Transitivity, Voting,
and Sports Ratings
Roland Minton
Roanoke College
A Violation of Transitivity
You pick a die, I’ll pick one.
We’ll play best-of-seven.
3
4
5
6
3
4
5
6
3
4
5
2
3
4
1
2
3
0
1
2
3
0
1
2
4 beats 3 (67%), 5 beats 4 (67%), 6 beats 5 (67%), and …
3 beats 6 (67%) !
More Violations of Transitivity
Oregon 59-20 Florida State 43-3 Wake Forest 6-3 Virginia Tech
Virginia Tech 35-21 Ohio State 42-20 Oregon
Voting for Jones or Miller or Barrett:
Person 1 ranks Jones-Miller-Barrett
Person 2 ranks Miller-Barrett-Jones
Person 3 ranks Barrett-Jones-Miller
The majority prefer Jones to Miller, and Miller to Barrett
and Barrett to Jones. So much for “majority rules”!
Some Voting Theory
Jones-Miller-Barrett form a Condorcet cycle.
1785: Marquis de Condorcet theory of voting published.
1948: Kenneth Arrow, RAND, “The Possibility of a
Universal Social Welfare Function.” Summary: not
There is no “rational” (transitive) ranking system that is
fairly determined by voter preference lists and obeys
non-dictatorship, monotonicity, and IIA (independence
of irrelevant alternatives).
Some Ranking Theory
Ranking: an ordered list of candidates, obeying transitivity.
Can be derived from a ratings system.
Some football ranking systems: Massey, Colley, Sagarin, Elo, …
Massey System (COMAP Module 725, 1992):
Ratings for teams A and B predict game score differential a – b.
If actual score is +5, the error is a – b – 5. Minimize
Σ (a-b-5)2 over all games and team ratings.
Some Hodge Theory
Visualize games scores
as a flow on a graph.
Define analogs to
Gradient flow;
Potential function;
Path independence;
Curl free; divergence free;
Harmonic flow.
Some Hodge Theory
Every flow graph can be decomposed into an orthogonal sum
of a gradient flow, a harmonic flow, and a curly flow.
Robert Sizemore, Wake Forest, 2013 (CSAM):
The gradient flow results from a potential function given by
the Massey ratings. The sum of the other two components
gives the residuals. A ratio of the norm of the gradient flow
to the norm of the residuals measures how transitive the
system is, or how reliable the ratings are.
Some Results
College football 2012: ratio about 1.
College basketball 2012: ratio about 1.
Soccer World Cup 2014: ratio about 1.5 (few upsets).
The ratings are very sensitive to the Germany-Brazil score.
The ratio is relatively stable.
7-1: Brazil #19, Mexico #20 (-4.82), ratio 1.53
3-1: Brazil #9, Mexico #17 (-2.38), ratio 1.48
France #2, Belgium/Argentina #3, USA #6
Conclusions
The Massey ratings can be obtained in several different ways:
season results, least squares, gradient flow
The Hodge decomposition separates the transitive part of
the results from the non-transitive.
Any Questions?