

As it stands, the formula for **we** should approximate the expected game points (**gp**).
However, it would be more interesting to instead estimate the probability of victory. This is accomplished by replacing 400 with an appropriate prediction constant (**k**), which will vary among the various sports. As an example, assume that k = 300, Ra = 1100, and Rb = 900. Then the probability that A would defeat B is

**1 / (1 + 10 ^ (900 - 1100)/300) = 0.8227 = 82.27%**
In general, to find the probability that A defeats B use the following prediction formula:

**1 / (1 + 10 ^ (Rb - Ra +/- h) / k))**

Eduardo Sauceda,
edited by Kenneth Massey
Massey's Ratings and Rankings |
Theory