本页为为准备孙广振老师而整理的较全的课程笔记，希望对所有同学有所帮助。由于是为了备考期末考试，因此针对性较强。

Section 1 Preliminary Game Theory

One-short games: finding Pure-Strategy Nash Equilibrium and Mixed-Strategy Nash Equilibrium

How to find Pure-Strategy Nash Equilibrium?

通常情况下，划线法就可以解决，只要两条线都划上，就一定是纯策略

How to find Mixed-Strategy Nash Equilibrium

One of the most important fact of Mixed-Strategy Nash Equilibrium is that it is about possibility

Sometimes a game could not have Pure-Strategy Nash Equilibrium, but it is always possible to get MSNE in a well-defined game (from lecture note by Professor Sun)

And during the class, we have a very interesting but complicated example see here: Complicated NE question

Also noted that the number of Mixed-Strategy Nash Equilibrium is always odd-number

Example

In this case:

	Heads	Tails
Heads	1,-1	-1,1
Tails	-1,1	1,-1

We found that there is no such Pure-Strategy Nash Equilibrium . So we assume the player randomly chooses heads or tails, so the probability would be 1/2 heads and 1/2 tails.

Simple Conclusion

The Mixed-Strategy Nash Equilibrium is always come with the probability.

Basic Model: Cournot Equilibrium, Bertrand Model and Stackelberg

Cournot Equilibrium

古诺模型建立的基础是， $Q = q_{A} + q_{B}$ 例如，假设 $Q = 120 - P$ 我们就可以得到它的反函数， $P = 120 - Q$ 。事实上，得到反函数这一步是非常重要的，因为我们通过这一步得到了一 Game Theory 模型， $P$ 是 Payoffs, $Q$ 是 Strategy。

继续尝试解方程： $P = 120 - q_{A} - q_{B}$ 所以对于 A 而言，总收益就是 $q_{A} (120 - q_{A} - q_{B})$ 由于两家企业都一样，所以会得到相同的收益函数，

他们的交点就是均衡点。

需要注意的是，Cournot Equilibrium 的结果介于 Perfect Competition 和 Monopoly 之间，而 Monopoly 等于他们都在尝试 Cartel

Bertrand Model

伯川德模型是一个价格竞争模型 ，它和 Cournot Equilibrium 主要的冲突点在于，它并不相信企业能够自由的选择产量。

伯川德假设生产不需要成本，只需要制定价格 $P_{A}$ 和 $P_{B}$

Elaborations on basic models (Capacity Choice Game, Product Differentiation)

Capacity Choice Game

Section 2-2 The Economics of Asymmetric Information

Hidden information: Akerlof’s The Market for Lemons

Section 3 Externality and Public Goods

Government Solution (Pigou Tax and Regulation)

Market Solution (The Coase Theorem)

Pareto Solution and the Samuelson Condition

The Lindahl Equilibrium

The Lindahl Equilibrium focuses on how to maximize the people’s welfare, like the Samuelson Condition, however, they have little differences.

We assume there are only 2 people in the society, here is Smith and Jones.

As an example for Smith, the demand for the Quantity of public good will decrease since the share of cost increase (No one want to pay too much)

The intersection point, which is $C$ , is the Lindahl Equilibrium for Smith and Jones

The Numerical Path

Also consider in a 2 player model, the utility function for player 1 is

u_{1} = x_{1} G^{α}

For player 2, it is

u_{2} = x_{2} G

Notice that $G = g_{1} + g_{2}$ and $x_{i}$ denotes to the players’ private consumption, so

x_{1} + g_{1} = x_{2} + g_{2} = 1

The Core Equation (important)

We want to maximize:

ma x u_{i} (M_{i} - τ_{i} pG, G)

The $M$ is the endowment, $M_{i} - τ_{i} pG$ is the $x$ (private goods) from my point of view.

Then we have to come up with the FOC, that is

τ_{i} p = \frac{u _{G}^{i}}{u _{X}^{i}}

And this is the most important part. Also remember $\sum_{i = 1}^{n} τ_{i} = 1$

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