国外博弈论lecture11
博弈论完整版PPT课件
2-阶理性： C相信R相信C是理性的，C会将R4从R的战略空间中剔除, 所以 C不会选择C1；
3-阶理性： R相信C相信R相信C是理性的， R会将C1从C的战略空间中剔 除, R不会选择R1；
基本假设：完全竞争，完美信息
个人决策是在给定一个价格参数和收入的条 件下最大化自己的效用，个人的效用与其他人 无涉，所有其他人的行为都被总结在“价格”参数 之中
一般均衡理论是整个经济学的理论基石 和道义基础，市场机制是完美的，帕累托 最优成立，平等与效率可以兼顾。
.
3
然而在以下情况,上述结论不成立：
.
19
理性共识
0-阶理性共识：每个人都是理性的，但不知道其 他人是否是理性的；
1-阶理性共识：每个人都是理性的，并且知道其 他人也是理性的，但不知道其他人是否知道自己 是理性的；
2-阶理性共识：每个人都是理性的，并且知道其
他人也是理性的，同时知道其他人也知道自己是
理性的；但不知道其他人是否知道自己知道他们
如果你预期我会选择X，我就真的会选择X。
如果参与人事前达成一个协议，在不存在外部强 制的情况下，每个人都有积极性遵守这个协议，这 个协议就是纳什均衡。
.
28
应用1——古诺的双寡头垄断模型(1938)
假定：
只有两个厂商 面对相同的线形需求曲线，P(Q)=a－Q， Q=q1+q2 两厂商同时做决策； 假定成本函数为C(qi)＝ciqi
劣策略：如果一个博弈中，某个参与人有占优策略，那么
该参与人的其他可选择策略就被称为“劣策略”。
国外博弈论课件lecture
完全信息博弈
总结词
所有参与者都拥有完全相同的信息， 即每个参与者都了解其他参与者的策 略和收益。
详细描述
完全信息博弈中，所有参与者都拥有 完全相同的信息，没有任何信息不对 称的情况。这种类型的博弈通常用于 分析具有高度透明度的竞争环境。
不完全信息博弈
总结词
至少有一个参与者不完全了解其他参与者的信息和策略。
博弈论广泛应用于经济学、政治学、社会学等领域，用于解释和预测行为主体的策 略选择和结果。
博弈论的发展历程
1944年，冯·诺依曼和摩根斯坦合著 的《博弈论与经济行为》标志着博弈 论的诞生。
20世纪70年代，泽尔腾提出了子博 弈精炼纳什均衡的概念，进一步丰富 了博弈论的理论体系。
20世纪50年代，纳什提出了纳什均 衡的概念，为博弈论的发展奠定了基 础。
斗鸡博弈
总结词
描述两个斗鸡在独木桥上相遇的情景，揭示了竞争中的策略和心理因素。
详细描述
斗鸡博弈是一个著名的博弈论案例，描述了两个斗鸡在独木桥上相遇的情景。每个斗鸡都有两种选择：前进或后 退。如果两只斗鸡都选择前进，它们将相撞并可能受伤；如果都选择后退，它们将保持安全但失去面子。这个案 例揭示了竞争中的策略和心理因素，如恐惧、面子和竞争意识等。
市场进入博弈
总结词
描述新企业和已存在的企业在决定是否进入市场时的 策略和利益关系。
详细描述
市场进入博弈是一个经典的博弈论案例，描述了新企 业和已存在的企业在决定是否进入市场时的策略和利 益关系。在这个情景中，已存在的企业可以选择保持 市场地位或进行阻挠，而新企业可以选择进入或不进 入市场。如果新企业决定进入市场，已存在的企业可 能会进行阻挠以保护自己的市场份额和利润。这个案 例揭示了市场进入和竞争中的策略和利益关系。
博弈论Lecture 11_Bargaining
1Lecture 11Bargaining 2Readings •Watson: Strategy_ An introduction to gametheory–Ch19.2–Exercise: 33Outline•The ultimatum game•A finite horizon game with alternating offers and impatient players2•An infinite horizon game with alternating offers and impatient players4The ultimatum game☐Players：the two players；☐Timing：player 1 proposes a division(x1,x2) of a pie, where x1+x2=1. If 2 accepts this division, she receives x2and player 1 receives x1; if she rejects it, neither player receives any pie.☐Preferences：Each person’s preferences are represented by payoffs equal to the division of pie she receives.x12,x x 120,0Y NThe ultimatum gameBackward induction☐First consider the optimal strategy of player 2☐Player 2 either accepts all offers (including 0), or accepts all offers and rejects the offer .☐Now consider the optimal strategy of player 1:☐If player 2 accepts all offers (including 0), then player 1’s optimal offer is 0;☐If player 2 accepts all offers except zero, then no offer of player 1 is optimal.20x >20x =Conclusion☐The only equilibrium of the game is the strategy pair in which player 1 offers 0 and player 2 accepts all offers.☐In this equilibrium, player 1’s payoff is all the pie and player 2’s payoff is zero.Extensions of the ultimatum gamex12,x x 0,012Y NNY 12,y y y 21Backward induction☐In this game, player 1 is powerless; Her proposal at the start of the game is irrelevant.☐Every subgame following player 2’s rejection of a proposal of player 1 is a variant of the ultimatum game in which player 2 moves first.☐Thus every such subgame has a unique equilibrium, in which player 2 offers nothing to player 1, andplayer 1 accepts all proposal.