  In this work we develop a new algorithm for rating of teams (or players) in
one-on-one games by exploiting the observed difference of the game-points (such
as goals), also known as a margin of victory (MOV). Our objective is to obtain
the Elo-style algorithm whose operation is simple to implement and to
understand intuitively. This is done in three steps: first, we define the
probabilistic model between the teams' skills and the discretized MOV variable:
this generalizes the model underpinning the Elo algorithm, where the MOV
variable is discretized into three categories (win/loss/draw). Second, with the
formal probabilistic model at hand, the optimization required by the maximum
likelihood rule is implemented via stochastic gradient; this yields simple
on-line equations for the rating updates which are identical in their general
form to those characteristic of the Elo algorithm: the main difference lies in
the way the scores and the expected scores are defined. Third, we propose a
simple method to estimate the coefficients of the model, and thus define the
operation of the algorithm; it is done in a closed form using the historical
data so the algorithm is tailored to the sport of interest and the coefficients
defining its operation are determined in entirely transparent manner. The
alternative, optimization-based strategy to find the coefficients is also
presented. We show numerical examples based on the results of the association
football of the English Premier League and the American football of the
National Football League.
