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博弈论代写代考| Legislative Voting and Political Power 数学代写

博弈论代考

When it comes to mathematics applied to politics, there are two distinct types of voting that can be studied. When voting for candidates for an election, that is called social choice voting. There is much that can be said about the mathematics of social choice voting, and it is a fascinating area of study. We don’t have the space here to cover it, so we refer the interested reader to Hodge and Klima (2005), Robinson and Ullman (2011), Taylor and Pacelli (2008), and chapter 20 of Straffin (1993).

In this final section of the chapter, we focus on legislative voting, where a legislative body (group of people) votes to pass a proposition, resolution, or law, based on its rules of passage. This type of voting is also called yes-no voting. We introduce this type of voting because such systems are examples of games in characteristic function form. Ours is a brief introduction. More details can be found in Taylor and Pacelli (2008), chapters 26-28 of Straffin (1993), and Taylor and Zwicker (1999), although the last text is for mathematically mature readers.
$410 \quad$ Sensitivity Analysis, Ordinal Games, and $n$-Person Games
8.5.1 Legislative Voting Systems
We begin with a hypothetical example.
Example 8.16 The Holding Company Big Sister has an executive committee that decides the company’s policies. The executive committee has three members: Janis, James, and Sam. Janis started the company, so her votes carries more weight than the other members; her weight is the combination of the other two members’ weights. So James and Sam each have one vote, and Janis has two votes (or think of it as one vote with a weight of 2 ). In order for a proposal to pass, it must have the support of at least three votes.

This is an example of a weighted voting system. In such a system, each voter has a weight attached to his or her vote, and passage requires a coalition whose total weight equals or surpasses a fixed quota $q$. Such a voting system can always be regarded as a game in characteristic function form. The set $N$ is the set of voters, and if the total weight of a coalition $X$ is $\geq q$, so it is a winning coalition, we assign $v(X)=1$; otherwise, it has value 0 .

REMARK $8.17$ We make the assumption throughout Section $8.5$ that in any voting system, the empty coalition is losing (so $v(\varnothing)=0$ as usual) and the grand coalition is winning, so $v(N)=1$.
Here is the characteristic function for the Big Sister example, where $q=3$.
\begin{tabular}{l}
\hline Coalitions and their values \
$v(\varnothing)=0$ \
$v($ Janis $)=v($ James $)=v($ Sam $)=0$ \
$v({$ James, Sam $)=0, v{$ Janis, James $}=v{$ Janis, Sam $})=1$ \
$v($ Janis, James, Sam $)=1$
\end{tabular}
We define two voting systems to be equivalent if they have the same value function $v$. This is the same thing as saying they have exactly the same winning coalitions. As an example, suppose the weights in the preceding example are 5 for Janis, 2 for James, and 3 for Sam, and we reset the quota at $q=6$. It should be clear that this weighted system has exactly the same winning coalitions as the preceding example – namely, {Janis, James}, {Janis, Sam }, and the grand coalition of all three voters. Thus, even though the weights and quota are different, this modified We define two voting systems to be $e q u i v a t e n t$ (Janis, James, Sam $}$ ) $=1$ the same thing as saying they have exactly the weights in the preceding example are 5 the quota at $q=6$. It should be clear that coalitions as the preceding example – namely, coalition of all three voters. Thus, even though the system is equivalent to the one in Example $8.16$.

In fact, the system need not even be described as a weighted system at all. Suppose we stipulate that in the executive committee, passage requires the support of at least two voters, but Janis has veto power, which means Janis must be a part of every winning coalition. If she does not support a proposal, it doesn’t pass. Again, we find the exact same three winning coalitions, so this system is equivalent to the example. So although this third case is not described as a weighted system (passage in a weighted system only requires enough weight to exceed the quota and does not have any extra stipulations such as veto power), it is actually equivalent to a weighted system. There are real-life examples of voting systems which are not described as weighted, but they are equivalent to a weighted system, such as the United Nations Security Council (see Exercise 1 at the end of this section).

$410 \quad$ 敏感性分析、Ordinal Games 和 $n$-Person Games
8.5.1 立法投票系统

\开始{表格}{l}
\hline 联盟及其价值观 \
$v(\varnothing)=0$ \
$v($贾尼斯$)=v($詹姆斯$)=v($山姆$)=0$\
$v({$ 詹姆斯, 山姆 $)=0, v{$ 詹尼斯, 詹姆斯 $}=v{$ 詹尼斯, 山姆 $})=1$ \
$v($ 詹尼斯、詹姆斯、山姆 $)=1$
\end{表格}

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复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。