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ECOS3012 代写-ECOS 3012
时间：2022-09-05
HD EDUCATION
HD
EDUCATION
ECOS 3012
Week 4
TUTOR: Jerry
HD EDUCATION
Week4 Dynamic games with complete information I 完全信息的动态博弈
--subgame perfect Nash equilibrium (SPE)
（1）引入：Consider the following two-stage game（两个阶段，存在先后的动态博弈）:
Player 1 has $1000 and chooses between two options: keep the money or give all the money
to player 2. 参与者 1有 1000刀，他有两个选择：留着这笔钱，或者把这笔钱给参与者 2
After observing player 1’s move: player 2 chooses whether to explode a grenade that will kill
both players. 在观察到参与者 1的选择后，参与者 2选择是否爆炸手榴弹把两人都杀死
Assume that both players prefer having money to not having money if they are alive, and both prefer
to be alive. 假设（1）如果两位玩家都存活，他们都更偏向拥有这笔钱，而不是没有这笔钱；假设（2）
两位玩家都更偏向于存活，而不是死亡。
分析：
（Give, not explode）：a Nash equilibrium outcome: if player 2 chooses the strategy: “I’ll explode the
grenade unless you give me the money!” 如果参与者 2选择的战略：我将会爆炸手榴弹，除非你把
钱给我。
In this equilibrium, giving the money is player 1’s best response to player 2’s strategy, and when
player 1 indeed gives the money, not to explode is player 2’s best response.
参与者 1的最优反应是：给参与者 2钱；而当参与者 2确实给参与者 2钱的时候，参与者 2的最优反
应是不爆炸。
However, this Nash equilibrium is based on a non-credible threat: If player 1 does not give the
money in stage 1, player 2, in fact, does not want to explode the grenade and kill himself.
然而这个 NE是基于一个不可信的空洞威胁：如果参与者 1在阶段 1的时候没有给钱，实际上参与者
2在阶段 2也不会选择爆炸手榴弹：因为爆炸手榴弹也将把自己杀死。
Although (give, not explode) is a Nash equilibrium, it is not a very likely prediction.因此，尽管(give,
not explode)是一个 NE，但是这并不是一个很有可能发生的预测。
（2） （SPE） We can make a better prediction using backward induction（反向归纳法）:
– Start from player 2. His best response is: regardless of player 1’s strategy, player 2 should never
explode the grenade 从参与者 2开始考虑，对于参与者 2而言，无论参与者 1的战略是什么，参与
者 2都永远不会爆炸手榴弹
– Go back to player 1: knowing that player 2 never explodes the grenade, player 1 should not give
the money. 然后考虑参与者 2，知道参与者 2永远不会爆炸手榴弹，参与者一就永远不会给钱
– This leads to (not give, not explode)
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– (not give, not explode) is not only a Nash equilibrium, but it is also a subgame perfect Nash
equilibrium（子博弈精炼纳什均衡）: player 2’s strategy (never explode) is a best response to
every possible action that player 1 can take (each of player 1’s action leads to a subgame子博
弈).
