### 经济代写|博弈论代写Game Theory代考|ECON3503

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• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|The ultimatum game

In the ultimatum game player 1 makes an offer $a \in{0,1,2,3,4}$ to player 2. Player 2 either accepts or rejects. If player 2 accepts then she receives $a$ dollars and player 1 receives $4-a$ dollars. If 2 rejects then both get nothing. This is how this game can be written in extensive form:

1. $N={1,2}$.
2. $A={0,1,2,3,4, a, r}$.
3. $Z={0 a, 1 a, 2 a, 3 a, 4 a, 0 r, 1 r, 2 r, 3 r, 4 r}$.
1. $O={(0,0),(0,4),(1,3),(2,2),(3,1),(4,0)}$. Each pair corresponds to what players 1 receives and what player 2 receives.
2. For $b \in{0,1,2,3,4}, o(b a)=(4-b, b)$ and $o(b r)=(0,0)$.
3. $P(\varnothing)=1, P(0)=P(1)=P(2)=P(3)=P(4)=2$.
4. For $a_1, b_1, a_2, b_2 \in{0,1,2,3,4},\left(a_1, a_2\right) \leq_1\left(b_1, b_2\right)$ iff $a_1 \leq b_1$, and $\left(a_1, a_2\right) \leq_2\left(b_1, b_2\right)$ iff $a_2 \leq b_2$.

A strategy for player 1 is just a choice among ${0,1,2,3,4}$. A strategy for player 2 is a map from ${0,1,2,3,4}$ to ${a, r}$ : player 2 ‘s strategy describes whether or not she accepts or rejects any given offer.

Remark 2.3. A common mistake is to think that a strategy of player 2 is just to choose among ${a, r}$. But actually a strategy is a complete contingency plan, where an action is chosen for every possible history in which the player has to move.

## 经济代写|博弈论代写Game Theory代考|Subgames and subgame perfect equilibria

A subgame of a game $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ is a game that starts after a given finite history $h \in H$. Formally, the subgame $G(h)$ associated with $h=\left(h_1, \ldots, h_n\right) \in H$ is $G(h)=$ $\left(N, A, H_h, O, o_h, P_h,\left{\leq_i\right}_{i \in N}\right)$, where
$$H_h=\left{\left(a_1, a_2, \ldots\right):\left(h_1, \ldots, h_n, a_1, a_2, \ldots\right) \in H\right}$$
and
$$o_h\left(h^{\prime}\right)=o\left(h h^{\prime}\right) \quad P_h\left(h^{\prime}\right)=P\left(h h^{\prime}\right) .$$
A strategy $s$ of $G$ can likewise used to define a strategy $s_h$ of $G(h)$. We will drop the $h$ subscripts whenever this does not create (too much) confusion.

A subgame perfect equilibrium of $G$ is a strategy profile $s^$ such that for every subgame $G(h)$ it holds that $s^$ (more precisely, its restriction to $H_h$ ) is a Nash equilibrium of $G(h)$. We will prove Kuhn’s Theorem, which states that every finite extensive form game with perfect information has a subgame perfect equilibrium. We will then show that Zermelo’s Theorem follows from Kuhn’s.

As an example, consider the following Cold War game played between the USA and the USSR. First, the USSR decides whether or not to station missiles in Cuba. If it does not, the game ends with utility 0 for all. If it does, the USA has to decide if to do nothing, in which case the utility is 1 for the USSR and $-1$ for the USA, or to start a nuclear war, in which case the utility is $-1,000,000$ for all.

Exercise 2.7. Find two equilibria for this game, one of which is subgame perfect, and one which is not.

Exercise 2.8. Find two equilibria of the ultimatum game, one of which is subgame perfect, and one which is not.

An important property of finite horizon games is the one deviation property. Before introducing it we make the following definition.

Let $s$ be a strategy profile. We say that $s_i^{\prime}$ is a profitable deviation from $s$ for player $i$ at history $h$ if $s_i^{\prime}$ is a strategy for $G$ such that
$$o_h\left(s_{-i}, s_i^{\prime}\right)>_i o_h(s) .$$
Note that a strategy profile has no profitable deviations if and only if it is a subgame perfect equilibrium.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|The ultimatum game

1. $N=1,2$.
2. $A=0,1,2,3,4, a, r$.
3. $Z=0 a, 1 a, 2 a, 3 a, 4 a, 0 r, 1 r, 2 r, 3 r, 4 r$.
4. $O=(0,0),(0,4),(1,3),(2,2),(3,1),(4,0)$. 每对对应于玩家 1 收到什么和玩家 2 收到什么。
5. 为了 $b \in 0,1,2,3,4, o(b a)=(4-b, b)$ 和 $o(b r)=(0,0)$.
6. $P(\varnothing)=1, P(0)=P(1)=P(2)=P(3)=P(4)=2$.
7. 为了 $a_1, b_1, a_2, b_2 \in 0,1,2,3,4,\left(a_1, a_2\right) \leq_1\left(b_1, b_2\right)$ 当且仅当 $a_1 \leq b_1$ ，和 $\left(a_1, a_2\right) \leq_2\left(b_1, b_2\right)$ 当且仅当 $a_2 \leq b_2$.
玩家 1 的策略只是一个选择 $0,1,2,3,4$. 玩家 2 的策略是来自 $0,1,2,3,4$ 至 $a, r:$ 玩家 2 的策略描述了她 是接受还是拒绝任何给定的提议。
备注 2.3。一个常见的错误是认为参与者 2 的策略只是在其中进行选择 $a, r$. 但实际上，策略是一个完整的 应急计划，其中为玩家必须移动的每一个可能的历史选择一个动作。

## 经济代写|博弈论代写Game Theory代考|Subgames and subgame perfect equilibria

$h \in H$. 形式上，子博亦 $G(h)$ 有关联 $h=\left(h_1, \ldots, h_n\right) \in H$ 是 $G(h)=$ left(N, A, H_h, O, o_h, P_h,Veft{ㄴleq_ilright}_{i in N}Yright), 在哪里

$$o_h\left(h^{\prime}\right)=o\left(h h^{\prime}\right) \quad P_h\left(h^{\prime}\right)=P\left(h h^{\prime}\right) .$$

o_h\left(s_{-i}, s_i^{\prime}\right)>_i o_h(s) .


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