Pure Strategy Game

纯粹的 strategy，例如石头剪刀布，要么出石头、要么出布、要么出剪刀，是离散的。

Game Theory博弈论

博弈论中有一些数学性质，要把握其中的逻辑，知道其是什么、能够解决什么类型的问题。

博弈论研究的是人以及人与人之间的关系。

什么是博弈？

任何需要顾及到个体利益的决策过程，都是博弈。

博弈的特点

分布式：一般没有中央控制单元，博弈者各自为政；
多成员：至少包含两名或以上博弈者；
相互联系：一名博弈者的决定，可能会影响其他博弈者。

基础是稳定（纳什均衡），目标是利益最大化。

完整博弈过程分析

博弈过程：例如大国博弈（贸易出口），或个人博弈（股票投资）
一种逻辑：每个博弈者（Game Player）都是理性的获取其最大利益。
分析：如何制定博弈规则，从而形成最佳的系统？每个博弈者，应当如何制定对其最优的策略，从而最大化其收益？

Prisoners’ Dilemma

Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. Both suspects are told the following policy:
1、If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail.
2、If both confess then both will be sentenced to jail for six months.
3、If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months.

The battle of the sexes（Non-zero-sum game）

At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening.
Both Chris and Pat know the following:
1、Both would like to spend the evening together.
2、But Chris prefers the opera.
3、Pat prefers the prize fight.

Matching pennies（Zero-sum game）

Each of the two players has a penny. Two players must simultaneously choose whether to show the Head or the Tail. Both players know the following rules:
1、If two pennies match (both heads or both tails) then player 2 wins player 1’s penny.
2、Otherwise, player 1 wins player 2’s penny.

（零和博弈不存在纳什均衡节点）

Static games of complete information含义

Static == simultaneous-move
Each player chooses his/her strategy without knowledge of others’ choices.

Complete information (on game’s structure)
Each player’s strategies and payoff function are common knowledge(I know he knows I know) among all the players.

Assumptions on the players
Rationality.
1、Players aim to maximize their payoffs;
2、Players are perfect calculators.
Each player knows that other players are rational.

The players cooperate?
No. Only non-cooperative games;
Methodological individualism.

The timing
Each player chooses his/her strategy without knowledge of others’ choices;
Then each player receives his/her payoff;
The game ends.

static game组成

1、A set of players (at least two players)；
2、For each player, a set of strategies/actions；
3、Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies.

normal-form representation

A finite set of players {1, 2, …, n};
players’ strategy spaces S1 S2 … Sn and
their payoff functions u1 u2 … un, where ui : S1 × S2 × …× Sn→R.

Bi-matrix representation

2 players: Player 1 and Player 2
Each player has a finite number of strategies, S1={s11, s12, s13}, S2={s21, s22}.

Prisoners’ Dilemma: normal-form representation

Set of players: {Prisoner 1, Prisoner 2}
Sets of strategies: S1 = S2 = {Mum, Confess}
Payoff functions: u1(M, M)=-1, u1(M, C)=-9, u1(C, M)=0, u1(C, C)=-6; u2(M, M)=-1, u2(M, C)=0, u2(C, M)=-9, u2(C, C)=-6

strictly dominated strategy

自己的多个策略中，无论博弈对手如何改变都有个别策略是永远比其他策略要好的，那么这几个策略被称为严格占优的策略。

si” is strictly better than si’
regardless of other players’ choices

Example
Two firms, Reynolds and Philip, share some market.
Each firm earns $60 million from its customers if neither do advertising.
Advertising costs a firm $20 million, advertising captures $30 million from competitor.

