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Problem of the Month (February 2020)

The problem used for Problem #1283 of the Macalester College Problem of the Week at my suggestion was the case K=4, N=8 of the following problem: Suppose K players enter a tournament, that the players are strictly ordered in ability, and that the better player wins 2/3 of the games. How should one construct a tournament of N games in series to maximize the chance of the best player winning the tournament. The answers are surprising, and can be found below.

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.

You can see all the best known results here. Submit your answers here.


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