Gaming the Chess Rating System

Before I get to my point, let me go over how chess ratings work for players that already have an established rating. There is another mechanism used for rating new players, that is beyond the scope of this post.

First, one point is given for a win, 0.5 for a draw, and 0 for a loss.

If the ratings of the two players are A and B, then the estimated probability A wins is given as:

Pr(A wins) = 1/(1+10^((B-A)/400)).

For example, if A = 1400 and B = 1600, the probability A wins is 0.240253073.

The new rating after a game is defined as:

New rating = Old rating + 32*(earned points - estimated probability of winning)

In our example, if A won, his rating would go up to 1424.312 and B would drop to 1575.688.

Had B won, the new ratings would be: A = 1392.312 and B = 1607.688

With that out of the way, let me get to my argument.

First, I assume that the winner of a game can be simulated by each player getting a random number with a mean of their rating and a standard deviation of 200. The player who gets the higher number wins.

This 200 figure is somewhat arbitrary, but done to mimic the probabilities produced by the formula above for closely ranked players. I'd be very interested to see a database of actual games to fine tune this figure.

That said, I believe there is an advantage to be gained by playing significantly worse players. Let's take the case of a difference in rating of 500. To be specific, let:

A = 1250
B = 1750

The formula for the probability of winning above gives B a probability of winning of 0.946760.

However, my own standard deviation formula results in a probability of 0.961450 (math available upon request).

Under the current rules, if B wins his rating goes up 1.703687 and if he loses it goes down by 30.296313.

Let's calculate the expected rating after the game:

Winner	Probability	New Rating	Exp Rating
A	0.038550	1719.703687	66.294467
B	0.961450	1751.703687	1684.175622
Total	1.000000		1750.470089

In conclusion, I show player B can expect to gain 0.5 points to just play the game.

That said, I welcome your comments.

The question for the poll is which statements do you agree with? Multiple votes allowed.

I spent hours on this post and get crickets.

Quote: Wizard

I spent hours on this post and get crickets.
link to original post

I'll contribute!

I didn't know I had been missing, but I voted for that anyway!

As for the OP, right away you go into something that has to be studied. That *is* a formula for getting crickets

Is there a setup where a group of players of various ratings can pre agree who will win/lose a set of matches, after which each player in the group has an equal or higher rating than before the matches?

Quote: Wizard

For example, if A = 1400 and B = 1600, the probability A wins is 0.240253073.

Isn't this 0.24 the probability of winning all non-drawn games? At the last 10 world championships, the eventual winner only won 18% of games. With those type of results, it's the lower rated player gaining points in about 80% of all games.

Quote: TomG

Isn't this 0.24 the probability of winning all non-drawn games? At the last 10 world championships, the eventual winner only won 18% of games. With those type of results, it's the lower rated player gaining points in about 80% of all games.
link to original post

Good question. For the sake of simplicity, I had to ignore draws. You might say the 0.24 is the probability of A winning assuming they were forced to play again if there was a draw.

Quote: unJon

Is there a setup where a group of players of various ratings can pre agree who will win/lose a set of matches, after which each player in the group has an equal or higher rating than before the matches?
link to original post

I don't think so. Any one game is a zero-net-sum event, where one player's increase in rating is another's decrease. Basically, every game is a bet where points are at stake. Much like poker, for every dollar won there is a dollar lost.

Quote: Wizard

I spent hours on this post and get crickets.
link to original post

It isn't clear what your argument is.

To quote Chris Rock "People don't laugh because they don't understand the premise of the joke".

I think your point is that good players will benefit disproportionately from playing weak players.

However I have no opinion on that because I don't know the practical implications of that, whether it is exploitable as your title implies, or is just an issue of structural unfairness. Only people on a chess forum who participate in tourneys would know that. This is a gambling forum.

How does all this information fit in your head? It's too complicated for me:)

Quote: Wizard

Poll option:
I miss odiousgambit

What happened to odiousgambit?

He rage quit this forum?
Got banned like Mickey?
Is in the same place as the guy who had 11 yo's in a row?

Quote: Wizard

I spent hours on this post and get crickets.
link to original post

I suck in chess.
After the first 6 moves of developing pieces, i'm lost.

What youtube videos do you recommend for a beginner to learn tactics and strategy instead of memorizing lines?

Quote: 100xOdds

What happened to odiousgambit?
He rage quit this forum?
Got banned like Mickey?
Is in the same place as the guy who had 11 yo's in a row?
link to original post

it could be I'm supposed to "get the message" , yikes!

Quote: 100xOdds

What youtube videos do you recommend for a beginner to learn tactics and strategy instead of memorizing lines?
link to original post

I'm not a good one to ask. As far as I know, OdiousGambit is the strongest active member here. When I was more active with chess, in the 90's, I read a lot Bruce Pandolfini's books. I tend to think they helped somewhat, perhaps making an increase of 100 in my rating, if forced to guess. I strongly believe you have to be born gifted to be great at chess and I wasn't.

A little chess humor:

Dog Hand

Quote: Wizard

I strongly believe you have to be born gifted to be great at chess and I wasn't.
link to original post

Judit Polgar's success would tend to contradict that opinion.

I only know one great chess player. He was a multi-time published author, did special projects for Bell Labs, and played bass for two bands. He had multiple orbits of friends and was highly respected but just slightly off. I had two friends in high school who were considered proteges but we lost contact almost fifty years ago,.

