Rating differences at any given point in time are comparable to rating differences at another given point in time. That is, A being 200 points better than B in 2013 means the same thing as C being 200 points better than D in 1963. This is a fundamental underpinning of Arpad Elo's model.

It is absolutely true that the Elo model is a relative one and does not guarantee that a given rating number represents the same ability over time, but the belief that modern ratings are inflated is not unanimous. Ratings have certainly gone up over time, but computer analyses of games have indicated that intrinsic play quality has gone up over time as well. See the work of Ken Regan, for example (search for "ken regan ipr").

Is there a mathematical proof that the Elo ratings have interval properties (so that "Rating differences at any given point in time are comparable to rating differences at another given point in time" is a surely true statement) and not just ordinal properties? I ask, because this is a hot issue in psychology, which I study, and it appears that almost all psychological "measurements" have, at best, merely ordinal properties ("A is smarter than B") and not interval properties ("A is as much smarter than B as B is smarter than C"). Constructing interval measurements properly is HARD, and yet assuming numbers have interval properties when they have only rank-order properties is easy, so I wonder if there is really a rigorous proof on this issue as to chess ratings.

It is mathemtatically proven that the Elo Rating system converges to the Bradley Terry interval rating system over time.

The difference between Elo and Bradley-Terry is that Bradley-Terry assumes to have "all" the data. So Bradley Terry is used in Psychology studies for that reason. (Eventually, you stop taking data).

Elo on the other hand, is mathematically equivalent to Bradley-Terry except with infinite data. Elo allows you to incrementally "add data" to an existing Bradley-Terry-esque model and make it more accurate.

BTW: Elo has been superceeded by superior models today, but all (good) ranking models are "interval properties" that have been derived from the Bradley Terry model from Psychology research.

See TrueSkill (which is based on Gaussian Curves, so they aren't technically Bradley-Terry but instead the Thurstone scale), Glicko, and Glicko2.

It's built into the definition of the rating. A 200 point difference in rating always means that there's a 75% expected chance of the higher ELO winning. In less fluid arenas, the actual ratings can diverge from ideal ratings but there's enough matchups in chess that this isn't a huge concern.

In addition to the other comments, I'd add that in the context of Chess, we have a contest in which there is a winner or a loser or a draw, and we can relatively easily verify that the system is functioning as designed with simple statistics. We also have a system that broadly speaking produces a fairly uniform ordering. Yes, at the very top of chess there's some question of who might win or whether styles have an effect, but broadly speaking in the population at large, a 50 point difference will have a certain meaning; at the macro level, we're measuring something with one dimension.

I would imagine many psychology studies are measuring something a great deal less well defined and more multidimensional in practice than "A will beat B at chess."

The Elo rating system is based on the model that a rating difference gives x an expected result as a function of x. Therefore, the Elo ratings "have interval properties". For example, the Elo rating model states that if A's rating is 200 points higher than B's, A will get in average 0.76 points against B.

However, of course the Elo model might not be perfect, and thus a lot of other things contribute into the rating difference of players A and B, not only the expected result of the players playing only against each other.

dfan, you make good points. But also we need to keep in mind that today's top players on an absolute level, really probably are better than previous era's top players. They simply have access to the past, and technology.

More databases, more games played, internet giving opportunity to play many, many more games. Programs to analyze their play, coaching in the modern era, etc.

That is something that Anand didn't really benefit from when he was young.

Yes, exactly! It is no insult to players from earlier eras that today's players have surpassed them, just as it is no insult to track and field athletes from earlier eras that their records don't still stand. Today's athletes have better equipment, better training, and better medicine, just as today's chess players have 24-hour access to computers better than any human and databases with millions of games. It would be surprising if today's top players weren't better than they used to be.

Given today's level of obsession required to become #1 in any professional sport, I wonder what kind will be needed to be #1 in, say, 25 years. Have we reached some sort of pinnacle? I doubt it, but we probably thought these kind of athletes 30 years ago were already as perfectionists as they could be. P.S. I mean obsession in regards to hours devoted to training, practicing, building etc, not as a description of their personality, although it probably is present there in varying quantities.

Also at chess, new players can learn from the previous top players.

Precisely why he measured difference to next player, rather than absolute rating.

Rating difference should be comparable, elo is designed so that a given rating difference is a given probability of beating your opponent.

yes, i believe that we can agree that it gives us some idea of where the players were relative to each other.

but, at the same time, it's messy. and comparing the top two players only will probably not be as meaningful as looking at their games and their play.

the interesting question is not that you can be dominant against bad competition, but you can dominate over great competition. I think we can agree on that point.