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Ideally, everyone would get to play everyone else. This is desirable from a social perspective, because it allows everyone to meet everyone. It is also desirable from the perspective of fairness, because it is more fair to declare someone the winner if he has played everyone else. This means that he has played the same opponents as others and that he has played against each of the losers. With large numbers of players, having everyone play everyone else becomes less desirable, because it may overload the players with too many games to play. In that case, the social goal may be compromised, since it is not critical. Instead of fully meeting it, it may be maximally met by giving a player a new opponent for each game. But the goal of fairness still remains as important as before, and it should not be compromised if at all possible. The advantage of the subtournament method is that it works best for maximizing the fairness of the tournament. But since it is not doable with most numbers, I have proposed a different method for more than eight players. It is a three-round method with eliminations after each of the first two rounds. Four games are played in the first round, four more in the second round, then three in the last round. The first round reduces the players to eight, and the second round reduces the players to four. In the first round, it would not be fair to immediately give the win to the highest scoring player, because there would be some players who have not played each other, including some who have not played the highest scoring player. But it does seem fair to eliminate all but the top eight players. Odds are good that the best player will be among these eight. In the second round, players play four more games, as much as possible against new opponents. At the end of this round, it is still possible that some of the players in the second round have not played each other, though it is no longer an inevitability. So, while it may not be fair to declare the highest scoring player in this round the winner, it does seem fair to eliminate the four lowest ranking players. Finally, in the third round, all remaining players play each other at one game apiece, and the winner of the tournament is the winner of this round. This method is also described in the text I am adding to this page, and it includes details on tie-breaking, which I have not included here. In the comments section, I am focusing more on the justification for this method.