$$g(\mathrm{RD}) = \frac{1}{\sqrt{1 + 3q^2 \mathrm{RD}^2 / \pi^2}}, \qquad q = \frac{\ln 10}{400} \approx 0.0057565$$

Equation 1 · Pokémon Formal Measurement

A Showdown Ladder Rating Measures the Ladder, Not the Player

This equation gives an approximation: it relates the quantities while allowing an approximation.

The mechanism is two coupled numbers, not one. A player’s rating R changes only from game outcomes; the rating deviation RD changes from game outcomes and from the simple passage of time — RD shrinks as a player competes and grows as a player sits idle, because “the more games played, the more information is learned about a player’s ability… as time passes, we become more uncertain about the player’s strength” [ 3 ] . The update equations, stated with Showdown’s own configured constants substituted in, are $g(\mathrm{RD}) = \frac{1}{\sqrt{1 + 3q^2 \mathrm{RD}^2 / \pi^2}}, \qquad q = \frac{\ln 10}{400} \approx 0.0057565$. where $d^2$ is a variance term built from the same g($\mathrm{RD}_j$) and expected-score factors, $s_j$ $\in$ \{0, 0.5, 1\} is the outcome of game j , and between rating…

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