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Assume we want to have a player rating system with the following conditions:

  1. For simplicity, no draws.
  2. If A wins against B with ratings $a,b$, their new ratings are $a'=f(a,b),b'=g(a,b)$.
  3. Most important: Order of games shall play no role. Say, A wins twice (rest analogous): $a''=f(f(a,b),c)=f(f(a,c),b)$ for all $a,b,c$.
  4. If possible, win=lose: $a+b=f(a,b)+g(a,b)$.

I'm quite sure condition(s) 3 is easily to fulfil and has been researched already; can condition 4 and 5 also be fulfilled? At least one trivial solution exists: $a'=a+k,b'=b-k$ with constant $k$. But I'd rather have an Elo-like $f$ that considers winning is easier the larger $|a-b|$ gets, so $k$ should get smaller.

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  • $\begingroup$ For chess , already $1$ is no good assumption. Between two strong players , a draw is a very likely result. Usually , the calculation is done after a tournament (the intermediate ratings are live-ratings) in which case the order does not matter. The current rating system has a particular strange problem with edge cases. For the first rating , the player cannot have a $0$% or $100$% score. This should be repaired ! $\endgroup$
    – Peter
    Commented Oct 15, 2023 at 11:39
  • $\begingroup$ @GerryMyerson: Wait. I think I botched a minus sign. I edit the question. 5 deleted since "win" already states that f and g must be 2 functions. $\endgroup$
    – Hauke Reddmann
    Commented Oct 15, 2023 at 14:28
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    $\begingroup$ Worth noting that for order of games not to matter, potentially everyone’s rating must be adjusted after every game, not just the two players. $\endgroup$
    – M W
    Commented Oct 15, 2023 at 14:51
  • $\begingroup$ To add to the comment by @Peter, a World Championship match hasn't ended in checkmate in a long, long time. Players would play to draw or resign after a lost position. $\endgroup$
    – Sean Roberson
    Commented Oct 15, 2023 at 18:53
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    $\begingroup$ 'Order should play no role' is a constraint, and due to this constraint, the functions should be $a'= f(a,b_1, r_1, b_2, r_2, b_k, r_k, b_n , r_n )$ where $r_k$ is the result (+1 or -1) of match $(a, b_k)$ ; Usually, another important parameter is the "confidence" : if you play against player X who has a ranking 1000, obtained after 5 games, or after 200 games, it is not the same ; If player X has only 5 games in his historic, his ranking is just a random number, and I should not be punished if I loose against him. $\endgroup$
    – Lourrran
    Commented Oct 15, 2023 at 23:16

1 Answer 1

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Unfortunately, condition 3 is not easy to fulfill at all.

In fact, unless you are planning on discriminating against certain players from the very beginning, or restricting the total number of players, every system under your criterion 3 must have $f$ depend only on $a$, and $g$ only on $b$, which in turn means that a player's rating depends only on their total number of wins and losses, and not on their opponent's rating.

That is, if every player begins at the same initial rating $I$, and there are an unbounded number of players, then we have $f(a,b)=f(a,I)$, $g(a,b)=g(I,b)$ for all possible rating combinations $a$ and $b$.

To see this, note that if $A$ and $B$ play against each other in a game $G$, and before $G$ they have ratings $a$ and $b$ respectively, and $A$ wins, then since there are an unbounded number of players, we may suppose with no loss of generality that $A$ and $B$ have never faced each other before, instead obtaining their ratings by playing in disjoint leagues of players $L_A$ and $L_B$, respectively, and that no one from $L_A$ had ever played against $L_B$ ever before.

Then by condition 3, we also may rearrange the order of the games so that all other games played in $L_A$ occur first, then game $G$ occurs, then all the games in $L_B$. Since the games in $L_B$ do not affect player $A$'s rating, player $A$ must end with the rating he obtains after playing $G$ against $B$, who under this rearrangement is a novice at game time. $A$'s rating when he wins is therefore $f(a,I)$. A similar argument, rearranging so that all of $L_B$'s games are played first, shows that $B$'s rating after he loses is $g(I,b)$.

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  • $\begingroup$ This "no-go" result doesn't surprise me at all, but at least I now know why Elo officials constantly have nervous breakdowns (if tournament results come in delayed - this is more a problem for national tournaments, but still). $\endgroup$
    – Hauke Reddmann
    Commented Oct 16, 2023 at 8:06

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