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

Is there something wrong with the forums today?

There is an empirical assumption in the Elo formula. It is that if you have three players A, B and C then the scores they expect to achieve against each other are related in a way that can be derived from the Elo formula. This is in no sense obvious and doesn't need to be precisely correct for the rating system to work (but it seems pretty accurate anyhow).

If the above is confusing, suppose A gets 60% against B and B gets 60% against C then how much should A get against C? To estimate this from the Elo ratings, you just looking what rating difference achieves a 60% score and then look up what score twice that rating difference achieves.

Not double but 60% of 60% from A to C, it's geometric, not linear, if it is transitive that way.

Quite true I agree

Lower end of the spectrum is weirder. How can you make a move worse then the worst possible moves If a 500 rated player always falls for a mate in 1, how could anyone be less than 500?

Depend on who fall mate within 5 moves vs 10 moves out of opening.

I saw a lot of < 500 rated people, although average game length among lower rated people is low like <30, they dont always die within 10 moves. The one who blunder worst usually lose.

( In comparision, average game length human OTB 2200-2800 is 40, among top engines is 60-70 depending on opening book and hardwares).

My original point was that it's hard to believe ratings are linear if exponential figures, such as square roots, are used in calculations after tournaments games. It also depends on which system is sued, rating changes after each game added up, or an average opponent rating calculation after a 5 game swiss system...etc, am not sure if both approaches are commutative or not.

??
I don't quite get what you mean, 60% of 60% is 36%. If A wins B most of the time, and B wins C most of the time, shouldn't A also win C most of the time? This isn't simple probability. 

The rating formula is geometric, not linear.

Maybe we should narrow the question to whether a 3501 is better than a 3500?

We tend to say players 1 point apart are even, yet one should still theoretically win .3% more games than the other! The question is when does the difference in rating points drop below the significance of the different choice of moves? If there are 40 moves in a given position, and say 10 of them win, 20 of them draw, and 10 lose, and there is a rating difference of less than 40 points, at some point the level of play must overlap with the same move choice with different ratings!

For context, the choice of color alone gives about 80 rating points’ difference (40 more for White and 40 less for Black).

The difference in rating for the first move is slightly higher in middle rating ranges because beginners don’t know how to take advantage of the first move while people don’t learn how to defend properly until they are generally above about 1800.

I really don't believe white has an advantage by moving first. White may be in zugzwang from the very beginning of the game for all we know, but I can't see a reason chess is not a theoretical draw, despite this not being provable. Reverse Logic. What makes people think it is not a draw in the first place. If White had 2 first moves then maybe I could see that theory, but then again Magnus played Knight back to G8 and A3 wasting a couple moves and still won, let alone drew

yeah idunno about that

No problem, I have racing thought problems too.

"Chess without mistakes is nothing." ~ K_RICHTER®

That's another whole discussion. Even if we know what the best moves are with perfect play, we do not the best moves from positions achieved with non-perfect play. A 32 piece tablebase would be needed for that.

agree

No, it is not geometric, exponential or linear. It is what the Elo formula for expected score says.

To be specific, here is the Elo calculation for the expected score for two players with rating RA and RB.

So if EA = 0.6 (i.e. 60%), RB-RA = 400*log10(1/0.6-1) = 70.43 points. 

This means the rating difference of A and C = 70.43 + 70.43 = 140.86, so from the same formula, the expected score of A against C is 69.23%. 

There is nothing obvious about this relationship between the expected scores, it just comes from the formula, and the formula has been found to be empirically quite accurate, so the rating system works.