Beating the Mogul Game — An Exercise in Applied Mathematics

I have often wondered about how to rank sports teams.  This goes way back to when I was 10 years old, when I ran across a magazine at summer camp that purported to do this for NFL football.  And so I wondered for many years, looking at similar problems and wondering how a ranking of teams could be generated from a win-loss history.  I finally came to a conclusion when I played the Mogul Game.

The Mogul Game has 148 rich people, and they vary from the super-rich (Gates, Buffett, Ellison) to the not-so-rich (I think they got a kick out of putting Donald Trump at/near the bottom of the list, much as he boasts to Forbes that he is much wealthier than they calculate).

After playing the game idly for a little while, I concluded that if I wanted to win, I would have to capture and analyze data from the game in order to win it.  And so I did, recording who was richer than whom.  I went through four phases:

  • Doing qualitative comparisons when I wasn’t certain of who was richer.  Who had the two parties beaten and lost to?
  • Comparing the trial ranks when the difference was greater than 10.
  • Looking at the highest ranked persons that a given set of contestants had won against, and the lowest ranked that they had lost to.
  • Looking at the average of the highest rank won against and the lowest rank lost to as the best proxy for a contestant’s own rank, unless it violated the results of an actual contest.  In hindsight, I should have adopted that rule much earlier.

It took three days of off-and-on playing to master the game.  Not all that important, but as I mentioned above, the method can be applied with some modifications for ranking sports teams in an unbiased way.  The same could be applied to any competitive activity where there is a win/loss result.  There are two changes for other activities, though.  Games are not necessarily transitive.  Rich person A is richer than B.  B is richer than C.  A will always be richer than C.  In competitions, Team A can beat team B one day, and lose the next.  Also, Team A can beat team B, which can beat Team C, but C can beat A.  So, if I were doing this for baseball teams, my ranks would drive probabilities of one team beating another.

Why would this be necessary when one can simply inspect the win-loss percentages?  Teams with good records may have weak schedules, and this takes account of the strength of the teams played in assessing the strength of a team.  I’m not sure what they do with ranking College Football or Basketball teams, but this would be a more bloodless way of making the comparison.  Granted, it takes a certain number of contests before there is enough density of information to create a ranking, but given a list of wins and losses from an entire season, this method should be capable of ranking an entire league.

I know this is an odd post for me, but I found it to be an interesting project, and it does have other applications.  Thoughts?