Marginal utility and leverage

Last month, Chris Anderson of Soccer by the Numbers wrote an excellent post on the concept of "leverage" in soccer matches, which is defined as the likelihood of winning the match given the score and the time remaining in the game.  I hope Chris will forgive me for writing this, but I have to admit that I didn't read Chris' post when it was originally published.  After writing my post on marginal utility last night, and then reading his initial post on leverage, I am impressed at how we arrived at the same basic concept from different directions.  His data go a long way toward backing up my intuition.

Essentially, a goal that proves to be of greatest marginal utility occurs when the game is much in balance and neither team has leverage over another.  As one team acquires leverage, additional goals scored by the team will further increase its leverage and be of reduced marginal utility.  If the scoreline is a blowout, leverage will be minimized or maximized, and any additional goals will be of diminished marginal utility.

One thing to note in Chris' post is that even though the relation between goal differential and win probability follows a logistic function, the data do not reveal a symmetric relationship.  Such an observation does go a way toward backing up the point of view expressed on Twitter this morning that a goal scored to tie the match should have greater marginal utility than one scored to give the team a two-goal lead, even though both have equal marginal utility in my expression.

An interesting extension for this work is to use the numerical derivative of Chris' leverage expression and use that to express marginal utility.