☐Hence in every equilibrium player 2 obtains all thepie.Conclusion☐In the extension of this game in which the players alternate offers over many periods, a similar result holds:☐In every equilibrium, the player who makes theoffer in the last period obtains all the pie.A finite horizon game with alternatingoffers and impatient players0,0x12,x x 12Y NNY1122,y y δδy2110y =22x δ=Two-period deadlineBackward Induction☐The game has a unique equilibrium in which ：☐Player 1’s initial proposal is .☐Player 2 accepts all proposals in which she receivesat least and rejects all proposals in which shereceives less than . ☐Player 2 proposes (0,1) after any history in which she rejects a proposal of player 1.☐Player 1 accepts all proposals of player 2 at the end of the game (after a history in which player 2 rejects player 1’s opening proposal).22(1,)δδ-2δ2δConclusion☐The outcome of this equilibrium is that player1 proposes，which player 2 accepts.☐Player 1’s payoff is and player 2’s payoffis .22(1,)δδ-21δ-2δ1x12,x x 0,012YN NY1122,y y δδy2Third-period deadline1x '2NY221122,x x δδ''20x '=11y δ=221(1)x δδ=-This game has a unique equilibrium:.2121(1(1)),(1))δδδδ---An infinite horizon game with alternatingoffers and impatient players: The is the following extensive game with perfect information.: Two negotiators, say 1 Definitio and 2n bargaining game of alternating offers Players Te : Every sequenc rminal historie e of the s form (1212,,,,,,)for 1, and every (infinite) sequence of the form (,,,,),where each is a division of the pie (a pair of numbers that sums to 1Player function ).: P()=1 (player 1 trx N x N x Y t x N x N x ≥∅ 1212121makes the first offer), and 1 if is even(,,,,,)(,,,,,,) 2 if is odd: For 1,2, player 's payoff to Preference the terminal history (,,,,,,) is s tttt it P x N x N x P x N x N x N t i i x N x N x Y δ-⎧==⎨⎩= 12, where 01, and her payoff to every (infinite) terminal history (,,,,) is 0.ti i x x N x N δ<<Stationary☐The structure of the game is stationary .☐The stationary structure of the game makes it reasonable to guess that the game has anequilibrium in which each player always makes the same proposal and always accepts the same set of proposals—that is, each player’s strategy is stationary .☐A stationary strategy pair:**11**22Player 1 always proposes and accepts a proposal if and only if Player 2 always proposes and accepts a proposal if and only if x y y yy x x x≥≥x**12,x x12YNNY**1122,y yδδy21y**12,y y21YNNY**1122,x xδδx12…………Properties of subgame perfect equilibrium: Player 2 accepts player 1's first offer, so that agreement is reached immediately; no resources are w aste Efficiency d in delay.Properties of subgame perfect equilibrium 2*11: For a given value of , the value of , the equilibrium payoff of player 1, increases as increases to 1. That is, fixing the patience of player 2, player 1's sha Effect of changes in patie r nce x δδ1e increases as she becomes more patient.Further, as player 1 becomes extremely patient ( close to 1), her share approaches 1. Symmetrically, fixing the patience of player 1,player 2's share increase δs to 1 as she becomes more patient.。
LectureEquilibrium Play(博弈论,Vanderbilt University)
Two main types of bidders
private value common value
14
Private Value
Dinner
What others know does not effect your willingness to pay
15
Source: Photograph courtesy of Erik Dungan, http://sxc.hu (accessed August 10, 2004). Used with permission.
