SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $G_{i}(\mathbf{p},\mathbf{r})$  is the scaled partial derivative introduced in Eq. (25). Then, it always holds that  $\mathcal{G}(\mathbf{p}) > 0, \forall \mathbf{p} \in \mathcal{P}^2 \backslash \{\mathbf{p}^{\star \star}\}$ , where  $\mathbf{p}^{\star \star} = (p_H^{\star \star}, p_L^{\star \star})$  denotes the unique SNE, and the function  $\mathrm{sign}(\cdot)$  is defined in Eq. (27).
Furthermore, for every  $\epsilon > 0$ , there exists  $M_{\epsilon} > 0$  such that  $\mathcal{G}(\mathbf{p}) \geq M_{\epsilon}$  if  $\varepsilon(\mathbf{p}) \geq \epsilon$ , where  $\varepsilon(\mathbf{p})$  is the weighted  $\ell_1$ -distance function defined in Eq. (24).
Proof. Firstly, by the first-order condition at the SNE (see Eq. (16)), we have  $G_{H}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star}) = G_{L}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star}) = 0$ , which also implies  $\mathcal{G}(\mathbf{p}^{\star \star}) = 0$ . Then, we recall the definition of four regions  $N_{1},N_{2},N_{3}$ , and  $N_{4}$  in Eq. (28). We show that  $\mathcal{G}(\mathbf{p}) > 0$  when  $\mathbf{p}$  belongs to either one of the four regions.
1. When  $\mathbf{p} \in N_1$ , i.e.,  $p_H > p_H^{\star \star}$  and  $p_L \geq p_L^{\star \star}$ , the function  $\mathcal{G}(\mathbf{p})$  becomes
$$
\begin{array}{l} \mathcal {G} (\mathbf {p}) = - G _ {H} (\mathbf {p}, \mathbf {p}) - G _ {L} (\mathbf {p}, \mathbf {p}) \\ = - \frac {1}{\left(b _ {H} + c _ {H}\right) p _ {H}} - \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L}} - \left(d _ {H} (\mathbf {p}, \mathbf {p}) + d _ {L} (\mathbf {p}, \mathbf {p})\right) + 2 \\ = - \frac {1}{\left(b _ {H} + c _ {H}\right) p _ {H}} - \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L}} + d _ {0} (\mathbf {p}, \mathbf {p}) + 1 \\ = - \frac {1}{\left(b _ {H} + c _ {H}\right) p _ {H}} - \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L}} + \frac {1}{1 + \exp \left(a _ {H} - b _ {H} p _ {H}\right) + \exp \left(a _ {L} - b _ {L} p _ {L}\right)} + 1, \tag {87} \\ \end{array}
$$
where  $d_0(\mathbf{p},\mathbf{r}) = 1 - d_H(\mathbf{p},\mathbf{r}) - d_L(\mathbf{p},\mathbf{r})$  denotes the no purchase probability. We observe from Eq. (87) that  $\mathcal{G}(\mathbf{p})$  is strictly increasing in  $p_H$  and  $p_L$ . Together with the fact that  $p_H$  and  $p_L$  are lowered bounded by  $p_H^{\star \star}$  and  $p_L^{\star \star}$ , respectively, and  $\mathcal{G}(\mathbf{p}^{\star \star}) = 0$ , we verify that  $\mathcal{G}(\mathbf{p}) > 0$  when  $\mathbf{p} \in N_1$ . With the similar approach, we show that when  $\mathbf{p} \in N_3$ , i.e.,  $p_H < p_H^{\star \star}$  and  $p_L \leq p_L^{\star \star}$ , it also follows that  $\mathcal{G}(\mathbf{p}) > 0$ .
