Question

Suppose we have 5 peewee teams in a certain hockey tournament.

Vipers

Hawks

Bruins

Sharks

Flames

They played a round robin tournament with tied games allowed. Based on the
results we have to rank the teams. We rank the teams based on the number of wins
and if two teams A and B have the same number of wins then A is ranked above B
if A beats B. No other game information is allowed.

Round Robin Results:

Vipers beat Hawks ;

Vipers tie with Bruins ;

Vipers beat Sharks

Hawks beat Bruins ;

Hawks beat Flames ;

Bruins beat Sharks

Bruins beat Flames ;

Sharks beat Hawks ;

Sharks beat Flames ;

Flames beat Vipers

If the team i beats the team j then the entry (i , j) in the matrix A is 1 and the
(j , i) entry in the matrix A is 0.

If the team i ties with the team j then the entry (i , j) in the matrix A is 1/2 and
the (j , i) entry in the matrix A is also 1/2.

What do the row sums in the matrix A represent?

4) Compute the matrix A^2?

What do the entries in A^2 represent? Explain.

What are the row sums in the matrix A^2 representing? Explain.

Can we use these row sums in A^2 to rank the teams? Explain.

Answer 1

(1)

"A --> B" means A beats B.

V --> 2.

H --> 2.

B --> 2.

S --> 2.

F --> 1.

V ties with B.

H --> B, B -->S, S --> H.

H, B, S forms a circle of beating with equal numbers of win.

And, from the figure, below in (2), the graph cuts the two lines.

So, it is not a planar graph.

Hence, it is not possible to rank them.

(2)

SK