# Maximum Elo Rating.

I'm trying to implement a variant of the Elo system, for a game I'm working on. Giving two players $A$ and $B$ with ratings $R_A$ and $R_B$ respectively, the expectation of $A; E_A$ is given by the formula: $$E_A = \frac{1}{1+10^{\frac{R_B-R_A}{Y}}}\tag{*}$$

The expectation of $B$ is similarly given by: $$E_B = \frac{1}{1+10^{\frac{R_A-R_B}{Y}}}$$

The different possible game outcomes are given scores: A win is $1.0$, a loss is: $0.0$, and a draw is $0.5$. The actual score of $A$ is $S_A$.

After a match between $A$ and $B$, the new ranking of $A; R'_A$ is given by: $$R'_A = R_A + K(S_A - E_A)$$

Question:

Given a sample with a population size of $n$ in which each player reveives an initial rating of $c$. Assuming that the players actively cooperate, what's the maximum possible rating that one player can receive?

A simpler question is: Is it possible to increase the rating of all players in the sample? I think the answer is 'No' (since the rating gain of each player is deducted from another player). However if it was 'Yes' then there would be no limit, since by induction it will be possible to prove that if a strategy to increase all player's rankings from $c$ to $g \{g: c \lt g\}$ existed, then same strategy could be used to improve all rankings from $a$ to $b$ $\{a,b: g \le a \lt b\}$ and there would be no maximum value.

My system doesn't implement the concept of floors or ceilings. $K$ is constant throughout all rankings. The maximum possible ranking is trivially $\le n\cdot c$, but I want a better upper bound. (As @Sil points down below:

$R'_A+R'_B = R_A+R_B$ (since $S_A+S_B=1$ and $E_A+E_B=1$), so the overall ELO sum should remain the same after match as it was before it. $\tag{**}$

The overall ELO sum in this case being: $n \cdot c$

As for a lower bound for max rankings, I'm thinking of a binary search: $n/2$ people win the first $n$ matches.

Their new rating will be calculated as $$R' = R + K(1-0.5)$$ $$R' = R + .5K$$

The set will be reduced to $n/2$, and this will continue till set $= 1$.

The number of matches $m$ played this way wilk be approx $\lg n$.

Max ranking through this method, will be: $$c + \frac{K}{2}\cdot \lg n$$

While that provides a lower bound, thus is one scenario in which I think a binary search is suboptimal in increasing rankings.

I think increasing average ratings, may work, but I'm not sure.

I would appreciate an answer where the maximum rating is given as a function of $K, c \& n$.

I'm currently using $K = \sqrt{Y}$

$(*)$ Most chess Elo algorithms use a value of $Y = 400$

$(**)$ By this argument, it is possible to prove that $\not\exists$ a strategy to increase ratings of all players.

It isalso possible to prove that it is impossible to improve average player rankings. (The average rating will always remain $1000$)

• If I am not missing something you have $R'_A+R'_B = R_A+R_B$ (since $S_A+S_B=1$ and $E_A+E_B=1$), so the overall ELO sum should remain the same after match as it was before it. – Sil Dec 18 '16 at 2:48
• Yup. Which is why I said the maximum ELO is trivially $\lt n\cdot c$. $n \cdot c$ being the ELO sum. – Tobi Alafin Dec 18 '16 at 2:53
• I thought you are asking if overall ELO can increase (the second question), which it can't if the sum remains the same. – Sil Dec 18 '16 at 2:59
• This should prove that $\not\exists$ a strategy to increase the ratings of all players. Didn't think of that. – Tobi Alafin Dec 18 '16 at 3:04
• For clarification, I wasn't really asking if it was possible to increase all player's ELO, (I didn't believe it). I just didn't think of your proof. – Tobi Alafin Dec 18 '16 at 3:07