2. When  $\mathbf{p} \in N_2$ , i.e.,  $p_H \leq p_H^{\star \star}$  and  $p_L > p_L^{\star \star}$ , the function  $\mathcal{G}(\mathbf{p})$  becomes
$$
\begin{array}{l} \mathcal {G} (\mathbf {p}) = G _ {H} (\mathbf {p}, \mathbf {p}) - G _ {L} (\mathbf {p}, \mathbf {p}) \\ = \frac {1}{\left(b _ {H} + c _ {H}\right) p _ {H}} - \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L}} + d _ {H} (\mathbf {p}, \mathbf {p}) - d _ {L} (\mathbf {p}, \mathbf {p}) \tag {88} \\ = \frac {1}{\left(b _ {H} + c _ {H}\right) p _ {H}} - \frac {1}{\left(b _ {L} + c _ {L}\right) p _ {L}} + \frac {\exp \left(a _ {H} - b _ {H} p _ {H}\right) - \exp \left(a _ {L} - b _ {L} p _ {L}\right)}{1 + \exp \left(a _ {H} - b _ {H} p _ {H}\right) + \exp \left(a _ {L} - b _ {L} p _ {L}\right)}. \\ \end{array}
$$
By Eq. (88), we notice that  $\mathcal{G}(\mathbf{p})$  under region  $N_{2}$  is strictly decreasing in  $p_H$  and strictly increasing in  $p_L$ . Meanwhile, since  $p_H \leq p_H^{\star \star}$ ,  $p_L > p_L^{\star \star}$ , and  $\mathcal{G}(\mathbf{p}^{\star \star}) = 0$ , it implies that  $\mathcal{G}(\mathbf{p}) > 0$  when  $\mathbf{p} \in N_2$ . Moreover, by similar reasoning, we show that when  $\mathbf{p} \in N_4$ , i.e.,  $p_H \geq p_H^{\star \star}$  and  $p_L < p_L^{\star \star}$ , the inequality  $\mathcal{G}(\mathbf{p}) > 0$  also holds.
Finally, we are left to establish the existence of  $M_{\epsilon}$ . It suffices to show that
$$
\min  _ {\varepsilon (\mathbf {p}) = \epsilon_ {1}} \mathcal {G} (\mathbf {p}) > \min  _ {\varepsilon (\mathbf {p}) = \epsilon_ {2}} \mathcal {G} (\mathbf {p}). \tag {89}
$$
for every  $\epsilon_1 > \epsilon_2 > 0$ . Suppose  $\mathbf{p}^{\epsilon_1} \coloneqq \arg \min_{\varepsilon(\mathbf{p}) = \epsilon_1} \mathcal{G}(\mathbf{p})$ . Define  $\mathbf{p}^{\epsilon_2} \coloneqq (\epsilon_2 / \epsilon_1) \mathbf{p}^{\epsilon_1} + (1 - \epsilon_2 / \epsilon_1) \mathbf{p}^{\star \star}$ , which satisfies that  $\varepsilon(\mathbf{p}^{\epsilon_2}) = \epsilon_2$ . Then, we have that
$$
\mathcal {G} \left(\mathbf {p} ^ {\epsilon_ {2}}\right) = \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {\epsilon_ {2}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {\epsilon_ {2}}, \mathbf {p} ^ {\epsilon_ {2}}\right) + \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {\epsilon_ {2}}\right) \cdot G _ {L} \left(\mathbf {p} ^ {\epsilon_ {2}}, \mathbf {p} ^ {\epsilon_ {2}}\right)
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {1})} {=} \operatorname {s i g n} \left(\frac {\epsilon_ {2}}{\epsilon_ {1}} (p _ {H} ^ {\star \star} - p _ {H} ^ {\epsilon_ {1}})\right) \cdot G _ {H} (\mathbf {p} ^ {\epsilon_ {2}}, \mathbf {p} ^ {\epsilon_ {2}}) + \operatorname {s i g n} \left(\frac {\epsilon_ {2}}{\epsilon_ {1}} (p _ {L} ^ {\star \star} - p _ {L} ^ {\epsilon_ {1}})\right) \cdot G _ {L} (\mathbf {p} ^ {\epsilon_ {2}}, \mathbf {p} ^ {\epsilon_ {2}}) \\ = \operatorname {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {\epsilon_ {1}}\right) \cdot G _ {H} \left(\mathbf {p} ^ {\epsilon_ {2}}, \mathbf {p} ^ {\epsilon_ {2}}\right) + \operatorname {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {\epsilon_ {1}}\right) \cdot G _ {L} \left(\mathbf {p} ^ {\epsilon_ {2}}, \mathbf {p} ^ {\epsilon_ {2}}\right) \\ \end{array}
