Lunchtime Sports Science: Introducing tanh5

As I mentioned in a previous article on ratings systems, the log5 estimate for participant 1 beating participant 2 given respective success probabilities $p_1, p_2$ is

$$\begin{align} p &= \frac{p_1 q_2}{p_1 q_2+q_1 p_2}\\ &= \frac{p_1/q_1}{p_1/q_1+p_2/q_2}\\ \frac{p}{q} &= \frac{p_1}{q_1} \cdot \frac{q_2}{p_2} \end{align}$$ where $q_1=1-p_1, q_2=1-p_2, q=1-p$.

Where does this come from? Assume that both participants each played average opposition. In a Bradley-Terry setting, this means

$$\begin{align} p_1 &= \frac{R_1}{R_1 + 1}\\ p_2 &= \frac{R_2}{R_2 + 1}, \end{align}$$ where $R_1$ and $R_2$ are the (latent) Bradley-Terry ratings; the $1$ in the denominators is an estimate for the average rating of the participants they've played en route to achieving their respective success probabilities.

In a Bradley-Terry setting, it's true that the product of the ratings in the entire pool is taken to equal 1. But participants don't play themselves! Thus, if participant 1 played every participant but itself, the average opponent would have rating $R$, where $R_1 \cdot R^{n-1} = 1$. Here $n$ is the number of participants in the pool.

Our strength estimate for the average opponent faced is then

$$\begin{align} R &= {R_1}^{-\frac{1}{n-1}}. \end{align}$$

There are two extreme cases. If $n=2$, then $R = \frac{1}{R_1}$; as $n \to +\infty$, $R \to 1$.

The limit extreme case is log5; the $n=2$ extreme case I call tanh5. We compute

$$\begin{align} p_1 &= \frac{R_1}{R_1 + 1/R_1}\\ p_2 &= \frac{R_2}{R_2 + 1/R_2}\\ \textrm{tanh5} = p &= \frac{ \sqrt{p_1 q_2} }{ \sqrt{p_1 q_2} + \sqrt{q_1 p_2} }. \end{align}$$

Why tanh5? We can think of log5 as derived from the logistic function by setting $\log(R)=0$ for the opponent's rating; analogously, tanh5 is derived from the hyperbolic tangent function by setting $\log(R)=0$ for the opponent's rating.

Note that we have a spectrum of estimates corresponding to each value for $n$, but these are the two extremes. This also gives a new spectrum of activation functions for neural networks, but I'll explore this application later.