最新2博弈论第二讲-Mixed-Strategies-复旦大学-王永钦课件ppt
合集下载
博弈论ppt(复旦大学中国经济研究中心)
Proposition A In the n -player normal form game
G {S1,..., Sn ; u1,..., un}
if iterated elimination of strictly dominated strategies
eliminates all but the strategies (s1*,..., sn* ) , then these
g* i 1
g* i 1
...
g
* n
)
Summing up all n farmers’ FOC and then dividing by n yields
v(G*) 1 G*v '(G*) c 0 (3) n
Cont’d
In contrast, the social optimum G** should resolve
Cont’d
A maximum number of goats : Gmax : v(G) 0 ,
for G Gmax but v(G) 0 for G Gmax
Also v '(G) 0, v ''(G) 0
The villagers’ problem is simultaneously choosing how many
i (3) The payoff to firm is given by the profit function
i (qi , q j ) qi[P(Q) c]
P(Q) a Q is the inverse demand function, Q q1 q2 , and
c is the constant marginal cost of production (fixed cost being zero).
G {S1,..., Sn ; u1,..., un}
if iterated elimination of strictly dominated strategies
eliminates all but the strategies (s1*,..., sn* ) , then these
g* i 1
g* i 1
...
g
* n
)
Summing up all n farmers’ FOC and then dividing by n yields
v(G*) 1 G*v '(G*) c 0 (3) n
Cont’d
In contrast, the social optimum G** should resolve
Cont’d
A maximum number of goats : Gmax : v(G) 0 ,
for G Gmax but v(G) 0 for G Gmax
Also v '(G) 0, v ''(G) 0
The villagers’ problem is simultaneously choosing how many
i (3) The payoff to firm is given by the profit function
i (qi , q j ) qi[P(Q) c]
P(Q) a Q is the inverse demand function, Q q1 q2 , and
c is the constant marginal cost of production (fixed cost being zero).
mixed strategy equilibrium混合策略均衡课件
TABLE 1. The percentage of times Rod successfully returns. A non-cooperative, zero-sum game. (Rod, Stefan). No Nash equilibrium in pure strategies.
Zou Yasheng Game Theory & Business Strategy 6
Stefan’s task
Stefan wants to keep the successful return percentage as low as possible; Rod has the exact opposite interest: as high as possible . If the two players decide on their strategies before the match, knowing the above probabilities, what should their strategies be? To help answer this question, we now plot: the percentage of times Rod returns serve against the proband.
Zou Yasheng Game Theory & Business Strategy
2
Unpredictability
A critical element of strategy whenever one side likes a coincidence of actions while the other wishes to avoid it. • The ATO wants to audit tax evaders; tax cheaters hope to avoid an audit. • The elder sister wants to rid herself of the younger brother, who wants to be included. • The invaders want choice of the place of attack to surprise , the defenders want to concentrate the forces on the place of attack. • The beautiful people want exclusivity, the hoipolloi want to be up with the latest trends. (As Yogi Berra said, “That night club is so crowded, no-one goes there anymore .”) • What is the best amount of a fine given a frequency of detection?
大学课程《博弈论及其应用》PPT课件：第二章(1234节)
博弈方2
左
中
右
上 博弈方1
下
1，0 0，4
1，3 0，2
0，1 2，0
图 2-7 划线法
博弈的相对优势策略位置在图2-7标出，策略组合{上，中}格 子中的两个数字下面都划了短线，这个格子对应的策略组合 就是由划线法得到的纳什均衡。