当 Reynolds 选择 No Ad 时，Philip 选择 Ad 可以带来更多的收益；而当 Reynolds 选择 Ad 时，Philip 选择 Ad 同样是明智的，所以无论 Reynolds 选择的策略是什么，Philip 的策略中 Ad 永远是最好的策略，所以 Ad 为严格占优策略 strictly dominated strategy。

Iterated elimination of strictly dominated strategies
1、If a strategy is strictly dominated, eliminate it.
2、The size and complexity of the game is reduced.
3、Eliminate any strictly dominated strategies from the reduced game.
4、Continue doing so successively.
5、Rationalizable equilibrium.
在 Bi-matrix game 中有一个策略被其他策略严格优于，就可以将此策略删除来简化策略，只剩下有优有劣的策略，才有比较的价值。

weakly dominated strategy

A rational player never chooses a strictly dominated strategy. Hence, any strictly dominated strategy can be eliminated.
A rational player may choose a weakly dominated strategy.
The order of elimination does not matter for strict dominance elimination (pin down the same equilibrium), but does for weak one.

best response function

给定其他人策略的条件下，能够给予 i 最大利益的策略，叫做 best response function（对其他人选定策略的最佳应答）。

Nash Equilibrium

互为 best response 的策略组，即为纳什策略。
起始于随机点，每次一个 player 改变策略，如果存在纳什均衡节点，会最终趋于该点达到平衡

A set of strategies, one for each player, such that each player’s strategy is best for her, given that all other players are playing their equilibrium strategies.

Using best response function to define Nash equilibrium

A set of strategies, one for each player, such that each player’s strategy is best for her, given that all other players are playing their strategies, or
A stable situation that no player would like to deviate if others stick to it.

Cournot model of duopoly

A product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. Each firm chooses the quantity without knowing the other firm has chosen.
The market priced is P(Q)=a-Q, where a is a constant number and Q=q1+q2.
The cost to firm i of producing quantity qi is Ci(qi)=cqi.

The normal-form representation:
Set of players: { Firm 1, Firm 2}
Sets of strategies: S1=[0, +∞), S2=[0, +∞)
Payoff functions: u1(q1, q2)=q1(a-(q1+q2)-c); u2(q1, q2)=q2(a-(q1+q2)-c)

How to find a Nash equilibrium
1、Find the quantity pair (q1*, q2*) such that q1* is firm 1’s best response to Firm 2’s quantity q2* and q2* is firm 2’s best response to Firm 1’s quantity q1*
2、That is, q1* solves
Max u1(q1, q2*)=q1(a-(q1+q2*)-c)
subject to 0<=q1<=+∞
and q2* solves
Max u2(q1*, q2)=q2(a-(q1*+q2)-c)
subject to 0<=q2<=+∞

3、Solve
Max u1(q1, q2*)=q1(a-(q1+q2*)-c)
subject to 0<=q1<=+∞
FOC( first order condition): a - 2q1 - q2*- c = 0
q1 = (a - q2*- c)/2
Max u2(q1*, q2)=q2(a-(q1*+q2)-c)
subject to 0<=q2<=+∞
FOC: a - 2q2 – q1* – c = 0
q2 = (a – q1* – c)/2

4、The quantity pair (q1*, q2*) is a Nash equilibrium if
q1* = (a – q2* – c)/2
q2* = (a – q1* – c)/2

5、Solving these two equations gives us
q1* = q2* = (a – c)/3

Best response function
Firm 1’s best function to firm 2’s quantity q2:
R1(q2) = (a – q2 – c)/2 if q2 < a– c; 0, othwise
Firm 2’s best function to firm 1’s quantity q1:
R2(q1) = (a – q1 – c)/2 if q1 < a– c; 0, othwise

The Tragedy of the Commons

n farmers in a village. Each summer, all the farmers graze their goats on the village green.
Let gi denote the number of goats owned by farmer i.
The cost of buying and caring for a goat is c, independent of how many goats a farmer owns.
The value of a goat is v(G) per goat, where G = g1 + g2 + … + gn
There is a maximum number of goats that can be grazed on the green. That is, v(G)>0 if G < Gmax, and v(G)=0 if G >= Gmax.
Assumptions on v(G): v’(G) < 0 and v”(G) < 0.
Each spring, all the farmers simultaneously choose how many goats to own.

Pareto Efficiency

An allocation in which no change that can be made to make some player better off without making other players worse off.
无法在不使别人更差的情况下，提升自己的资源分配方案，叫Pareto Optimal
Nash equilibrium represents the stable state (or practical state) of a game
Pareto optimal represents the efficient (or ideal which may not be achievable) state of a game

设计一个资源分配方案时，若既可以使其为纳什均衡节点，又可以使其为 Pareto Efficiency 节点，那么这样的资源分配方案就是最优的。