Understanding Rating Increases: What +100 Elo Actually Means

Most chess players have rating goals.  We want to gain rating and hit some nice round numbers: 1000, 1500, and 2000 are frequent milestones.  I distinctly remember starting out online and getting a rating of 1400 ...which soon fell to the 1300s. I was young and distraught, so I created a new account ... and immediately fell back to 1300.

I played a bunch of games against the same pool of players and started improving, eventually hitting (and staying) over 1400.  Despite this, I still lost games to those 1300 players.  A lot of games, in fact, and this rankled me.  I've improved, I'm the better player, I have the higher rating, why am I still losing?

Fundamentally, I had a mistaken understanding of ratings, one that a number of people share.  I've written previously how ratings are merely predictions, not objectively statements about you and your chess. Today I want to tackle this from a different angle: what does gaining +100 rating actually mean?

The Theory

Assume two people are perfectly matched.  Neither person has an advantage, and if we had to predict results, it would be a coin-flip.  Each side has a 50% chance of winning (or it might be a draw).  Even if they played 100 games, we'd expect each side to score 50 points by the end.  That might be 100 draws, 50 wins each or some combination thereof, but at the end of the day, it's equal.

As soon as one person is better, the odds change.  This makes sense: the better player is ... well, better, and so he or she will win more often.  The greater the disparity in skill, the greater the odds shift.  All of this makes intuitive sense, so now let's go from the general to the specific: what does a +100 elo difference mean?

Rather than doing the math myself, I am indebted to this site, which allows you to play with different rating distributions and see the potential outcomes. Here's a graph of the expected score:

If even ratings give 50% odds, then a 100 rating difference amounts to 64% odds.  Said another way, in a 100 game match, you are expected to score 64 points, whether that's 64 wins and 36 losses, 28 wins and 72 draws or anything in between.

What does this mean?  Well, suppose you are 1500 and play 10 games against other 1500s.  You might score 5 wins and 5 losses.  Now suppose you improve to 1600 and play 10 games against those same 1500s.  It's entirely conceivably, if we round down, to score 6 wins and 4 losses.

This blows my mind.  In chess, 100 rating points is a big deal.  It's a sign of massive improvement ... yet, in practice, that only amounts to one extra win!

I've rigged the example to exaggerate the effect.  The score should be 64%, which is closer to 6 wins, 3 losses and 1 draw.  Further, if you play a best of 10 match, you have a 72% chance of winning ... but that might include losing 4 of those 10 games.  It's a very narrow margin.

In Praxis

Most of us don't wake up and magically improve by 100 rating points.  Improvement comes slowly.  For sake of argument, let's use 25 points.  That's still improvement, and it's worth celebrating ... and it amounts to a 53% expected score. In other words, a 3% change. That barely registers.

Understanding this can help reign in our expectations.  It's possible to improve a lot and yet have our results remain more or less the same.  You won't see the difference between 50% and 53% in one game, five games, not even 15 games.  You might only see it after 100 games.  

The point: you can put a lot of time in chess, hone your skills, and not see immediate results.  That's perfectly normal; indeed, that's expected behaviour.  Even a considerable jump in skill, +100 points, has a relatively minor result.

Certainly, when I was younger, I didn't want to win 64% of the time against my rivals.  I wanted to win every time.  I never wanted to lose against John or Ted down at the club.  How much would I have to improve to do this?  Well, even a 400+ point increase is "only" 92%, and it takes approximately 650 points to hit 99%.  That's absolutely massive and, for many people, unachievable. It was also my ego talking, and it could have set me up for pretty big disappointment.

Final Thought

As I improved and marched towards 1800, I remember scoring ridiculously well against the under-1700 crowd.  As in, winning almost every game.  My results against equally rated players, though, was much lower than 50%, and I very rarely beat anyone higher than 1800.  Certainly it was a big deal when it did happen.

For whatever reason, my skillset let me feast on those even just moderately weaker than me, but that same skillset completely failed against equal opposition.  I don't have the hard data in front of me, and I'm not terribly interested in converting 1,500+ games into a spreadsheet to check, but certainly my performance did not match my predicted performance.  And that makes sense: there are many factors beyond our objective Elo that influence our play.

Conclusion

It's possible to improve and not see it in your games.  Indeed, it's possible to improve a lot and still lose a lot, even against your old opposition.   Improvement is rarely seen in one game and takes a long-term view... sometimes a very long-term view.  Don't worry about individual games, or even individual groups of games.  Focus on the long-term.

Also, if you are like young Smithy and your ego demands that you win every game, recognize that that's likely not possible (nor healthy).  Improvement isn't about winning every game.  It's simply doing a little better than you were doing before.