SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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Appendix A Discussion about alternative methods
As we briefly mentioned in Section 3, our problem can also be translated into a four-player online game or a dynamical system. In this section, we will discuss these alternative methods and explain why existing tools from the literature of online game and dynamical system cannot be applied.
A.1 Four-player online game formulation
By viewing the underlying state variables  $(r_H, r_L)$  as virtual players that undergo deterministic transitions, we are able to convert our problem into a general four-player game. Specifically, we can construct revenue functions  $R_i(p_i, r_i)$  for  $i \in \{H, L\}$  and a common sequence of step-sizes  $\{\eta_r^t\}_{t \geq 0}$  for the two virtual players such that when firms  $H, L$  and virtual players implement OPGA using the corresponding step-sizes, the resulting price path  $\{\mathbf{p}^t\}_{t \geq 0}$  and reference price path  $\{\mathbf{r}^t\}_{t \geq 0}$  recover those generated by Algorithm 1. The functions  $R_i(\cdot, \cdot), \forall i \in \{H, L\}$  and step-sizes  $\{\eta_r^t\}_{t \geq 0}$  are specified as follows
$$
R _ {i} \left(p _ {i}, r _ {i}\right) = - \frac {1}{2} r _ {i} ^ {2} + r _ {i} \cdot p _ {i}, \quad \forall i \in \{H, L \}; \tag {11}
$$
$$
\eta_ {r} ^ {t} \equiv 1 - \alpha , \quad \forall t \geq 0.
$$
Then, given the initial reference price  $\mathbf{r}^0$  and initial price  $\mathbf{p}^0$ , the four players respectively update their states through the online projected gradient ascent specified as follows, for  $t \geq 0$ :
$$
\left\{ \begin{array}{l} p _ {H} ^ {t + 1} = \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {H} ^ {t} + \eta^ {t} \cdot \frac {\partial \log (\Pi_ {H} (\mathbf {p} ^ {t} , \mathbf {r} ^ {t}))}{\partial p _ {H}}\right) = \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {H} ^ {t} + \eta^ {t} D _ {H} ^ {t}\right), \\ p _ {L} ^ {t + 1} = \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {L} ^ {t} + \eta^ {t} \cdot \frac {\partial \log (\Pi_ {L} (\mathbf {p} ^ {t} , \mathbf {r} ^ {t}))}{\partial p _ {L}}\right) = \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {L} ^ {t} + \eta^ {t} D _ {L} ^ {t}\right), \\ r _ {H} ^ {t + 1} = \operatorname {P r o j} _ {\mathcal {P}} \left(r _ {H} ^ {t} + \eta_ {r} ^ {t} \cdot \frac {\partial R _ {H} (p _ {H} ^ {t} , r _ {H} ^ {t})}{\partial r _ {H}}\right) = \operatorname {P r o j} _ {\mathcal {P}} \left(\alpha r _ {H} ^ {t} + (1 - \alpha) p _ {H} ^ {t}\right), \\ r _ {L} ^ {t + 1} = \operatorname {P r o j} _ {\mathcal {P}} \left(r _ {L} ^ {t} + \eta_ {r} ^ {t} \cdot \frac {\partial R _ {L} (p _ {L} ^ {t} , r _ {L} ^ {t})}{\partial r _ {L}}\right) = \operatorname {P r o j} _ {\mathcal {P}} \left(\alpha r _ {L} ^ {t} + (1 - \alpha) p _ {L} ^ {t}\right). \end{array} \right. \tag {12}
$$
It can be easily seen that the pure strategy Nash equilibrium of this four-player static game is equivalent to the SNE of the original two-player dynamic game, as defined in Definition 3.1. However, even after converting to the static game, no general convergence results are readily applicable in this setting. One obstacle on the convergence analysis comes from the absence of critical properties, such as monotonicity of the static game [14, 16] or variational stability at its Nash equilibrium [26, 15].
Meanwhile, another obstacle stems from the asynchronous updates for real firms (price players) and virtual firms (reference price players). While the real firms have the flexibility in adopting time-varying step-sizes, the updates for the two virtual firms in Eq. (12) stick to the constant step-size of  $(1 - \alpha)$ . As a result, this rigidity of virtual players perplexes the analysis, given that the typical convergence results of online games call for the step-sizes of multiple players to have the same pattern (all diminishing or constant step-sizes) [47, 48, 14, 15].