$$
$$
\stackrel {(\Delta_ {2})} {\leq} \operatorname {s i g n} (p _ {H} ^ {\star \star} - p _ {H} ^ {\epsilon_ {1}}) \cdot G _ {H} (\mathbf {p} ^ {\epsilon_ {1}}, \mathbf {p} ^ {\epsilon_ {1}}) + \operatorname {s i g n} (p _ {L} ^ {\star \star} - p _ {L} ^ {\epsilon_ {1}}) \cdot G _ {L} (\mathbf {p} ^ {\epsilon_ {1}}, \mathbf {p} ^ {\epsilon_ {1}}) = \mathcal {G} (\mathbf {p} ^ {\epsilon_ {1}}),
$$
where  $(\Delta_1)$  follows from substituting  $p_i^{\epsilon_2}$  in  $\mathrm{sign}(p_i^{\star \star} - p_i^{\epsilon_2})$  with  $p_i^{\epsilon_2} = (\epsilon_2 / \epsilon_1)p_i^{\epsilon_1} + (1 - \epsilon_2 / \epsilon_1)p_i^{\star \star}$ . To see why  $(\Delta_2)$  holds, recall that we have shown in Eq. (87) and Eq. (88) that when two prices are in the same region (see four regions defined in Eq. (28)), the price closer to  $\mathbf{p}^{\star \star}$  in terms of the metric  $\varepsilon(\cdot)$  has a greater value of  $\mathcal{G}(\mathbf{p})$ . Since  $\mathbf{p}^{\epsilon_1}$  and  $\mathbf{p}^{\epsilon_2}$  are from the same region and  $\epsilon_1 > \epsilon_2$ , we conclude that  $\min_{\varepsilon(\mathbf{p})=\epsilon_1} \mathcal{G}(\mathbf{p}) = \mathcal{G}(\mathbf{p}^{\epsilon_1}) > \mathcal{G}(\mathbf{p}^{\epsilon_2}) \geq \min_{\varepsilon(\mathbf{p})=\epsilon_2} \mathcal{G}(\mathbf{p})$ .
Lemma F.3 Define function  $\mathcal{H}(\mathbf{p})$  as follows
$$
\mathcal {H} (\mathbf {p}) := \left(b _ {H} + c _ {H}\right) \cdot G _ {H} (\mathbf {p}, \mathbf {p}) \cdot \left(p _ {H} ^ {\star \star} - p _ {H}\right) + \left(b _ {L} + c _ {L}\right) \cdot G _ {L} (\mathbf {p}, \mathbf {p}) \cdot \left(p _ {L} ^ {\star \star} - p _ {L}\right) \tag {91}
$$
Then, there exist  $\gamma > 0$  and a open set  $U_{\gamma} \ni \mathbf{p}^{\star \star}$  such that
$$
\mathcal {H} (\mathbf {p}) \geq \gamma \cdot \| \mathbf {p} - \mathbf {p} ^ {\star \star} \| _ {2} ^ {2}, \quad \forall \mathbf {p} \in U _ {\gamma}. \tag {92}
$$
Proof. According to the partial derivatives in Eq. (98a) and Eq. (98b) from Lemma F.5, we have
$$
\begin{array}{l} \frac {\partial G _ {i} (\mathbf {p} , \mathbf {p})}{\partial p _ {i}} = - \frac {1}{(b _ {i} + c _ {i}) p _ {i} ^ {2}} - b _ {i} \cdot d _ {i} (\mathbf {p}, \mathbf {p}) \cdot (1 - d _ {i} (\mathbf {p}, \mathbf {p})); \\ \frac {\partial G _ {i} (\mathbf {p} , \mathbf {p})}{\partial p _ {- i}} = b _ {- i} \cdot d _ {i} (\mathbf {p}, \mathbf {p}) \cdot d _ {- i} (\mathbf {p}, \mathbf {p}). \tag {93} \\ \end{array}
$$
Then, to compute the gradient  $\nabla \mathcal{H}(\mathbf{p}) = [\partial \mathcal{H}(\mathbf{p}) / \partial p_H,\partial \mathcal{H}(\mathbf{p}) / \partial p_L]$ , we utilize partial derivatives of  $G_{i}(\mathbf{p},\mathbf{p})$  in Eq. (93) and obtain the partial derivatives of  $\mathcal{H}(\mathbf{p})$  for  $i\in \{H,L\}$ :
$$
