Match order

Two teams hold a knock out match. Each game consists of one player from each team, after the game, the loser got knocked out, the winner stays. The loser's team send out another player, another game begins. If a team run out of players, i.e. all the players have been knocked out, the team lost the match. We know all the players have a power rating. When a player with power A is matched with a player with power B, his winning chance is A/(A+. Suppose each team consists of 10 players, with power ratings 1,2,3,...,10. Suppose one team knows the other team will send out its players in increasing order, i.e. player with rating 1 goes up first, then, player with rating 2 goes up, and so on. What order should this team send its players so that to maximize its chance of winning.

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Very nice solution, Robert!!

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Nice!

Yes, indeed, this is a wonderful solution. Thank you very much.

I was thinking along a different line. If you go to science museum in Boston, in the mathematics hall, there's a verticle grid with has balls dropping from the top, the balls have 50-50 chance going left or right each time they pass a grid pole, at the bottom, you see a nice gaussian (or normal) shape after a while. What I am thinking is instead of 50-50, we have p(i,j) going left, somehow prove that the number of balls drop to the left will not be affected by the order of the pole's arrangement. Obviously, since this is not true for general p(i,j), we need special construction about this p(i,j)=f(i)/(f(i)+g(j)). That's where I got stuck. But I still think this might be a good approach.