研究生课程 博弈论 英文课件2
博弈论(modified)PPT课件
The Determination of Financial Structure
------The Incentive-Signalling Approach
小组成员
主讲 刘银 火建宏 李自光 王刚 徐雄博
From HNU
•本文研究的问题 •背景知识以及已存理论的介绍 •对局模型的建立与分析 讨论 how to find a balance？
Singal A？ Singal B？
From HNU
对于均衡的讨论
公司A
γ0a+ γ1a> γ0b+ γ1a
恒成立 .
即0< γ0(a-b) 管理者肯定的会选
择传递A类型的信号
γ0a+ γ1(b-L) < γ0b+ γ1
两边对F求导，可以得到：
由 t = a(F)可以得到：
From HNU
ROSS模型
求解微分方程 可以得到：
当t = c时，也就是当利润为市场上面的最小值的时候，则 说明该公司没有任何信号传递的优势，则有FC = 0
， 结论：1、质量越高的企业负债水平越高，投资者通过负债 水平来预计公司类型。
2、每个管理者都有自己最优的资本策略。
2
业与B企业在T0时的市场价值分别为V0A
与 V0B，在T1时的市场价值分别为
V1A =a和V1B =b，且a>b
3
假设风险中立，即预期收益率为固定值
From HNU
Case1.在A与B公司的类型确定
结论
公司的负债结构与其预期利润不具有相关性， 也就是MM模型的结论。
From HNU
Case2.混同战略模型
From HNU
lecture_2（博弈论讲义GameTheory（MIT））
Last Time:Defined knowledge, common knowledge, meet (of partitions), and reachability.Reminders:• E is common knowledge at ω if ()I K E ω∞∈.• “Reachability Lemma” :'()M ωω∈ if there is a chain of states 01,,...m 'ωωωωω== such that for each k ω there is a player i(k) s.t. ()()1()(i k k i k k h h )ωω+=:• Theorem: Event E is common knowledge at ωiff ()M E ω⊆.How does set of NE change with information structure?Suppose there is a finite number of payoff matrices 1,...,L u u for finite strategy sets 1,...,I S SState space Ω, common prior p, partitions , and a map i H λso that payoff functions in state ω are ()(.)u λω; the strategy spaces are maps from into . i H i SWhen the state space is finite, this is a finite game, and we know that NE is u.h.c. and generically l.h.c. in p. In particular, it will be l.h.c. at strict NE.The “coordinated attack” game8,810,11,100,0A B A B-- 0,010,11,108,8A B A B--a ub uΩ= 0,1,2,….In state 0: payoff functions are given by matrix ; bu In all other states payoff functions are given by . a upartitions of Ω1H : (0), (1,2), (3,4),… (2n-1,2n)... 2H (0,1),(2,3). ..(2n,2n+1)…Prior p : p(0)=2/3, p(k)= for k>0 and 1(1)/3k e e --(0,1)ε∈.Interpretation: coordinated attack/email:Player 1 observes Nature’s choice of payoff matrix, sends a message to player 2.Sending messages isn’t a strategic decision, it’s hard-coded.Suppose state is n=2k >0. Then 1 knows the payoffs, knows 2 knows them. Moreover 2 knows that 1knows that 2 knows, and so on up to strings of length k: . 1(0n I n K n -Î>)But there is no state at which n>0 is c.k. (to see this, use reachability…).When it is c.k. that payoff are given by , (A,A) is a NE. But.. auClaim: the only NE is “play B at every information set.”.Proof: player 1 plays B in state 0 (payoff matrix ) since it strictly dominates A. b uLet , and note that .(0|(0,1))q p =1/2q >Now consider player 2 at information set (0,1).Since player 1 plays B in state 0, and the lowest payoff 2 can get to B in state 1 is 0, player 2’s expected payoff to B at (0,1) is at least 8. qPlaying A gives at most 108(1)q q −+−, and since , playing B is better. 1/2q >Now look at player 1 at 1(1,2)h =. Let q'=p(1|1,2), and note that '1(1)q /2εεεε=>+−.Since 2 plays B in state 1, player 1's payoff to B is at least 8q';1’s payoff to A is at most -10q'+8(1-q) so 1 plays B Now iterate..Conclude that the unique NE is always B- there is no NE in which at some state the outcome is (A,A).But (A,A ) is a strict NE of the payoff matrix . a u And at large n, there is mutual knowledge of the payoffs to high order- 1 knows that 2 knows that …. n/2 times. So “mutual knowledge to large n” has different NE than c.k.Also, consider "expanded games" with state space . 0,1,....,...n Ω=∞For each small positive ε let the distribution p ε be as above: 1(0)2/3,()(1)/3n p p n ee e e -==- for 0 and n <<∞()0p ε∞=.Define distribution by *p *(0)2/3p =,. *()1/3p ∞=As 0ε→, probability mass moves to higher n, andthere is a sense in which is the limit of the *p p εas 0ε→.But if we do say that *p p ε→ we have a failure of lower hemi continuity at a strict NE.So maybe we don’t want to say *p p ε→, and we don’t want to use mutual knowledge to large n as a notion of almost common knowledge.So the questions:• When should we say that one information structure is close to another?• What should we mean by "almost common knowledge"?This last question is related because we would like to say that an information structure where a set of events E is common knowledge is close to another information structure where these events are almost common knowledge.Monderer-Samet: Player i r-believes E at ω if (|())i p E h r ω≥.()r i B E is the set of all ω where player i r- believesE; this is also denoted 1.