10
300MHz
3GHz
A
1850MHz
D
B
E
F
C
A
D
B
F
C
1990MHz
99 licenses (corresponding to the red circles) were sold to 18 companies for a total price of $7.7 billion
Images of spectrum courtesy of U.S. Department of Commerce’s National Telecommunications and Information Administration (/, accessed 16 July 2004).
Slide courtesy of Mike Shor, Vanderbilt University.
13
Types of Bidders
Auctions have rules and bidders
基础博弈论大学英文讲义
Week 11: Game TheoryRequired Reading: Schotter pp. 229 - 260Lecture Plan1. The Static Game TheoryNormal Form GamesSolution Techniques for Solving Static Games Dominant StrategyNash Equilibrium2. Prisoner’s Dilemma3. Decision AnalysisMaximim CriteriaMinimax Criteria4. Dynamic One-Off GamesExtensive Form GamesThe Sub-Game Perfect Nash Equilibrium1. The static Game TheoryStatic games: the players make their move in isolation without knowing what other players have done1.1 Normal Form GamesIn game theory there are two ways in which a game can be represented.1st) The normal form game or strategic form game2nd) The extensive form gameA normal form game is any game where we can identity the following three things:1. Players:2. The strategies available to each player.3. The Payoffs. A payoff is what a player will receive at the endof the game contingent upon the actions of all players in the game.Suppose that two people (A and B) are playing a simple game. A will write one of two words on a piece of paper, “Top” or “Bottom”. At the same time, B will independently write “left” or “right” on a piece of paper. After they do this, the papers will be examined and they will get the payoff depicted in Table 1.Table 1If A says top and B says left, then we examine the top-left corner of the matrix. In this matrix, the payoff to A(B) is the first(Second) entry in the box. For example, if A writes “top” and B writes “left” payoff of A = 1 payoff of B = 2.What is/are the equilibrium outcome(s) of this game?1.2Nash Equilibrium Approach to Solving Static GamesNash equilibrium is first defined by John Nash in 1951 based on the work of Cournot in 1893.A pair of strategy is Nash equilibrium if A's choice is optimal given B's choice, and B's choice is optimal given A's choice. When this equilibrium outcome is reached, neither individual wants to change his behaviour.Finding the Nash equilibrium for any game involves two stages.1) identify each optimal strategy in response to what the other players might do.Given B chooses left, the optimal strategy for A isGiven B chooses right, the optimal strategy for A isGiven A chooses top, the optimal strategy for B isGiven A chooses bottom, the optimal strategy for B isWe show this by underlying the payoff element for each case.2) a Nash equilibrium is identified when all players are player their optimal strategies simultaneouslyIn the case of Table 2,If A chooses top, then the best thing for B to do is to choose left since the payoff to B from choosing left is 1 and the payoff from choosing right is 0. If B chooses left, then the best thing for A to do is to choose top as A will get a payoff of 2 rather than 0.Thus if A chooses top B chooses left. If B chooses left, A chooses top. Therefore we have a Nash equilibrium: each person is making optimal choice, given the other person's choice.If the payoff matrix changes as:Table 2then what is the Nash equilibrium?Table 3If the payoffs are changed as shown in Table 32. Prisoners’ dilemm aPareto Efficiency: An allocation is Pareto efficient if goods cannot be reallocated to make someone better off without making someone else worse off.Two prisoners who were partners in a crime were being questioned in separate rooms. Each prisoner has a choice of confessing to the crime (implicating the other) or denying. If only one confesses, then he would go free and his partner will spend 6 months in prison. If both prisoners deny, then both would be in the prison for 1 month. If both confess, they would both be held for three months. The payoff matrix for this game is depicted in Table 4.Table 4The equilibrium outcome3. Decision AnalysisLet N=1 to 4 a set of possible states of nature, and let S=1 to 4be