### 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|Mathematical modelling

Mathematics is the powerful human instrument to analyze and to structure observations and to possibly discover natural “laws”. These laws are logical principles that allow us not only to understand observed phenomena (i.e., the so-called real world) but also to compute possible evolutions of current situations, and thus to attempt a “look into the future”.

Why is that so? An answer to this question is difficult if not impossible. There is a wide-spread belief that mathematics is the language of the universe. ${ }^{1}$ So everything can supposedly be captured by mathematics and all mathematical deductions reveal facts about the real world. I do not know whether this is true. Even if it were, one would have to be careful with real-world interpretations of mathematics, nonetheless. A simple example may illustrate the difficulty:
While apples on a tree are counted in terms of natural numbers, it would certainly be erroneous to conclude: for every natural number $n$, there exusts a tree with $n$ apples. In other words, when we use the set of nonnegative integers to describe the number of apples on a tree, our mathematical model will comprise mathematical objects that have no real counterparts.

Theoretically, one could try to get out of the apple dilemma by restricting the mathematical model to those numbers $n$ that are realized by apple trees. But such a restricted model would be of no practical use as neither the set of such apple numbers $n$ nor its specific algebraic structure is explicitly known. Indeed, while the sum $m+n$ of two natural numbers $m$ and $n$ is a natural number, it is not clear whether the existence of two apple trees with $\mathrm{m}$ resp. $n$ apples guarantees the existence of an apple tree with $m+n$ apples.

In general, a mathematical model of a real-world situation is, alas, not necessarily guaranteed to be absolutely comprehensive. Mathematical conclusions are possibly only theoretical and may suggest objects and situations which do not exist in reality. One always has to double-check real-world interpretations of mathematical deductions and ask whether an interpretation is “reasonable” in the sense that it is commensurate with one’s own personal experience.

In the analysis of a game-theoretic situation, for example, one may want to take the psychology of individual players into account. A mathematical model of psychological behavior, however, is typically based on assumptions whose accuracy is unclear. Consequently, mathematically established results within such models must be interpreted with care, of course.

## 经济代写|博弈论代写Game Theory代考|Algebra of functions and matrices

While the coefficients of data vectors or matrices can be quite varied (colors, sounds, configurations in games, etc.), we will typically deal with numerical data so that coordinate vectors have real numbers as their component values. Hence we deal with coordinate spaces of the type
$$\mathbb{R}^{S}={f: S \rightarrow \mathbb{R}}$$
Addition and scalar multiplication. The sum $f+g$ of two coordinate vectors $f, g \in \mathbb{R}^{S}$ is the vector of component sums $(f+g){s}=f{s}+g_{s}$, i.e.,
$$f+g=\left(f_{s}+g_{s} \mid s \in S\right) .$$
For any scalar $\lambda \in \mathbb{R}$, the scalar product $\lambda f$ multiplies each component of $f \in \mathbb{R}^{S}$ by $\lambda$ :
$$\lambda f=\left(\lambda f_{s} \mid s \in S\right) .$$
WARNING. There are many – quite different – notions for “multiplication” operations with vectors.

Products. The (function) product $f \bullet g$ of two vectors $f, g \in \mathbb{R}^{S}$ is the vector with the componentwise products, i.e.,
$$f \bullet g=\left(f_{s} g_{s} \mid s \in S\right)$$
In the special case of matrices $A, B \in \mathbb{R}^{X \times Y}$ the function product of $A$ and $B$ is called the HADAMARD ${ }^{5}$ product
$A \bullet B \in \mathbb{R}^{X \times Y} \quad\left(\right.$ with coefficients $\left.(A \bullet B){x y}=A{x y} B_{x y}\right)$
WARNING. The HADAMARD product is quite different than the standard matrix multiplication rule (3) below.

## 经济代写|博弈论代写Game Theory代考|Algebra of functions and matrices

$$\mathbb{R}^{S}=f: S \rightarrow \mathbb{R}$$

$$f+g=\left(f_{s}+g_{s} \mid s \in S\right) .$$

$$\lambda f=\left(\lambda f_{s} \mid s \in S\right)$$

$$f \bullet g=\left(f_{s} g_{s} \mid s \in S\right)$$

$A \bullet B \in \mathbb{R}^{X \times Y} \quad\left(\right.$ 有系数 $\left.(A \bullet B) x y=A x y B_{x y}\right)$

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## MATLAB代写

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