Challenging Matrix Question

G'day

this is a question my teacher gave to us in class the other day.

We will consider a simplified situation, involving 5 players- A,B,C,D and E

In a round robin tournament the results were:

A beat C and D

B beat A

C beat B,D and E

D beat B

E beat A,B and D

1. produce a matrix M, and find the current ranking for these players

M= (5x5)

00110

10000

01011

01000

11010

M^{2} =

02011

00110

22010

10000

11110

RANKKING

M+M^{2}=

02121=6=Third

10110=3=Fourth

23021=8=**First**

11000=2=Fifth

22120=7=Second

2. One player is not happy with the outcome, and suggests to you, as the officail ranker, that is unfair to give equal importance to first, second and thrid- order influence, and perhaps arbitrary constats could be allocated to weight these influences, that is use:

M+aM^{2}+bM^{3}.... and so forth

If a little money were to change hands, is it possible for you to produce a different ranking, which you can and must justify mathematically?

I have tried many, many, many arbitrary constants.

some of them are

M+.75M^{2}+.5625M^{3}....

M+.5M^{2}+.25M^{3}....

M+1/3M^{2}+1/9M^{3}....

Would anybody please be able to exaplin to me if you can make arbitrary canstants that would change to order of the results? i was thinking if the constants are arbitrary then would the order of the athelets stay the same ??

please help cheers(Wink)(Wink)(Wink)

Re: Challenging Matrix Question

Quote:

**If a little money were to change hands,** is it possible for you to produce a different ranking, which you can and must justify mathematically?

This made me laugh

If you look at the sum of the rows in M, M^{2}, M^{3}, M^{4}; when put in order of highest to lowest they always have the same order so unless the constants are negative which would make no sense.

I only checked up to M^{4}, I didn't prove that in the general case their sums always have the same order but there is certainly a trend.