A 3000 could easily beat a 2000, but could a 4000 easily beat a 3000?

exactly, and also statistics don't mean much when it comes to actually playing the game. Even if the game is easy, you get to practice your tactics. See how fast you can force mate ya know.

Yeah, they are clearly rounded to the nearest 1%.

Firstly, I should correct the formula. The stronger players score is actually this.

1 / (1 + 10^( -k / 400)

I slightly garbled it in an earlier post by rearranging it in my head. Ahem.

The only assumption of the Elo rating system is that the probabilities are near enough consistent. For the formula to work well when used to rate players, there is the assumption that it works regardless of which players actually meet. This is not a given.

If you have three players A, B and C, you have three scores for A v B, B v C and A v C, but there are only 2 degrees of freedom for ratings differences. The thing you need is that if the rating differences for A-B and B-C are worked out, and if you add them together, the formula gives the actual expected score for A against C.  It turns out it works rather well. Professor Elo was well aware of this issue and checked the statistics were ok.  There is another function that can also be used, which is quite different in form but only slightly different in shape. 

If you invert the formula, you get that if you score S against a player, then s = 1 / (1 + 10^(D/400)) where D is the rating difference.

This means that D = 400 * (log_10( (S -1)/S) )

In this form you can see that D is actually a constant times the logarithm of the odds of winning ( because S -1 is the score for your opponent, S is your score).

So the fact that rating differences can be added is the same as saying odds can be multiplied.

If the odds for A against B are x and the odds for B against C are y then it is assumed that the odds for A against C are x * y.  This seems quite intuitively reasonable, and turns out to be quite accurate.

I always thought rating acted like an exponential function. Think of it this way, two 300 rated players playing will be chatotic, complete random moves, random wins and losses, where as 2 grandmasters will play extremely hard games, resulting in many draws, despite the differences between a "weak" a grandmaster and a strong grandmaster.

As I pointed out, the rating system is completely unconcerned with draws, only with scores. A draw just counts as half a win and half a loss.

The reason is that it is only about averages. It includes no model of the variance of results. This is what goes down with stronger players.

The extreme cases are the two extremely strong players of the same rating just draw all the time, with variance 0, two weak players of the same rating virtually never draw, each winning half the time, with variance of 1/4 in the score.

Basically, a good player will almost always beat a bad player, but a great player will have a hard time beating a good player, cause they are both good, will see each other's tactics, understand the endgame better..etc, where as a 1000 player can be winning a game, then screw up the endgame and loose.

Again, no. If you have the same rating differences, the good player does win more games against the weak player than the great player against the good player, but he loses more as well to exactly cancel it out in the score.

What I mean is that games between good players will be harder even if the difference is the same. You agree that a 1400 will beat a 1000 way easier than a 2800 will beat a 2400, right?

I don't have a way of quantifying ease beyond results.

Just Logic does the trick. A 1500 will probably see a mate in 2, a 1000 may not. but BOTH a 2000 and a 2500 will DEFINITELY see the mate in 2.

You could say players at a higher level have to show more effort to get the same results. However It's relative since being where they are they're naturally more comfortable with the extra work. It doesn't feel any different in principle.

Any recent evidence that a centaur is better than a pure engine! Who will win  depends on how strong their engines are.

Official Nakamura+Rybka= lose to stockfish 3-1.

https://www.chess.com/news/view/stockfish-outlasts-nakamura-3634

I would bet any human with stockfish 16 cores will definately lose vs stockfish 32 cores ! (must play about 20-30 games to narrow statistical error) 

Please note, centaurs may win vs engines decades ago.  Current engines are stronger and stronger everyday. Within 6 months, Stockfish 10 dev version becomes 25+ elo stonger than Stockfish 9, that mean Stockfish 10 dev on 12 cores computer has now equal strength vs Stockfish 9 on 16 cores computer.

What 3000, 4000? numbers cannot play chess

So that's easy for both of them.

It's like sports such as running. It's not really harder for a top marathon runner to run 5 minute miles than for an average guy to run, say, 8 minute miles. Both may be totally drained if they run at a pace which they can just about manage. What is hard just varies with how fit and naturally fast you are.

For fitness sports, a popular scale for setting training regimes is called "perceived effort". Whether you are fast or slow, training at a speed that is "hard" for you has a similar role. Training at a level that is "fairly easy" has a different role.

Unfortunately, the price for always being in your head with all these delicious facts and figures is you don`t feel, don`t allow yourself to show emotions, have trouble interacting with people in the flesh and so by keeping yourself in the head makes you feel safe, heard and loved

Elo rating is supposed to be linear, it's not exponential. In theory, a 3000 beats a 2000 player in the exactly same proportion a 4000 beats a 3000 player.

It's not intuitive, but that's how it works.

edit: 

Good topic! But then how would the 4000 get there without beating 3000's?

Good point, all rating does is compare you to other ratings. What does a rating in itself mean? A 1000 beats a 700, how good is a 700? Better than a 500? How good is a 500..etc. How much better is a 2500 than a 2000? See what I'm saying.

Not really.