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This month's problem is to generalize this further. What if the tournament's purpose is not just to specify a winner, but to order all the players in ability. What tournament maximizes the probability that we are correct?
For example, if there are K=3 players and N=2 games, the best we can do is A vs B and (regardless of who wins the first time) A vs C, and then to guess an ordering that is consistent with those results. 1/3 of the time, A will be the middle player in ability, and then we will guess correctly if both games are accurate. 2/3 of the time, A will be the best or worst player, and then will guess half the time both games are accurate. So we will succeed with probability (1/3)(2/3)2+(2/3)(1/2)(2/3)2=8/27.
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