\begin{array}{l} \frac {\partial \mathcal {H} (\mathbf {p})}{\partial p _ {i}} = (b _ {i} + c _ {i}) \left[ \frac {\partial G _ {i} (\mathbf {p} , \mathbf {p})}{\partial p _ {i}} (p _ {i} ^ {\star \star} - p _ {i}) - G _ {i} (\mathbf {p}, \mathbf {p}) \right] + (b _ {- i} + c _ {- i}) (p _ {- i} ^ {\star \star} - p _ {- i}) \frac {\partial G _ {- i} (\mathbf {p} , \mathbf {p})}{\partial p _ {i}} \\ = - \frac {p _ {i} ^ {\star \star}}{p _ {i} ^ {2}} - (b _ {i} + c _ {i}) \left(d _ {i} (\mathbf {p}, \mathbf {p}) - 1\right) - b _ {i} (b _ {i} + c _ {i}) \cdot d _ {i} (\mathbf {p}, \mathbf {p}) \cdot \left(1 - d _ {i} (\mathbf {p}, \mathbf {p})\right) \left(p _ {i} ^ {\star \star} - p _ {i}\right) \\ + b _ {i} \left(b _ {- i} + c _ {- i}\right) \cdot d _ {i} (\mathbf {p}, \mathbf {p}) \cdot d _ {- i} (\mathbf {p}, \mathbf {p}) \cdot \left(p _ {- i} ^ {\star \star} - p _ {- i}\right). \tag {94} \\ \end{array}
$$
From the definition of  $G_{i}(\cdot ,\cdot)$  in Eq. (31) and the system equations of first-order condition in Eq. (16), we have  $G_{i}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star}) = 1 / \bigl [(b_{i} + c_{i})p_{i}^{\star \star}\bigr ] + (d_{i}^{\star \star} - 1) = 0$ , where  $d_{i}^{\star \star}:= d_{i}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star})$  denotes the market share of product  $i$  at the SNE. Thereby, it follows that  $\nabla \mathcal{H}(\mathbf{p}^{\star \star}) = 0$ .
Next, the Hessian matrix  $\nabla^2\mathcal{H}(\mathbf{p})$  evaluated at  $\mathbf{p}^{\star \star}$  can be computed as
$$
\begin{array}{l} \nabla^ {2} \mathcal {H} (\mathbf {p} ^ {\star \star}) \\ = \left[ \begin{array}{c c} \frac {\partial^ {2} \mathcal {H} (\mathbf {p} ^ {\star \star})}{\partial p _ {H} {} ^ {2}}, & \frac {\partial^ {2} \mathcal {H} (\mathbf {p} ^ {\star \star})}{\partial p _ {H} \partial p _ {L}} \\ \frac {\partial^ {2} \mathcal {H} (\mathbf {p} ^ {\star \star})}{\partial p _ {L} \partial p _ {H}}, & \frac {\partial^ {2} \mathcal {H} (\mathbf {p} ^ {\star \star})}{\partial p _ {L} {} ^ {2}} \end{array} \right] \\ = \left[ \begin{array}{l l} \frac {2 p _ {H} ^ {\star \star}}{(p _ {H} ^ {\star \star}) ^ {3}} + 2 b _ {H} (b _ {H} + c _ {H}) \cdot d _ {H} ^ {\star \star} (1 - d _ {H} ^ {\star \star}), & - [ b _ {H} (b _ {L} + c _ {L}) + b _ {L} (b _ {H} + c _ {H}) ] \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star} \\ - [ b _ {H} (b _ {L} + c _ {L}) + b _ {L} (b _ {H} + c _ {H}) ] \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star}, & \frac {2 p _ {L} ^ {\star \star}}{(p _ {L} ^ {\star \star}) ^ {3}} + 2 b _ {L} (b _ {L} + c _ {L}) \cdot d _ {L} ^ {\star \star} (1 - d _ {L} ^ {\star \star}) \end{array} \right] \\ = \left[ \begin{array}{c} 2 (b _ {H} + c _ {H}) \cdot \big (1 - d _ {H} ^ {\star \star} \big) \cdot \big [ (b _ {H} + c _ {H}) - c _ {H} d _ {H} ^ {\star \star} \big ],   - \big [ b _ {H} (b _ {L} + c _ {L}) + b _ {L} (b _ {H} + c _ {H}) \big ] \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star} \\ - \big [ b _ {H} (b _ {L} + c _ {L}) + b _ {L} (b _ {H} + c _ {H}) \big ] \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star}, \qquad 2 (b _ {L} + c _ {L}) \cdot \big (1 - d _ {L} ^ {\star \star} \big) \cdot \big [ (b _ {L} + c _ {L}) - c _ {L} d _ {L} ^ {\star \star} \big ] \end{array} \right]. \\ \end{array}
$$
Note that the last equality results again from substituting in the identity  $G_{i}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star}) = 1 / \left[(b_{i} + c_{i})\cdot p_{i}^{\star \star}\right] + (d_{i}^{\star \star} - 1) = 0.$
