## Simpler Elo Rating System

For the discussion of math. Duh.

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Fine Man
Posts: 13
Joined: Thu Aug 16, 2012 6:20 pm UTC

### Simpler Elo Rating System

One thing that bother me about rating systems in general, and particularly the Elo rating system, is how confusing they seem to be. Particularly the K factors, and 1500 baseline.

Why aren't there any simpler rating systems?

For example, have a scale from 0 to 1 with scores asymptotically approaching either extreme.
Initially set a player to 0.5.
Have it only depend on wins, losses, ties and the competitors scores.
Have it take into account extraordinary data (i.e. losing to a much weaker player)
Encourage higher level competition (so a player can't just increase their score arbitrarily by defeating many weaker opponents)

Obviously this doesn't just have to be for chess. It could really be for any competition. It just seems somehow "messy" the way scores are rated now.

korona
Posts: 495
Joined: Sun Jul 04, 2010 8:40 pm UTC

### Re: Simpler Elo Rating System

One practical problem with such a system would be that is inconvenient (for humans) to compare a score of 0.778 to a score of 0.779.
Elo is actually pretty simple. There are a few numbers that can be tuned but it basically boils down to just one simple formula involving elementary arithmetic only. It archives all the goals you listed. If you want to have scores between 0 and 1 with a baseline of 0.5 you could take something like arctan(EloScore - 1500)/pi + 0.5 but you would loose the nice property of the Elo system that you can easily predict winning chances from rating differences.

Fine Man
Posts: 13
Joined: Thu Aug 16, 2012 6:20 pm UTC

### Re: Simpler Elo Rating System

Well ideally, my hypothetical system would also be able to predict the odds from the rating differences. I know one of the great things about Elo is it is relatively simple, computation-wise, but with modern technology computation isn't really a problem. I guess by "simpler" I really mean "less arbitrary". It's like imperial vs metric units. Why is average 1500, and 200 points represent an expected score of .75? I mean, the principle behind it is fine, but I guess the implementation just irks me.

And while we are on the subject, why are some rating systems time-dependent?

Turiski
Posts: 31
Joined: Sat Aug 02, 2008 2:23 pm UTC

### Re: Simpler Elo Rating System

EDIT: I should clarify that I really don't know very much about ELO so take all this for what it's worth.

Fine Man wrote:WAnd while we are on the subject, why are some rating systems time-dependent?

Well, speaking my experience, I'm significantly worse at tennis after not training for six months (a reality I've faced for the last several Octobers now). Obviously a lot of that skill loss is muscle memory, but if I'm not thinking about the game as much, then I'm not doing as well strategically when I start back up again. I haven't played chess seriously for a couple years but I imagine the problem there is the same.

Another reason for this might be that, theoretically, the field is continually improving at any single ranking. That might not be as big a factor in chess as, say, poker, but I can't imagine that (especially at the highest levels) the same thing isn't happening at least a little.
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tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Simpler Elo Rating System

If you want a simpler rating system, then try to make a system that works for two players first, then try to extend this to many players. The simplest way (and the way which will make sure there is no inflation, is if player 1 has x points, and player 2 has y points, and if player 1 wins in a game, then player 1 should get f(x,y) points, and player 2 should have f(x,y) points take away. This can extend to more players, and if all players enter with the same starting score, then there will be no inflation or deflation. Note that f(x,y)<f(y,x) if x>y, and f(x,y)>0 for all x,y. To simplify even further, and to make the rating system more relative, is to make f(x,y)=g(y-x). If you do something like f(x,y)=g(y-x)=sqrt(ln(ey-x+1)), which is defined and positive for all values of y-x. This way, f(x,y) is very small if y-x is extremely negative, and it won't completely change the ratings if y-x is positive.

