## 3 New Offensive-Defensive Plus-Minus Ratings
### Rating Model  
Following [PERSON] and [PERSON] (2016), the new offensive-defensive plus-minus ratings for soccer are calculated by solving an unconstrained quadratic program. Let \(M\) be a set of matches, with each match \(m\in M\) divided into segments \(s\in S_{m}\), defined as a maximal period of time without changes to the players appearing on the pitch. The duration of segment \(s\) is \(d(m,s)\) minutes. Let \(h=h(m)\) and \(a=a(m)\) be the two teams involved in match \(m\), where \(h\) is the home team. In the case that the match is played on neutral ground, one of the two teams is arbitrarily assigned as \(h\) and the other as \(a\). Let \(g(m,s,h)\) be the number of goals scored by team \(h\) in segment \(s\) of match \(m\), whereas the number of goals scored by \(a\) is \(g(m,s,a)\). Past plus-minus rating models have used \(g(m,s)=g(m,s,h)-g(m,s,a)\) as the dependent variable of the observation associated to the segment. Here, to facilitate both offensive and defensive ratings, each segment corresponds to two observations. One observation is from the perspective of team \(h\), and has \(g(m,s,h)\) as the dependent variable, whereas the other is from the perspective of \(a\), with \(g(m,s,a)\) as the dependent variable. In addition, define \(g^{S}(m,s)\) as the goal difference in favour of \(h\) at the beginning of the segment, and \(g^{E}(m,s)\) as the goal difference at the end of the segment. The change of the goal difference in favour of \(h\) in segment \(s\) of match \(m\) then becomes \(g(m,s)=g^{E}(m,s)-g^{S}(m,s)\). Let \(t^{MATCH}(m)\) denote the time that match \(m\) is played, and let \(T\) be the time at which ratings are calculated, as illustrated in Figure 1.  
Let \(P\) be the set of all players. The set of players for team \(t\) that appears on the pitch for a given segment \(s\) are denoted by \(P_{mst}\). The set of players that are involved in offensive contributions is denoted by \(P_{mst}^{D}\), and the set of players involved in defensive contributions is denoted by \(P_{mst}^{D}\). For \(n=1,\ldots,4\), define \(r(m,s,n)=1\) if team \(h\) has received \(n\) or more red cards before the beginning of segment \(s\) and team \(a\) has not, \(r(m,s,n)=-1\) if team \(a\) has received \(n\) or more red cards and team \(h\) has not, and \(r(m,s,n)=0\) otherwise. If a team has made all its available substitutions and a player on the team becomes injured and must leave the pitch, the situation is modelled in the same way as a red card. Each match \(m\) belongs to a competition organized by an association. Let \(c^{A}(m)\) be the association organizing the match. This could be a national organization, a continental federation such as UEFA, or FIFA. The model allows the home field advantage of match \(m\) to depend on \(c^{A}(m)\). Furthermore, a set of domestic league competitions \(C\) is considered, with \(C_{p}\) being the subset of such competitions in which player \(p\) has participated. For example, a given player could have appeared in the French Ligue 2, the English Championship, and the English Premier League, resulting in these three leagues being members of \(C_{p}\). Each player \(p\) is associated to a set \(P_{p}^{SIM}\) of players that are considered to be similar. This set is based on which players have appeared together on the same team for the most minutes of playing time. The time of the last match where players \(p\) and \(p^{\prime}\) appeared together is denoted by \(t^{SIM}(p,p^{\prime})\) (see Figure 1). The quality of players is assumed to depend on their age, allowing the model to capture their typical improvement in early years as well as their decline when getting older. Let \(t^{BIRTH}(p)\) be the time of birth for player \(p\). The age of player \(p\) at the time of match \(m\) is then \(\Delta^{AGE}(m,p)=t^{MATCH}(m)-t^{BIRTH}(p)\), as illustrated in Figure 1. The average effect of age on the ratings of players is modelled as a piecewise linear function. To this end, an ordered set of \(k\) age values \(Y=\{y_{1}=y^{MIN},y_{2},\ldots,y_{k}=y^{MAX}\}\) is defined.