The diagonal entries of  $\nabla^2\mathcal{H}(\mathbf{p}^{\star \star})$  are clearly positive. For  $i\in \{H,L\}$ , define  $k_{i}\coloneqq b_{i} / (b_{i} + c_{i})\in [0,1]$ . Then, the determinant of  $\nabla^2\mathcal{H}(\mathbf{p}^{\star \star})$  can be computed as follows:
$$
\begin{array}{l} \det \left(\nabla^ {2} \mathcal {H} (\mathbf {p} ^ {\star \star})\right) \\ = 4 \left(b _ {H} + c _ {H}\right) \left(b _ {L} + c _ {L}\right) \cdot \left(1 - d _ {H} ^ {\star \star}\right) \left(1 - d _ {L} ^ {\star \star}\right) \cdot \left[ \left(b _ {H} + c _ {H}\right) - c _ {H} d _ {H} ^ {\star \star} \right] \left[ \left(b _ {L} + c _ {L}\right) - c _ {L} d _ {L} ^ {\star \star} \right] \\ \left. - \left(\left[ b _ {H} \left(b _ {L} + c _ {L}\right) + b _ {L} \left(b _ {H} + c _ {H}\right) \right] \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star}\right) ^ {2} \right. \\ = (b _ {H} + c _ {H}) ^ {2} (b _ {L} + c _ {L}) ^ {2} \cdot \left(4 \left(1 - d _ {H} ^ {\star \star}\right) \left(1 - d _ {L} ^ {\star \star}\right) \cdot \left[ 1 - \left(1 - k _ {H}\right) d _ {H} ^ {\star \star} \right] \left[ 1 - \left(1 - k _ {L}\right) d _ {L} ^ {\star \star} \right] \right. \\ \left. - \left(k _ {H} + k _ {L}\right) ^ {2} \left(d _ {H} ^ {\star \star} d _ {L} ^ {\star \star}\right) ^ {2}\right) \\ \end{array}
$$
$$
\stackrel {(\Delta_ {1})} {\geq} 4 (b _ {H} + c _ {H}) ^ {2} (b _ {L} + c _ {L}) ^ {2} \cdot \left(\left(1 - d _ {H} ^ {\star \star}\right) \left(1 - d _ {L} ^ {\star \star}\right) \cdot \left[ 1 - (1 - k _ {H}) d _ {H} ^ {\star \star} \right] \left[ 1 - (1 - k _ {L}) d _ {L} ^ {\star \star} \right] - \left(d _ {H} ^ {\star \star} d _ {L} ^ {\star \star}\right) ^ {2}\right)
$$
$$
\stackrel {(\Delta_ {2})} {\geq} 4 (b _ {H} + c _ {H}) ^ {2} (b _ {L} + c _ {L}) ^ {2} \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star} \left(\left[ 1 - (1 - k _ {H}) d _ {H} ^ {\star \star} \right] \left[ 1 - (1 - k _ {L}) \cdot d _ {L} ^ {\star \star} \right] - d _ {H} ^ {\star \star} d _ {L} ^ {\star \star}\right)
$$
$$
\begin{array}{l} \stackrel {(\Delta_ {3})} {\geq} 4 \left(b _ {H} + c _ {H}\right) ^ {2} \left(b _ {L} + c _ {L}\right) ^ {2} \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star} \left[ \left(1 - d _ {H} ^ {\star \star}\right) \left(1 - d _ {L} ^ {\star \star}\right) - d _ {H} ^ {\star \star} d _ {L} ^ {\star \star} \right] \\ = 4 \left(b _ {H} + c _ {H}\right) ^ {2} \left(b _ {L} + c _ {L}\right) ^ {2} \cdot d _ {H} ^ {\star \star} d _ {L} ^ {\star \star} \left(1 - d _ {H} ^ {\star \star} - d _ {L} ^ {\star \star}\right) \\ \end{array}
$$
$$
\begin{array}{c} (\Delta_ {4}) \\ > 0, \end{array}
$$
(95)
where inequalities respectively result from the following facts  $(\Delta_1)\colon k_H + k_L\leq 2$ $(\Delta_{2})\colon 1 - d_{i}^{\star \star} > d_{-i}^{\star \star}$  for  $i\in \{H,L\}$ $(\Delta_3)\colon 1 - k_i\leq 1$  for  $i\in \{H,L\}$ $(\Delta_4)\colon d_H^{\star \star} + d_L^{\star \star} < 1$  . Thus, we conclude that  $\nabla^2\mathcal{H}(\mathbf{p}^{\star \star})$  is positive definite.
