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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|Common knowledge

Consider now a set $N=\left{p_1, \ldots, p_n\right}$ of $n$ players $p_i$ with respective information functions $P_i$ and knowledge functions $K_i$. We say that the event $E \subseteq \mathfrak{S}$ is evident for $N$ if $E$ is evident for each of the members of $N$, i.e., if
$$E=K_1(E)=\ldots=K_n(E) .$$
More generally, an event $E \subseteq \mathfrak{S}$ is said to be common knowledge of $N$ in the state $\sigma$ if there is an event $F \subseteq E$ such that
$F$ is evident for $N$ and $\sigma \in F$.

Proposition 4.1. If the event $E \subseteq \mathfrak{S}$ is common knowledge for the n players $p_i$ with information functions $P_i$ in state $\sigma$, then
$$\sigma \in K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(E)\right) \ldots\right)\right)$$
holds for all sequences $i_1 \ldots i_m$ of indices $1 \leq i_j \leq n$.
Proof. If the event $E$ is common knowledge, it comprises an evident event $F \subseteq E$ with $\sigma \in F$. By definition, we have
$$\in K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(F)\right) \ldots\right)\right)=F$$
By property (K.2) of a knowledge function (Lemma 4.2), we thus conclude
$$K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(E)\right) \ldots\right)\right) \supseteq K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(F)\right) \ldots\right)\right)=F \ni \sigma$$
As an illustration of Proposition 4.1, consider the events
$$K_1(E), K_2\left(K_1(E)\right), K_3\left(K_2\left(K_1(E)\right)\right) .$$
$K_1(E)$ are all the states where player $p_1$ is sure that $E$ has occurred. The set $K_2\left(K_1(E)\right)$ comprises those states where player $p_2$ is sure that player $p_1(E)$ is sure that $E$ has occurred. In $K_3\left(K_2\left(K_1(E)\right)\right)$ are all the states where player $p_3$ is certain that player $p_2$ is sure that player $p_1$ believes that $E$ has occurred. And so on.

经济代写|博弈论代写Game Theory代考|Different opinions

Let $p_1$ and $p_2$ be two players with information functions $P_1$ and $P_2$ relative to a finite system $\mathfrak{S}$ and assume:

• Both players have the same probability estimates $\operatorname{Pr}(E)$ on the
• Can there be common knowledge among the two players in a certain state $\sigma^*$ that they have different likelihood estimates $\eta_1$ and $\eta_2$ for an event $E$ having occurred?

Surprisingly(?), the answer can be “yes” as Ex. $4.13$ shows. For the analysis in the example, recall that the conditional probability of an event $E$ given the event $A$, is
$$\operatorname{Pr}(E \mid A)=\left{\begin{array}{lll} \operatorname{Pr}(E \cap A) / \operatorname{Pr}(A) & \text { if } \operatorname{Pr}(A)>0 \ 0 & \text { if } \operatorname{Pr}(A)=0 \end{array}\right.$$
Ex. 4.13. Let $\mathfrak{S}=\left{\sigma_1, \sigma_2\right}$ and assume $\operatorname{Pr}\left(\sigma_1\right)=\operatorname{Pr}\left(\sigma_2\right)=$ 1/2. Consider the information functions
\begin{aligned} &P_1\left(\sigma_1\right)=\left{\sigma_1\right} \quad \text { and } P_1\left(\sigma_2\right)=\left{\sigma_2\right} \ &P_2\left(\sigma_1\right)=\left{\sigma_1, \sigma_2\right}=P_2\left(\sigma_2\right) \end{aligned}
For the event $E=\left{\sigma_1\right}$, one finds
$\operatorname{Pr}\left(E \mid P_1\left(\sigma_1\right)\right)=1 \quad$ and $\operatorname{Pr}\left(E \mid P_1\left(\sigma_2\right)\right)=0$
$\operatorname{Pr}\left(E \mid P_2\left(\sigma_1\right)\right)=1 / 2 \quad$ and $\quad \operatorname{Pr}\left(E \mid P_2\left(\sigma_2\right)\right)=1 / 2$
The ground set $\mathcal{S}=\left{\sigma_1, \sigma_2\right}$ corresponds to the event “the two players differ in their estimates on the likelihood that $E$ has occurred”. $\mathfrak{S}$ is (trivially) common knowledge in each of the two states $\sigma_1, \sigma_2$

经济代写|博弈论代写博弈论代考|常识

$$E=K_1(E)=\ldots=K_n(E) .$$
，更一般地说，一个事件$E \subseteq \mathfrak{S}$在$\sigma$的状态下被认为是$N$的常识，如果有一个事件$F \subseteq E$，如
$F$对于$N$和$\sigma \in F$是明显的

$$\sigma \in K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(E)\right) \ldots\right)\right)$$

$$\in K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(F)\right) \ldots\right)\right)=F$$

$$K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(E)\right) \ldots\right)\right) \supseteq K_{i_1}\left(K_{i_2}\left(\ldots\left(K_{i_m}(F)\right) \ldots\right)\right)=F \ni \sigma$$

$$K_1(E), K_2\left(K_1(E)\right), K_3\left(K_2\left(K_1(E)\right)\right) .$$
$K_1(E)$是玩家$p_1$确信$E$已经发生的所有状态。集合$K_2\left(K_1(E)\right)$包含玩家$p_2$确信玩家$p_1(E)$确信$E$已经发生的状态。在$K_3\left(K_2\left(K_1(E)\right)\right)$中是玩家$p_3$确信玩家$p_2$确信玩家$p_1$相信$E$已经发生的所有状态。

经济代写|博弈论代写博弈论代考|不同的意见

.

• 两个玩家在

$$\operatorname{Pr}(E \mid A)=\left{\begin{array}{lll} \operatorname{Pr}(E \cap A) / \operatorname{Pr}(A) & \text { if } \operatorname{Pr}(A)>0 \ 0 & \text { if } \operatorname{Pr}(A)=0 \end{array}\right.$$
Ex。4.13. 设$\mathfrak{S}=\left{\sigma_1, \sigma_2\right}$，设$\operatorname{Pr}\left(\sigma_1\right)=\operatorname{Pr}\left(\sigma_2\right)=$ 1/2。考虑信息函数
\begin{aligned} &P_1\left(\sigma_1\right)=\left{\sigma_1\right} \quad \text { and } P_1\left(\sigma_2\right)=\left{\sigma_2\right} \ &P_2\left(\sigma_1\right)=\left{\sigma_1, \sigma_2\right}=P_2\left(\sigma_2\right) \end{aligned}

$\operatorname{Pr}\left(E \mid P_1\left(\sigma_1\right)\right)=1 \quad$和$\operatorname{Pr}\left(E \mid P_1\left(\sigma_2\right)\right)=0$
$\operatorname{Pr}\left(E \mid P_2\left(\sigma_1\right)\right)=1 / 2 \quad$和$\quad \operatorname{Pr}\left(E \mid P_2\left(\sigma_2\right)\right)=1 / 2$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师