()ri B ENow do an iterative definition in the style of c.k.: 11()()rr I i i B E B E =Ç (everyone r-believes E) 1(){|(()|())}n r n ri i I B E p B E h r w w -=³ ()()n r n rI i i B E B =ÇEE is common r belief at ω if ()rI B E w ¥ÎAs with c.k., common r-belief can be characterized in terms of public events:• An event is a common r-truism if everyone r -believes it when it occurs.• An event is common r -belief at ω if it is implied by a common r-truism at ω.Now we have one version of "almost ck" : An event is almost ck if it is common r-belief for r near 1.MS show that if two player’s posteriors are common r-belief, they differ by at most 2(1-r): so Aumann's result is robust to almost ck, and holds in the limit.MS also that a strict NE of a game with knownpayoffs is still a NE when payoffs are "almost ck” - a form of lower hemi continuity.More formally:As before consider a family of games with fixed finite action spaces i A for each player i. a set of payoff matrices ,:l I u A R ->a state space W , that is now either finite or countably infinite, a prior p, a map such that :1,,,L l W®payoffs at ω are . ()(,)()w u a u a l w =Payoffs are common r-belief at ω if the event {|()}w l w l = is common r belief at ω.For each λ let λσ be a NE for common- knowledgepayoffs u .lDefine s * by *(())s l w w s =.This assigns each w a NE for the corresponding payoffs.In the email game, one such *s is . **(0)(,),()(,)s B B s n A A n ==0∀>If payoffs are c.k. at each ω, then s* is a NE of overall game G. (discuss)Theorem: Monder-Samet 1989Suppose that for each l , l s is a strict equilibrium for payoffs u λ.Then for any there is 0e >1r < and 1q < such that for all [,1]r r Î and [,1]q q Î,if there is probability q that payoffs are common r- belief, then there is a NE s of G with *(|()())1p s s ωωω=>ε−.Note that the conclusion of the theorem is false in the email game:there is no NE with an appreciable probability of playing A, even though (A,A) is a strict NE of the payoffs in every state but state 0.This is an indirect way of showing that the payoffs are never ACK in the email game.Now many payoff matrices don’t have strictequilibria, and this theorem doesn’t tell us anything about them.But can extend it to show that if for each state ω, *(s )ω is a Nash (but not necessarily strict Nash) equilibrium, then for any there is 0e >1r < and 1q < such that for all [,1]r r Î and [,1]q q Î, if payoffs are common r-belief with probability q, there is an “interim ε equilibria” of G where s * is played with probability 1ε−.Interim ε-equilibria:At each information set, the actions played are within epsilon of maxing expected payoff(((),())|())((',())|())i i i i i i i i E u s s h w E u s s h w w w w e-->=-Note that this implies the earlier result when *s specifies strict equilibria.Outline of proof:At states where some payoff function is common r-belief, specify that players follow s *. The key is that at these states, each player i r-believes that all other players r-believe the payoffs are common r-belief, so each expects the others to play according to s *.*ΩRegardless of play in the other states, playing this way is a best response, where k is a constant that depends on the set of possible payoff functions.4(1)k −rTo define play at states in */ΩΩconsider an artificial game where players are constrained to play s * in - and pick a NE of this game.*ΩThe overall strategy profile is an interim ε-equilibrium that plays like *s with probability q.To see the role of the infinite state space, consider the"truncated email game"player 2 does not respond after receiving n messages, so there are only 2n states.When 2n occurs: 2 knows it occurs.That is, . {}2(0,1),...(22,21,)(2)H n n =−−n n {}1(0),(1,2),...(21,2)H n =−.()2|(21,2)1p n n n ε−=−, so 2n is a "1-ε truism," and thus it is common 1-ε belief when it occurs.So there is an exact equilibrium where players playA in state 2n.More generally: on a finite state space, if the probability of an event is close to 1, then there is high probability that it is common r belief for r near 1.Not true on infinite state spaces…Lipman, “Finite order implications of the common prior assumption.”His point: there basically aren’t any!All of the "bite" of the CPA is in the tails.Set up: parameter Q that people "care about" States s S ∈,:f S →Θ specifies what the payoffs are at state s. Partitions of S, priors .i H i pPlayer i’s first order beliefs at s: the conditional distribution on Q given s.For B ⊆Θ,1()()i s B d =('|(')|())i i p s f s B h s ÎPlayer i’s second order beliefs: beliefs about Q and other players’ first order beliefs.