\begin{aligned} \sum_i v_i(S_i) & \geq \frac{1}{2} \sum_i \sum_{j \in S_i} v_i(j \vert S_i^{<j}) + \frac{1}{2} \sum_k \sum_{j \in O_k}  v_k(j \vert S_k^{<j} \cup O_k^{<j}) \\ & \geq \frac{1}{2} \sum_i \sum_{j \in S_i} v_i(j \vert S_i^{<j}  \cup O_i^{<j}) + \frac{1}{2} \sum_k \sum_{j \in O_k}  v_k(j \vert  S_k^{<j} \cup O_k^{<j}) \\ & \geq \frac{1}{2} \sum_i \sum_{j \in S_i \cup O_i} v_i(j \vert S_i^{<j} \cup O_i^{<j}) \\ & = \frac{1}{2} \sum_i v_i(S_i \cup O_i) \geq \frac{1}{2} \sum_i v_i(O_i) \end{aligned}