### 经济代写|博弈论代写Game Theory代考|ECON3503

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Numerical Example

Consider a game that has two states. In an MTD scenario, the defender will have two actions representing selecting between each of these two states. Similarly, the attacker is assumed to have two actions of attacking profiles (both can be applied at either state).

A defender’s strategy can then be given as $\boldsymbol{f}=[a b]$, where $a$ is the action of moving to state $s_1$ and $b$ is the action of moving to state $s_2$.
The defender’s and attacker’s strategies’ permutation will be given as:
$$\boldsymbol{F}=\left[\begin{array}{ll} 1 & 1 \ 1 & 2 \ 2 & 1 \ 2 & 2 \end{array}\right] \boldsymbol{G}=\left[\begin{array}{ll} 1 & 1 \ 1 & 2 \ 2 & 1 \ 2 & 2 \end{array}\right]$$
where $\boldsymbol{F}$ and $\boldsymbol{G}$ represent the defender’s and the attacker’s permutations, respectively.
Since each player has four different permutations, each of the formulated matrices (bimatrix) will be of size $4 \cdot 4$ representing all the possible combinations of the players’ permutations. The elements of these matrices will be the accumulated utilities for each player resulting from starting at combination and considering all the future transitions.

Now suppose that the mixed Nash equilibrium for the bimatrix game, calculated from any numerical algorithm such as Lemke and Howson (1964), is given by:
$$\boldsymbol{x}^=\left[\begin{array}{l} 0.2 \ 0.1 \ 0.3 \ 0.4 \end{array}\right] \boldsymbol{y}^=\left[\begin{array}{l} 0.0 \ 0.1 \ 0.4 \ 0.5 \end{array}\right]$$

where each row in $\boldsymbol{x}^$ and $\boldsymbol{y}^$ represents the players’ probabilities of selecting a strategy in $\boldsymbol{F}$ and $\boldsymbol{G}$, respectively.

Finally, the stochastic game equilibrium strategies can be calculated by summing the probabilities of choosing each action over the corresponding states. For example, the defender can choose action 1 at state 1 twice in $\boldsymbol{F}$ with probabilities 0.2 and 0.1 (from $\boldsymbol{x}^$ ). When summed, the defender knows that when the game is at state 1 , it will choose action 1 with a probability 0.3 . Following the same approach, we can compute the full equilibrium matrices as follows: $$\boldsymbol{E}^=\left[\begin{array}{ll} 0.3 & 0.5 \ 0.7 & 0.5 \end{array}\right] \boldsymbol{H}^=\left[\begin{array}{ll} 0.1 & 0.4 \ 0.9 & 0.6 \end{array}\right]$$ where $\boldsymbol{E}^$ and $\boldsymbol{H}^*$ are the defender’s and the attacker’s equilibrium solutions, respectively, and that the rows of the matrices represent players’ actions and the columns represent the game states.

## 经济代写|博弈论代写Game Theory代考|A Case Study for Applying Single-Controller

Consider a wireless sensor network that consists of a BS and a number of wireless nodes. The network is deployed for sensing and collecting data about some phenomena in a given geographic area. Sensors will collect data and use multi-hop transmissions to forward this data to a central receiver or BS. The multiple access follows a slotted Aloha protocol. Time is divided into slots and the time slot size equals the time required to process and send one packet. Sensor nodes are synchronized with respect to time slots. We assume that nodes are continuously working and so every time slot there will be data that must be sent to the BS.

All packets sent over the network are assumed to be decrypted using a given encryption technique and a previously shared secret key. All the nodes in the system are pre-programmed with a number of encryption techniques along with a number of encryption keys per technique, as what is typically done in sensor networks (Casola et al. 2013). The BS chooses a specific encryption technique and key by sending a specific control signal over the network including the combination it wants to use. We note that the encryption technique and key sizes should be carefully selected in order not to consume a significant amount of energy when encrypting or decrypting packets. Increasing the key size will increase the amount of consumed energy particularly during the decryption (Lee et al. 2010). Since the BS is mostly receiving data, it spends more time decrypting packets rather than encrypting them and, thus, it will be highly affected by key size selection.

In our model, an eavesdropper is located in the communication field of the BS and it can listen to packets sent or received by the BS. As packets are encrypted, the attacker will seek to decrypt the packets it receives in order to get information. The attacker knows the encryption techniques used in the network and so it can try every possible key on the received packets until getting useful information. This technique is known as brute-force attack.

The idea of using multiple encryption techniques was introduced in Casola et al. (2013). However, in this work, each node individually selects one of these technique to encrypt transmitted packets. The receiving node can know the used technique by a specific field in the packet header. Large encryption keys were used which require a significant amount of power to be decrypted. Nonetheless, these large keys are highly unlikely to be revealed using a brute-force attack in a reasonable time. Here, we propose to use small encryption keys to save energy and, in conjunction with that, we enable the BS to change the encryption method in a way that reduces the chance that the encryption key is revealed by the attacker. This is the main idea behind MTD. In this model, the encryption key represents the attack surface, and by changing the encryption method, the BS will make it harder for the eavesdropper to reveal the key and get the information from the system. Naturally, the goals of the eavesdropper and the BS are not aligned. On the one hand, the BS wants to protect the data sent over the network by changing encryption method. On the other hand, the attacker wants to reveal the used key in order to get information. To understand the interactions between the defender and the attacker, one can use game theory to study their behavior in this MTD scenario. The problem is modeled as a game in which the attacker and the defender are the players. As the encryption method should be changed over time and depending on the attacker’s actions, we must use a dynamic game.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Numerical Example

$$\boldsymbol{x}^{=}\left[\begin{array}{llll} 0.2 & 0.1 & 0.3 & 0.4 \end{array}\right] \boldsymbol{y}=\left[\begin{array}{llll} 0.0 & 0.1 & 0.4 & 0.5 \end{array}\right]$$

## 经济代写|博弈论代写Game Theory代考|A Case Study for Applying Single-Controller

Casola 等人介绍了使用多种加密技术的想法。(2013)。然而，在这项工作中，每个节点单独选择其中一种技术来加密传输的数据包。接收节点可以通过包头中的特定字段知道所使用的技术。使用了需要大量能量才能解密的大型加密密钥。尽管如此，这些大密钥极不可能在合理的时间内使用暴力攻击来泄露。在这里，我们建议使用小的加密密钥来节省能源，与此同时，我们使 BS 能够以一种降低加密密钥被攻击者泄露的机会的方式改变加密方法。这是 MTD 背后的主要思想。在这个模型中，加密密钥代表攻击面，通过改变加密方式，BS 将使窃听者更难泄露密钥并从系统中获取信息。自然地，窃听者和 BS 的目标并不一致。一方面，BS 想通过改变加密方法来保护通过网络发送的数据。另一方面，攻击者想要泄露使用的密钥以获取信息。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。窃听者和 BS 的目标不一致。一方面，BS 想通过改变加密方法来保护通过网络发送的数据。另一方面，攻击者想要泄露使用的密钥以获取信息。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。窃听者和 BS 的目标不一致。一方面，BS 想通过改变加密方法来保护通过网络发送的数据。另一方面，攻击者想要泄露使用的密钥以获取信息。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。

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## MATLAB代写

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