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Note

I am a first-time user of this site, and am not sure what determines how broad a question is. I consciously limited the question to general principles in order to avoid making it too broad (by asking for examples, etc.).

Nevertheless, if it is too broad, let me know in the comments, and do suggest how to narrow it down.

The question

I am aware (through chess) of the Elo rating system and how it works. Basically if you know the ratings of two players, you can calculate the expected score of both of them, and adjust their ratings based on the actual score.

However, I was wondering how this system is adapted to work for games in which the margin of victory matters.

Specifically, I was wondering how the rating system can be extended so that one can calculate not only the expected score but also the expected margin of victory or defeat from the ratings of two players/teams.

I'd also expect the system to take into consideration the actual score as well as the actual margin of victory when adjusting the ratings after a match.

A hypothetical example

The actual details needn't be the very same as what I mention here, but the general idea goes like this:

Consider a rating system where Chelsea and Man City have ratings 2000 and 2100. I'm looking for a rating system which not only predicts the score (around 0.64 for City) but also the margin of victory.

Considering that the rating somehow gives us an expected margin of +3.2 for Manchester City, and the team wins 2-0, I'd also expect the system to reduce City's rating for not winning by a large enough margin.

But I do wonder if two variables (expected score and expected margin of victory) are needed or just one (expected score) does the job.

In short

What are the general principles and methods involved for extending the Elo rating system to games where the margin of victory matters?

Community
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Myungjin Hyun
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2 Answers2

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A simple version of ELO can be cast as a logistic regression: for players $i,j$ with ratings $R_i,R_j$,

$$P(i\mbox{ beats }j)=\frac{1}{1+\exp(\beta(R_i-R_j))}.$$

So you could just as easily predict score instead by using a different link function, for example a lorentzian or gaussian:

$$P(\mbox{Game score}=x)=a\exp(-\alpha|\beta(R_i-R_j)-x|^\gamma)$$,

where the game score can be positive (in favor of $i$) or negative (in favor of $j$). So you don't need to calculate the probability of beating and just directly optimize the game score.

Alex R.
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  • That sounds promising. I thought of something similar, where the expected margin of victory $m_e$ is given by $m_e=A({2S_e}-1)$ and $S_e$ is the expected score, calculated as usual $\left(S_e=\frac{1}{1+10^\frac{-d_R}{400}}\right)$. Then, $\Delta R=K(m_a-m_e)$, where $m_a$ is the actual margin of victory. The constants $A$ and $K$ can, I guess, be obtained by some experimentation. – Myungjin Hyun Jan 06 '18 at 15:54
  • Could you edit your answer to elaborate a little on how the ratings are updated once the result of the game is known? I'm not sure if I understand fully. – Myungjin Hyun Jan 07 '18 at 13:38
  • @HarryWeasley: The most straightforward way to update result for players $(i,j)$ with a new datapoint is to recalculate $R_i,R_j$ by updating the loss function, via gradient descent. – Alex R. Jan 08 '18 at 18:15
  • Sorry to be so bothersome, but I'm not an expert, and didn't quite understand some of the terms (especially 'loss function' and 'gradient descent') you used in your last comment. Could you rephrase the same in simpler terms? :) – Myungjin Hyun Jan 08 '18 at 18:28
  • @AlexR. what is the function represented by β? – user160104 Nov 27 '18 at 09:33
  • @user160104: that’s a learned parameter of the model. – Alex R. Nov 28 '18 at 01:54
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There are some works intended to include the margin of victory in rating system (e.g. FiveThirtyEight for NFL), but usually ranking systems (e.g. Elo, Glicko, or our rankade - here's a comparison) don't incorporate the margin of victory.

In most sports/games the margin of victory is not significant. In chess the goal is to checkmate your opponent's king (and it doesn't matter how many pieces you and your opponent have on the board when you're able to do this), in basketball - like in most sports - winning 89-60, or 86-85, or 90-23 gives the team just a victory (and the score doesn't matter - except for mostly unused tiebreaker), and so on.

Consider a rating system where Chelsea and Man City have ratings 2000 and 2100. I'm looking for a rating system which not only predicts the score (around 0.64 for City) but also the margin of victory. Considering that the rating somehow gives us an expected margin of +3.2 for Manchester City, and the team wins 2-0, I'd also expect the system to reduce City's rating for not winning by a large enough margin.

Opposite to rugby, in which you get a (little) bonus if you score 4+ tries, in soccer City gets same 3 points even if it wins 8-0 (and probably, while leading 4-0, City coach wants their best players to rest for next matches...). Margin of victory could be significant (showing that there's a big difference between teams), but it also could not, for many reasons. And, in a structure in which the goal is winning (no matter for the score), it's not a good idea to build a ranking system that rewards an 'unuseful' large victory (3 points for championship ranking) and subtracts points for a 1-0 win with last team in ranking (same 3 points!).

Eventually, you can somehow reward a bigger than expected win, but you can't 'punish' a team for not winning by a large enough margin. They won, so they did their job.

Sure there are (few) games in which the margin of victory matters, but soccer (and nearly all sports, both with round robin or brackets) is not in this list.

Tomaso Neri
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  • Thanks for your answer (+1)! I found it quite informative, and liked the idea of the ree algorithm. While I do agree that the margin is insignificant in many games, my question was about the games in which it actually matters. – Myungjin Hyun Jan 07 '18 at 13:45
  • You're welcome! On rankade you can easily manage this issue (i.e. incorporating margin of victory) using weight feature (e.g. normal weight for matches with normal margin, mid-light weight for low margin, heavy weight for high margin), tuning point rewards (that will be always given to the winner, anyway). – Tomaso Neri Jan 07 '18 at 20:45