a set of strategy decided by you. Now you have to decide which strategy you have to choose given the following payoff matrix.Table 5S=YouN=OpponentIn this case you don't care the payoff of your opponent i.e. nature.3.1 The Maximin Decision Rule or Wald criterionWe look for the minimum pay-offs in each choice and then maximising the minimum pay-offLet us highlight the mimimum payoff for each strategy.3.2 The Minimax Decision Rule or Savage criterionOn this rule we need to compute the losses or regret matrix from the payoff matrix. The losses are defined as the difference between the actual payoff and what that payoff would have been had the correct strategy been chosen.Regret/Loss = Max. payoff in each column – actual payoffFor example of N=1 occurs and S=1 is chosen, the actual gain = 2 from Table 3. However, the best action given N=1 is also to choose S=1 which gives the best gain = 2. For (N=1, S=1) regret = 0.If N=1 occurs but S=2 is chosen, the actual gain = 1. However, the best action given N=1 is also to choose S=1 which gives the best gain = 2. For (N=1, S=2) regret = 2-1.Following the similar analysis, we can compute the losses for each N and S and so can compute the regret matrix.Table 6: Regret MatrixAfter computing the regret matrix, we look for the maximum regret of each strategy and then taking the minimum of these.Minimax is still very cautious but less so than the maximin.4. Dynamic one-off GamesA game can be dynamic for two reasons. First, players may be ableto observe the actions of other players before deciding upon theiroptimal response. Second, one-off game may be repeated a number of times.4.1 Extensive Form GamesDynamic games cannot be represented by payoff matrices we have touse a decision tree (extensive form) to represent a dynamic game.Start with the concept of dynamic one-off game the game can beplayed for only one time but players can condition their optimal actions on what other players have done in the past.Suppose that there are two firms (A and B) that are considering whether or not to enter a new market. If both firms enter the market,then they will make a loss of $10 mil. If only one firm enters the market, it will earn a profit of $50 mil. Suppose also that Firm B observes whether Firm A has entered the market before it decides what to do.Any extensive form game has the following four elements in common:Nodes: This is a position where the players must take a decision.The first position is called the initial node, and each node is labelled so as to identify who is making the decision.Branches: These represent the alternative choices that the person faces and so correspond to available actions.Payoff Vectors: These represent the payoffs for each player, with the payoffs listed in the order of players. When we reach a payoffvector the game ends.In period 1, Firm A makes its decisions. This is observed by Firm B which decides to enter or stay out of the market in period 2. In this extensive form game, Firm B’s decision nodes are the sub-game. This means that firm B observes Firm A’s action before making its own decision.4.2 Subgame Perfect Nash EquilibriumSubgame perfect Nash equilibrium is the predicted solution to a dynamic one-off game. From the extensive form of this game, we can observe that there are two subgames, one starting from each of Firm B’s decision nodes.How could we identify the equilibrium outcomes?In applying this principle to this dynamic game, we start with the last period first and work backward through successive nodes until we reach the beginning of the game.Start with the last period of the game first, we have two nodes. At each node, Firm B decides whether or not entering the market based on what Firm A has already done.For example, at the node of “Firm A enters”, Firm B will either make a loss of –$10mil (if it enters) or receive “0” payoff (if it stays out); these are shown by the payoff vectors (-10, -10) and (50, 0). If Firm B is rational, it will stays outThe node “Firm A enters” can be replaced by the vector (50, 0).At the second node “Firm A stays out”, Firm A h as not entered the market. Thus, Firm B