One informative benchmark when ranking and rating sports teams is how many times the ranking has been violated. A ranking violation occurs when a team beats a higher ranked team. Ideally no violations would occur, but in practice this rarely happens. In many cases it is unavoidable, for example in this three team competition: Team A beats team B, team B beats team C, and team C beats team A. In this case, for any of the 6 possible rankings of these three teams at least one violation would occur.

Inspired by this, one could try to construct a ranking with as few violations as possible. A minimum violations ranking (MVR), as it is called. The idea is simple and intuitive, and has been put to use in ranking American college sport teams. The MinV ranking by Jay Coleman is one example.

MV rankings have some other nice properties other than being just an intuitive measure. A MV ranking is the best ranking in terms of backwards predictions. It can also be a method for combining several other rankings, by using the other rankings as the data.

Despite this, I don’t think MV rankings are that useful in the context of football. The main reason for this is that football has a large number of draws and as far as I can tell, a draw has no influence on a MV ranking. A draw is therefore equivalent with no game at all and provides no information.

MV rankings also has another problem. In many cases there can be several rankings that satisfies the MV criterion. This of course depends on the data, but it seems nevertheless to be quite common, such as in the small example above.

Unfortunately, I have not found any software packages that can find a MV ranking. One algorithm is described in this paper (paywall), but I haven’t tried to implemented it myself. Most other MVR methods I have seen seem to be based on defining a set of mathematical constraints and then letting some optimization software search for solutions. See this paper for an example.