Combating Online Counterfeits: A Game-Theoretic Analysis of Cyber Supply Chain Ecosystem

Abstract

Counterfeiting has been a pervasive threat to the security of supply chains. With the development of cyber technologies, traditional supply chains move their logistics to the cyberspace for better efficiency. However, counterfeiting threats still exist and may even cause worse consequences. It is imperative to find mitigating strategies to combat counterfeiting in the cyber supply chain. In this paper, we establish a games-in-games framework to capture the interactions of counterfeiting activities in the cyber supply chain. Specifically, the sellers in the cyber supply chain play a Stackelberg game with consumers, while sellers compete with each other by playing a Nash game. All sellers and consumers aim to maximize their utilities. We design algorithms to find the best response of all participants and analyze the equilibrium of the supply chain system. Finally, we use case studies to demonstrate the equilibrium behavior and propose effective anti-counterfeit strategies.

This research is partially supported by award 2015-ST-061-CIRC01, U. S. Department of Homeland Security, awards ECCS-1847056, CNS-1544782, CNS-2027884, and SES-1541164 from National Science of Foundation (NSF), and grant W911NF-19-1-0041 from Army Research Office (ARO).

Appendices

A Proof of Theorem 1

Let $f(x) = px^2 + qx + r$. The concavity of f indicates that $p < 0$, and let $h(x) = f(x) g(x)$. Then

$$\begin{aligned} h^{\prime \prime }(x) = 2p g(x) + f(x) \frac{-2a}{(x+a)^3} + \frac{2a}{(x+a)^2}(2px + q). \end{aligned}$$

Note that g(x) is concave and increasing on [b, c]. We discuss three possibilities.

• f(x) is increasing on [b, c], i.e., $-\frac{q}{2p} \ge c$. Since f(x) and g(x) are both increasing and positive on [b, c], thus $x^* = \arg \max h(x) = c$.

• f(x) is decreasing on [b, c], i.e., $-\frac{q}{2p} \le b$. We check the Hessian of h(x). The first two term are clearly negative. As f(x) is decreasing, we have $f'(x) = 2px + q < 0$ on [b, c]. Therefore, the Hessian $h''(x) < 0$ and h(x) is concave on [b, c]. The maximizer of h(x) can characterized by the first-order condition $h'(x_{foc}) = 0$. Thus $x^*$ is the argument of $\max \{h(b), h(c), h(x_{foc})\}$.

• f(x) is both increasing and decreasing on [b, c], i.e., $b< -\frac{q}{2p} < c$. We split [b, c] into two subintervals $[b, -\frac{q}{2p}]$ and $(-\frac{q}{2p}, c]$. In the first interval, h(x) is increasing. In the second interval, h(x) is concave. Note that h(x) is continuously differentiable on [b, c], which means $h'(x)$ is continuous. Since

  $$\begin{aligned} h'(-\frac{q}{2p}) = f(-\frac{q}{2p}) g'(-\frac{q}{2p}) > 0, \end{aligned}$$

  $h'(x)$ is positive in a small neighborhood of $x = -\frac{q}{2p}$, which means h(x) is still increasing in that small neighborhood. Therefore, we obtain that the maximizer is either $x=c$ or the point which satisfies the first-order condition, i.e., $x^*$ is the argument of $\max \{h(c), h(x_{foc}) \}$.

Since $h'(x)$ is continuous and is nonzero constant on any subintervals of [b, c], the uniqueness of the maximizer is guaranteed.

B Proof of Theorem 2

From (14), The nonsmoothness of $D_b$ occurs when $q_3$ crosses $\mathcal {C}_{12}$. Let $\mathcal {I}_1$ and $\mathcal {I}_{23}$ denote the interval such that $D_b = \frac{k-p_b}{ka}$ when $q_3 \in \mathcal {I}_1$ and $D_b = \frac{p_o - p_b}{\eta ka}$ when $q_3 \in \mathcal {I}_{23}$. Note that $\mathcal {I}_1$ and $\mathcal {I}_{23}$ are parameterized by $q_1$ and $q_2$. We write $\mathcal {I}_{23} = [0, q_{3,s}]$ and $\mathcal {I}_{1} = [q_{3, s}, 1]$. When $q_{3,s} = 0$ or 1, $\mathcal {I}_{23}$ or $\mathcal {I}_1$ is empty; when $q_{3, s} \in (0, 1)$, both $\mathcal {I}_{1}$ and $\mathcal {I}_{23}$ are nonempty. We call $q_{3, s}$ the crossing point.