()21()(){'|(('),('))}|()i i j i s B p s f s s B h d d =Îs and so on.The main point can be seen in his exampleTwo possible values of an unknown parameter r .1q q = o 2qStart with a model w/o common prior, relate it to a model with common prior.Starting model has only two states 12{,}S s s =. Each player has the trivial partition- ie no info beyond the prior.1122()()2/3p s p s ==.example: Player 1 owns an asset whose value is 1 at 1θ and 2 at 2θ; ()i i f s θ=.At each state, 1's expected value of the asset 4/3, 2's is 5/3, so it’s common knowledge that there are gains from trade.Lipman shows we can match the players’ beliefs, beliefs about beliefs, etc. to arbitrarily high order in a common prior model.Fix an integer N. construct the Nth model as followsState space'S ={1,...2}N S ´Common prior is that all states equally likely.The value of θ at (s,k) is determined by the s- component.Now we specify the partitions of each player in such a way that the beliefs, beliefs about beliefs, look like the simple model w/o common prior.1's partition: events112{(,1),(,2),(,1)}...s s s 112{(,21),(,2),(,)}s k s k s k -for k up to ; the “left-over” 12N -2s states go into 122{(,21),...(,2)}N N s s -+.At every event but the last one, 1 thinks the probability of is 2/3.1qThe partition for player 2 is similar but reversed: 221{(,21),(,2),(,)}s k s k s k - for k up to . 12N -And at all info sets but one, player 2 thinks the prob. of is 1/3.1qNow we look at beliefs at the state 1(,1)s .We matched the first-order beliefs (beliefs about θ) by construction)Now look at player 1's second-order beliefs.1 thinks there are 3 possible states 1(,1)s , 1(,2)s , 2(,1)s .At 1(,1)s , player 2 knows {1(,1)s ,2(,1)s ,(,}. 22)s At 1(,2)s , 2 knows . 122{(,2),(,3),(,4)}s s s At 2(,1)s , 2 knows {1(,2)s , 2(,1)s ,(,}. 22)sThe support of 1's second-order beliefs at 1(,1)s is the set of 2's beliefs at these info sets.And at each of them 2's beliefs are (1/3 1θ, 2/3 2θ). Same argument works up to N:The point is that the N-state models are "like" the original one in that beliefs at some states are the same as beliefs in the original model to high but finite order.(Beliefs at other states are very different- namely atθ or 2 is sure the states where 1 is sure that state is2θ.)it’s1Conclusion: if we assume that beliefs at a given state are generated by updating from a common prior, this doesn’t pin down their finite order behavior. So the main force of the CPA is on the entire infinite hierarchy of beliefs.Lipman goes on from this to make a point that is correct but potentially misleading: he says that "almost all" priors are close to a common. I think its misleading because here he uses the product topology on the set of hierarchies of beliefs- a.k.a topology of pointwise convergence.And two types that are close in this product topology can have very different behavior in a NE- so in a sense NE is not continuous in this topology.The email game is a counterexample. “Product Belief Convergence”:A sequence of types converges to if thesequence converges pointwise. That is, if for each k,, in t *i t ,,i i k n k *δδ→.Now consider the expanded version of the email game, where we added the state ∞.Let be the hierarchy of beliefs of player 1 when he has sent n messages, and let be the hierarchy atthe point ∞, where it is common knowledge that the payoff matrix is .in t ,*i t a uClaim: the sequence converges pointwise to . in t ,*i t Proof: At , i’s zero-order beliefs assignprobability 1 to , his first-order beliefs assignprobability 1 to ( and j knows it is ) and so onup to level n-1. Hence as n goes to infinity, thehierarchy of beliefs converges pointwise to common knowledge of .in t a u a u a u a uIn other words, if the number of levels of mutual knowledge go to infinity, then beliefs converge to common knowledge in the product topology. But we know that mutual knowledge to high order is not the same as almost common knowledge, and types that are close in the product topology can play very differently in Nash equilibrium.Put differently, the product topology on countably infinite sequences is insensitive to the tail of the sequence, but we know that the tail of the belief hierarchy can matter.Next : B-D JET 93 "Hierarchies of belief and Common Knowledge”.Here the hierarchies of belief are motivated by Harsanyi's idea of modelling incomplete information as imperfect information.Harsanyi introduced the idea of a player's "type" which summarizes the player's beliefs, beliefs