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Data shows white has a non-trivial advantage (Does white have an advantage?) but has anyone ever done any analysis what ratings point advantage/handicap?
Put another way if player W plays white against player B playing black, how big a rating advantage (FIDE or ELO) would player B require to have a 50% chance of victory?
I do not know whether any statistical analyses have been conducted using real games, but let me suggest the following (theoretical) estimation:
My Stockfish 15 gives White an advantage of 0.3. Calculating the expected score gives 50 + 50 * (2 / (1 + exp(-0.004 * 30)) - 1) = 53, i.e. 0.53 expected points.
This is consistent with the commonly accepted 52-56% as seen in both computer and human player's games.
Using the Elo model's expected points formula, we can see, that this translates to an approx. Elo difference of 21 points (0.53 expected points) to 42 points (0.56 expected points).
Kaggle held a contest called the "Deloitte/FIDE Chess Rating Challenge". As part of the benchmarks for the challenge, they say the following (emphasis mine):
(2) White Advantage Benchmark - It is always a slight advantage to have the white pieces in a game of chess, because White always gets the first move. The advantage of that extra move is highest among closely-matched top players, and less advantageous among weaker or mismatched players. It is worth roughly 25-40 rating points on average, giving White an expected score of about 54% in general. The White Advantage Benchmark is another extremely simplistic entry, using an expected white score of 53.875% for all games. This is slightly more accurate than the All Draws Benchmark, and so it performs better, but only marginally so.
I don't know of an empirical analysis either. Some years ago I attempted to do one on my own but ended up frustrated. The 54% advantage mentioned in the answers above is just an average from all recorded games and most probably conceals important correlations. My hypothesis was that white's ELO-advantage is low for weak players and high for masters. I have the feeling there is a peak somewhere below the elite level. Who knows? But even with a database of ~10 million games the results remained inconclusive.
You have to group players according to their strength and then look at how they perform against other groups. For example: What performance do players from the group (2500, 2510] achieve with white against the group (2510, 2520]? The problem is, that if you make these intervals too big, the hypothetical rating advantage is lost in the noise. If you choose them too small you end up with too little games.
Your second formulation of the question might have a definitive answer although it probably depends on the ELO of W. But I doubt this formulation is equivalent to your first one. A player with a rating advantage of 100 points might profit more from the white advantage than a player with a mere 50 point advantage, while a player with a 500 point advantage will win almost every game independent of the colour they are playing.
Unfortunately you have to make numerous theoretical decisions before you can hand down your question to statistics.