By the continuity of  $\nabla^2\mathcal{H}(\mathbf{p})$ , there exists some constant  $\gamma > 0$  and a open set  $U_{\gamma} \ni \mathbf{p}^{\star \star}$  such that  $\nabla^2\mathcal{H}(\mathbf{p}) \succeq 2\gamma I_2$ ,  $\forall \mathbf{p} \in U_{\gamma}$ , where  $I_2$  is the  $2 \times 2$  identity matrix. Using the second-order Taylor expansion at  $\mathbf{p}^{\star \star}$ , for all  $\mathbf{p} \in U_{\gamma}$ , there exists  $\widetilde{\mathbf{p}} \in U_{\gamma}$  such that
$$
\begin{array}{l} \mathcal {H} (\mathbf {p}) = \mathcal {H} (\mathbf {p} ^ {\star \star}) + \nabla \mathcal {H} (\mathbf {p} ^ {\star \star}) \cdot (\mathbf {p} - \mathbf {p} ^ {\star \star}) + \frac {1}{2} (\mathbf {p} - \mathbf {p} ^ {\star \star}) ^ {\top} \cdot \nabla^ {2} \mathcal {H} (\widetilde {\mathbf {p}}) \cdot (\mathbf {p} - \mathbf {p} ^ {\star \star}) \\ = \frac {1}{2} \left(\mathbf {p} - \mathbf {p} ^ {\star \star}\right) ^ {\top} \cdot \nabla^ {2} \mathcal {H} (\widetilde {\mathbf {p}}) \cdot \left(\mathbf {p} - \mathbf {p} ^ {\star \star}\right) \tag {96} \\ \geq \frac {1}{2} \left(\mathbf {p} - \mathbf {p} ^ {\star \star}\right) ^ {\top} \cdot 2 \gamma I _ {2} \cdot \left(\mathbf {p} - \mathbf {p} ^ {\star \star}\right) \\ = \gamma \| \mathbf {p} - \mathbf {p} ^ {\star \star} \| _ {2} ^ {2}, \\ \end{array}
$$
where the second equality arises from that  $\mathcal{H}(\mathbf{p}^{\star \star}) = 0$  and  $\nabla \mathcal{H}(\mathbf{p}^{\star \star}) = 0$ .
Lemma F.4 For any product  $i\in \{H,L\}$  , let
$$
\widehat {G} _ {i} (\mathbf {p}) := \operatorname {s i g n} \left(p _ {i} ^ {\star \star} - p _ {i}\right) \cdot G _ {i} (\mathbf {p}, \mathbf {p}), \tag {97}
$$
where  $G_{i}(\mathbf{p},\mathbf{r})$  is the scaled partial derivative defined in Eq. (25). Then,  $\widehat{G}_i(\mathbf{p})$  is always increasing as  $|p_i^{\star \star} - p_i|$  increases, and
1. when  $\mathbf{p} \in N_1 \cup N_3$  (see the definition in Eq. (28)),  $\widehat{G}_i(\mathbf{p})$  is decreasing as  $|p_{-i}^{\star i} - p_{-i}|$  increases;
2. when  $\mathbf{p} \in N_2 \cup N_4$ ,  $\widehat{G}_i(\mathbf{p})$  is increasing as  $|p_{-i}^{\star i} - p_{-i}|$  increases.
Proof. Without loss of generality, consider the case that product  $i = H$  and product  $-i = L$ .
Apparently, we have  $\mathrm{sign}\left(p_H^{\star \star} - p_H\right) \leq 0$  in  $N_1 \cup N_4$ , and  $\mathrm{sign}\left(p_H^{\star \star} - p_H\right) \geq 0$  in  $N_2 \cup N_3$  by definitions (see Eq. (28)). On the other hand, by Eq. (98b) in Lemma F.5, it holds that  $\partial G_H(\mathbf{p}, \mathbf{p}) / \partial p_H < 0$  and  $\partial G_H(\mathbf{p}, \mathbf{p}) / \partial p_L > 0, \forall \mathbf{p} \in \mathcal{P}^2$ . Thus,
1. in  $N_{1}\cup N_{4},G_{H}(\mathbf{p},\mathbf{p})$  is decreasing as  $\left|p_H^{\star \star} - p_H\right|$  increases; in  $N_{2}\cup N_{3},G_{H}(\mathbf{p},\mathbf{p})$  is increasing as  $\left|p_H^{\star \star} - p_H\right|$  increases;
2. in  $N_{1} \cup N_{2}$ ,  $G_{H}(\mathbf{p},\mathbf{p})$  is increasing as  $|p_L^{\star \star} - p_L|$  increases; conversely,  $G_{H}(\mathbf{p},\mathbf{p})$  is decreasing as  $|p_L^{\star \star} - p_L|$  increases in  $N_{3} \cup N_{4}$ .
