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Ratings

yes there are rapid and blitz FIDE but I understood the original poster was referring to standard FIDE as the shorter time control FIDE ratings are far fewer and based on far fewer games and not generally regarded as highly as the standard FIDE rating.

And still the games are played in a completely different environment.

So I re-iterate there is no comparison possible between the two ratings. You cannot even infer that if player A has a higher Lichess rating than Player B that if they played in a FIDE tournament that the higher Lichess rated player would be more likely to win.
I think unless i am an extremely baddo at fide but not at lichess, there is no way to compare the grades, because one of my fides are 14--, but my ecf =119 and my lichess grades on my @yifanluo account are no way near 14-- fide.
@tpr analysis on top list players is meaningful only for blitz as the strong player hardly play rapid on classic here. and since classic pool is so weak they would have play hundreds of games to for rating to settle. quite bit better in rapid but classic pool is really weak compared to others. Blitz will have best correleation with fide for sure

another problem for weak players is that their fide rating is in accurate. In most countries only a fraction of tournaments are fide sanctioned. It does not cost anythign but required FIDE arbitrer or was it two. getting national arbitret tends to be easier
One thing that is always neglected when comparing fide and online rating is that in online, there is no competitive preparation. For example you can master the King's Gambit and always surprise your opponents, because no one cares as much to prepare against you and you rarely play the same opponent twice. Whereas in fide tournaments your opponents analyze your previous games (and therefore they know your strengths and weaknesses as well as your opening choices) and are booked up to the brim against you.

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