about beliefs etc- that is, the infinite belief hierarchy we were working with in Lipman's paper.In Lipman we were taking the state space Ω as given.Harsanyi argued that given any element of the hierarchy of beliefs could be summarized by a single datum called the "type" of the player, so that there was no loss of generality in working with types instead of working explicitly with the hierarchies.I think that the first proof is due to Mertens and Zamir. B-D prove essentially the same result, but they do it in a much clearer and shorter paper.The paper is much more accessible than MZ but it is still a bit technical; also, it involves some hard but important concepts. (Add hindsight disclaimer…)Review of math definitions:A sequence of probability distributions converges weakly to p ifn p n fdp fdp ®òò for every bounded continuous function f. This defines the topology of weak convergence.In the case of distributions on a finite space, this is the same as the usual idea of convergence in norm.A metric space X is complete if every Cauchy sequence in X converges to a point of X.A space X is separable if it has a countable dense subset.A homeomorphism is a map f between two spaces that is 1-1, and onto ( an isomorphism ) and such that f and f-inverse are continuous.The Borel sigma algebra on a topological space S is the sigma-algebra generated by the open sets. (note that this depends on the topology.)Now for Brandenburger-DekelTwo individuals (extension to more is easy)Common underlying space of uncertainty S ( this is called in Lipman)ΘAssume S is a complete separable metric space. (“Polish”)For any metric space, let ()Z D be all probability measures on Borel field of Z, endowed with the topology of weak convergence. ( the “weak topology.”)000111()()()n n n X S X X X X X X --=D =´D =´DSo n X is the space of n-th order beliefs; a point in n X specifies (n-1)st order beliefs and beliefs about the opponent’s (n-1)st order beliefs.A type for player i is a== 0012(,,,...)()n i i i i n t X d d d =¥=δD0T .Now there is the possibility of further iteration: what about i's belief about j's type? Do we need to add more levels of i's beliefs about j, or is i's belief about j's type already pinned down by i's type ?Harsanyi’s insight is that we don't need to iterate further; this is what B-D prove formally.Coherency: a type is coherent if for every n>=2, 21marg n X n n d d --=.So the n and (n-1)st order beliefs agree on the lower orders. We impose this because it’s not clear how to interpret incoherent hierarchies..Let 1T be the set of all coherent typesProposition (Brandenburger-Dekel) : There is a homeomorphism between 1T and . 0()S T D ´.The basis of the proposition is the following Lemma: Suppose n Z are a collection of Polish spaces and let021201...1{(,,...):(...)1, and marg .n n n Z Z n n D Z Z n d d d d d --´´-=ÎD ´"³=Then there is a homeomorphism0:(nn )f D Z ¥=®D ´This is basically the same as Kolmogorov'sextension theorem- the theorem that says that there is a unique product measure on a countable product space that corresponds to specified marginaldistributions and the assumption that each component is independent.To apply the lemma, let 00Z X =, and 1()n n Z X -=D .Then 0...n n Z Z X ´´= and 00n Z S T ¥´=´.If S is complete separable metric than so is .()S DD is the set of coherent types; we have shown it is homeomorphic to the set of beliefs over state and opponent’s type.In words: coherency implies that i's type determines i's belief over j's type.But what about i's belief about j's belief about i's type? This needn’t be determined by i’s type if i thinks that j might not be coherent. So B-D impose “common knowledge of coherency.”Define T T ´ to be the subset of 11T T ´ where coherency is common knowledge.Proposition (Brandenburger-Dekel) : There is a homeomorphism between T and . ()S T D ´Loosely speaking, this says (a) the “universal type space is big enough” and (b) common knowledge of coherency implies that the information structure is common knowledge in an informal sense: each of i’s types can calculate j’s beliefs about i’s first-order beliefs, j’s beliefs about i’s beliefs about j’s beliefs, etc.Caveats:1) In the continuity part of the homeomorphism the argument uses the product topology on types. The drawbacks of the product topology make the homeomorphism part less important, but theisomorphism part of the theorem is independent of the topology on T.2) The space that is identified as“universal” depends on the sigma-algebra used on . Does this matter?(S T D ´)S T ×Loose ideas and conjectures…• There can’t be an isomorphism between a setX and the power set 2X , so something aboutmeasures as opposed to possibilities is being used.• The “right topology” on types looks more like the topology of uniform convergence than the product topology. (this claim isn’t meant to be obvious. the “right topology” hasn’t yet been found, and there may not be one. But Morris’ “Typical Types” suggests that something like this might be true.)•The topology of uniform convergence generates the same Borel sigma-algebra as the product topology, so maybe B-D worked with the right set of types after all.。
