My favorite talk at ISERC 2014 (the IIE conference) was “A new approach to ranking using dual-level decisions” by Baback Vaziri, Yuehwern Yih, Mark Lehto, and Tom Morin (Purdue University) [Link]. They used a Markov chain to rank Big Ten football teams in their ability to recruit prospective players. Players would accept one of several offers. The team that got the player was the “winner” and the other teams were losers. We end up with a matrix P where element (i,j) in P is the number of times team j beats team i.
The Markov chain is then normalized so that each row sums to 1 and solved for the limiting distribution. The probability of being in team j in the limit was interpreted as meaning the proportion of time that team j is the best. Therefore, the limiting distribution can be used to rank teams from best to worst.
They found that using this method with 2001 – 2012 data, Wisconsin was ranked fourth, which was much higher than it was ranked by experts and explains why they have been to 12 bowl games in a row. Illinois (my alma mater) was ranked second to last, only above lowly Indiana.
I used this method regular season 2014 Big Ten basketball wins and ended up with the following ranking. I also have the official ranking based on win-loss record for comparison. We see large discrepancies for only two teams: Michigan State (which is over-ranked according to its win-loss record) and Indiana (which is under-ranked according to its win-loss record). The Markov chain method ranks these two teams differently because Indiana had high quality wins despite not winning so frequently and because Michigan State lost to a few bad teams when they were down a few players due to injuries.
|Ranking||MC Ranking||W-L record Ranking|
|10||Penn St||Penn State|
Sophisticated methods are a little more complex than this. Paul Kvam and Joel Sokol estimate conditional probabilities in the transition probability matrix for the logistic regression Markov chain (LRMC) model using logistic regression [Paper link here]. The logistic regression yields an estimate for the probability that a team with a margin of victory of x points at home is better than its opponent, and thus, looks at margin of victory not just wins and losses.