# Why do Elo rating system use wrong update rule?

Elo rating system use a gradient descent minimization algorithm of the cross-entropy loss function between the expected and observed probability of an outcome in paired comparisons. We can write the general loss functions as

$$E=-\sum_{n,i} p_i Log (q_i)$$

where the sum is performed over all outcomes $i$ and all opponents $n$. $p_i$ is the observed frequency of the event $_i$ and $q_i$ the expected frequency.

In the case of only two possible outcome (win or loose) and one opponent we have

$$E=-p Log (q)-(1-p)Log(1-q)$$

If $\pi_i$ is the ranking of the player $i$ and $\pi_j$ is the ranking of the player $j$ we can built the expected probability as $$q_i=\frac{e^{\pi_i}}{e^{\pi_i}+e^{\pi_j}}$$ $$q_j=\frac{e^{\pi_j}}{e^{\pi_i}+e^{\pi_j}}$$ then the gradient descent update rule tell use

$$\pi_i'=\pi_i-\eta (q_i-p_i)$$

$$\pi_j'=\pi_j-\eta (q_j-p_j)$$

where $q_i$ and $p_i$ are the expected and observed probability of win of the player $i$ against the player $j$. This is the two outcomes update rules.

In the presence of draws we can generalize the above model including and third outcome with probability

$$q(d)=\frac{\nu e^{\frac{\pi_i+\pi_j}{2}}}{e^{\pi_i}+e^{\pi_j}+\nu e^{\frac{\pi_i+\pi_j}{2}}}$$ $$q_i(w)=\frac{ e^{\pi_i}}{e^{\pi_i}+e^{\pi_j}+\nu e^{\frac{\pi_i+\pi_j}{2}}}$$ $$q_j(w)=\frac{ e^{\pi_j}}{e^{\pi_i}+e^{\pi_j}+\nu e^{\frac{\pi_i+\pi_j}{2}}}$$

And we can build the Loss function as

$$E=-p(w)Log(q(w))-(1-p(w)-p(d))Log(q(l))-p(d)Log(q(d))$$

where $p(w),p(l),p(d)$ are respectively the observed likelihood of win,loose and draw and $q(w),q(l),q(d)$ expected likelihood of win,loose and draw. In the latter case the update rule would be

$$\pi_i'=\pi_i-\eta (q_i(w)+\frac{q_i(d)}{2}-p_i(w)-\frac{p_i(d)}{2})$$

$$\pi_j'=\pi_j-\eta (q_j(w)+\frac{q_j(d)}{2}-p_j(w)-\frac{p_j(d)}{2})$$

where $q_j(w)$ and $q_j(d)$ are the expected probability of the player $i$ to win and to draw against the player $j$. And where $p_i(w)$ and $p_i(d)$ are the observed probability of the player $i$ to win and to draw against the player $j$. This is the three outcome update rule.

The question is, why do Elo rating system use the two outcomes update rules even in the presence of draws?

The probability of drawing, as I said, is not specified, and it leads to a simple two outcome update rule, $R_A^\prime = R_A + K(S_A - E_A)$, in which $S_A=1 \cdot (n_w + 0.5 \cdot n_d ) + 0 \cdot (0.5 \cdot n_d + n_l)$, so, after a single match, $S_A=1$ (win), or $S_A=0.5$ (draw, as half a win), or $S_A=0$ (loss).