A.2 Dynamical system formulation
The study of the limiting behavior of a competitive gradient-based learning algorithm is related to dynamical system theories [64]. In fact, the update of Algorithm 1 can be viewed as a nonlinear dynamical system. Assume a constant step-size is employed, i.e.,  $\eta^t\equiv \eta ,\forall t\geq 0$  . Then, Lines 5 and 6 in Algorithm 1 are equivalent to the dynamical system
$$
\left(\mathbf {p} ^ {t + 1}, \mathbf {r} ^ {t + 1}\right) = \mathbf {f} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right), \quad \forall t \geq 0, \tag {13}
$$
where  $\mathbf{f}(\cdot)$  is a vector-valued function defined as
$$
\mathbf {f} (\mathbf {p}, \mathbf {r}) := \left( \begin{array}{c} \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {H} + \eta \left(\frac {1}{p _ {H}} + (b _ {H} + c _ {H}) \cdot d _ {H} (\mathbf {p}, \mathbf {r}) - (b _ {H} + c _ {H})\right)\right) \\ \operatorname {P r o j} _ {\mathcal {P}} \left(p _ {L} + \eta \left(\frac {1}{p _ {L}} + (b _ {L} + c _ {L}) \cdot d _ {L} (\mathbf {p}, \mathbf {r}) - (b _ {L} + c _ {L})\right)\right) \\ \operatorname {P r o j} _ {\mathcal {P}} (\alpha r _ {H} + (1 - \alpha) p _ {H}) \\ \operatorname {P r o j} _ {\mathcal {P}} (\alpha r _ {L} + (1 - \alpha) p _ {L}) \end{array} \right). \tag {14}
$$
Under the assumption that  $\mathbf{p}^{\star \star} \in \mathcal{P}^2$ , it is evident that  $\mathbf{p}^{\star \star}$  is the unique fixed point of the system in Eq. (13). Generally, fixed points can be categorized into three classes:
- asymptotically stable, when all nearby solutions converge to it,
- stable, when all nearby solutions remain in close proximity,
- unstable, when almost all nearby solutions diverge away from the fixed point.
Hence, if we can demonstrate the asymptotic stability of  $\mathbf{p}^{\star \star}$ , we can at least prove the local convergence of the price and reference price.
Standard dynamical systems theory [49, 50] states that  $\mathbf{p}^{\star \star}$  is asymptotically stable if the spectral radius of the Jacobian matrix  $\nabla \mathbf{f}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star})$  is strictly less than one. However, computing the spectral radius is not straightforward. The primary challenge stems from the fact that, while the entries of  $\nabla \mathbf{f}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star})$  contain  $\mathbf{p}^{\star \star}$  and  $d_{i}(\mathbf{p}^{\star \star},\mathbf{p}^{\star \star})$ , there is no closed-form expression for  $\mathbf{p}^{\star \star}$ .
Apart from the above issue, it is worth noting that the function  $f(\cdot)$  is not globally smooth due to the presence of the projection operator. Furthermore, the function  $f(\cdot)$  also depends on the step-size  $\eta$ . When firms adopt time-varying step-sizes, the dynamical system in Eq. (13) becomes non-stationary, i.e.,  $(\mathbf{p}^{t + 1},\mathbf{r}^{t + 1}) = \mathbf{f}^t (\mathbf{p}^t,\mathbf{r}^t)$ . Although the sequence of functions  $\mathbf{f}_{t\geq 0}^t$  shares the same fixed point, verifying the convergence (stability) of the system requires examining the spectral radius of  $\nabla \mathbf{f}^t (\mathbf{p}^{\star \star},\mathbf{p}^{\star \star})$  for all  $t\geq 0$ .
Most importantly, even if asymptotic stability holds, it can only guarantee local convergence of Algorithm 1. Our goal, however, is to prove global convergence, such that both the price and reference price converge to the SNE for arbitrary initializations.
