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.
翻译:在这项工作中,我们为一对一游戏中球队(或球员)的评级开发了一种新的算法,利用观察到的游戏点差异(如目标)的观察差异,也称为胜利幅度(MOV)。我们的目标是获得Elo式算法,其操作简单,可以执行和直观理解。这分三个步骤完成:首先,我们界定了球队技能和分散的MOV变量之间的概率模型:这概括了埃洛算法的模型,其中MOV变量分为三类(双向/亏损/拖动)。第二,随着正式的概率模型的掌握,最大可能性规则所要求的优化通过随机梯度梯度得到实施;这为评级更新提供了简单的在线方程式,其总体形式与埃洛算法的特征相同:主要差异在于分数和预期分的界定方式。第三,我们提出了一种简单的方法来估计模型的系数,从而界定了算法的运行方式;第二,在正式的概率模型模型模型模型模型模型模型模型模型模型模型的运行方式上,我们用一种封闭的比值模型,并且用一种封闭的货币比值来界定国家足球比值计算利率的汇率的计算方法,因此,我们用一种固定的公式的计算得出了美国货币比值。