博弈论完整课件[浙江大学]GAME_Cha(3)
![博弈论完整课件[浙江大学]GAME_Cha(3)](https://img.taocdn.com/s3/m/2e96a44002d276a201292e62.png)
In this chapter we introduce dynamic games. We again restrict attention to games with complete information(i.e.,games in which the players’ payoff functions are common knowledge).We analyze dynamic games that have not only complete but also perfect information, by which we mean that at each move in the game the p可l编a辑yppet r with the move 1
now analyze dynamic games by representing
such games in extensive form. This expositional
approach may make it seem that static games
must be represented in normal form and
采取蕴涵可信威胁的策略。
可编辑ppt
6
设想有一家寡占企业（在位者）在市场上享有
丰厚的利润，另一家企业（进入者）企图进入
分享；为了进入该行业，进入者必须付出4000
万元的（沉没）成本建一个工厂。在位者当然
希望进入者别进入。如果进入者不进入，在位
者能继续定高价，享受垄断利润10000万元。
如果进入者进入，在位者可以“容忍”，维持高
博弈论讲义入门 slides2
学生的成绩就是 其中， 是学生报出的数字。
VNM公理
• 公理A2（独立性）：对任意p,q,r∈P，和 任意a∈(0,1]，
VNM公理
• 公里A3（连续性）：对任意p,q,r∈P，如 果 p > q > r，那么存在a,b∈(0,1) 使
定理—VNM表示
• P上的关系≥可用VNM效用函数u：Z→R 表示的充分必要条件是它满足公理A1-A3。 • u和v表示≥的充分必要条件是 其中a＞0，b∈R
习题
• 考虑一正实数上可用VNM效用函数 u(x)=x2表示的关系≥。 这种关系可用VNM函数 表 示吗？ 用 表示又如何呢？
对风险的态度
• 公平赌博：
• 参与者是风险中性的 iff 他对所有公平赌博无 所谓。 • 他是(严格)厌恶风险的 iff 他从不想参与公平赌 博。 • 他是(严格)追逐风险的 iff 他总是想参与公平赌 博。
基本概念：偏好 • 关系≥ (X上)是X×X上的任一子集 • 例如， • T≥C≡(T,C)∈≥ • ≥是完全的iff 要么X≥Y，要么Y≥X • ≥是传递的iff ［X≥Y和Y≥Z］
偏好关系
• 定义：一种关系是偏好关系的充分必要 条件是，它是完全和传递的。
实例
• 在本班学生中定义下列关系 • xTy表示x至少和y一样高； • xMy表示x在课程14.04的期末成绩至少和 y一样高； • xHy表示x和y 上的同一所中学； • xYy表示x比y年龄小； • xSy表示x与y 同龄；
实例
可由函数u**表示，其中
练习
• 设想一群学生围绕一张圆桌坐下。定义 一种关系R，写成xRy，表示x坐在y的右 边。你能用一效用函数表示R吗？ • 考虑一在正实数上以u表示的关系≥，其 中u(x)=x2。 • 这种关系可表示为 吗？ • u**(x)=1/x 又 如何呢？
博弈论最全完整ppt-讲解
导论
二、博弈论与诺贝尔经济学奖获得者
1994年诺贝尔经济学奖获得者
美国人约翰-海萨尼(John C. Harsanyi) 和美国人 约翰-纳什(John F. Nash Jr.)以及德国人莱因 哈德-泽尔腾(Reinhard Selten)
获奖理由：在非合作博弈的均衡分析理论方面做 出了开创性的贡献，对博弈论和经济学产生了重 大影响 。
如果一个博弈在所有各种对局下全体参与人之得 益总和总是保持为一个常数，这个博弈就叫常和 博弈；
相反，如果一个博弈在所有各种对局下全体参与 人之得益总和不总是保持为一个常数，这个博弈 就叫非常和博弈。
常和博弈也是利益对抗程度最高的博弈。 非常和（变和）博弈蕴含双赢或多赢。
导论
四、主要参考文献
课程主要内容
第一章 完全信息静态博弈 第二章 完全信息动态博弈 第三章 不完全信息静态博弈 第四章 不完全信息动态博弈 第五章 委托-代理理论 第六章 逆向选择与信号传递
第一章 完全信息静态博弈
博弈论的基本概念及战略式表述 纳什均衡
纳什均衡应用举例 混合战略纳什均衡 纳什均衡的存在性与多重性
第一节 博弈论的基本概念
与战略式表述
博弈论的基本概念与战略式表述
博弈论（game theory）是研究决策主体的行为发生直 接相互作用时候的决策以及这种决策的均衡问题。
博弈的战略式表述：G={N,(Si)iN,(Ui)iN} 有三个基本要素： （1）参与人（players）iN={1,2,…,n} ； （2）战略（strategies）,siSi(战略空间)； （3）支付（payoffs）,ui=ui(s-i,si)。
Because We Had a Flat Tire”
gametheory2博弈论英文精品PPT课件
Review
Elimination
Nash Equilibrium
Summary
Games
Course topics:
• Games of complete and perfect information • Static Games (Nash Equilibrium) • Dynamic Games (Backward Induction)
6 / 43
Review
Elimination
Nash Equilibrium
Summary
Games
• Consider the following game:
• Two players • Each player chooses between two actions: A and B • Payoff for all outcomes is in the table below:
1
2
A
B
your ID keyword
A 50,50 0,200
B 200,0 80,80
1
2
A
B
A 50,50 0,200
B 200,0 80,80
7 / 43
Review
Elimination
Nash Equilibrium
Summary
Games
• Game Participation:
• you can win up to 200 CZK • send SMS with your action • phone numbers + IDs will be strictly protected • participant will be matched randomly in pairs, one pair will be
section2(博弈论讲义(Harvard University))
Section 2: Externalities
Alexis Diamond adiamond@
Agenda
• Key terms and definitions • Complementarity and cross-partial derivatives)
• πi = 2(i + j + cij) - i2 • πj = 2(i + j + cij) - j2
• Best response functions:
– d (πi)/di = 2+2cj - 2i – • i*=1+cj … This is our BR function for agent i
Nash Eqm (TR & DC)
strategy profiles
Pareto Optima
Set of Rationalizable Strategies: {T,D} x {C,R} Set of Weakly Congruent Strategies: {T,D} x {C,R} Set of Best Response Complete: {T,D} x {C, R} or {T,D} x {L,C,R} Set of Congruent Strategies: {T,D} x {C, R}
Cournot Oligopoly: Explanation
Positive Cross-Partial Derivatives = Complementarity
π1 = (1000 - q1 - q2)q1 - 100q1 d (π1)/dq1 = 1000 - 2q1 - q2 - 100 d ((π1)/dq1)/dq2 = -1 < 0 • Why? What does this mean?
