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Strategic setting

Strategic setting is a setting where the outcomes that affect you depend on actions, not just on your own actions ,but on actions of others

Strictly dominant strategy

My strategy Alpha strictly dominates my strategy Beta (Alpha is strictly dominant strategy, Beta is strictly dominated strategy) if my payoff from Alpha is strictly greater than that from Beta regardless of what others do.

Lesson 1 : Do not play a strictly dominated strategy.
Lesson 2 : Rational choice in Prisoner's Dilemma lead to inefficient outcomes.
Lesson 3 : Payoffs matter. What people care about matter.
Lesson 4 : Put yourself in others' shoes and try to figure out what they will do

2

The ingredients of a game

ingredient	notation
players	$i$
strategies	$s_i$ (a particular strategy of $i$ ) , $S_i$ (the set of possible strategies of $i$ ) , $s$ (a strategy profile where one strategy for each player in the game) , $s-i$ (a strategy choice for everybody except $i$ )
payoffs	$u_i(s)$ (the payoff of $i$ depend on the strategy profile)

In the course, these ingredients are assumed to be known which means everyone knows the possible strategies everyone else could choose and everyone knows everyone else's payoffs, etc.

Strictly dominated strategy

$i$ 's strategy $s_i'$ is strictly dominated by his strategy $s_i$ if $u_i(s_i,s-i)>u_i(s_i',s-i)$ for all $s-i$ ( if $s_i$ always yields a higher payoff for $i$ no matter what the other people do )

Weakly dominated strategy

$i$ 's strategy $s_i'$ is weakly dominated by his strategy $s_i$ if $u_i(s_i,s-i) \geq u_i(s_i',s-i)$ for all $s-i$ and $u_i(s_i,s-i)>u_i(s_i',s-i)$ for at least one $s-i$

3

Iterative deletion of dominated strategies

Putting self in someone else's shoes, and trying to figure out what they are going to do, then think about them putting themselves in your shoes figuring out what you are going to do and so on.

Best response

To think of a strategy that is the best you can do, given your belief about what the other people are doing.(在你对别人如何行动有一定信念时，你能做出的最佳策略)
信念->所有可能的概率最佳策略->某一信念下达到最大数学期望的策略

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Lesson 5 : Do not choose a strategy that is never a best response to any belief

Best response

$i$ 's strategy $\hat{s_i}$ is a best response to the strategies $s-i$ of the other players if $u_i(\hat{s_i},s-i) \geq u_i(s_i',s-i)$ for all $s_i' \in S_i$ ——决定论——
which also means the best response $\hat{s_i}$ solves $Max \text{ } u_i(s_i,s-i)$
估计对方选择哪一策略可能性更大，根据上式决定己方最佳对策 ——直觉可能性——
$i$ 's strategy $\hat{s_i}$ is a best response to the belief $p$ about the other players' choice if $Eu_i(\hat{s_i},p) \geq Eu_i(s_i',p)$ for all $s_i' \in S_i$ ——概率论——
$Eu_i(s_i,p)=u_i(s_i,(s-i)_1)p((s-i)_1)+...+u_i(s_i,(s-i)_n)p((s-i)_n)$

Nash Equilibrium

In a partnership game, 2 players share 50% of the profits each and they need to choose the effort level as his strategy.

players' continuum of strategies $s_i \in [0,4]$
total profit $4(s_1+s_2+bs_1s_2)$ synergy rate $b \in [0,\frac {1}{4}]$
payoffs $u_i(s_1,s_2)=\frac {1}{2} [4(s_1+s_2+bs_1s_2)]-s_i^2$

$u_1(s)=2(s_1+s_2+bs_1s_2)-s_1^2$
$Max\,u_1\mid _{s_1}$
$\frac {\delta u_1}{\delta s_1}=2(1+bs_2)-2s_1=0$
$\frac {\delta ^2 u_1}{\delta s_1\,^2}=-2<0\implies \text{一阶导函数斜率为负，原函数在一阶导函数零点处取得最大值}$
$2(1+bs_2)-2\hat{s_1}=0\implies BR_1(s_2)=\hat{s_1}=1+bs_2 \text{，同理得}BR_2(s_1)=\hat{s_2}=1+bs_1$
Iterative deletion of dominated strategies $\implies$ Nash Equilibrium

b=1/4

5

Nash Equilibrium

In a strategy profile $NE(s_1^*,s_2^*,...,s_n^*)$ , $\forall i$ , $s_i^*$ is a best response to $s^*-i$

Motivation 1 : No regrets. In NE, holding everyone else's actions fixed, no individual strategy can do strictly better by moving away.
Motivation 2 : A NE can be thought of as self-fulfilling belief.

Lesson 6 : NE can be a self-enforcing agreement. So in coordination problems, unlike prisoner's dilemma, just communication can help.(协同谬误中存在多个NE，可以利用沟通导向更有利的NE)
Lesson 7 : Coordination games are games where there is a "Scope for leadership".

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Games of strategic complements 策略互补博弈

In games of strategic complements, the more the other person does, the more I want to do.

Cournot Duopoly 古诺双寡头模型

players : 2 firms
strategies : the quantities of an identical product $q_i$
constant marginal cost : $c$
cost of prodution : $c \times q$
market price : $p=a-b(q_1+q_2)$

Textbook

Dutta's Strategy and Games
Joel Watson's Strategies

《博弈论》笔记

《博弈论》笔记

1

Strategic setting

Strictly dominant strategy

2

The ingredients of a game

Strictly dominated strategy

Weakly dominated strategy

3

Iterative deletion of dominated strategies

Best response

4

Best response

Nash Equilibrium

5

Nash Equilibrium

6

Games of strategic complements 策略互补博弈

Cournot Duopoly 古诺双寡头模型

Textbook