第四节 箭头方法
还有一种寻找纳什均衡的方法，和划线法的分析理念的出发 点不同，这种方法的思路是对博弈中的每个策略组合进行分 析，判断各博弈方是否能够通过单独改变自己的策略而改善 自己的得益，如果可以，则从所考察的策略组合的得益引一 个箭头到改变策略后的策略组合对应的得益。这样对每个可 能的策略组合都分析考察过以后，根据箭头反映的情况来判 断博弈的结果。
博弈方2
Hale Waihona Puke 左中上 博弈方1
下
1，0 0，4
1，3 0，2
右
0，1 2，0
图 2-8 箭头法
观察图2-8，在策略组合{上，中}中只有指向的箭头，没有指 出的格子所代表的就是纳什均衡。
略“上”改变的倾向，用一个竖着的箭头表示这个倾向；横 着比较后面的得益，4比2大，4比0大，博弈方2没有改变的 动力。在策略组合{上，左}中，横着比较后面，分析博弈方2 的得益，3比0大，1比0大，所以博弈方2有从策略“左”向
策略“中”和策略“右”改变的倾向，用两个横向的箭头表 示这两个改变的倾向。
在策略组合{上，中}中，竖着比较前面的得益，还是横着比较后 面的得益，博弈方1和博弈方2都没有改变的倾向。在策略组合 {上，右}中，竖着比较前面，2比0大，博弈方1有从策略“上”
向策略“下”改变的倾向，用一个竖向的箭头表示这个倾向； 横着比较后面，3比1大，博弈方2有从策略“右”向策略“中” 改变的倾向，用一个横向的箭头表示这个倾向。
博弈论-混合策略纳什均衡.ppt
游闲
政府
救济 不救济
政府和流浪汉的博弈
• 思考：政府会采用纯策略吗？流浪汉呢？这 个博弈有没有纯策略的纳什均衡？
• ——跟你玩剪子石头布游戏一样，你会一直 采用纯策略吗？
• 那么政府和流浪汉最有可能采用什么策略？ • ——使自己的预期支付最大化。 • ——若能够猜的对方的策略，就可以采用针
对性的策略，使自己的支付增加。
§剪刀、石头、布的游戏
• 因此，秘决在于—— • 自己的策略选择不能预先被对手方知道或猜
测到，在该博弈的多次重复中，博弈方一定 要避免自己的选择具有规律性； • 观察对手方策略选择是否具有规律或者偏好， 预先猜测对手策略，从而采用针对性策略赢 得这个博弈。
§ 第三章 混合策略纳什均衡
• 纯策略(pure strategies)：如果一个策略规 定参与人在一个给定的信息情况下只选择一 种特定的行动。
求解混合策略纳什均衡
1、假定政府采用混合策略：
G,1 即政府 的以 概率选1择 的 救概 济率 ，选择不
2、流浪汉的混合策略为：
L,1 即流浪 的汉 概以 率选择 1寻 的找 概工 率作 选， 择
解一:支付最大化
那么，政府的期望效用函数为：
v G G , L 3 1 1 1 0 1 5 1
• 政府选择救济策略
• 政府选择不救济策略
1
期望效用
0 期望效用
vG1, 3 11 vG0, 1 01
4 1
如果一个混合策略是流浪汉的最优选择，那一定意味 着政府在救济与不救济之间是无差异的，即：
v G 1 ,4 1 v G 0 , 0 .2
• 解二:支付等值法
如果一个混合策略是政府的最优选择，那一定意 味着流浪汉在寻找工作与游闲之间是无差异的， 即：
2博弈论第二讲-Mixed-Strategies-复旦大学-王永钦
➢ v1((0.4, 0.6), (0.3, 0.7))=0.40.4+0.6(-0.4)=-0.08
Player 2:
➢ EU2(H, (0.4, 0.6)) = 0.4×1+0.6×(-1) = -0.2 ➢ EU2(T, (0.4, 0.6)) = 0.4×(-1)+0.6×1 = 0.2 ➢ v2((0.4, 0.6), (0.3, 0.7))=0.3×(-0.2)+0.7×0.2=0.08
Player 1’s expected payoffs
➢ If Player 1 chooses Head, -q+(1-q)=1-2q ➢ If Player 1 chooses Tail, q-(1-q)=2q-1
Fall, 2007, Fudan
2021/2/4
5
Solving matching pennies
Fall, 2007, Fudan
2021/2/4
7
Solving matching pennies
Head Player 1
Tail
Player 2
Head
Tail
-1 , 1
1 , -1
1 , -1 -1 , 1
r 1-r
Expected payoffs
1-2q
2q-1
Expected payoffs
8
Solving matching pennies
Player 1’s best response B1(q):
➢ For q<0.5, Head (r=1)
➢ For q>0.5, Tail (r=0)
Player 1
➢ For q=0.5, indifferent (0r1)
Player 2:
➢ EU2(H, (0.4, 0.6)) = 0.4×1+0.6×(-1) = -0.2 ➢ EU2(T, (0.4, 0.6)) = 0.4×(-1)+0.6×1 = 0.2 ➢ v2((0.4, 0.6), (0.3, 0.7))=0.3×(-0.2)+0.7×0.2=0.08
Player 1’s expected payoffs
➢ If Player 1 chooses Head, -q+(1-q)=1-2q ➢ If Player 1 chooses Tail, q-(1-q)=2q-1
Fall, 2007, Fudan
2021/2/4
5
Solving matching pennies
Fall, 2007, Fudan
2021/2/4
7
Solving matching pennies
Head Player 1
Tail
Player 2
Head
Tail
-1 , 1
1 , -1
1 , -1 -1 , 1
r 1-r
Expected payoffs
1-2q
2q-1
Expected payoffs
8
Solving matching pennies
Player 1’s best response B1(q):
➢ For q<0.5, Head (r=1)
➢ For q>0.5, Tail (r=0)
Player 1
➢ For q=0.5, indifferent (0r1)
Mixed Strategies混合的策略 30页PPT文档
= (2p - 1) + q(2 – 4p) Maximize E2(Payoff) choosing q. If 2 - 4p < 0 [p > ½] q = 0 (Tail) is best
response If 2 - 4p > 0 [p < ½] q = 1 (Head) is best
U ia aiaiEiai,ai ai Ai
