### 经济代写|博弈论代写Game Theory代考|ECON 6025

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Evaluation of Heuristics

Our goal is to evaluate the heuristics we used to reduce the number of possible ways HPs can be allocated in the network to make the algorithm more scalable compared to our previous experiments. In the first experiment, we evaluate the solution quality of the algorithm with and without heuristics by comparing how early the defender can identify an attacker. We used different sizes of networks with different limits on the number of rounds the players can play. Each of the three attackers we considered, $a, b, c$, has some unique and some shared exploits in their possession. Attacker $a$ has exploits $\phi_{0}, \phi_{1}, \phi_{2}$. Attacker $b$ has exploits $\phi_{2}, \phi_{3}, \phi_{1}$. Attacker $c$ has exploits $\phi_{4}, \phi_{5}, \phi_{2}$. We picked the attacker $b$ as the acting attacker. We kept the edge density and shared vulnerabilities between nodes to $40 \%$. Results are averaged over 20 game instances. In the Figure $3.9$, the first row, Figure $3.9 \mathrm{a}$ and $\mathrm{b}$ computes the solution without using the heuristics and the second row, Figure $3.9 \mathrm{c}$ and d computes the solution using the heuristics to reduce the action space. As we can see, the solution quality did not degrade for the experiments we conducted. However, currently, we have no proof to guarantee an optimal solution with the use of heuristics.

In the next experiment, we compare the run time between the algorithms using without and with heuristics for different sizes of networks and the same setup as the previous experiment. Run times are averaged over all the attackers for a particular size of games. As shown in Figure $3.10$, for smaller instances the algorithm with heuristics can compute the solution much quicker than if we do not use any heuristics. For larger instances of games if we do not heuristics, the algorithm runs out of memory very quickly (as shown by ” $x$ “), whereas using the heuristic we can compute the solutions.

## 经济代写|博弈论代写Game Theory代考|Conclusions and Future Direction

Identification of an attacker is an even harder problem in many ways than detection, especially when many attackers use similar TTP in the early stages of attacks. However, any information that can help to narrow down the intention and likely TTP of an attacker can also be of immense value to the defender if it is available early on.

We present several case studies and a formal game model showing how we can use deception techniques to identify different types of attackers represented by the different AGs they use in planning optimal attacks based on their individual goals and capabilities. We show that strategically using deception can facilitate significantly earlier identification by leading attackers to take different actions early in the attack that can be observed by the defender. Our simulation results show this in a more general setting. However, the optimal algorithm does not scale very well since it considers all the possible action space to allocate HPs. So, we presented a scalable version of the optimal algorithm that reduces the action space by a reasonable margin to handle larger instances of games. In future work, we plan to explore how this model can be extended to different types of deception strategies, integration with other IDS techniques, as well as larger and more diverse sets of possible attacker types to make it operational in the real world.

## 经济代写|博弈论代写Game Theory代考|Reward Function

As introduced earlier, there is a cost associated with each action. The network defender incurs a fixed cost for placing a new honeypot on an edge in the network. Let $P_{\mathrm{c}}$ denote the honeypot placement cost. On the attacker side, there is a cost per attack denoted as $A_{\mathrm{c}}$. The attack cost reflects the risk taken by the attacker as mentioned earlier.

If the defender placed honeypot on the same edge the attacker exploits, the defender gains a capturing reward. Otherwise, if the attacker exploited another safe edge, the attacker gains a successful attack reward. Let Cap and Esc denote the defender capture reward and the attacker successful reward, respectively.

To account for different nodes in the network, we adopt a reward function that takes into account the importance of the network nodes. Therefore, both capturing reward and the successful attack reward are weighted by the value of the secured or attacked node value, $w_{v}$. We start by expressing the reward matrix for the game illustrated in Figure $4.1$ and present the general reward matrix afterward.
$$R_{1}=\left[\begin{array}{ccc} -P_{\mathrm{c}}+A_{\mathrm{c}}+\operatorname{Cap} * w_{b} & -P_{\mathrm{c}}+A_{\mathrm{c}}-\operatorname{Esc} * w_{c} & -P_{\mathrm{c}} \ -P_{\mathrm{c}}+A_{\mathrm{c}}-\mathrm{Esc} * w_{\mathrm{b}} & -P_{\mathrm{c}}+A_{\mathrm{c}}+\operatorname{Cap} * w_{\mathrm{c}} & -P_{\mathrm{c}} \ A_{\mathrm{c}}-\operatorname{Esc} * w_{b} & A_{\mathrm{c}}-\mathrm{Esc} * w_{c} & 0 \end{array}\right] .$$
The attacker reward matrix is $R_{2}=-R_{1}$. The reward, $R_{1}(1,1)$ and $R_{1}(2,2)$, represents a captured attacker as the defender installed a honeypot at the attacked node. Therefore, the defender pays placement cost $P_{\mathrm{c}}$ and gains a capturing reward weighted by the value of the defended node. The attacker incurs an attack cost, $A_{\mathrm{c}}$, which represents a reward for the defender in a zero-sum game. On the other hand, $R_{1}(1,2)$ and $R_{1}(2,1)$ represent a successful attack, where the honeypot is allocated at a different node. Therefore, there is a Esc loss weighted by the compromised node value. In $R_{1}(3,1)$ and $R_{1}(3,2)$, the defender decides not to place any honeypot to save placement

cost $P_{\mathrm{c}^{+}}$Similarly, in $R_{1}(1,3), R_{1}(2,3)$, and $R_{1}(3,3)$, the attacker backs off to either avoid capture cost or attack cost $A_{\mathrm{c}}$.

The reward function can easily be generalized to an arbitrary number of possible edges as follows.
$$R_{1}\left(a_{1}, a_{2}\right)=\left{\begin{array}{clr} -P_{\mathrm{c}}+A_{\mathrm{c}}+\operatorname{Cap} * w_{v} ; & a_{1}=e_{a, w}, a_{2}=v & \forall v \in \mathcal{V} \ -P_{\mathrm{c}}+A_{\mathrm{c}}-\mathrm{Esc} * w_{u} ; & a_{1}=e_{a, v}, a_{2}=u & \forall u \neq v \in \mathcal{V} \ -P_{\mathrm{c}} ; & a_{1}=e_{a, v}, a_{2}=0 & \forall v \in \mathcal{V} \ 0 ; & a_{1}=0, a_{2}=0 & \end{array}\right.$$
where $a_{1}=0$ denotes the defender is not allocating any new honeypot. Similarly, $a_{2}=0$ denotes the attacker decided to back off.

## 经济代写|博弈论代写Game Theory代考|Reward Function

R1=[−磷C+一个C+帽∗在b−磷C+一个C−Esc键∗在C−磷C −磷C+一个C−和sC∗在b−磷C+一个C+帽∗在C−磷C 一个C−Esc键∗在b一个C−和sC∗在C0].

$$R_{1}\left(a_{1}, a_{2}\right)=\left{ −磷C+一个C+帽∗在在;一个1=和一个,在,一个2=在∀在∈在 −磷C+一个C−和sC∗在在;一个1=和一个,在,一个2=在∀在≠在∈在 −磷C;一个1=和一个,在,一个2=0∀在∈在 0;一个1=0,一个2=0\正确的。$$

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