The final results directly follows by combining the above pieces together.
Lemma F.5 Let  $G_{i}(\mathbf{p},\mathbf{r})$  be the scaled partial derivative defined in Eq. (25), then partial derivatives of  $G_{i}(\mathbf{p},\mathbf{r})$  with respect to  $\mathbf{p}$  and  $\mathbf{r}$  are given as
$$
\frac {\partial G _ {i} (\mathbf {p} , \mathbf {r})}{\partial p _ {i}} = - \frac {1}{\left(b _ {i} + c _ {i}\right) p _ {i} ^ {2}} - \left(b _ {i} + c _ {i}\right) \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot \left(1 - d _ {i} (\mathbf {p}, \mathbf {r})\right); \tag {98a}
$$
$$
\frac {\partial G _ {i} (\mathbf {p} , \mathbf {r})}{\partial p _ {- i}} = \left(b _ {- i} + c _ {- i}\right) \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot d _ {- i} (\mathbf {p}, \mathbf {r}); \tag {98b}
$$
$$
\frac {\partial G _ {i} (\mathbf {p} , \mathbf {r})}{\partial r _ {i}} = c _ {i} \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot (1 - d _ {i} (\mathbf {p}, \mathbf {r})); \tag {98c}
$$
$$
\frac {\partial G _ {i} (\mathbf {p} , \mathbf {r})}{\partial r _ {- i}} = - c _ {- i} \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot d _ {- i} (\mathbf {p}, \mathbf {r}). \tag {98d}
$$
Meanwhile,  $G_{i}(\mathbf{p},\mathbf{r})$  and its gradient are bounded as follows
$$
\left| G _ {i} (\mathbf {p}, \mathbf {r}) \right| \leq M _ {G}, \quad \left\| \nabla_ {\mathbf {r}} G _ {i} (\mathbf {p}, \mathbf {r}) \right\| _ {2} \leq \ell_ {r}, \quad \forall \mathbf {p}, \mathbf {r} \in \mathcal {P} ^ {2} a n d \forall i \in \{H, L \}, \tag {99}
$$
where the upper bound constant  $M_G$  and the Lipschitz constant  $\ell_r$  are defined as
$$
M _ {G} := \max  \left\{\frac {1}{\left(b _ {H} + c _ {L}\right) \underline {{p}}}, \frac {1}{\left(b _ {L} + c _ {L}\right) \underline {{p}}} \right\} + 1, \quad \ell_ {r} := \frac {1}{4} \sqrt {c _ {H} ^ {2} + c _ {L} ^ {2}}. \tag {100}
$$
Proof. We first verify the partial derivatives from Eq. (98a) to Eq. (98d):
$$
\begin{array}{l} \frac {\partial G _ {i} (\mathbf {p} , \mathbf {r})}{\partial p _ {i}} = \frac {1}{(b _ {i} + c _ {i}) p _ {i} ^ {2}} + \frac {\partial d _ {i} (\mathbf {p} , \mathbf {r})}{\partial p _ {i}} \\ = \frac {1}{\left(b _ {i} + c _ {i}\right) p _ {i} ^ {2}} - \frac {\left(b _ {i} + c _ {i}\right) \cdot \exp \left(u _ {i} \left(p _ {i} , r _ {i}\right)\right) \cdot \left(1 + \exp \left(u _ {- i} \left(p _ {- i} , r _ {- i}\right)\right)\right)}{\left(1 + \exp \left(u _ {i} \left(p _ {i} , r _ {i}\right)\right) + \exp \left(u _ {- i} \left(p _ {- i} , r _ {- i}\right)\right)\right) ^ {2}} \\ = \frac {1}{\left(b _ {i} + c _ {i}\right) p _ {i} ^ {2}} - \left(b _ {i} + c _ {i}\right) \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot \left(1 - d _ {i} (\mathbf {p}, \mathbf {r})\right). \tag {101} \\ \end{array}
$$
$$
\begin{array}{l} \frac {\partial G _ {i} (\mathbf {p} , \mathbf {r})}{\partial p _ {- i}} = \frac {\partial d _ {i} (\mathbf {p} , \mathbf {r})}{\partial p _ {- i}} \\ = \frac {\left(b _ {- i} + c _ {- i}\right) \cdot \exp \left(u _ {i} \left(p _ {i} , r _ {i}\right)\right) \cdot \exp \left(u _ {- i} \left(p _ {- i} , r _ {- i}\right)\right)}{\left(1 + \exp \left(u _ {i} \left(p _ {i} , r _ {i}\right)\right) + \exp \left(u _ {- i} \left(p _ {- i} , r _ {- i}\right)\right)\right) ^ {2}} \\ = \left(b _ {- i} + c _ {- i}\right) \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot d _ {- i} (\mathbf {p}, \mathbf {r}). \\ \end{array}
$$
Then, the partial derivatives with respect to  $\mathbf{r}$ , as shown in Eq. (98c) and Eq. (98d), can be similarly computed.
