What do we know about the statistics of 1-vs-team chess games, like Kasparov vs The World?

Question

Kasparov vs the World was a chess game where Gary Kasparov played against many people on the internet. Gary won. I am basically wondering if 1-vs-many chess games have been done systematically, and if there are some statistical results from this? e.g.:

• Practically speaking, if we have N players of Elo rating X each, what is the Elo rating of the combined team (typically)?

• Practically speaking, if we have N players of Elo rating ranging between X - 500 and X + 500, what is the Elo rating of the combined team, typically?

The Kasparov vs the world match suggests to me that Elo (team) is approximately equal to Elo (best player in team) + a small number, but I'd like to know if there are statistics.

Also, there are some limitations to what the Kasparov vs The World match tells us:

1. Gary apparently watched the forum in which his opponents communicated, which wasn't clearly forbidden beforehand.
2. it's not clear to me how serious the opponents took the game, as it was maybe a bit informal.
3. It's only one game.

Answer 1

Carlsen had a similar match (https://www.chess.com/news/view/raw-chess-challenge-all-the-info) and there may be well more, but while the questions you ask are very interesting, even if there were millions of such games they would not provide an answer to your questions since 'The World' is anonymous (we do not know the Elo distribution of the players who participated). Another, subtler limitation that you don't mention is that there is a difference between how normal chess games are played (to which Elo ratings apply directly) and these games where players are allowed to take notes, browse forums and written materials, etc. (even if we imagine that they stick to the condition of not using an engine; if you model a certain small proportion of the voting team as using an engine, that could also be an interesting question how much that would impact the combined team's Elo).

It's likely a very different approach is appropriate for huge N (like 'The World' as a team) compared to small N (where they may be more data or example games). For tiny N, you might look into consultation games (where two players discuss their moves: here's a list, which for some reason misses what I thought to be the most famous pair of consultation games of all-time, Morphy vs Staunton), which for N=2 player teams seems the best solution. (Obviously, quantifying Elo will be very non-trivial, since most of these are not 2v1 games; you might have to use some kind of play-quality estimate using engine analysis, then follow it up with Elo correlations with play-quality.)

I'm not sure there is a reasonable approach for N>2, although there is the 'variant' Business chess where apparently teams discuss each move. I'm a little unclear on whether it ultimately comes down to a single game of chess being played from start to finish. Apparently tournaments were held in this variant, so maybe you can look for any available games.