This may or may not be an interesting addition to this discussion, so fair warning if not.

I'm fascinated by the topic of chess ratings for several reasons. First and foremost it's a sound application of mathematics to rate real world human performance with all it's quirks. Second it has a sound history. Though known to be flawed, it basically works. The Wizard points out a flaw that I accept.

But how can a flaw be exploited? Is there any advantage of driving your rating a little higher or lower from its proper mathematically calculated value? I'm not a chess player so I'm not aware of all uses of the rating in handicapping and screening players in tournaments.

However I am very active in competitive pool. And pool has a similar but better (I believe) rating system called Fargorate.

Fargorate is better (as I understand anyway) because it does not use this known inaccuracy that the chess system uses, namely that when two players compete they "exchange" rating points, the winner's rising and loser's falling based on some arbitrary volatility factor, the "32" in the formula the Wizard cites.

Instead in Fargorate a database stores every game played between all players along with their rating and robustness (robustness is very simply the number of games a player has recorded in the database). Then periodically (each night) all the ratings are calculated from scratch, a process known as ab initio global optimization. The end ratings are simply the best estimation of every player's true rating based on the most
likely outcomes being all those game results. The branch of mathematics used to calculate the ratings based on the results is called MLE which stands for maximum likelihood estimation. (Sorry my math expertise is limited so I can't provide the math).

There's another difference that is worth mentioning though it's superficial. In chess a rating difference of 400 points by definition reflects that a player is ten times as good, for example a 1400 player playing a 1000 player will win ten times as many games on average head to head. In pool, 100 rating points means a player will win twice as many games on average.

There could be a good discussion about the relative merits of the two systems in terms of mathematics, practical application, and vulnerability to abuse. Unfortunately I'm not qualified to discuss it in great depth. ...but I do believe Wizard is right and it's an issue with the chess ELO system as currently in use.

In both cases, regardless of how accurately either system updates ratings, both aim to do the same thing, to reflect with a rating for each player the historical relative performance (games won and lost) between players. The history and math can deliver that result accurately with the assumption that the data is real.

However, just knowing the ratings that reflect the historical results is not really what the goal is. Players want to use their ratings for more practical reasons. We might want to conclude that player A is better than player B because they have a higher rating (a fair conclusion though it's limited to the accuracy that can be drawn from past performance).

Another use of ratings is in handicapping. One assumes that the public ratings accurately reflect innate skill (as well as anything can) and predicts future results Thus if you use ratings as a handicap, you can balance competition by for example rewarding players who win by more than expected, or having a lower entry fee for lower rated players and so on, to add an incentive for lower rated participants to play. This is one way that artificially propping up or down one's rating might be advantageous. A good example of this is a cash tournament I play where a player has to win an extra game per match against every 100 point difference in rating.

And another use for comparing ratings is to presume they reflect a certain level of skill or accomplishment. It's undeniable that historical performance is highly correlated with your skill level even though that is not what's measured directly.

For example, a player might be proud to be thought of as a 600 Fargorate player because it is considered the level where they can run many racks and play at a certain skill level. Likewise, in chess, a 2500 player would perhaps be expected to rarely miss a mate in two, or rarely miss a fork, or blunder an end game etc.

So a player could prefer a higher or lower rating than their history equates for all those reasons.

I hear from a lot of players that they want to increase their rating (to reflect that they deserve more regard as a good player) or decrease their rating (so they can qualify to play in tournaments with lower rated players and be more likely to win money).

Their ideas on how to accomplish rising or lowering are usually wrong.

For example, a player usually thinks in simplistic terms, that more wins will raise their rating and more losses will lower their rating. So to raise their ratings they think they should play lots of lower rated players. But of course that's wrong. (In Wizard's example of exploiting a flaw in the system, it is right however!). But in general it's wrong. Your rating improves when you exceed your past performance relative to each competitor.

But it is true that each player's actual rating may reflect their abilities with difference accuracies. For example, players with thousands of games on record are likely to have highly accurate ratings, where as players with only a few hundred games on record may be under or over rated temporarily.

So the question comes up, how can one raise or lower their rating, legitimately or otherwise. Everyone wants a shortcut. The real answer assuming the ratings models are accurate is simple to perform better against all rated players.

But in more depth, the answer is to play relatively better than expected against any opponent of any rating.

It doesn't matter if you play players with higher ratings and lose more or play players with lower ratings and win more.

As an example from pool. Suppose you are rated 600. If you play a 500 rated player you are expected to win twice as many games against them. So if you beat them 20 games to 5, you have played like how a 700 rated player would play and you expect your rating to increase (of course it won't increase to 700 because it will be weighted by both player's robustness.) Now consider the same 600 rated player plays against an 800 rated player. They play 100 games and the higher rated player wins 75 to 25. The 600 feels humiliated by such a massive loss. However they won a third as many games yet their relative ratings predicted - they were only expected to win a fourth as many games, so the 600 will notice their rating increase (presuming the 800 rated player has greater robustness).

So actually, I feel there's very little advantage to artificially attempting to manipulate your rating, and it's very hard to do.

The way to do it in chess might be to play a lot of players rated <500 pts below you, exploiting the Wizard's loophole. I'd suggest the same approach would not work in pool. Instead you should target specific players who you think are underrated or overrated that you can beat more than expected. However both systems "work", because in both cases the best way to raise your rating is to improve and then win more against everyone.