Problem Set 9, Due: Thursday, Nov. 21, 2008
(late papers OK until 1:00 Friday)

1. A	0	3	1
   B	2	0	3
   C	4	4	0

   [Kendall-Wei Ranking Method] The table on the right gives the number of times each of three teams won games from the other two teams. For instance, Team A beat Team B 3 times and Team C 1 time.
   Call the corresponding win-loss matrix M and let Z₀ = (1,1,1) be the initial neutural ranking vector (use it as a column vector) that ranks each team equally.
   a). Compute   Z₁ = MZ₀,   Z₂ = MZ₁,   Z₃ = MZ₂,   Z₄ = MZ₃, and Z₅ = MZ₄,
   How does this rank the teams?
   b). From this computation, it is difficult to see if the iterations are really converging. Moreover, the entries will soon grow too large for the computer. To avoid this, repeat the computation EXCEPT that after computing each Z_k we then normalize it so the sum of its entries is 1. Then use this new normalized vector, W_k, to compute Z_k+1 = MW_k.
   These normalized vectors W₁,..., W₅, show the convergence occurs rapidly.

2. At the conclusion of the regular (before playoffs) 2002 Ivy League Basketball season there were three teams tied for first place. The standing are in the win-loss table on the left.

   Because only one team would qualify for the NCAA tournament there were two playoff games to determine the team that would go to the tournament. An alternative would have been to use the methods discussed in class to rank the teams. In particular we may assume that the final ranking of a team is somehow related to the ranking of the teams that it has defeated during the regular season. The table on the right shows how many times each team beat the other teams. For instance, Penn won over Cornell twice while Cornell never beat Penn.

   Team	Conf.
   Penn	11 - 3
   Yale	11 - 3
   Princeton	11 - 3
   Brown	8 - 6
   Harvard	7 - 7
   Columbia	4 - 10
   Dartmouth	2 - 12
   Cornell	2 - 12

   	Pe	Y	Pr	B	H	Col	D	Cor
   Penn	0	1	2	2	1	1	2	2
   Yale	1	0	1	1	2	2	2	2
   Princeton	0	1	0	2	2	2	2	2
   Brown	0	1	0	0	1	2	2	2
   Harvard	1	0	0	1	0	2	2	1
   Columbia	1	0	0	0	0	0	1	2
   Dartmouth	0	0	0	0	0	1	0	1
   Cornell	0	0	0	0	1	0	1	0

   
   

   Use the Kendall-Wei Ranking Method discussed in class to rank these teams. [See: Problem 1 (above) and Basketball Teams Example ].

3. [This problem does not particularly use techniques discusssed in class.]
   Snow White distributed 21 quarts of milk among the seven dwarfs. The first dwarf then distributed the contents of his pail evenly to the pails of other six dwarfs. Then the second did the same, and so on. After the seventh dwarf distributed the contents of his pail evenly to the other six dwarfs, it was found that each dwarf had exactly as much milk in his pail as at the start.
   What was the initial distribution of the milk?
   Suggestion: You might find it useful first to try similar more modest problems with, say, two, or three or four dwarfs.