I tried to add the right amount of detail here but I think I have failed at that to the point where it has made the whole thing difficult to answer.

This is where I fear that I lacked clarity. In your three color, two round, three players, two games example, what would happen if the total number of players is 21 so that 14 games are required? Would you logic for estimating the third rank player hold?

In your example, if I was trying to estimate the value of the second ranked player, the simulation trial would be exactly correct. Obviously that is a coincidence due to your assumptions but it makes me wonder if it is problematic to estimate top or bottom ranks but ones in the middle might be more successful.

I can't help but think that there is more information available in my dataset than your example makes it seem. But I'm no expert and I probably failed to provide enough detail. Note that the score of each player is not correlated like that. That is, even though there is some skill involved in this game it is common for a player do well in one game but poorly in another.

@stats_model correct!

@stats_model each color in a game represents a different player. The object of the game (and the scoring) is unrelated to the color itself. Please let me know if I can clarify anything else. I'll try to edit for clarity.

Each experiment consists of picking k elements without replacement. But every time a new experiment begins we start with the same n items.