# Playing with models: quantitative exploration of life.

Posted in Sports, Statistics by Alexander Lobkovsky Meitiv on March 10, 2010

1986 Celtics

1997 Bulls

What would you do with a time machine? I bet some people would be chomping at the bit to pit two dominant teams from different eras against each other and have a grand old spectacle!
But alas, it is safe to say that a time machine will remain for the foreseeable future in the realm of magic.

Can we get a glimpse at what the outcome of such a magical game might be? Is there a scientifically sound way to rate sports teams in a way that judges their true strength. Most importantly, we need a method that yields ratings whose scale does not change with time so that a team that gets a rating of 2000 thirty years ago is as strong (in some sense) as a team that gets a rating of 2000 today.

We are indeed in luck! Such a system exists. It was proposed in the 1950’s by a Hungarian mathematician Arpad Elo (read about him on Wikipedia) and bears his name. His system is based on sound mathematical theory and ever since then dozens upon dozens of mathematical papers have been proving how reliable and reasonable the system is. Although Elo originally proposed his system to rate chess players, it has been adopted by a number of other sports bodies including FIDE, FIFA, MLB, EGF and others.

At the core of the ELO system is the ranking updating scheme which adjusts the ranking of the two teams (or players) after each match depending on the result. Given the rankings before the game, one can compute the probability of each outcome given that the actual performance has a certain probability distribution. If the stronger team wins its rating increases by a smaller amount than if the weaker team wins. There are many different specific incarnations of the system. While some are more accurate than others, even in its simplest form, the system is quite useful. In fact using publicly available match data we can resolve the question:

### If 1997 Chicago Bulls played a best of 7 series against the 1986 Boston Celtics, what are the chances of each team winning?

After downloading the match data (56,467 games over 64 years that involved a total of 53 franchises some of which changed names and cities a number of time) and computing the rating history I came up with the top ten highest rated franchises:

 Rank Team Year achieved Rating 1 Chicago Bulls 1997 2233.7 2 Boston Celtics 1986 2184.9 3 Los Angeles Lakers 1988 2163.3 4 Philadelphia 76ers 1983 2149.2 5 Detroit Pistons 1990 2137.4 6 Utah Jazz 1999 2129.9 7 Dallas Mavericks 2007 2126.5 8 San Antonio Spurs 2007 2089.4 9 Milwaukee Bucks 1971 2081.6 10 Seattle Supersonics 1996 2076.5

It is a telling sign that the NBA is a competitively healthy organization that the top 10 all time high ranking teams of all time pretty close to each other in rating. Also, it seems at least superficially, that there is no historical bias meaning the objective meaning of a rating does not change with time.

So, what would happen if the 1997 Bulls played a best of 7 series against the 1986 Celtics?
Home field advantage aside (the ranking I am using does not take that into account), the probability of the Bulls winning any particular game is $p=0.53$ The probability of winning a best of 7 series (I defined $q = 1 - p$ below)

$\displaystyle{\frac{p^4(1 + 4q _ 15q^2 + 4q^3}{p^4(1 + 4q + 15q^2 + 4q^3) + q^4(1 + 1p + 15p^2 +4p^3)} = 0.575}$

The Bulls would have a 57.5% chance of winning the series: an exciting spectacle indeed!

Finally I leave you with a graph of the historical ratings of six teams from large metropolitan areas from 1980 to present day. It seems that it is extremely difficult to maintain a dominant team for more than a few seasons (although the Lakers managed to do so in the 1980’s).

Historical season ending ELO ratings for six NBA teams from large metropolitan areas