Appendix B More details about SNE
B.1 Connection between equilibrium pricing policy and SNE
First, we further elaborate on the connection between equilibrium pricing policy and SNE. The work [22] investigates the properties of equilibrium pricing policy under the full information setting. They demonstrate that the policy  $\mathbf{p}^{\star}(\mathbf{r})$  is a well-defined function since it outputs a unique equilibrium price for every given reference price  $\mathbf{r}$ . Further, for any given reference price  $\mathbf{r}$ , the equilibrium price  $(p_H^{\star}(\mathbf{r}), p_L^{\star}(\mathbf{r}))$  is the unique solution to the following system of first-order equations [22, Eq. (B.10)]:
$$
\left\{ \begin{array}{l} p _ {H} ^ {\star} (\mathbf {r}) = \frac {1}{\left(b _ {H} + c _ {H}\right) \cdot \left(1 - d _ {H} \left(\mathbf {p} ^ {\star} (\mathbf {r}), \mathbf {r}\right)\right)}, \\ p _ {L} ^ {\star} (\mathbf {r}) = \frac {1}{\left(b _ {L} + c _ {L}\right) \cdot \left(1 - d _ {L} \left(\mathbf {p} ^ {\star} (\mathbf {r}), \mathbf {r}\right)\right)}. \end{array} \right. \tag {15}
$$
The intertemporal aspect of the memory-based reference effect prompts researchers to examine the long-term consequence of repeatedly applying the equilibrium pricing policy in conjunction with reference price updates. Theorem 4 in [22] confirms that in the complete information setting, the price paths and reference price paths generated by the equilibrium pricing policy will converge to the SNE in the long run. Moreover, based on Eq. (15) for the equilibrium pricing policy, the SNE  $\mathbf{p}^{\star \star}$  is
the unique solution of the following system of equations:
$$
\left\{ \begin{array}{l} p _ {H} ^ {\star \star} = \frac {1}{\left(b _ {H} + c _ {H}\right) \cdot \left(1 - d _ {H} \left(\mathbf {p} ^ {\star \star} , \mathbf {p} ^ {\star \star}\right)\right)} = \frac {1 + \exp \left(a _ {H} - b _ {H} \cdot p _ {H} ^ {\star \star}\right) + \exp \left(a _ {L} - b _ {L} \cdot p _ {L} ^ {\star \star}\right)}{\left(b _ {H} + c _ {H}\right) \cdot \left(1 + \exp \left(a _ {L} - b _ {L} \cdot p _ {L} ^ {\star \star}\right)\right)}, \\ p _ {L} ^ {\star \star} = \frac {1}{\left(b _ {L} + c _ {L}\right) \cdot \left(1 - d _ {L} \left(\mathbf {p} ^ {\star \star} , \mathbf {p} ^ {\star \star}\right)\right)} = \frac {1 + \exp \left(a _ {H} - b _ {H} \cdot p _ {H} ^ {\star \star}\right) + \exp \left(a _ {L} - b _ {L} \cdot p _ {L} ^ {\star \star}\right)}{(\left(b _ {L} + c _ {L}\right) \cdot \left(1 + \exp \left(a _ {H} - b _ {H} \cdot p _ {H} ^ {\star \star}\right)\right)}. \end{array} \right. \tag {16}
$$
It can be seen from Eq. (16) that the value of SNE depends only on the model parameters.
B.2 Proof of Proposition 3.1
Proof. Firstly, the uniqueness of SNE has been established in the discussion in Appendix B.1. Below, we show the boundedness of the SNE. By performing a transformation on Eq. (16), we obtain the following inequality for  $\mathbf{p}^{\star \star}$ :
$$
\frac {\exp \left(a _ {i} - b _ {i} \cdot p _ {i} ^ {\star \star}\right)}{\left(b _ {i} + c _ {i}\right) p _ {i} ^ {\star \star} - 1} = 1 + \exp \left(a _ {- i} - b _ {- i} \cdot p _ {- i} ^ {\star \star}\right) > 1, \quad \forall i \in \{H, L \}. \tag {17}
$$
To make the above inequality valid, it immediately follows that
$$
\frac {1}{b _ {i} + c _ {i}} <   p _ {i} ^ {\star \star} <   \frac {1 + \exp \left(a _ {i} - b _ {i} \cdot p _ {i} ^ {\star \star}\right)}{b _ {i} + c _ {i}}, \quad \forall i \in \{H, L \}. \tag {18}
$$
