Useful measure to quantify how competitive a sports match is

Question

I'm trying to put together a summary of data from a recreational sports league's most recent season. There are significant disparities in skill level between teams, so a large portion of the games result in one team winning by a huge margin. I am trying to support the idea that the league should be split into two divisions so that this happens less often and the less competitive teams can get more enjoyment out of playing. I put together a list of all the match scores labeled by team but with no grouping other than the winning/losing score for a game. I've done some simple calculations, and it's pretty obvious things are unbalanced, but I'm not sure how to best represent the data.

I did some basic plots on the absolute and relative differences $$\frac{|win - lose|}{\max(win, lose)}$$ between the scores for each game, but both measures seem to have drawbacks.

The significance of the absolute difference depends a bit on the magnitude of the scores; 10 to nothing is somewhat more unbalanced than 20 to 10. The relative difference seems skewed the opposite direction, where a score of 2 to nothing is significantly different from 20 to nothing (I feel kind of bad for the losing team of that game :-\ ).

I've done a bit of Googling around this, and most everything I've found is either advanced sports statistics and complicated statistical tests. Is there a good, simple measure that would be useful in this situation?

In case it helps, some general characteristics of the dataset:

• Teams score points in one-point increments.
• The scores in this dataset range from 0 to 21.
• Scores are not constrained by any upper limit, and ties are possible.
• A lot of games are obviously unbalanced, with about half the games ending with a difference of 10 points, and about half the games ending with a margin of at least 75%. There are also quite a few games where one team scored 0.

Answer 1

What about: $$ f(x,y) = \frac{(x-y)^2}{1+x+y} $$ where $x$ and $y$ are the scores.

Values of $f$ closer to zero indicate a more balanced game. Ties result in exactly zero, i.e., $f(x,x)=0$.

Regarding your examples, $f(10,0)\approx9.09$ and $f(20,10)\approx3.23$, so, $10$ to $0$ is more imbalanced than $20$ to $10$.

Answer 2

Because you are round-robin, you have a natural measure of balance. In economics, the problem you are discussing is called "competitive balance." The difficulty is that there is not a simple or easy measure points based measure for two distinct reasons.

The first is that the process is not random in the ordinary sense of the word. Imagine a Major League team plays a Little League team. The score will not be random. The Major League team will be choosing the score. They may decide to win. They may choose to lose. The outcome is not random. For highly skewed outcomes, where competitive balance is completely gone, randomness is not much of an issue.

The second is that what you are measuring if it were random, is called skew. At a certain level, skew is a comparatively simple measure; it sounds like it is more complex than you want to use.

Fortunately, if next season will be representative of this season, then you have the simplest measure of all, the number of wins.

You could use something like the Elo measure as a ranking system. It is pretty simple to calculate, but it is probably an unnecessary complication.

Most chess systems use the Elo ranking system because they also want to predict the likely winners and losers for use in Swiss system tournaments because most tournaments segment the tournament into levels based on rankings. For example, masters play masters; beginners play beginners; grandmasters play grandmasters.

Since you have three divisions, split your field by total wins into three groups in ascending order. You get to move between groups from year to year by changing your wins. The top of the bottom group moves to the middle. The bottom of the middle moves to the bottom. The top of the middle moves to the top, bottom of the top moves to the middle.

There is also a special advantage to this system. It is hard to rig.

If I know that I can rig my position in the rankings by controlling my final points, then I might choose to be the top of the middle by always winning by one point. If you win every game by exactly one point, then you should be in the top group even though you appear less good than the team that wins by 20 points. Also, imagine there is a team that wins every game but one by twenty points. That team was defeated that wins every game by one point.

You will get closer to balance if you eliminate the random sorting. Alternatively, you could create an overall Elo system to split the groups, but the gain is probably minimal. The difference is that many teams could break out of the middle or the bottom into another group if their characteristics changed a lot.

If the teams are subject to a lot of change, then you could use a forgetful Elo system so that the results from two years ago and before disappeared as if they never happened.

So your choices are a simple Elo ranking where you ignore the statistical components chess uses to estimate the chance of one team beating another, or you could use the number of wins as a ranking. In that system, the bottom third of the three leagues become a league, the middle third of the three leagues become a league, and the leftover becomes a league. Going forward, only the teams on the boundaries move.