Edit: I have tested my idea, and it seems to be relatively stable, but I have a better idea: f(x,y)=g(y-x)=1-(1/(1+ey-x))
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Fine Man
Posts: 13
Joined: Thu Aug 16, 2012 6:20 pm UTC

### Re: Simpler Elo Rating System

I completely understand that skill level is time dependent, but it isn’t the same for everyone. Say the best player in the world came out of a 20 year retirement and lost to some mediocre player. Maybe the best player had lost his touch, or maybe it was just a fluke. So if they play more games, the real skill levels will become apparent. And if they only play that one game, then I’m sure people would look more at the circumstances then the actual score itself. In which case, maybe put an asterisk a la Barry Bonds.

tomtom2357 wrote:if all players enter with the same starting score, then there will be no inflation or deflation.

But if this happened, then how would the original scores change?

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Simpler Elo Rating System

Even if two players have the same score, one player winning against another still changes the score of both players.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

roderik
Posts: 24
Joined: Fri Apr 10, 2009 4:16 pm UTC
Location: Below sealevel

### Re: Simpler Elo Rating System

tomtom2357 wrote:Even if two players have the same score, one player winning against another still changes the score of both players.

Yes, but it will change very little.

Besides, if a new (the ranking on him isn't that accurate yet, because he didn't have many matches yet) player has a skill rank of 1200 and a veteran has a rank of 1200, and the new player wins, then of course you want to adjust the new player his ranking since apparently he is better than the veteran.

CatOfGrey
Posts: 23
Joined: Sat May 15, 2010 12:25 am UTC

### Re: Simpler Elo Rating System

Fine Man wrote:One thing that bother me about rating systems in general, and particularly the Elo rating system, is how confusing they seem to be. Particularly the K factors, and 1500 baseline.

Why aren't there any simpler rating systems?

For example, have a scale from 0 to 1 with scores asymptotically approaching either extreme.
Initially set a player to 0.5.
Have it only depend on wins, losses, ties and the competitors scores.
Have it take into account extraordinary data (i.e. losing to a much weaker player)
Encourage higher level competition (so a player can't just increase their score arbitrarily by defeating many weaker opponents)

Obviously this doesn't just have to be for chess. It could really be for any competition. It just seems somehow "messy" the way scores are rated now.

Here's my critique:

1. In encouraging higher level competition, you need to have diminishing return for defeating weaker opponents. The Elo system takes care of this automatically, see my next comment.

2. Setting a (0,1) score makes things problematic. The Elo scale is an interval scale, where a given interval corresponds to a given probability of victory. An unknown player who scores 75% against a 1500 player has a performance of 1700, while another player who scores 75% against a 2200 player has a performance of 2400. I can't see how to put this in a (0,1) scale.

3. The Elo system handles extraordinary data in its system. On a single game (like your local chess server, or USCF, I recall), a master playing an amateur might only gain 1 point for a win, lose 16 for a draw, lose 32 for a loss. The amateur has the opposite, losing 1 for a loss, but winning 16 or 32 for a draw or win. This is limited by discrete changes in rating: the system might not allow changes of less than one point.

CatOfGrey
Posts: 23
Joined: Sat May 15, 2010 12:25 am UTC

### Re: Simpler Elo Rating System

roderik wrote:
tomtom2357 wrote:Even if two players have the same score, one player winning against another still changes the score of both players.

Yes, but it will change very little.

Besides, if a new (the ranking on him isn't that accurate yet, because he didn't have many matches yet) player has a skill rank of 1200 and a veteran has a rank of 1200, and the new player wins, then of course you want to adjust the new player his ranking since apparently he is better than the veteran.

My local chess server handles this by establishing 'provisional' ratings for 20-30 games. So if a 'fully rated' player is playing a new player, the outcome doesn't affect their rating as much. However, a provisional player's rating changes with much more magnitude, so it converges to a stable level quicker.

mike-l
Posts: 2758
Joined: Tue Sep 04, 2007 2:16 am UTC

### Re: Simpler Elo Rating System

Other systems to consider are Glicko and Glicko-2, and the proprietary TrueSkill (microsoft). These are all like ELO, except instead of a constant K factor within each range, they track a separate uncertainty factor with each player's rating, and your rating changes based on your, and inversely based on your opponent's, uncertainty. Uncertainty also changes game over game based on results, and possibly over time as well.

This makes some things less arbitrary, though you still have to decide on initial value and a scaling factor, but it turns out changing these causes straight linear changes to all values so it's no biggy.
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