Now, we derive the upper bound for  $p_i^{\star \star}$  from the second inequality in Eq. (18). Since the quantity on the right-hand side of Eq. (18) is monotone decreasing in  $p_i^{\star \star}$ , the value of  $p_i^{\star \star}$  must be upper-bounded by the unique solution to the following equation with respect to  $p_i$ :
$$
p _ {i} = \frac {1 + \exp \left(a _ {i} - b _ {i} \cdot p _ {i}\right)}{b _ {i} + c _ {i}}. \tag {19}
$$
Define  $x \coloneqq -b_{i} / (b_{i} + c_{i}) + b_{i} \cdot p_{i}$ . Then, one can easily verify that the above equation can be converted into
$$
x \exp (x) = \frac {b _ {i}}{b _ {i} + c _ {i}} \exp \left(a _ {i} - \frac {b _ {i}}{b _ {i} + c _ {i}}\right), \tag {20}
$$
$$
x = W \left(\frac {b _ {i}}{b _ {i} + c _ {i}} \exp \left(a _ {i} - \frac {b _ {i}}{b _ {i} + c _ {i}}\right)\right), \tag {21}
$$
where  $W(\cdot)$  is known as the Lambert  $W$  function [65]. For any value  $y \geq 0$ ,  $W(y)$  is defined as the unique real solution to the equation
$$
W (y) \cdot \exp (W (y)) = y. \tag {22}
$$
$$
p _ {i} ^ {\star \star} <   p _ {i} = \frac {1}{b _ {i} + c _ {i}} + \frac {1}{b _ {i}} W \left(\frac {b _ {i}}{b _ {i} + c _ {i}} \exp \left(a _ {i} - \frac {b _ {i}}{b _ {i} + c _ {i}}\right)\right). \tag {23}
$$
Together with the lower bound provided in Eq. (18), this completes the proof.
Appendix C Proof of Theorem 5.1
Proof. In this section, we present the proof of Theorem 5.1. By Lemma F.1, when the step-sizes in Algorithm 1 are non-increasing and  $\lim_{t\to \infty}\eta^t = 0$ , the difference between reference price and price converges to zero as  $t$  goes to infinity, i.e.,  $\lim_{t\to \infty}\left(\mathbf{r}^t -\mathbf{p}^t\right) = 0$ . Thus, we are left to verify that the price path  $\{\mathbf{p}^t\}_{t\geq 0}$  converges to the unique SNE, denoted by  $\mathbf{p}^{\star \star}$  (see Definition 3.1). Recall that the uniqueness of the SNE is established in Proposition 3.1.
Consider the following weighted  $\ell_1$ -distance between a point  $\mathbf{p}$  and the SNE  $\mathbf{p}^{\star \star}$ :
$$
\varepsilon (\mathbf {p}) := \frac {\left| p _ {H} ^ {\star \star} - p _ {H} \right|}{b _ {H} + c _ {H}} + \frac {\left| p _ {L} ^ {\star \star} - p _ {L} \right|}{b _ {L} + c _ {L}}. \tag {24}
$$
In a nutshell, our proof contains the following two parts:
- In Part 1 (Appendix C.1), we show that the price path  $\{\mathbf{p}^t\}_{t\geq 0}$  would enter the  $\ell_1$ -neighborhood  $N_{\epsilon_0}^1 \coloneqq \left\{\mathbf{p} \in \mathcal{P}^2 \mid \varepsilon(\mathbf{p}) < \epsilon_0\right\}$  infinitely many times for any  $\epsilon_0 > 0$ .
- In Part 2 (Appendix C.2), we show that when the price path  $\{\mathbf{p}^t\}_{t\geq 0}$  enters the  $\ell_2$ -neighborhood  $N_{\epsilon_0}^2 \coloneqq \{\mathbf{p} \in \mathcal{P}^2 \mid \| \mathbf{p} - \mathbf{p}^{\star \star}\|_2 < \epsilon_0\}$  for some sufficiently small  $\epsilon_0 > 0$  with small enough step-sizes, the price path will stay in  $N_{\epsilon_0}^2$  during subsequent periods.
Due to the diminishing step-sizes and the equivalence between  $\ell_1$  and  $\ell_2$  norms, Part 1 guarantees that the price path would enter any  $\ell_2$ -neighborhood  $N_{\epsilon_0}^2$  of  $\mathbf{p}^{\star \star}$  with arbitrarily small step-sizes. Together with Part 2, this proves that for any sufficiently small  $\epsilon_0 > 0$ , there exists some  $T_{\epsilon_0} > 0$  such that  $\| \mathbf{p}^t - \mathbf{p}^{\star \star}\|_2 \leq \epsilon_0$  for every  $t \geq T_{\epsilon_0}$ , which implies the convergence to the SNE.