In the next part, we show that  $G_{i}(\mathbf{p},\mathbf{r})$  is bounded for  $\mathbf{p},\mathbf{r}\in \mathcal{P}^2$ :
$$
\begin{array}{l} \left| G _ {i} (\mathbf {p}, \mathbf {r}) \right| = \left| \frac {1}{\left(b _ {i} + c _ {i}\right) p _ {i}} + d _ {i} (\mathbf {p}, \mathbf {r}) - 1 \right| \\ \leq \left| \frac {1}{\left(b _ {i} + c _ {i}\right) p _ {i}} \right| + \left| d _ {i} (\mathbf {p}, \mathbf {r}) - 1 \right| \tag {102} \\ \leq \frac {1}{(b _ {i} + c _ {i}) p} + 1. \\ \end{array}
$$
Hence, it follows that  $\left|G_{i}(\mathbf{p},\mathbf{r})\right|$  for every  $i\in \{H,L\}$  is upper bounded by
$$
\left| G _ {i} (\mathbf {p}, \mathbf {r}) \right| \leq \max  \left\{\frac {1}{(b _ {H} + c _ {L}) \underline {{p}}}, \frac {1}{(b _ {L} + c _ {L}) \underline {{p}}} \right\} + 1 =: M _ {G}, \quad \forall \mathbf {p}, \mathbf {r} \in \mathcal {P} ^ {2}, \forall i \in \{H, L \}. (1 0 3)
$$
Finally, we demonstrate that  $\| \nabla_{\mathbf{r}}G_i(\mathbf{p},\mathbf{r})\|$  is also bounded  $\forall \mathbf{p},\mathbf{r}\in \mathcal{P}^2$  and  $\forall i\in \{H,L\}$ . From Eq. (98c) and Eq. (98d), we have
$$
\begin{array}{l} \left\| \nabla_ {\mathbf {r}} G _ {i} (\mathbf {p}, \mathbf {r}) \right\| _ {2} ^ {2} = \Big (c _ {i} \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot \big (1 - d _ {i} (\mathbf {p}, \mathbf {r}) \big) \Big) ^ {2} + \Big (- c _ {- i} \cdot d _ {i} (\mathbf {p}, \mathbf {r}) \cdot d _ {- i} (\mathbf {p}, \mathbf {r}) \Big) ^ {2} \\ = c _ {i} ^ {2} \cdot \left(d _ {i} (\mathbf {p}, \mathbf {r}) \cdot \left(1 - d _ {i} (\mathbf {p}, \mathbf {r})\right)\right) ^ {2} + c _ {- i} ^ {2} \cdot \left(d _ {i} (\mathbf {p}, \mathbf {r}) \cdot d _ {- i} (\mathbf {p}, \mathbf {r})\right) ^ {2} \tag {104} \\ \leq \frac {1}{1 6} \left(c _ {i} ^ {2} + c _ {- i} ^ {2}\right), \\ \end{array}
$$
where the inequality follows from the fact that  $x \cdot y \leq 1/4$  for any two numbers such that  $x, y > 0$  and  $x + y \leq 1$ . Thus, it follows that  $\left\| \nabla_{\mathbf{r}} G_i(\mathbf{p}, \mathbf{r}) \right\|_2 \leq (1/4) \sqrt{c_H^2 + c_L^2} \coloneqq l_r$ ,  $\forall \mathbf{p}, \mathbf{r} \in \mathcal{P}^2$  and  $\forall i \in \{H, L\}$ .
Limitaions