Chap.2 Normal –Form 博弈论英文版教学课件
Given that each firm knows the market demand, Investment of New Product is a static game of complete information in which tow firms act simultaneously, or a dynamic game of complete information in which two firm act successively.
Information
In game-theoretic analysis, Information is just all the knowledge that one player owns, such as the other Players actions/strategy, the other players payoff, and so on.
In Investment of New Product , the firms’ actions are “Invest” and “No Invest”, e.g. ai=a (Invest) or ai=b ( No Invest) .
Player i’s action set, Ai={ai} , is the entire set of actions available to him.
We will occasionally write si S i to indicate that the strategy si is a member of the set of strategies Si.
A strategy combination is an ordered set s=(s1 ,… sn) consisting of one strategy for each of the n players in the game.
lecture23(博弈论讲义(Carnegie Mellon University))
June 20, 2003
73-347 Game Theory--Lecture 23
6
Cournot duopoly model of incomplete information (version one) cont'd
Firm 2's marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be
* * * Firm 1 chooses q1 which is its best response to firm 2's ( q2 (cH ) , q2 (cL ) ) (and the probability). * If firm 2's marginal cost is HIGH then firm 2 chooses q2 (cH ) which is its * best response to firm 1's q1 . * If firm 2's marginal cost is LOW then firm 2 chooses q2 (cL ) which is its * best response to firm 1's q1 .
June 20, 2003
73-347 Game Theory--Lecture 23
3
Static (or simultaneous-move) games of complete information
A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies All these are common knowledge among all the players.
gametheory6博弈论英文精品PPT课件
Review Dynamic Games Centipede Game Ultimatum Game Summary
• action is a decision in one particular node (confess, remain silent, head, tail,…) • strategy is a plan of actions for every possible situation that might occur, for every possible node (AF-Accept if Albert goes In, Fight if Albert plays Out) • strategy – it is deciding about the action in each decision node prior to the game • it is like as if you want your friend to play the game instead of you, you have to tell him in advance what to do in each situation
OUT
0 2
IN
FA
-3
2
-1
1
5 / 27
Review Dynamic Games Centipede Game Ultimatum Game Summary
Dynamic Game (tree):
OUT
IN
0 2
Static game (table):
IN OUT
FA
-3
2
-1
1
F
NEA:
-3,-1
博弈论9-2. Cheap Talk 与直接显示机制课件
PPT学习交流
4
Cheap Talk :why, how
• ISSUES
• 分析特定环境中 空谈博弈的效果； • 如何设计一个环境，以便发挥空谈博弈的优势；
• 在某些机制下，自利的立法者之间的争论，可以提高法案的社 会价值；
• ？网络 辩论 • ……
• 工会可以提高社会福利，因为工会可以提高工人与管理层之间 的沟通
• Sometimes there is no incentive to lie, and cheap talk will fully convey private information
PPT学习交流
7
Cheap Talk Game: Payment information: message vs. signal
• The key feature of such a cheap-talk game is that
the message has no direct effect on either the Sender's or the Receiver's payoff.
• The only way the message can matter is through its information content: by changing the Receiver's belief about the Sender‘s type, a message can change the Receiver's action, and thus indirectly affect both players' payoffs.
PPT学习交流
5
空谈如何起作用？
Chap.5 Extensive Form of Games 博弈论英文版教学课件
1
L
x1
R
2
2
L x2 R
L x3 R
1
1
1
L x4 R
L x5 R L x6 R
L x7 R
5.2 The Strategies of Extensive-Form Games
Definition 5.3 A strategy for a player is a complete plan of action—it specifies a feasible action for the player in very contingency in which the player might be called on to act.
In extensive-form games, a contingency in which the player should act is just a information set. So a strategy of extensive-form games specifies a feasible action for the player in very information set.
Consider the dynamic Investment in New Product game of complete information, in which firm 1 chooses first and firm 2 chooses after he/she observes the firm’s choice.
