### 经济代写|博弈论代写Game Theory代考|ECON2070

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Equilibrium Strategy Determination

In this game, the defender is considered the controller as it can select the actions which move the game from a state to another. The defender will also control when to apply the MTD, i.e. it will determine the duration between the time steps of the stochastic game. Assuming that the attacker has enough power, it can complete the brute-force attack in time $t_i$ for $i=1,2, \ldots, N$ for each one of the encryption techniques. Then, the defender should choose the time step $t$ to take the next action as follows:
$$t<\min \left(t_i\right), \quad i=1,2, \ldots, N .$$
By doing this, the defender can make sure that it takes a timely action before the attacker succeeds in revealing one of the keys.
The accumulated utility of player $i$ at state $s$ will be
$$\Phi_i(\boldsymbol{f}, \boldsymbol{g}, s)=\sum_{t=1}^{\infty} \beta^{t-1} \cdot U_i\left(f\left(s_t\right), g\left(s_t\right), s_t\right),$$
where $\boldsymbol{f}$ and $\boldsymbol{g}$ are the strategies adopted by the defender and attacker, respectively. The strategy specifies a vector of actions to be chosen at each of the states, e.g. $\boldsymbol{f}=\left[f\left(s_1\right), \ldots, f\left(s_K\right)\right]$ for all the $K$ states. Actions $f\left(s_t\right)$ and $g\left(s_t\right)$ are the actions chosen at $s_t$, which is the state of the game at time $t$, according to strategies $\boldsymbol{f}, \boldsymbol{g}$. State $s_t \in \mathcal{\delta}$ is determined by the defender’s action at time $t-1$. The game is assumed to start at a specific state $s=s_1$. Note that the utility in (10.4) is always bounded at infinity due to the fact that $0<\beta<1$.

When designing the bimatrix, the defender needs to calculate the accumulated utility when choosing each pure strategy against all of the attacker’s pure strategies. The defender, as a controller, can know the next state resulting from its actions, and, thus, it sums the utilities in all states using the discount factor $\boldsymbol{\beta}$. Let $\boldsymbol{X}$ be the defender’s accumulated utility matrix for all defender’s pure strategies’ permutations and all attacker’s pure strategies’ permutations. We let $\boldsymbol{F}{\boldsymbol{\bullet} \bullet}=\left[\boldsymbol{f}_1, \boldsymbol{f}_2, \ldots, \boldsymbol{f}{K^k}\right]$ be a matrix of all defender’s pure strategies’ permutation where each row represents actions in this strategy and similarly $\boldsymbol{G}{i \bullet}=\left[\boldsymbol{g}_1, \boldsymbol{g}_2, \ldots, \boldsymbol{g}{N^k}\right]$ the matrix of all attacker’s pure strategies’ permutation.

## 经济代写|博弈论代写Game Theory代考|Simulation Results and Analysis

For our simulations, we choose a system that uses two encryption techniques with two different keys per technique. Thus, the number of system states is four and the defender has four actions in each state. For the bimatrix, the attacker has $2^4=16$ different strategy permutations and the defender has $4^4=256$ different strategy permutations. The power values are set to 1 and 3 to pertain to the ratio between the power consumption in the two different encryption techniques. These values are the same for both players. We set $R_1$ and $R_2$ to be 10 and 5 depending on the opponent’s actions. We choose these values to be higher than the power values in order for the utilities to be positive. The transition reward is set to 5 and 10 for switching to another state defined by another key or another technique, respectively.

First, we run simulations when there is no transition cost, $q=0$. The equilibrium strategies for both the attacker and defender are shown in Table 10.2. Note that actions $a_1, a_2$ represent the selection of two keys for the same encryption technique and actions $a_3, a_4$ represent two keys for another technique. Table 10.1 shows the probabilities over all actions for each player. These probabilities show how players should select actions in every state. For the defender, if it starts in state $s_3$, then it should move to state $s_1$ with the highest probability and move to state $s_2$ with a very similar probability. This is because the defender will change the technique and so gets a higher transition reward. We can see that the probability of moving to the same state is always very low and can reach 0 as in state $s_1$. The probability of moving to a state that has a similar encryption key is always less than that of moving to a state with different technique as the transition reward will be lower. For the attacker, the probability of attacking the same technique that is used in the current state is always higher than attacking any other technique.

In Figure 10.3, we show the effect of the discount factor on the defender’s utility at equilibrium in every state. First, we can see that all utility values at all states increase as the discount factor increases. This is due to the fact that increasing the discount factor will make the defender care more about future rewards thus choosing the actions that will increase these future rewards. Figure 10.3 also shows that the defender’s values at states 1 and 2 are higher than at states 3 and 4 . This because states 1 and 2 adopt the first encryption technique which uses less power than the encryption technique used in states 3 and 4 . The difference mainly arises in the first state before switching to other states and applying the discount factor. Clearly, changing the discount factor has a big effect on changing the equilibrium strategy, and, thus, the game will move between states with different probabilities resulting in a different accumulated reward.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Equilibrium Strategy Determination

$$t<\min \left(t_i\right), \quad i=1,2, \ldots, N$$

$$\Phi_i(\boldsymbol{f}, \boldsymbol{g}, s)=\sum_{t=1}^{\infty} \beta^{t-1} \cdot U_i\left(f\left(s_t\right), g\left(s_t\right), s_t\right)$$

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