Where Ai is player i’s set of actions (pure strategies)
Mixed Strategies
MSNE Proposition: A mixed strategy profile a* in a strategic game in which each player has finitely many actions is a mixed strategy Nash equilibrium if and only if, for each player i,
Player 1
Mixed Strategies
Player 2
q
1-q
Head
Tail
Head
1, -1
-1, 1
Tail
-1, 1
1, -1
Where 0 q 1
Mixed Strategies
Head Player 1
Tail
Player 2
q
1-q
Head
Tail
1, -1
-1, 1
Mixed Strategies
Player 2
q
1-q
Head
Tail
p
Player 1
response If 2 - 4p > 0 [p < ½] q = 1 (Head) is best
U ia aiaiEiai,ai ai Ai
Where Ai is player i’s set of actions (pure strategies)
Mixed Strategies
MSNE Proposition: A mixed strategy profile a* in a strategic game in which each player has finitely many actions is a mixed strategy Nash equilibrium if and only if, for each player i,
Player 1
Mixed Strategies
Player 2
q
1-q
Head
Tail
Head
1, -1
-1, 1
Tail
-1, 1
1, -1
Where 0 q 1
Mixed Strategies
Head Player 1
Tail
Player 2
q
1-q
Head
Tail
1, -1
-1, 1
Mixed Strategies
Player 2
q
1-q
Head
Tail
p
Player 1
高级微观经济学博弈论讲义(复旦大学CCES Yongqin Wang)
13
Player 1 s12 s13
What is game theory?
We focus on games where: There are at least two rational players Each player has more than one choices The outcome depends on the strategies chosen by all players; there is strategic interaction Example: Six people go to a restaurant. Each person pays his/her own meal – a simple decision problem Before the meal, every person agrees to split the bill evenly among them – a 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:
Dec, 2006, Fudan University
Game Theory--Lecture 1
11
Definition: normal-form or strategicform representation
The normal-form (or strategic-form)
representation of a game G specifies:
Player 1 s12 s13
What is game theory?
We focus on games where: There are at least two rational players Each player has more than one choices The outcome depends on the strategies chosen by all players; there is strategic interaction Example: Six people go to a restaurant. Each person pays his/her own meal – a simple decision problem Before the meal, every person agrees to split the bill evenly among them – a 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:
Dec, 2006, Fudan University
Game Theory--Lecture 1
11
Definition: normal-form or strategicform representation
The normal-form (or strategic-form)
representation of a game G specifies:
Mixed Strategies混合的策略31页PPT
response If 4q – 2 = 0 [q = ½] any p in [0, 1] is a best
response
Mixed Strategies
E2(Payoff) = pq*(-1) + p(1 - q)*1 + (1 – p)q*1 + (1 – p)(1 – q) *(-1)
= (1 – 2q) + p(4q – 2) Maximize E1(Payoff) choosing p. If 4q – 2 < 0 [q < ½] p = 0 (Tail) is best
response If 4q – 2 > 0 [q > ½] p = 1 (Head) is best
• The expected payoff, given a*-i, to every action to which a*i assigns positive probability is the same,
• The expected payoff, given a*-i, to every action to which a*i assigns zero probability is at most the expected payoff to any action to which a*i assigns positive probability.