– Game tree （博弈树）：
（ SPNE, SPNE outcome, SPNE payoff）
（3） 关于“完全信息动态博弈“的四项定义
Definition 1. A subgame in an extensive-form game ：(a) begins at a decision node n that is a
singleton information set (defined later) (b) includes all the decision and terminal nodes following n
in the game tree(c) does not cut any information set
拓展式博弈中的子博弈 （a）从单节信息集的决策节开始 （b）包括博弈树中 n之下的所有的决策节
和终点节（terminal node）（3）不切割任何信息集
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Definition 2. A Nash equilibrium is subgame-perfect if the players’ strategies constitute a Nash
equilibrium in every subgame. 如果参与者在每个子博弈中的策略都构成纳什均衡，那么这个纳什均
衡就是子博弈精炼纳什均衡。(SPNE vs NE)
* Transform an extensive-form representation into a normal-form representation
Definition 3. A strategy for a player is a complete plan of action. It specifies a feasible action for the
player in every contingency in which the player might be called on to act. 参与者的一个战略是关于
行动的一个完整的计划。这个计划明确了参与者所有可能会遇到的偶发事件下，参与者作出的可行行
动的选择。
* 区别 Nash equilibrium和 equilibrium outcome
Definition 4. An information set for a player is a collection of decision nodes satisfying: (i) the player
has the move at every node in the information set, and (ii) when the play of the game reaches a
node in the information set, the player with the move does not know which node in the information
set has been reached 一个参与者的一个信息集是决策节的集合，决策节满足以下条件：（1）在这
个信息集中的每个节点都轮到该参与者行动 （2）当博弈进行到该信息集的某个节点时，该行动的参
与者都不知道自己正处于信息集的哪个节点
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（4） Stackelberg Model of Duopoly 双寡头的斯塔克尔伯格模型 （Sequentially choose quantity
有先后顺序地选择数量）
Players: two firms, 1 and 2
Stage 1: Firm 1 chooses quantity q1 ≥ 0
Stage 2: After observing q1, Firm 2 chooses quantity q2 ≥ 0
Payoffs: firms’ profits are determined in the following way
– Market demand has a downward slope: P (q1, q2) = 100 − q1 − q2
– Total cost to product qi for each firm is Ci(qi) = 10qi for i = 1, 2
– Profit for each firm is Revenue - Cost, i.e.
π1(q1,q2)=q1(100−q1 −q2)−10q1
π2(q1,q2)=q2(100−q1 −q2)−10q2
Set up similar to Cournot, except that the firms move sequentially
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（5）Sequential Bargaining -- Three-period bargaining
• Day 1: $10 to split 10刀平分
– Player 1 proposes (player 1’s share, player 2’s share) = (s1, 1 − s1)
– Player 2 either accepts (game ends) or rejects (proceed to day 2)
• Day 2: $9 to split 9刀平分
– Player 2 proposes (player 1’s share, player 2’s share) = (s2, 1 − s2) – Player 1 either accepts
(game ends) or rejects (proceed to day 3)
• Day 3: $8 in total, each player gets $4. 4刀平分
Assume that a player accepts an offer when he is indifferent between acceptance and rejection.（假
设如果接受和拒绝带来的 payoff相同，参与者将选择接受）
Use backward induction to find the SPE outcome:
HD
EDUCATION
ECOS 3012
Week 4
TUTOR: Jerry
HD EDUCATION
Week4 Dynamic games with complete information I 完全信息的动态博弈
--subgame perfect Nash equilibrium (SPE)
（1）引入：Consider the following two-stage game（两个阶段，存在先后的动态博弈）:
Player 1 has $1000 and chooses between two options: keep the money or give all the money
to player 2. 参与者 1有 1000刀，他有两个选择：留着这笔钱，或者把这笔钱给参与者 2
After observing player 1’s move: player 2 chooses whether to explode a grenade that will kill
both players. 在观察到参与者 1的选择后，参与者 2选择是否爆炸手榴弹把两人都杀死
Assume that both players prefer having money to not having money if they are alive, and both prefer
to be alive. 假设（1）如果两位玩家都存活，他们都更偏向拥有这笔钱，而不是没有这笔钱；假设（2）
两位玩家都更偏向于存活，而不是死亡。
分析：
（Give, not explode）：a Nash equilibrium outcome: if player 2 chooses the strategy: “I’ll explode the
grenade unless you give me the money!” 如果参与者 2选择的战略：我将会爆炸手榴弹，除非你把
钱给我。
In this equilibrium, giving the money is player 1’s best response to player 2’s strategy, and when
player 1 indeed gives the money, not to explode is player 2’s best response.
参与者 1的最优反应是：给参与者 2钱；而当参与者 2确实给参与者 2钱的时候，参与者 2的最优反
应是不爆炸。
However, this Nash equilibrium is based on a non-credible threat: If player 1 does not give the
money in stage 1, player 2, in fact, does not want to explode the grenade and kill himself.