C.1 Proof of Part 1
We argue by contradiction. Suppose there exists  $\epsilon_0 > 0$  such that  $\{\mathbf{p}\}_{t\geq 0}$  only visits  $N_{\epsilon_0}^1$  finitely many times. This is equivalent to say that  $\exists T^{\epsilon_0}$  such that  $\{\mathbf{p}^t\}_{t\geq 0}$  never visits  $N_{\epsilon_0}^1$  if  $t\geq T^{\epsilon_0}$ .
Firstly, let  $G_{i}(\mathbf{p},\mathbf{r})$  be the scaled partial derivative of the log-revenue, defined as
$$
G _ {i} (\mathbf {p}, \mathbf {r}) := \frac {1}{b _ {i} + c _ {i}} \cdot \frac {\partial \log (\Pi_ {i} (\mathbf {p} , \mathbf {r}))}{\partial p _ {i}} = \frac {1}{(b _ {i} + c _ {i}) p _ {i}} + d _ {i} (\mathbf {p}, \mathbf {r}) - 1, \quad \forall i \in \{H, L \}. \tag {25}
$$
For the ease of notation, we denote  $\mathcal{P}_i\coloneqq \{p / (b_i + c_i)\mid p\in \mathcal{P}\}$  as the scaled price range. Then, the price update in Line 5 of Algorithm 1 is equivalent to
$$
\frac {p _ {i} ^ {t + 1}}{b _ {i} + c _ {i}} = \operatorname {P r o j} _ {\mathcal {P} _ {i}} \left(\frac {p _ {i} ^ {t}}{b _ {i} + c _ {i}} + \eta^ {t} \frac {D _ {i} ^ {t}}{b _ {i} + c _ {i}}\right) = \operatorname {P r o j} _ {\mathcal {P} _ {i}} \left(\frac {p _ {i} ^ {t}}{b _ {i} + c _ {i}} + \eta^ {t} G _ {i} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t})\right). \qquad (2 6)
$$
Let  $\mathrm{sign}(\cdot)$  be the sign function defined as
$$
\operatorname {s i g n} (x) := \left\{ \begin{array}{l l} 1, & \text {i f} x > 0, \\ 0, & \text {i f} x = 0, \\ - 1, & \text {i f} x <   0. \end{array} \right. \tag {27}
$$
Then, an essential observation from Eq. (26) is that: if  $\mathrm{sign}(p_i^{\star \star} - p_i^t)\cdot G_i(\mathbf{p}^t,\mathbf{r}^t) > 0$ , we have that  $\mathrm{sign}(p_i^{t + 1} - p_i^t) = \mathrm{sign}\left(G_i(\mathbf{p}^t,\mathbf{r}^t)\right) = \mathrm{sign}(p_i^{\star \star} - p_i^t)$ , i.e., the update from  $p_i^t$  to  $p_i^{t + 1}$  is toward the direction of the SNE price  $p_i^{\star \star}$ . Conversely, if  $\mathrm{sign}(p_i^{\star \star} - p_i^t)\cdot G_i(\mathbf{p}^t,\mathbf{r}^t) < 0$ , the update from  $p_i^t$  to  $p_i^{t + 1}$  is deviating from  $p_i^{\star \star}$ .