When one of $\mathcal {I}_{1}$ and $\mathcal {I}_{23}$ is empty, $D_b$ is smooth on the entire [0, 1]. As the utility function $u_{3}$ is concave, the maximizer is unique. When $\mathcal {I}_1$ and $\mathcal {I}_{23}$ are both nonempty, $u_3$ comprises two concave and quadratic functions $u_{3, 1}, u_{3,23}$ on $\mathcal {I}_1$ and $\mathcal {I}_{23}$. Clearly, the concavity of $u_3$ is not guaranteed.

Let $\mathcal {W} = \{ (q_1, q_2, q_3) \ \vert \ 0 \le q_1, q_2 \le 1, q_3=0 \}$, and ${\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}$ be the projection of $\mathcal {C}_{12}$ onto $\mathcal {W}$. Note that if $\eta < 1$, the profit plane $\mathcal {C}_{12}$ is not parallel to the $q_3$ axis, and hence the projection forms a closed polytope: ${\text {proj}}_{\mathcal {W}} \mathcal {C}_{12} = \{(q_1, q_2) \ \vert \ (q_1, q_2, q_3) \in \mathcal {C}_{12}, \forall q_{3} \in [0,1] \}$. When $(q_1, q_2) \in {\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}$, $\mathcal {I}_1$ and $\mathcal {I}_{23}$ are both nonempty. When $(q_1, q_2) \not \in {\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}$, either $\mathcal {I}_1$ or $\mathcal {I}_{23}$ is empty. Next, we let $q_{3,u}^*$ and $q_{3,u}^{**}$ be the unconstrained maximizers of $u_{3, 1}$ and $u_{3, 23}$, respectively. Let $q_3^*$ be the maximizer of $u_3$ in [0, 1]. To guarantee the uniqueness of $q_3^*$, we set $q_3^*$ as the crossing point $q_{3,s}$. The following inequalities must hold:

$$\begin{aligned} q_{3,u}^* \le q_{3, s}, \quad q_{3, u}^{**} \ge q_{3, s} \quad \forall (q_1, q_2) \in {\text {proj}}_{\mathcal {W}} \mathcal {C}_{12} \end{aligned}$$

Further simplification tells for all $(q_1, q_2) \in {\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}$, we have

$$\begin{aligned} p_o \le \frac{3-\eta }{2} k+\frac{(1-\eta ) c_{3} k a}{2(1-\alpha ) a_{b}}, \quad p_o \ge \frac{2 \eta k}{1+\eta }-\frac{\eta (1-\eta ) c_{3} k a}{(1+\eta )(1-\alpha ) a_{b}} \end{aligned}$$

Since ${\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}$ is closed, $p_{o, \min }$ and $p_{o,\max }$ exist. By taking these two values into the inequalities above, we obtain the inequalities (17)-(18).

To prove the continuity, it is clear that ${\text {proj}}_{\mathcal {W}} \mathcal {C}_{12} \subset \mathcal {W}$. For the region $\{ (q_1, q_2, q_3) \ \vert \ (q_1, q_2) \in {\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}, q_3 \in [0,1] \}$, the best response is the crossing point $q_{3, s}$. All the crossing points form the plane $\mathcal {C}_{12}$, which is continuous in $(q_1, q_2)$. For the region $\{ (q_1, q_2, q_3) \ \vert \ (q_1, q_2) \in \mathcal {W} \backslash {\text {proj}}_{\mathcal {W}} \mathcal {C}_{12}, q_3 \in [0,1] \}$, the best response is either 0 or 1 or the unconstrained minimizer calculated by (15) or (16). All of them are continuous in $(q_1, q_2)$. This proves the continuity of the best response of the seller $S_3$.