Mixed Strategies
A player’s expected payoff to the mixed strategy profile a is a weighted average of her expected payoffs to all mixed strategy profiles of the type (ai, a-i), where the weight attached to (ai, a-i) is the probability ai(ai) assigned to ai by player i’s mixed strategy ai
response
Mixed Strategies
E2(Payoff) = pq*(-1) + p(1 - q)*1 + (1 – p)q*1 + (1 – p)(1 – q) *(-1)
= (1 – 2q) + p(4q – 2) Maximize E1(Payoff) choosing p. If 4q – 2 < 0 [q < ½] p = 0 (Tail) is best
response If 4q – 2 > 0 [q > ½] p = 1 (Head) is best
• The expected payoff, given a*-i, to every action to which a*i assigns positive probability is the same,
• The expected payoff, given a*-i, to every action to which a*i assigns zero probability is at most the expected payoff to any action to which a*i assigns positive probability.
Mixed Strategies
A player’s expected payoff to the mixed strategy profile a is a weighted average of her expected payoffs to all mixed strategy profiles of the type (ai, a-i), where the weight attached to (ai, a-i) is the probability ai(ai) assigned to ai by player i’s mixed strategy ai
博弈论混合策略纳什均衡 PPT
• 答案:用反应曲线法找到政府与流浪汉博弈 得混合策略纳什均衡
练习:混合策略得纳什均衡
下面得博弈就是否存在纯策略得纳什均衡,如果没有采用混合策
略纳什均衡分析。试用支付最大化法与支付等值法两种方法算
一算混合策略得纳什均衡就是多少？通过反应曲线,求得混合策
略得纳什均衡、
博弈方2
C
D
博
A
弈
方
1
B
2， 3 3， 1
0.2
• 解二:支付等值法
如果一个混合策略就是政府得最优选择,那一定意 味着流浪汉在寻找工作与游闲之间就是无差异得, 即:
vL 1, 1 3 vL 0,
0.5
政府与流浪汉得博弈
• 如果政府救济得概率小于0、5; • 则流浪汉得最优选择就是寻找工作; • 如果政府救济得概率大于0、5; • 则流浪汉得最优选择就是游闲等待救济。 • 如果政府救济得概率正好等于0、5; • 流浪汉得选择无差异。
析核心概念得根本原因之一。
§扑克牌对色游戏
• 甲乙玩扑克牌对色游戏,每人都有红黑两张 扑克牌,约定如果出牌颜色一样,甲输乙赢,如 果出牌颜色不一样,则甲赢乙输。
• 找到这个博弈得纳什均衡。
红 甲
黑
红 -1， 1 1， -1
乙 黑
1， -1 -1， 1
§ 反应函数法
• 假设甲、乙均采用混与策略,随机地以p得概率出 红牌与以(1-p)得概率出黑牌,而乙则随机地以q得 概率出红牌与以(1-q)得概率出黑牌。
• 混合策略(mixed strategies):如果一个策略 规定参与人在给定得信息情况下,以某种概 率分布随机地选择不同得行动。
• 在静态博弈里,纯策略等价于特定得行动,混 合策略就是不同行动之间得随机选择。
博弈论(复旦大学 王永钦)
* 2
a q1 c 2
).
(provided that
q1 a c
Cont’d
Now, firm 1’s problem
q1 arg max 1 ( q1 , R 2 ( q1 )) q1 [ a q1 R 2 ( q1 ) c ] q
* 1
ac 2
(4)
G* G * *
Implications for social and economic systems (Coase Theorem)
2. Dynamic Games of Complete Information
2.1 Dynamic Games of Complete and Perfect Information 2.1.A Theory: Backward Induction Example: The Trust Game General features:
least one NE, possibly involving mixed strategies.
See Fudenberg and Tirole (1991) for a rigorous proof.
1.4 Applications 1.4.1 Cournot Model
Two firms A and B quantity compete.
Cont’d
His payoff is
g i v ( g 1 ... g i 1 g i g i 1 ... g n ) cg i
* * In NE ( g 1 , ..., g n ) , for each
(1)
* i , gi
must maximize (1), given
a q1 c 2
).
(provided that
q1 a c
Cont’d
Now, firm 1’s problem
q1 arg max 1 ( q1 , R 2 ( q1 )) q1 [ a q1 R 2 ( q1 ) c ] q
* 1
ac 2
(4)
G* G * *
Implications for social and economic systems (Coase Theorem)
2. Dynamic Games of Complete Information
2.1 Dynamic Games of Complete and Perfect Information 2.1.A Theory: Backward Induction Example: The Trust Game General features:
least one NE, possibly involving mixed strategies.
See Fudenberg and Tirole (1991) for a rigorous proof.
1.4 Applications 1.4.1 Cournot Model
Two firms A and B quantity compete.