We separate the whole feasible price range into four quadrants with  $\mathbf{p}^{\star \star}$  being the origin:
$$
N _ {1} := \left\{\mathbf {p} \in \mathcal {P} ^ {2} \mid p _ {H} > p _ {H} ^ {\star \star} \text {a n d} p _ {L} \geq p _ {L} ^ {\star \star} \right\}; N _ {2} := \left\{\mathbf {p} \in \mathcal {P} ^ {2} \mid p _ {H} \leq p _ {H} ^ {\star \star} \text {a n d} p _ {L} > p _ {L} ^ {\star \star} \right\};
$$
$$
N _ {3} := \left\{\mathbf {p} \in \mathcal {P} ^ {2} \mid p _ {H} <   p _ {H} ^ {\star \star} \text {a n d} p _ {L} \leq p _ {L} ^ {\star \star} \right\}; N _ {4} := \left\{\mathbf {p} \in \mathcal {P} ^ {2} \mid p _ {H} \geq p _ {H} ^ {\star \star} \text {a n d} p _ {L} <   p _ {L} ^ {\star \star} \right\}. \tag {28}
$$
Below, we first show that when  $t$  is sufficiently large, prices in consecutive periods cannot always stay within a same region. We prove it by a contradiction argument. Suppose that there exists some period  $T_1^{\epsilon_0} > T^{\epsilon_0}$ , after which the price path  $\{\mathbf{p}^t\}_{t \geq T_1^{\epsilon_0}}$  stays within a same region, i.e.,
$\mathrm{sign}\left(p_i^{\star \star} - p_i^{t_1}\right) = \mathrm{sign}\left(p_i^{\star \star} - p_i^{t_2}\right), \forall t_1, t_2 \geq T_1^{\epsilon_0}, \forall i \in \{H, L\}$ . Then, for  $t \geq T_1^{\epsilon_0}$ , it holds that
$$
\varepsilon (\mathbf {p} ^ {t + 1}) = \frac {| p _ {H} ^ {\star \star} - p _ {H} ^ {t + 1} |}{b _ {H} + c _ {H}} + \frac {| p _ {L} ^ {\star \star} - p _ {L} ^ {t + 1} |}{b _ {L} + c _ {L}}
$$
$$
\begin{array}{l} = \left. \right.\left| \frac {p _ {H} ^ {\star \star}}{b _ {H} + c _ {H}} - \operatorname {P r o j} _ {\mathcal {P} _ {H}} \left(\eta^ {t} G _ {H} \left(\mathbf {p} ^ {t}, \mathbf {r} ^ {t}\right) + \frac {p _ {H} ^ {t}}{b _ {H} + c _ {H}}\right)\right| \\ + \left| \frac {p _ {L} ^ {\star \star}}{b _ {L} + c _ {L}} - \mathrm {P r o j} _ {\mathcal {P} _ {L}} \left(\eta^ {t} G _ {L} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) + \frac {p _ {L} ^ {t}}{b _ {L} + c _ {L}}\right) \right| \\ \stackrel {(\Delta_ {1})} {\leq} \left| \frac {p _ {H} ^ {\star \star}}{b _ {H} + c _ {H}} - \eta^ {t} G _ {H} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - \frac {p _ {H} ^ {t}}{b _ {H} + c _ {H}} \right| + \left| \frac {p _ {L} ^ {\star \star}}{b _ {L} + c _ {L}} - \eta^ {t} G _ {L} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - \frac {p _ {L} ^ {t}}{b _ {L} + c _ {L}} \right| \\ \stackrel {(\Delta_ {2})} {=} \mathrm {s i g n} (p _ {H} ^ {\star \star} - p _ {H} ^ {t}) \cdot \left(\frac {p _ {H} ^ {\star \star}}{b _ {H} + c _ {H}} - \eta^ {t} G _ {H} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - \frac {p _ {H} ^ {t}}{b _ {H} + c _ {H}}\right) \\ + \mathrm {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {t}\right) \cdot \left(\frac {p _ {L} ^ {\star \star}}{b _ {L} + c _ {L}} - \eta^ {t} G _ {L} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - \frac {p _ {L} ^ {t}}{b _ {L} + c _ {L}}\right) \\ = \mathrm {s i g n} \left(p _ {H} ^ {\star \star} - p _ {H} ^ {t}\right) \cdot \left(\frac {p _ {H} ^ {\star \star}}{b _ {H} + c _ {H}} - \eta^ {t} G _ {H} (\mathbf {p} ^ {t}, \mathbf {p} ^ {t}) - \frac {p _ {H} ^ {t}}{b _ {H} + c _ {H}}\right) \\ + \mathrm {s i g n} \left(p _ {L} ^ {\star \star} - p _ {L} ^ {t}\right) \cdot \left(\frac {p _ {L} ^ {\star \star}}{b _ {L} + c _ {L}} - \eta^ {t} G _ {L} (\mathbf {p} ^ {t}, \mathbf {p} ^ {t}) - \frac {p _ {L} ^ {t}}{b _ {L} + c _ {L}}\right) \\ + \eta^ {t} \left(\left| G _ {H} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - G _ {H} (\mathbf {p} ^ {t}, \mathbf {p} ^ {t}) \right| + \left| G _ {L} (\mathbf {p} ^ {t}, \mathbf {r} ^ {t}) - G _ {L} (\mathbf {p} ^ {t}, \mathbf {p} ^ {t}) \right|\right) \\ \end{array}
$$