Casino Rationale: Countering Attacker Deception in Zero-Sum Stackelberg Security Games of Bounded Rationality

Abstract

In this work, we consider a zero-sum game between an adaptive defender and a potentially deceptive attacker who is able to vary their degree of rationality as a deceptive ruse. Under this setup, we provide a complete characterization of the deception space of the attacker and uncover optimal strategies for adaptive defender against a deceptive attacker. In addition, we consider the setup in which both the attacker and defender are allowed to evolve their strategies over time. In this setting, one of our main results is to demonstrate that allowing the attacker to vary their degree of rationality can significantly impact the game in favor of the attacker.

A Proof of Lemma 3

Analogous to the approach in Lemma 2, we note that $\left( Tx - 1 \right) V \exp \left( \lambda (1-Tx)V \right) \leqslant \frac{1}{\lambda \exp (1)},$ which is achieved when $x = \frac{1}{T} + \frac{1}{T \lambda V}$. Therefore, $\sum _{i \in \{1,2,\ldots , T\}} q_i({\boldsymbol{x}}^{*}, \lambda ) U_i^{d}(x^{*}_i) \leqslant \frac{ (T-1) \frac{\exp (-1)}{\lambda } }{ \sum _{j \in \{1,2,\ldots , T\}} \exp \left( \lambda V_j \left( 1-T x_j \right) \right) }.$ In order to maximize the previous expression, we seek to minimize the convex function $\sum _{j \in \{1,2,\ldots , T\}} \exp \left( \lambda V_j \left( 1-T x_j \right) \right)$. To this end, consider the Lagrangian

$$\begin{aligned} \mathcal {L}(x_1,x_2, \ldots , x_T, \beta ) = \sum _{j \in \{1,2,\ldots , T\}} \exp \left( \lambda V_j \left( 1-T x_j \right) \right) + \beta \left( \sum _{j \in \{1,2,\ldots ,T\}} x_j - 1 \right) . \end{aligned}$$

For $i \in [T]$, $\frac{ \partial \mathcal {L}(x_1,x_2, \ldots , x_T, \beta )}{\partial x_i} = 0$ implies that

$$\begin{aligned} x_i = \frac{1}{T} - \frac{ \log \frac{\beta }{T \lambda V_i}}{T \lambda ^{\ell } V_i }. \end{aligned}$$

(21)

Then, since $\sum _{j \in \{1,2,\ldots ,T\}} x_j = 1$, either $\sum _{j \in \{1,2,\ldots ,T\}} \frac{ \log \frac{\beta }{T \lambda ^{\ell } V_j}}{T \lambda ^{\ell } V_j } = 0$ or $\sum _{j \in \{1,2,\ldots ,T\}} \frac{ \log \beta }{T \lambda ^{\ell } V_j } = \sum _{j \in \{1,2,\ldots ,T\}} \frac{ \log T \lambda ^{\ell } V_j }{T \lambda ^{\ell } V_j }.$ This implies that $\beta \geqslant T \lambda ^{\ell } V_{1}$ (recall $V_{1} = \min _{j \in [T]} V_j$). From (21), $\sum _{j \in \{1,2,\ldots , T\}} \exp \left( \lambda ^{\ell } V_j \left( 1-T x_j \right) \right) \geqslant \sum _{j \in \{1,2,\ldots , T\}} \frac{\beta }{T \lambda ^{\ell } V_j} \geqslant \sum _{j \in \{1,2,\ldots , T\}} \frac{V_{1}}{ V_j},$ which implies$\sum _{i \in \{1,2,\ldots , T\}} q_i({\boldsymbol{x}}^{*}, \lambda ^{\ell }) U_i^{d}(x^{*}_i) \leqslant \frac{(T-1) \frac{\exp (-1)}{\lambda ^{\ell }}}{\sum _{j \in \{1,2,\ldots , T\}} \frac{V_{1}}{ V_j}},$ as desired.