Trader Optimality Conditions

\[\begin{aligned} \left[r_{y} d_{m} \begin{aligned} \left(\begin{aligned} -{1}^{NC}_{tnc} \cdot \frac{\partial\tilde{P}^{T \rightarrow D}_{ncbmy}\biggl(Q^{T \rightarrow D}_{ncbmy}\biggr)}{\partial q^{T \rightarrow D}_{tncbomy} } \cdot q^{T \rightarrow D}_{tncbomy} \\ - {1}^{NC}_{tnc} \cdot \tilde{P}^{T \rightarrow D}_{ncbmy}\biggl(Q^{T \rightarrow D}_{ncbmy}\biggr) \\ - \left( 1 - {1}^{NC}_{tnc} \right) \cdot \pi^{T \rightarrow D}_{ncbmy} \\ \end{aligned}\right) \\ + \phi^{T}_{tncomy} + d_{m} \lambda^{T}_{tncoy} \end{aligned} \right] \geq 0 & \perp q^{T \rightarrow D}_{tncbomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, b \in \mathcal{DSB}, \\ o \in \mathcal{O}, m \in \mathcal{M}, \\ y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} r_{y} d_{m} \left(\begin{aligned} \pi^{P}_{ncomy} \end{aligned}\right) - \phi^{T}_{tncomy} \end{aligned}\right] \geq 0 & \perp q^{T \leftarrow P}_{tncomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}_p(t), \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} \phi^{T}_{tncomy} - r_{y} d_{m} \left(\begin{aligned} \pi^{T \rightarrow V}_{tncomy} \end{aligned}\right) \end{aligned}\right] \geq 0 & \perp q^{T \rightarrow V}_{tncomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ m \in \mathcal{M}, y \in \mathcal{Y}, \end{aligned} \\ % \left[\begin{aligned} r_{y} d_{m} \left(\begin{aligned} \pi^{V \rightarrow T}_{tncomy} \end{aligned}\right) - \phi^{T}_{tncomy} \end{aligned}\right] \geq 0 & \perp q^{T \leftarrow V}_{tncomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} \phi^{T}_{tncomy} - r_{y} d_{m} \left(\begin{aligned} \pi^{T \rightarrow S}_{tncomy} \end{aligned}\right) \end{aligned}\right] \geq 0 & \perp q^{T \rightarrow S}_{tncomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} r_{y} d_{m} \left(\begin{aligned} \pi^{S \rightarrow T}_{tncomy} \end{aligned}\right) - \phi^{T}_{tncomy} \end{aligned}\right] \geq 0 & \perp q^{T \leftarrow S}_{tncomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} r_{y} d_{m} \pi^{A}_{acmy} \\ + \sum_{n \in \mathcal{N} | a \in \mathcal{A}_s(n)} (1+l^{A}_{ac}) \phi^{T}_{tncomy} \\ - \sum_{n \in \mathcal{N} | a \in \mathcal{A}_e(n)} \phi^{T}_{tncomy} \end{aligned}\right] \geq 0 & \perp q^{T}_{tacomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, o \in \mathcal{O}, \\ (a,c) \in \mathcal{AC} , \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} \sum_{a \in \mathcal{A}_e(n) | (a,c) \in \mathcal{AC} } q^{T}_{tacomy} \\ + \sum_{n \in \mathcal{N}_p(t)} q^{T \leftarrow P}_{tncomy} \\ + q^{T \leftarrow V}_{tncomy} \\ + q^{T \leftarrow S}_{tncomy} \\ - \sum_{a \in \mathcal{A}_s(n) | (a,c) \in \mathcal{AC} } (1+l^{A}_{ac}) q^{T}_{tacomy} \\ - \sum_{b \in \mathcal{DSB}} q^{T \rightarrow D}_{tncbomy}\\ - q^{T \rightarrow V}_{tncomy} \\ - q^{T \rightarrow S}_{tncomy} \end{aligned}\right] \geq 0 & \perp \phi^{T}_{tncomy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} \Lambda^{T}_{tncoy} - \sum_{m \in \mathcal{M}} d_{m} \sum_{b \in \mathcal{DSB}} q^{T \rightarrow D}_{tncbomy} \end{aligned}\right] \geq 0 & \perp \lambda^{T}_{tncoy} \geq 0 & \begin{aligned} \forall t \in \mathcal{T}, n \in \mathcal{N}, \\ c \in \mathcal{C}, o \in \mathcal{O}, \\ y \in \mathcal{Y} \end{aligned} \\ \end{aligned}\]