1
Mum
x1
2
x2
Fink 2
x3
Mum
Fink
Mum
Fink
博弈论课件 2
Game Theory1Spring201511Lecture 2Dominance and Best Response22Readings•Watson: Strategy_ An introduction to game theory–Ch 6,1rd ed p.43-55;–Ch6, 3rd ed p.49-66.32Outline•Dominance.•Pareto Efficiency.•Best Response.•Undominated Strategy Sets.425262Consider a mixed strategy of player 1: σ1=(½, ½ , 0).72•A pure strategy s i of player i is dominated if there is a strategy (pure or mixed)σi ∈∆S i such that u i (σi , s -i )>u i (s i , s -i ),for all strategy profiles s -i ∈∆S -i of the other players.•An important component of the definition of dominance is the “strict” inequality.–That is, in mathematical terms, we have the expression u i (σi , s −i ) > u i (s i , s −i ) rather than u i (σi , s −i ) ≥u i (s i , s −i ).•Note: A rational player will never play a dominated strategy.8Game 1The “Grade Game”Without showing your neighbor what you are doing, write down on a form either the letter αor the letter β. Think of this as a ‘grade bid’. We will randomly pair your form with one other form. Neither you nor your pair will ever know with whom you were paired.10Game 1The “Grade Game”☐Here is how grades may be assigned for this course.☐If you put αand your pair putsβ, then you will getgrade A, and your pair grade C.☐if both you and your pair putα, then you both will get grade B-.☐if you put βand your pair putsα, then you will getgrade C, and your pair grade A.☐if both you and your pair putβ, then you will both get grade B+.11Definition☐Definition 1. My strategy αstrictly dominates my strategy βif my payoff from αis strictly higher than that from βregardless of others' choices.☐Definition 2. My strategy αweakly dominates my strategy βif my payoff from αis as high as that from βregardless of others' choices, and is strictly higher for at least one such choice.14Lesson 1. You should never play a strictly dominated strategy.15Lesson 2. Rational play by rational players can lead to bad outcomes.16Prisoners’ Dilemma217Prisoners’ Dilemma•The prisoners’ dilemma illustrates one of the major tensions in strategic settings:–the clash between individual and group interests.218Prisoners’ Dilemmas☐Examples☐Joint project: incentive to shirk☐Price competition: incentive to undercut price☐Common resource: incentive to “overfish” orpollute19Prisoners’ Dilemmas☐Remedies☐Contracts☐Treaties change payoffs☐Regulations☐Repeated play☐Education→ change payoffs22Pareto Efficiency•s is more efficient than s’ if all of the players prefer the outcome of s to the outcome of s’ and if the preference is strict for at least one player.–In mathematical terms, s is more efficient than s’ if u i(s) ≥u i(s’) for each player i and if the inequality is strict for atleast one player.• A strategy profile s is called Pareto efficient if there is no other strategy profile that is more efficient; that is, there is no other strategy profile s’such that u i(s’) ≥u i(s) for every player i and u j(s’) > u j(s) for some player j.•For prisoner's dilemma,CC is more efficient than DD. CC,CD and DC are Pareto efficient.•Suppose player i has a belief θ-i∈ΔS-i about the strategies played by the other players.•Player i's strategy s i∈S i is a best response to a beliefθif u i(s i,θ-i)≥u i(s i', θ-i)for all s i'∈S i.•Define BR i(-i)as the set of strategies that are best responses to the belief-i.•R is the BR when p <4/11.M is the BR when p >4/11.pu 24/11Dominance and Best Response Compared •Let B i be the set of strategies for player i that can berationalized as best responses to some beliefs.–B i={s i|there exists aθ-i∈∆S-i such that s i∈BR i(θ-i)}–For the previous example,B2={M, R}.•Let UD i be the set of undominated strategies for i.–UD i={s i∈S i|there is noσi∈∆S i that dominates s i}–For the previous example,UD2={M, R}•Result: In any finite two-player game,B i=UD i.Procedure for Calculating B i=UD i• 1. Look for strategies that are best responses to the simplest beliefs—those beliefs that put all probability on just one of the other player’s strategies. These best responses are obviously in the set B i so they are also in UD i.• 2. Look for strategies that are dominated by other pure strategies; these dominated strategies are not in UD i and thus they are also not in B i.• 3. Test each remaining strategy to see if it is dominated by a mixed strategy. This final step is the most difficult, but if you are lucky, then you will rarely have to perform it.。
吉本斯博弈论2课件
■ 与x0相邻的节点是x0的后 续节 （successors ）. x0的后续节点 是x1, x2
■ 对任何两个相邻的节点来说, 与 根相连接的路径更长的那个节点 是另一个节点的后续节.
■ 例3: x7 是x3的后续节点, 因为它
们相邻, 而且x7到 x0的路径比x3
到x0的路径更长
x4
x0 x1
x5
-1, 1
1 , -1
TT 1 , -1 -1, 1
Game theory-Chapter 2
17
Nash equilibrium
■完全信息动态博弈中的纳什均衡集（the set of Nash equilibrium）就是它的标准式的纳什均衡 集合.
Game theory-Chapter 2
18
弈可能的终点
x4
■ 例4: x4, x5, x6, x7, x8 都是终点 节
x0 x1
x5 x7
x2 x3
x6 x8
Game theory-Chapter 2
11
Game tree
■ 除终点节以外的任何节 点都代表了某个参与人.
■ 对于终点节以外的任意 后节续点节来的说边, 连缘接代它表和了它这的Player 2 个节点所代表的参与人 H 可能采取的行动
A
F
1, 2
A
F
2, 1
0, 0
2, 1
0, 0
Accommodate is the Nash equilibrium in this subgame.
