# Gaming Tree Based Evaluation Model for Badminton Tactic Benefit Analysis and Prediction
## 2 Materials and Methods
### Evaluation Model of Tactical Benefit  
#### 2.4.1 Tactical Benefit  
According to the feature of a badminton match, i.e., each rally has only one result and each stroke has a limited start placement and destination, the players have limited choices to conduct their techniques and tactics. Thus, if we consider each single rally as an individual game with different weights, we can count all the rallies together, and compute the benefit of each stroke. Given a number of rallies \(R=\{r_{1},r_{2},\ldots,r_{m}\}\), we construct a gaming tree \(T\) for all the strokes \(S_{i}=\{s^{i}_{1},s^{i}_{2},\ldots,s^{i}_{n}\}\) in each rally \(r_{i}\). and compute the benefit for each node in the gaming tree. Specifically, each node of \(T\) represent a possible stroke, and all the nodes of \(T\) covers all the strokes in the selected rallies.  
Figure 1 presents a simplified example (the specific technique and the destination for each stroke is not considered here) of building a gaming tree for three rallies \(r_{1},r_{2},r_{3}\), where \(S\) denotes strokes, \(P\) denotes the player, the \(Y\) denotes the placement, and \(N\) denotes the gaming tree node. To illustrate, consider the strokes \(\{s^{1}_{1},s^{2}_{1},s^{3}_{1}\}\), although they belong to different rallies, they are the first stroke for each rally and share the same player and placement (player \(p_{1}\) with the placement 1), leading to the same tree node position \(N_{1}\). Note that two strokes can be classified into the same tree node if and only if their player and placement are the same and their previous strokes (if existing) all have the same player and placement. For this reason, \(s^{1}_{2}\) and \(s^{2}_{2}\) can be classified to be node \(N_{2}\), as they succeed \(s^{1}_{1}\) and \(s^{3}_{1}\) separately (both can be regarded as \(N_{1}\)) and have the same player and placement. Meanwhile, although \(s^{1}_{3}\) and \(s^{2}_{3}\) also have the same player and placement, they belong to different tree nodes as their previous strokes are different, i.e., \(s^{1}_{2}\) and \(s^{2}_{2}\) have different placement.  
Based on the gaming tree constructed from each stroke \(s_{k}^{i}=(p_{k^{\prime}}^{i},x_{k^{\prime}}^{i},y_{k^{\prime}}^{i},y_{k+1}^{i})\), each tree node represents a possible stroke, and we can derive all the possible strokes for each single orally, as each rally starts from the root nodes (which have no predecessors) of the gaming tree and ends at leaf nodes (which have no successors). Meanwhile, the benefit of each possible stroke can be obtained by considering Formula (3) and the gaming tree, i.e., we summarize the score of each stroke \(s_{i}\) that can be reduced to the same gaming tree node \(N\) (denoted as \(s_{i}\in N\)) and regard it as the benefit of that stroke. Generally, the benefit of each Node N can be computed as follows.  
\[Score(N)=\sum_{s_{i}\in N}Score(s_{i}).\bigg{\{}\begin{array}{cl}1&\text{the server player wins}\\ -1&\text{the server player loses}\end{array} \tag{4}\]  
For example, given two strokes \(s_{k}^{i}\) and \(s_{k}^{i}\) that can be reduced to the same gaming tree node \(N\), and \(s_{k}^{i}\) leads to the server player wins while \(s_{k}^{i}\) is the opposite, we calculate the benefit of \(N\) as \(Score(N)=Score(s_{k}^{i})-Score(s_{k}^{i})\). Note that according to the Formulas (1)-(3), we can obtain that \(\sum Score(s_{k}^{i})=1\). Thus, the benefit range of each node is \([-1,1]\).