1. Mathematical Rating Systems Who’s #1?
Dr. Roland Minton
with
Gregory Minton
Kenneth Massey
Kevin Bacon

2. How to Rank Candidates
I decide. (dictatorship)

4. How to Rank Candidates I decide. (dictatorship)
Plurality, other voting methods.

5. How to Rank Candidates
Too close to call: 4 candidates and 37 voters. Their preferences are
What is the right choice?

6. Answers Plurality: A wins (A-14, C-11, D-8, B-4)
Borda: B wins (B-69, C-67, D-44, A-42)
Run-off: D wins (drop B, 4 votes to D, drop C, 11 votes to D)
Condorcet: C wins (23-14,19-18,25-12)

7. An Unfair Outcome
Suppose 4 of A’s voters change to D. Now, who wins the run-off?

8. New Answer
Run-off: D wins (drop B, 4 votes to D, drop C, 11 votes to D)
Run-off after change: D now loses to C ! (drop B, 4 votes to D, drop A, 10 v’s to C)

9. How to Rank Candidates I decide. (dictatorship)
Plurality, other voting methods.
Arrow’s Impossibility Theorem.
Experts decide. (Borda count, approval voting)
Judges assign points.
Bowl Championship System (BCS) with computers.

10. Which system is the best?
What are the criteria?
Match common beliefs.
Avoid fallacies (figure skating switch, 2001 Miami vs. Nebraska).
Get the best championship game.
Predict outcomes of games.

11. A Simple System
Clemson has played 11 games, scoring 377 points and giving up 151 points. They are +226 points.
On the average, they win by 20.6 points per game.
Clemson’s rating should be 20.6 points higher than the average of their opponents’ ratings.

12. Implementing the System
C = (FA+BC+FS+…+NS) /
11 C–FA–BC-FS–…-NS = 226
One corresponding equation for each of the A teams.
Solve them.

13. Solving the Equations
Small league: A beat B by 5, B beat C by 6, C beat D by 7
A – B = 5
2B – A – C = 1
2C – B – D = 1
D – C = -7
How do you solve such systems?

14. Solving the Equations One more game: D upsets B by 2 A – B = 5
3B – A – C – D = -1
2C – B – D = 1
2D – B - C = -5
How much changes?

15. Solving the Equations Set up as matrix A – B = 5 3B – A – C – D = -1
2C – B – D = 1
2D – B - C = -5

16. Solving the Equations Add first two lines (equations).
Place result in second row.

17. Solving the Equations
Add ½ row 2 to row 3.
Place result in row 3.

18. Solving the Equations
Divide row 3 by 1.5.
Place result in row 3.

19. Solving the Equations Keep “reducing” the matrix
to “row echelon form.”

20. Solving the Equations This gives the ratings in terms of D.
Ratings are in the right column.

21. Implementing the System
Use the computer (Mathematica) to reduce the matrix (119 by 120)
Here are the results this week.
What is wrong with this system?
Wins and losses are important. In fact, the BCS computers are not allowed to use points.

22. Implementing the System
Replace points with wins – losses
11 C–FA–BC-FS–…-NS = 8-3
Solve this.
Which one is better?
20*wins is slightly better than points
60% (20*wins) + 40% (points) ± 4 is the best predictive combination according to (old) regressions.

23. Implementing the System
Mathematica solves the equations.
Weekly results at link off of
Some math (linear algebra) details:
Load results, row reduce the matrix
The matrix is not invertible
Each row gives xn – x119 = rn
So relative ratings are uniquely determined … if

24. Schedule Networks
Teams A and B can’t be compared if there’s not a schedule path between them ( a basis for comparison).
This is the familiar game of linking teams together. In 2004 …

25. Schedule Networks 618 teams in list 310,900 out of 381,924 links work
Most indispensable teams:
Albion College (245,967) Michigan
Adrian College (248,018)
Wisconsin Lutheran (249,676)
Which form a transitive loop

26. Other Networks Me -> Dennis Smith (VCU physics lab)
Dennis Smith -> Stephen Furst (VCU apartment)
Stephen Furst -> Kevin Bacon (Animal House)
Six Degrees of Kevin Bacon
Oracle of Kevin Bacon

27. Small World Phenomenon
Oracle of Bacon (3.65 links)
1-Rod Steiger (2.53 links)
876-Kevin Bacon (2.79 links)
Stanley Milgram’s letters (6 links)
Paul Erdos (over 1500 papers with 507 co-authors)
Erdos number

28. Small World Phenomenon
Billion nodes in WWW (19 links)
Molecules in cell (3 reactions)
Neurons in worm (14 synapses)
Baseball players, Elvis, Arnold, musicians, ….

29. Small World Phenomenon
Present if network has hubs
Examples: airlines, phone networks, internet, power lines, movie casts
Present if network has localized links with a small number of random long distance links
Examples: personal relationships, spread of disease, football schedules

30. Schedule Networks
Back to sports: Harvey Mudd beat Virginia Tech by 357 points over a 34-game link. So Harvey Mudd was 357 points better than Virginia Tech. But …
Virginia Tech was 149 points better than Harvey Mudd over a 10-game link. So Tech was 149 points better. And ….

31. Schedule Networks These links can be used to rank teams. (Massey)
The more links from A to B, the more likely that A is better than B.
The same system is used by Google to rank matched web sites.
The math involved includes Markov chains and stochastic processes.

32. Schedule Networks
Start at a random web site and pick a random link to follow. After many links, what is the probability that you are at site A?
The higher the probability, the more important the site is.
Problem : what about “dumb” sites that have no outward links?
With probability p (=.25) link to a random site.

33. Schedule Networks for Sports
Team A links to team B if A lost to B.
Start at a random team and pick a random link to follow. After many links, what is the probability that you are at A?
The higher the probability, the better the team is.
Problem : undefeated teams !
With probability p (=.25) link to a random site.

34. Predictions Virginia Tech over Virginia by 21
Maryland over Wake Forest by 1
Southern Cal over Notre Dame by 12
Middle Tenn over Troy by 10
Hawaii over Purdue by 13