然而这个 NE是基于一个不可信的空洞威胁：如果参与者 1在阶段 1的时候没有给钱，实际上参与者
2在阶段 2也不会选择爆炸手榴弹：因为爆炸手榴弹也将把自己杀死。
Although (give, not explode) is a Nash equilibrium, it is not a very likely prediction.因此，尽管(give,
not explode)是一个 NE，但是这并不是一个很有可能发生的预测。
（2） （SPE） We can make a better prediction using backward induction（反向归纳法）:
– Start from player 2. His best response is: regardless of player 1’s strategy, player 2 should never
explode the grenade 从参与者 2开始考虑，对于参与者 2而言，无论参与者 1的战略是什么，参与
者 2都永远不会爆炸手榴弹
– Go back to player 1: knowing that player 2 never explodes the grenade, player 1 should not give
the money. 然后考虑参与者 2，知道参与者 2永远不会爆炸手榴弹，参与者一就永远不会给钱
– This leads to (not give, not explode)
HD EDUCATION
– (not give, not explode) is not only a Nash equilibrium, but it is also a subgame perfect Nash
equilibrium（子博弈精炼纳什均衡）: player 2’s strategy (never explode) is a best response to
every possible action that player 1 can take (each of player 1’s action leads to a subgame子博
弈).
– Game tree （博弈树）：
（ SPNE, SPNE outcome, SPNE payoff）
（3） 关于“完全信息动态博弈“的四项定义
Definition 1. A subgame in an extensive-form game ：(a) begins at a decision node n that is a
singleton information set (defined later) (b) includes all the decision and terminal nodes following n
in the game tree(c) does not cut any information set
拓展式博弈中的子博弈 （a）从单节信息集的决策节开始 （b）包括博弈树中 n之下的所有的决策节
和终点节（terminal node）（3）不切割任何信息集
HD EDUCATION
Definition 2. A Nash equilibrium is subgame-perfect if the players’ strategies constitute a Nash
equilibrium in every subgame. 如果参与者在每个子博弈中的策略都构成纳什均衡，那么这个纳什均
衡就是子博弈精炼纳什均衡。(SPNE vs NE)
* Transform an extensive-form representation into a normal-form representation
Definition 3. A strategy for a player is a complete plan of action. It specifies a feasible action for the
player in every contingency in which the player might be called on to act. 参与者的一个战略是关于
行动的一个完整的计划。这个计划明确了参与者所有可能会遇到的偶发事件下，参与者作出的可行行
动的选择。
* 区别 Nash equilibrium和 equilibrium outcome
Definition 4. An information set for a player is a collection of decision nodes satisfying: (i) the player
has the move at every node in the information set, and (ii) when the play of the game reaches a
node in the information set, the player with the move does not know which node in the information
set has been reached 一个参与者的一个信息集是决策节的集合，决策节满足以下条件：（1）在这
个信息集中的每个节点都轮到该参与者行动 （2）当博弈进行到该信息集的某个节点时，该行动的参
与者都不知道自己正处于信息集的哪个节点
HD EDUCATION
（4） Stackelberg Model of Duopoly 双寡头的斯塔克尔伯格模型 （Sequentially choose quantity
有先后顺序地选择数量）
Players: two firms, 1 and 2
Stage 1: Firm 1 chooses quantity q1 ≥ 0
Stage 2: After observing q1, Firm 2 chooses quantity q2 ≥ 0
Payoffs: firms’ profits are determined in the following way
– Market demand has a downward slope: P (q1, q2) = 100 − q1 − q2
– Total cost to product qi for each firm is Ci(qi) = 10qi for i = 1, 2
– Profit for each firm is Revenue - Cost, i.e.
π1(q1,q2)=q1(100−q1 −q2)−10q1
π2(q1,q2)=q2(100−q1 −q2)−10q2
Set up similar to Cournot, except that the firms move sequentially
HD EDUCATION
（5）Sequential Bargaining -- Three-period bargaining
• Day 1: $10 to split 10刀平分
– Player 1 proposes (player 1’s share, player 2’s share) = (s1, 1 − s1)
– Player 2 either accepts (game ends) or rejects (proceed to day 2)
• Day 2: $9 to split 9刀平分
– Player 2 proposes (player 1’s share, player 2’s share) = (s2, 1 − s2) – Player 1 either accepts
(game ends) or rejects (proceed to day 3)
• Day 3: $8 in total, each player gets $4. 4刀平分
Assume that a player accepts an offer when he is indifferent between acceptance and rejection.（假
设如果接受和拒绝带来的 payoff相同，参与者将选择接受）
Use backward induction to find the SPE outcome:
