Posted on Categories:Game theory , 博弈论, 经济代写

# 经济代写|博弈论代考Game theory代写|ECON7062 Imperfect information, many rounds

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## 经济代写|博弈论代考Game theory代写|Imperfect information, many rounds

We now expand this Bayesian game to a game with $n$ rounds, where, as before, in each round the mobster engages a new grocer. We number the rounds ${0,1, \ldots, n-1}$. We associate with each round $k$ the number $b_k=b^{n-k}$.

The mobster’s type is picked once in the beginning of the game, and is not observed by any of the grocers. We think of $\varepsilon$ as small but fixed, and of $n$ of being very large, so that $b_0=b^n<\varepsilon$

We describe an assessment that is sequentially rational, and is in fact a perfect Bayesian equilibrium, according to the definition in Fudenberg and Tirole [14]. In this equilibrium the mobster will (obviously) punish if he is crazy. However, even if he is sane, he will punish (except towards the end), to maintain the grocers’ beliefs that he might be crazy. The result is that the grocers all comply, except for the last few.

Note that the mobster observes his own type and all the grocers’ actions, and so there is no need to define beliefs for him. The grocers’ beliefs are probability distributions over the mobster’s type, and so we denote the belief of grocer $P(h)$ after a history $h$ by $\mu(h)$, which we will take to denote the belief that the mobster is crazy (rather than having it denote the entire distribution).

The game will have two phases. In the first phase, the mobster will always punish any grocer that does not comply. This will last $k^$ rounds, where $$k^=\max \left{k \in \mathbb{N}: \varepsilon>b_k\right}$$

## 经济代写|博弈论代考Game theory代写|Repeated games

Let $G_0=\left(N,\left{A_i\right},\left{u_i\right}\right)$ be a strategic form game, and, as usual, let $A=\prod_i A_i$. We will only consider games in which $A$ is finite.

A repeated game $G$, with base game $G_0$, and lasting $T$ periods (where $T$ can be either finite or infinity) is a game in which, in each period, the players simultaneously choose an action from $G_0$, and then observe the other’s actions. Their utility is some function of the stage utilities: the utilities they get in the base game in each period. Formally, the game is given by $G=\left(G_0, T,\left{v_i\right}_{i \in N}\right)$, defined as follows.

• The action of player $i$ in period $t$ is denoted by $a_i^t$. The action profile in period $t$ is $a^t=\left(a_i^t\right)_{i \in N}$
• The history of the game at period $t$ is the tuple $\left(a^1, \ldots, a^t\right)$ of action profiles played until and including period $t$. The history of the entire game is $a=\left(a^1, a^2, \ldots\right)$.
• The stage utility at period $t$ is $u_i\left(a^t\right)$.
• A strategy $s_i$ for player $i$ is a map that assigns to each history $\left(a^1, \ldots, a^t\right)$, with $t<T$, the action $a_i^{t+1}$.
• For each player $i$, the utility or payoff in the game is $v_i=v_i\left(u_i\left(a^1\right), u_i\left(a^2\right), \ldots\right)$. As usual, given a strategy profile $s$, we denote by $v_i(s)$ the utility for $i$ when the players play $s$. We will also write $v_i(a)$ to denote the utility player $i$ gets when the history of the game is $a=\left(a^1, a^2, \ldots\right)$.

# 博弈论代写

## 经济代写博弈论代考Game theory代写|Imperfect information, many rounds

\left 缺少或无法识别的分隔符

## 经济代与写博弈论代考Game theory代写|Repeated games

• 玩家的动作 $i$ 在期间 $t$ 表示为 $a_i^t$. 期间行动概兄 $t$ 是 $a^t=\left(a_i^t\right)_{i \in N}$
• 游戏历史时期 $t$ 是元组 $\left(a^1, \ldots, a^t\right)$ 播放直到并包括期间的动作配置文件 $t$. 整个游戏的历史是 $a=\left(a^1, a^2, \ldots\right)$
• 时期的舞台效用 $t$ 是 $u_i\left(a^t\right)$.
• 一策略 $s_i$ 对于玩家 $i$ 是分配给每个历史记录的地图 $\left(a^1, \ldots, a^t\right)$ ，和 $t<T$ ，那个行动 $a_i^{t+1}$
• 对于每个玩家 $i$ ，游戏中的效用或收益是 $v_i=v_i\left(u_i\left(a^1\right), u_i\left(a^2\right), \ldots\right)$. 像往常一样，给定 一个策略配置文件 $s$, 我们用 $v_i(s)$ 实用程序 $i$ 球员上场的时候 $s$. 我们也会写 $v_i(a)$ 表示效用播放 器 $i$ 当游戏的历史是 $a=\left(a^1, a^2, \ldots\right)$.

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