### 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|Probabilities, information and entropy

Consider $n$ mutually exclusive events $E_1, \ldots, E_n$, and expect that any one of these, say $E_i$, indeed occurs “with probability” $p_i=\operatorname{Pr}\left(E_i\right)$. Then the parameters $p_i$ form a probability distribution $p \in \mathbb{R}^{\mathcal{E}}$ on the set $\mathcal{E}=\left{E_1, \ldots, E_n\right}$, i.e., the $p_i$ are nonnegative real numbers that sum up to 1 :
$$p_1+\cdots+p_n=1 \quad \text { and } \quad p_1, \ldots, p_n \geq 0 .$$
If we have furthermore a measuring or observation device $f$ that produces the number $f_i$ if $E_i$ occurs, then these numbers have the expected value
$$\mu(f)=f_1 p_1+\cdots+f_n p_n=\sum_{k=1}^n f_i p_i=\langle f \mid p\rangle .$$
In a game-theoretic context, a probability is often a subjective evaluation of the likelihood for an event to occur. The gambler, investor, or general player may not know in advance what the future will bring, but has more or less educated guesses on the likelihood of certain events. There is a close connection with the notion of information.

Intensity. We think of the intensity of an event $E$ as a numerical parameter that is inversely proportional to its probability $p=\operatorname{Pr}(E)$ with which we expect its occurrence to be: the smaller $p$, the more intensely felt is an actual occurrence of $E$. For simplicity, let us take $1 / p$ as our objective intensity measure.

Remark $1.7$ (Fechner’s law). According to Fechner, ${ }^{11}$ the intensity of a physical stimulation is physiologically felt on a logarithmic scale. Well-known examples are the Richter scale for earthquakes or the decibel scale for the sound.

Following FECHNER, we feel the intensity of an event $E$ that we expect with probability $p$ on a logarithmic scale and hence according to a function of type
$$I_a(p)=\log _a(1 / p)=-\log _a p,$$
where $\log _a p$ is the logarithm of $p$ relative to the basis $a>0$ (see Ex. 1.7). In particular, the occurrence of an “impossible” event, which we expect with zero probability, has infinite intensity
$$I_a(0)=-\log _a 0=+\infty .$$

## 经济代写|博弈论代写Game Theory代考|Systems

A system is a physical, economic, or other entity that is in a certain state at any given moment. Denoting by $\mathfrak{S}$ the collection of all possible states $\sigma$, we identify the system with $\mathfrak{S}$. This is, of course, a very abstract definition. In practice, one will have to describe the system states in a way that is suitable for a concrete mathematical analysis. To get a first idea of what is meant, let us look at some examples.

Chess. A system arises from a game of chess as follows: A state of chess is a particular configuration $C$ of the chess pieces on the chess board, together with the information which of the two players ( ” $B$ ” or ” $W$ “) is to draw next. If $\mathfrak{C}$ is the collection of all possible chess configurations, a state could thus be described as a pair
$$\sigma=(C, p) \quad \text { with } C \in \mathfrak{C} \text { and } p \in{B, W} .$$
In a similar way, a card game takes place in the context of a system whose states are the possible distributions of cards among the players together with the information which players are to move next.

Economies. The model of an exchange economy involves a set $N$ of agents and a set $\mathcal{G}$ of certain specified goods. A bundle for agent $i \in N$ is a data vector
$$b=\left(b_G \mid G \in \mathcal{G}\right) \in \mathbb{R}^{\mathcal{G}},$$
where the component $b_G$ indicates that the bundle $b$ comprises $b_G$ units of the good $G \in \mathcal{G}$. Denoting by $\mathcal{B}$ the set of all possible bundles, we can describe a state of the exchange economy by a data vector
$$\beta=\left(\beta_i \mid i \in N\right) \in \mathcal{B}^N$$
that specifies each agent $i$ ‘s particular bundle $\beta_i \in \mathcal{B}$.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Probabilities, information and entropy

$$p_1+\cdots+p_n=1 \quad \text { and } \quad p_1, \ldots, p_n \geq 0 .$$

$$\mu(f)=f_1 p_1+\cdots+f_n p_n=\sum_{k=1}^n f_i p_i=\langle f \mid p\rangle .$$

$$I_a(p)=\log _a(1 / p)=-\log _a p,$$

$$I_a(0)=-\log _a 0=+\infty$$

## 经济代写|博弈论代写Game Theory代考|Systems

$$\sigma=(C, p) \quad \text { with } C \in \mathfrak{C} \text { and } p \in B, W .$$

$$b=\left(b_G \mid G \in \mathcal{G}\right) \in \mathbb{R}^{\mathcal{G}},$$

$$\beta=\left(\beta_i \mid i \in N\right) \in \mathcal{B}^N$$

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