Cont’d
His payoff is
g i v ( g 1 ... g i 1 g i g i 1 ... g n ) cg i
* * In NE ( g 1 , ..., g n ) , for each
(1)
* i , gi
must maximize (1), given
复旦博弈论课件 (2)
1, 1, 1
n What is the subgame perfect Nash equilibrium?
4
Find subgame perfect Nash equilibria:
backward induction
Player 1
L
R
Player 2 L’
3
L”
R”
R’
3
L”
R”
L’ 3
L”
n Two investors, 1 and 2, have each deposited D with a bank.
n The bank has invested these deposits in a long-term project. If the bank liquidates its investment before the project matures, a total of 2r can be recovered, where D > r > D/2.
n Imperfect information
Ø A player may not know exactly Who has made What choices when she has an opportunity to make a choice.
2
Subgame-perfect Nash equilibrium
Dynamic Games of Complete Information
Dynamic games of complete information
n Perfect information
Ø A player knows Who has made What choices when she has an opportunity to make a choice
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Tail is Player 1’s best response to Player 2’s strategy Head Head is Player 2’s best response to Player 1’s strategy Head
➢Hence, NO Nash equilibrium
➢ (3/4, 0, ¼) is a mixed strategy. That is, 1(T)=3/4, 1(M)=0 and 1(B)=1/4.
Player 2:
➢ (0, 1/3, 2/3) is a mixed strategy. That is, 2(L)=0, 2(C)=1/3 and 2(R)=2/3.
➢ Player 2 chooses Head and Tail with probabilities q and 1-q, respectively.
Mixed Strategy:
➢ Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities.
1-q 1-2r
r
1
1/2
➢ For r>0.5, Head (q=1) ➢ For r=0.5, indifferent (0q1)
Fall, 2007, Fudan
1/2
1q
Solving matching pennies
Player 1’s best response B1(q):
➢ For q<0.5, Head (r=1)
Fall, 2007, Fudan
Expected payoffs: 2 players each with two pure strategies
Player s11 ( r )
✓ Check
r = 0.5 B1(0.5) q = 0.5 B2(0.5)
Fall, 2007, Fudan
Player 2
Head
Tail
Head -1 , 1 1 , -1 r
Tail 1 , -1 -1 , 1 1-r
q
1-q
r
1
Mixed strategy Nash equilibrium
Head Player 1
Tail
Player 2
Head
Tail
-1 , 1
1 , -1
1 , -1 -1 , 1
r 1-r
Expected payoffs
1-2q
2q-1
Expected payoffs
q 2r-1
Player 2’s best response B2(r):
➢ For r<0.5, Tail (q=0ed strategy: example
Matching pennies
Player 1 has two pure strategies: H and T ( 1(H)=0.5, 1(T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively. ( 1(H)=0.3, 1(T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively.
Fall, 2007, Fudan
Solving matching pennies
Head Player 1
Tail
Expected payoffs
Player 2
Head
Tail
-1 , 1
1 , -1
1 , -1 -1 , 1
q 2r-1
1-q 1-2r
r 1-r
Expected payoffs
Fall, 2007, Fudan
Mixed strategy: example
Player 1
T (3/4) M (0) B (1/4)
L (0) 0, 2 4, 0 3, 4
Player 2 C (1/3) 3, 3 0, 4 5, 1
R (2/3) 1, 1 2, 3 0, 7
Player 1:
1-2q
2q-1
Player 2’s expected payoffs
➢ If Player 2 chooses Head, r-(1-r)=2r-1 ➢ If Player 2 chooses Tail, -r+(1-r)=1-2r
Fall, 2007, Fudan
Solving matching pennies
2博弈论第二讲-MixedStrategies-复旦大学-王永
钦
Matching pennies
Player 1
Head Tail
Player 2
Head
Tail
-1 , 1 1 , -1
1 , -1 -1 , 1
Head is Player 1’s best response to Player 2’s strategy Tail Tail is Player 2’s best response to Player 1’s strategy Tail
Fall, 2007, Fudan
Solving matching pennies
Player 2
Head Player 1
Tail
Head
-1 , 1 1 , -1
Tail
1 , -1 -1 , 1
r 1-r
q
1-q
Randomize your strategies
➢ Player 1 chooses Head and Tail with probabilities r and 1-r, respectively.
➢ For q>0.5, Tail (r=0)
Player 1
➢ For q=0.5, indifferent (0r1)
Player 2’s best response B2(r):
➢ For r<0.5, Tail (q=0)
➢ For r>0.5, Head (q=1)
➢ For r=0.5, indifferent (0q1)