### 经济代写|博弈论代写Game Theory代考|ECON2112

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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|ORDINAL UTILITIES

In this scenario, there are four possible outcomes: Scarlett leaves with vanilla ice cream, which we will label $V$, leaves with chocolate ice cream, labeled $C$, leaves with strawberry ice cream, labeled $S$, or leaves with no ice cream, labeled $N$. Our outcome set is then ${V, C, S, N}$. The rules dictate that Scarlett may choose any outcome that is available.
To model Scarlett’s preferences and build a utility function, we consider the choices she would make when certain outcomes are available. Supposing that all potential outcomes are available, Scarlett would choose $V$; if $C$ were not an available outcome, Scarlett would choose $S$; if only $V$ and $N$ were available outcomes, Scarlett would flip a coin to determine whether to select $V$ or $N$; and if only $C$ and $S$ were available outcomes, Scarlett would be unable to make a choice. While this behavior would be possible, most would find it bizarre: Scarlett’s choice of $V$ when presented with the outcome set ${V, C, S, N}$ suggests she prefers $V$ to $S$, but Scarlett’s choice of $S$ when presented the outcome subset ${V, S, N}$ suggests she prefers $S$ to $V$. To avoid such absurd possibilities, we will assume that individual behavior is governed by self-consistent internal preferences over the outcomes, which is reflected in the mathematical definition of ordinal preferences below.

Definition 2.1.1. A player $i$ is said to have ordinal preferences among outcomes if there exists a utility function $u_i$ from the set $O$ of outcomes into the real numbers, $\mathbb{R}$, such that whenever presented with a subset $O^{\prime} \subseteq O$ of outcomes, player $i$ chooses any of the outcomes that maximize $u_i$ over all outcomes $o \in O^{\prime}$.

To ensure that ordinal preferences align with the players real-world choice behavior, we note that whenever player $i$ prefers outcome $o_j$ over outcome $o_k$, we should have $u_i\left(o_j\right)>$ $u_i\left(o_k\right)$, and when player $i$ is indifferent between $o_j$ and $o_k$ we should have $u_i\left(o_j\right)=u_i\left(o_k\right)$. We now ask under what conditions Scarlett’s outcome choice behavior can be modeled by ordinal preferences and describe three reasonable properties for a self-consistent set of choices. Since it is usually easier for a player to choose between two rather than among many outcomes, these properties will focus on pairwise choices.

First, we want Scarlett’s pairwise choices to be complete, meaning that whenever she is presented with a pair of outcomes, Scarlett is able to make a choice. Equivalently, for each pair of outcomes, $A$ and $B$, exactly one of the following conditions holds: (a) Scarlett chooses $A$ over $B$, (b) Scarlett chooses $B$ over $A$, or (c) Scarlett is willing to flip a coin to determine which outcome to choose (in this case we will often say Scarlett chooses either $A$ or $B$ ). Hence, this condition excludes the following option as a possibility: (d) Scarlett chooses neither $A$ nor $B$. (When we want the rules to allow a player to choose none of the options, we must include that as an outcome, as we did in the Ice Cream Parlor scenario.) For (a), we will assign utilities so that $u(A)>u(B)$. Likewise for (b) we will assign utilities so that $u(B)>u(A)$. Finally for (c), since Scarlett is willing to flip a coin to determine the outcome, we assume she is indifferent between $A$ and $B$ and assign $u(A)=u(B)$.

## 经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

A significant limitation of ordinal preferences and their associated utility functions is that they cannot describe the intensity of a player’s preference for a particular outcome. That is, they cannot capture the difference between Scarlett preferring vanilla ice cream over chocolate ice cream and Scarlett so strongly preferring vanilla that she would pay for it rather than have a free serving of chocolate. Notice how we have once again translated our intuitive sense of internal preference intensity into something that is observable (a real-world choice) so we can create a utility function based on these choices. While asking players to choose among outcomes that include the receipt or payment of money would be one observable way to determine intensity of preferences, we will take an approach that does not rely on the availability of money.

We begin by introducing a new, probability-based outcome called a lottery. Suppose that when a second customer, Regis, enters the ice cream parlor, he encounters a college student conducting a taste test involving different flavors of ice cream. The college student offers Regis the choice of either a sample of chocolate ice cream (his second-most favorite) or an unknown sample that is either vanilla (his favorite) or strawberry (his least favorite). The second option in this example is a simple lottery.

Definition 2.2.1. Given a set of outcomes, $O$, a simple lottery is a probability distribution over this set. When $O=\left{o_1, o_2, \ldots, o_m\right}$, a finite set, a simple lottery can be denoted by $p_1 o_1+p_2 o_2+\ldots+p_m o_m$ where $p_i$ is the probability of outcome $o_i$. A compound lottery is a probability distribution over other lotteries. Because an outcome $o \in O$ can be written as the simple lottery $1 o$, we see that $O \subset \mathcal{L}$, the set of all (simple and compound) lotteries.
Tó reveál thè strength of a player’s préference for one outcomé over another, wè must examine not nnly chnires hetween single nutromes hut chnices hetween single nutcomes and lotteries. Suppose Regis prefers vanilla $V$ over chocolate $C$ and prefers $C$ over strawberry $S$. The choice Regis makes between $C$ and the lottery $0.5 S+0.5 V$ tells us about the strength of his preference for $V$ over $C$ and for $C$ over $S$. If he would choose either of the two possibilities, then the strength of Regis’s preference for $C$ is exactly halfway between $S$ and $V$. If Regis were to choose $C$ over $0.5 S+0.5 \mathrm{~V}$, it reveals that his preference intensity for $C$ is closer to $V$ than to $S$. If Regis were willing to choose either $C$ or the lottery $0.1 S+0.9 \mathrm{~V}$, it would reveal that his preference for $V$ over $C$ is very small and his preference for $C$ over $S$ is relatively large. However, if he were instead willing to choose $C$ or the lottery $0.9 S+0.1 \mathrm{~V}$, it would reveal a strong preference for $V$ over $C$ and that his preferences for $C$ over $S$ is small. When a player is willing to choose either of two lotteries, this reveals the player is indifferent between these choices. This motivates the following generalization of the utility function concept.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

O=\left{0_1，o_2, Vdots, o_m\right } } \text { ，一个有限集，一个简单的彩票可以表示为 }
$p_1 o_1+p_2 o_2+\ldots+p_m o_m$ 在哪里 $p_i$ 是结果的概率 $o_i$. 复合彩票是其他彩票的概率分布。因为一个结 果 $o \in O$ 可以写成简单的彩票 $l o$ ，我们看到 $O \subset \mathcal{L}$ ，所有（简单和复合） 彩票的集合。

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## MATLAB代写

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