Game theory-Chapter 2
26
Find subgame perfect Nash equilibria backward induction
博弈论 SPE 复旦大学 王永钦PPT课件
Player H
1
T
HH -1 , 1
1 , -1
Player 2
HT
TH
-1 , 1 1 , -1
-1 , 1 1 , -1
TT 1 , -1 -1 rium
• The set of Nash equilibria in a dynamic game of complete information is the set of Nash equilibria of its normal-form.
8
Game tree
• If a node x is a
successor of another
node y then y is called a predecessor of x.
• In a game tree, any node other than the root has a unique predecessor.
Definition: extensive-form representation
• The extensive-form representation of a game specifies:
➢ the players in the game ➢ when each player has the move ➢ what each player can do at each of his or her
• Player 2’s strategies
➢H if player 1 plays H, H if player 1 plays T ➢H if player 1 plays H, T if player 1 plays T ➢T if player 1 plays H, H if player 1 plays T ➢T if player 1 plays H, T if player 1 plays T
- 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)。
Example 2.4 (Prisoners’ Dilemma, [Tucker(1950)])
In the celebrated Prisoners’ Dilemma game, two people have collaborated in a crime. The police has managed to arrest them but it lacks sufficient evidence to convict them. Therefore, it interrogates them in separate rooms, offering each of the two suspects a deal according to which he will face a lighter sentence if he provides incriminating evidence against his accomplice. Each criminal is given two choices, to remain cooperative with the other player (C) or to defect and betray him (D).
6 / 26
The payoffs, in terms of time in prison, are as follows:
Prisoners’ dilemma - game matrice (1)
1\2 C D C −2, −2 −1, −9 D −9, −1 −8, −8
Equivalently, one could calculate how many of the following 10 years the criminals will be out of prison:
◮
When each player has finitely many choices, these games are fully represented by game matrices. We will call such games normal form games or strategic form games.
Note (Prisoners’ Dilemma)
◮ ◮
Game theory predicts that both prisoners defect The incentive to reduce one’s own sentence by one additional year makes the jointly optimal outcome unattainable One must consider indefinitely repeated interaction between the two players, for cooperation to emerge as a rational outcome. Typical example of tension between social optimality and individual incentives.
8 / 26
Notation 2.2 (Inducement correspondence of column-player)
Consequently, the dotted lines represent the inducement correspondence of the column-player, i.e. it represents the set of outcomes that are induced by player 1 and accessible to player 2.
9 / 26
Example 2.5 (Metro game I)
Two players, 1 and 2, try to meet at one of two metro exits, A or B
Game matrice
1\2 P R S P 0, 0 −1, 1 1, −1 R 1, −1 0, 0 −1, 1 S −1, 1 1, −1 0, 0
In fact, these are instances of strictly competitive games, that is, of games in which the two players have opposite preferences over the possible outcomes: Player 1 prefers outcome (a1 , a2 ) over outcome (b1 , b2 ) if and only if player 2 prefers outcome (b1 , b2 ) over outcome (a1 , a2 ).
magnus.hoffmann@ovgu.de
1 / 26
Chapter 2 - Games with Simultaneous Moves and Perfect Information
In games with simultaneous moves and perfect information:
Game matrice
1\2 H T H 1, −1 −1, 1 T −1, 1 1, −1
This is an example of a zero-sum game.
3 / 26
Example 2.2 (Serving in Tennis)
In tennis, in every point, the server (player 1) can direct his service left or right. The receiver (player 2) anticipates to receive the ball left or right. If the receiver guesses the server’s action correctly, the point is split (i.e., each player may win with equal probability). Otherwise, the server wins the point.
Microeconomic Analysis
Lecture in the Graduate Program in International Economics and Finance OTTO-VON-GUERICKE-UNIVERSITY MAGDEBURG
———–
Part 2
Dr. Magnus Hoffmann
Notation 2.1 (Inducement correspondence of row-player)
The solid lines represent the inducement correspondence (see [Greenberg(1990)]) of the row-player, here: player 1. The inducement correspondence of the row-player is the set of outcomes that are induced by the column-player (player 2) and accessible to the row-player. For example the left solid line, links the possible outcomes of the game, once player 2 has chosen D.
◮
All players choose their actions simultaneously or, equivalently, each player chooses his action without knowing the choices of the other players. There is no private information. All aspects of the game are known to the players.
Example 2.1 (Matching Pennies:)
Two players, 1 and 2, choose independently “heads” or “tails” (H or T ). If their choices match, player 1 wins $1 from player 2; otherwise, player 2 wins $1.
◮
◮
◮
If no criminal defects, then they are both imprisoned for two years, for some other (relatively minor) offence. If both criminals defect, then they are both imprisoned for eight years. If only one criminal defects, then he is imprisoned for only one year; while the other criminal is imprisoned for nine years (since obstruction of justice is added to the crime charges).
Example of certain classes of games
In the following three games, the players’ incentives are diametrically opposed. The fundamental strategic issue is that of competition.