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# 博弈论代写代考| Ordinal Games 数学代写

## 博弈论代考

Une problem with the example about MIr. Holly s umbrella is that the payott points didn $t$ mean
anything. Is a payoff of 6 really worth twice what a payoff of 3 is worth? There are many games
where the payoffs are either subjective or difficult to measure numerically. In this section, we’ll
see that we can still retain something of the dynamics of the game if, instead of actual numerical
payoffs, all we know is in what order we prefer the outcomes of the game.
So assume there are two players, and we retain all the other assumptions (rational behavior, both
players have the same information about the payoffs, and the players move simultaneously and
independently). Suppose the row player has $m$ strategies, and the column player has $n$ strategies,
so that there are $r=m n$ outcomes in the game. Each player ranks the $r$ outcomes in order of his
or her preference. Mimicking what is done in Taylor and Pacelli (2008), we will assign the most
preferred outcome with the highest number $r$, because we are already accustomed to thinking that
a higher number is better for a player. Thus, the most preferred outcome is denoted $r$, the next most preferred is denoted $r-1$, etc., until the least preferred outcome, which is labeled 1. We
assume there are no two outcomes that the players prefer equally – no ties in the rankings. Then
we form ordered pairs of these preference rankings and use them to fill in the payoff matrix.
For example, suppose the row player has two strategies and the column player has three. The
payoff matrix might look like this:
$\left[\begin{array}{lll}(3,5) & (2,6) & (4,1) \ (6,2) & (1,4) & (5,3)\end{array}\right]$.
So, for example, in the $(1,1)$ position, the “payoff” $(3,5)$ indicates that this outcome is the
player’s second choice. The $(4,1)$ “payoff” indicates it is the row player’s third choice and the
column player’s last choice, etc. Because “first” (choice),
are ordizal mumbers, we call these games ordinal gand
First, we see what concepts from the usual theory carry over to ordinal games. Note that the
notions of “zero-sum” or “variable-sum” are meaningless. The “payoffs” are preference rankings

• so ordinal numbers – for which an addition is not even defined. However, some concepts carry
over just fine. The notion of a movement diagram, for example, still makes sense and is drawn
the same way as for variable-sum games. Vertical arrows point to the highest first coordinate in
• Sensituity Amalysis, Ordinal Games, and n-Person Games
a column, and horizontal arrows point to the highe diagram for the matrix is shown in Figure $8.6$
diagram for the matrix is shown in Figure 8.6.
diagram for the matrix is shown in Figure 8.6.
The concept of one strategy dominating another also carries over.
DEFINITION 8.4 A strategy $S$ dominates a strategy $T$ if the player prefers every outcome in $S$
more than the corresponding outcome in $T$.
For the row player, that means every first coordinate of $S$ is higher than the corresponding first
coordinate in $T$; for the column player, every second coordinate in $S$ is higher than the corre-
sponding coordinate in $T$. In the preceding example, neither row dominates the other. However,
the middle column dominates both of the other two columns. As usual, you can read dominance in
the movement diagram since all the arrows point to the entries in the dominant row or column. (It’s
not obvious directly from the movement diagram, however, that neither the first column dominates
the third nor the reverse. One must look more closely at the payoff matrix.)
Similarly, the way to detect stability carries over as well.
DEFINITION $8.5$ An outcome of an ordinal game is called a Nash equilibrium if neither player
can improve the preference ranking of their payoff by a unilateral change of strategy.
As usual, Nash equilibria are evident from the movement diagram by virtue of being an outcome
to which both vertical and horizontal arrows point. In the preceding example, the outcome in the
1,2 position, with payoff $(2,6)$ is a Nash equilibrium.
The concept of a payoff polygon does not carry over because the payoffs are not real numbers.
Nevertheless, the concept of Pareto efficiency still makes sense:
DEFINITION $8.6$ An outcome of an ordinal game is Pareto efficient if there is no other outcome
that improves both players’ preferences.
374
Notice that in this definition it is not necessary to consider the case of just one player improving
because we have assumed there are no ties in the preference rankings for each player.
In the game just described, the Nash equilibrium is necessarily Pareto efficient because one of the players obtains his or her most preferred outcome 6 , so it is not possible to improve his or her
ranking.
8.2.2 Prisoners’ Dilemma and 0ther Dilemmas
Unfortunately, some of the difficulties we had with variable-sum games also carry over. There
are ordinal games where Nash equilibria are not Pareto efficient and games where they are not
Unfortunately, some of the difficulties we had with variable-sum games also carry over. There are ordinal games where Nash equilibria are not Pareto efficient and games where they are not

br>所以假设有两个玩家，我们保留所有其他假设（理性行为，两个

$\left[\begin{array}{lll}(3,5) & (2,6) & (4,1) \ (6,2) & (1,4) & (5,3)\end{array}\right]$.

• so 序数 – 甚至没有定义加法。但是，有些概念可以很好地继承
。例如，运动图的概念仍然有意义，并且绘制方式与可变和游戏相同。垂直箭头指向
• Sensituity Amalysis、Ordinal Games 和 n-Person Games
a 列中的最高第一个坐标，水平箭头指向矩阵的最高图，如图 $8.6$
矩阵图如图 8.6 所示。
矩阵图如图 8.6 所示。
一种策略支配另一种策略的概念也得以延续。
定义 8.4 策略 $S$如果玩家更喜欢 $S$ 中的每个结果
比 $T$ 中的相应结果更喜欢策略 $T$。
对于行玩家，这意味着 $S$ 的每个第一个坐标都高于$T$中对应的第一个
坐标；对于列播放器，$S$ 中的每一秒坐标都高于 $T$ 中相应的
对应坐标。在前面的示例中，两行都没有支配另一行。但是，
中间的列在其他两列中占主导地位。像往常一样，您可以在
运动图中阅读优势，因为所有箭头都指向优势行或列中的条目。 （但是，
从运动图中直接看不出来，第一列都不占优势
第三列也不是相反的。必须更仔细地查看收益矩阵。）
同样，检测方法稳定性也会延续。
定义 $8.5$ 如果没有参与者
不能通过单方面改变策略来提高其收益的偏好排名，则序数博弈的结果称为纳什均衡。
像往常一样，纳什均衡从运动图中显而易见，因为它是垂直和水平箭头所指向的结果
。在前面的示例中，
1,2 位置的结果，收益 $(2,6)$ 是纳什均衡。
收益多边形的概念不会延续，因为收益不是真实的
尽管如此，帕累托效率的概念仍然有意义：
定义 $8.6$ 如果没有其他结果可以改善两个玩家的偏好，则序数博弈的结果是帕累托有效的。
374
请注意，在此定义中，没有必要考虑只有一名玩家提高的情况
因为我们假设每个玩家的偏好排名没有关系。
在刚刚描述的游戏中，纳什均衡必然是帕累托有效的，因为其中一个参与者获得了他或她最喜欢的结果 6 ，

br>所以假设有两个玩家，我们保留所有其他假设（理性行为，两个

$\left[\begin{array}{lll}(3,5) & (2,6) & (4,1) \ (6,2) & (1,4) & (5,3)\end{array}\right]$.

## 博弈论代写

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## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码

## 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。