will either make a profit of $50mil (if it enters) or receive “0” payoff (if it stays out); these are shown by the payoff vectors (0, 50) and (0, 0). If Firm B is rational, it will enter thus we could rule out the possibility of both firms stay outWe can now move back to the initial node. Here Firm A has to decide whether or not to enter. If Firm B is rational, it is known that the game will never reach the previously “crossed” vectors. Firm A also knows that if it enters, the game will eventually end at (A enters, B stays out) where A gets 50 and B gets 0. On the other hand, if Firm A stays out, the game will end at (A stays out, B enters) where A gets 0 and B gets 50 Firm A should enter the market at the first stage. The eventual outcome is (A enters, B stays out)How to find a subgame perfect equilibrium of a dynamic one-off game?1. Start with the last period of the game cross out the irrelevant payoff vectors.2. Replace the preceding nodes by the uncrossed payoff vectorsuntil you reach the initial node.3. The only uncrossed payoff vector(s) is the subgame perfect Nash equilibrium.。
国外博弈论课件lecture
s22 ( 1- q ) u1(s11, s22), u2(s11, s22) u1(s12, s22), u2(s12, s22)
Mixed strategy Nash equilibrium:
A pair of mixed strategies ((r*, 1-r*), (q*, 1-q*))
is a Nash equilibrium if (r*,1-r*) is a best response to (q*, 1-q*), and (q*, 1-q*) is a best response to (r*,1-r*). That is,
v1((r*, 1-r*), (q*, 1-q*)) v1((r, 1-r), (q*, 1-q*)), for all 0 r 1 v2((r*, 1-r*), (q*, 1-q*)) v2((r*, 1-r*), (q, 1-q)), for all 0 q 1
Introduction to games Normal-form (or strategic-form)
representation Iterated elimination of strictly dominated
strategies Nash equilibrium Review of concave functions, optimization Applications of Nash equilibrium Mixed strategy Nash equilibrium
May 30, 2003
h
4
Mixed strategy Nash equilibrium: 2player each with two pure strategies
国外博弈论课件lecture(10)
针对复杂系统的博弈研究将进一步深化，探索如 何利用博弈论分析复杂系统中的相互作用和演化 规律。
人工智能与博弈论的结合
随着人工智能技术的发展，博弈论将进一步应用 于机器学习、数据挖掘等领域，为人工智能提供 更好的决策支持。
博弈论对人类社会的影响
促进社会公平和合作
博弈论的研究有助于揭示社会现象背后的利益关系和互动机制， 推动社会公平和合作，减少冲突和矛盾。
纳什均衡的求解方法
迭代法
01
通过不断迭代计算每个参与者的最优策略，最终收敛到一个稳
定的策略组合。
线性规划法
02
将博弈问题转化为线性规划问题，通过求解线性规划来找到纳
什均衡。
代数法
03
通过代数方法求解博弈问题，找到纳什均衡。
纳什均衡的应用
经济学
纳什均衡可以用来解释市场中的竞争行为和价格形成机制，以及 企业之间的竞争策略。
立法博弈
在立法过程中，博弈论可以帮助分析各方利 益集团的博弈行为，预测法案的通过与否以 及最终的立法结果。这有助于利益集团制定
最佳的游说策略和立法策略。
社会问题
环境保护博弈
在环境保护方面，博弈论可以用于分析政府、企业和个人之间的环保博弈行为。通过建 立博弈模型，可以预测各方的行为反应，从而为政府制定有效的环保政策提供依据。
国外博弈论课件 Lecture(10)
目 录
• 博弈论简介 • 博弈论的基本概念 • 博弈的类型 • 纳什均衡 • 博弈论的实际应用 • 结论与展望
01
CATALOGUE
博弈论简介
博弈论的定义
博弈论：研究决策主体在给定 信息结构下如何决策以最大化 自己的效用，以及不同决策主
博弈论最全完整-讲解课件
• 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零，这个博弈就叫零和博 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零，这个博弈就 叫非零和博弈。
• 零和博弈是利益对抗程度最高的博弈。
• 即使决策或行动有先后，但只要局中人在决策 时都还不知道对手的决策或者行动是什么，也 算是静态博弈
学习交流PPT
28
完全信息博弈与不完全信息博弈
(games of complete information and games of incomplete information)
• 按照大家是否清楚对局情况下每个局中人 的得益。
供万无一失的应对办法。
学习交流PPT
5
例1：无谓竞争（The GPA Rat Race）
• 你所注册的一门课程按照比例来给分：无论卷 面分数是多少，只有40％的人能够得优秀，40 ％的人能得良好。
• 所有学生达成一个协议，大家都不要太用功， 如何？想法不错，但无法实施!稍加努力即可 胜过他人，诱惑大矣。
• 某些博弈中，由于偶然的外因可以对策略贴标 签，或者参与者之间拥有某些共同的知识体验， 导致了焦点的存在。
• 没有某个这样的暗示，默契的合作就完全不可 能。
学习交流PPT
9
例3：为什么教授如此苛刻？
• 许多教授强硬地规定，不进行补考，不允许迟 交作业或论文。
• 教授们为何如此苛刻？
• 如果允许某种迟交，而且教授又不能辨别真伪， 那么学生就总是会迟交。
• 王则柯、李杰编著，《博弈论教程》，中国人民大学 出版社，2004年版。
- 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)。
2p11+2 p12= 3p11+1 p12 3p11+2 p12 and p11+p12=1.
We should have
2anpd21+p201+
pp2222++1p23p=23
=3p21+1 1
p22+0
p23
Solve these. If we can get a solution that satisfies p11>0, p12>0, p21>0, p22>0, p23=0 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does not satisfies
not satisfies p11>0, p12>0, p21>0, p22>0, p23>0, then we do not have such a mixed strategy Nash equilibrium.
June 3, 2003
73-347 Game Theory--Lecture 11
3
Exercise 138.1 of Osborne
Player 1
T (p11) B (p12)
L (p21) 2, 2 3, 2
Player 2 M (p22) 0, 3 1, 1
R (p23) 1, 3 0, 2
We first consider pure-strategy Nash equilibria. How many can you find?
5
Exercise 138.1 of Osborne
Player 2
Player 1
T (p11) B (p12)
L (p21) 2, 2 3, 2
M (p22) 0, 3 1, 1
R (p23) 1, 3 0, 2
Case 2: check whether there is a mixed strategy in which p11>0, p12>0, p21>0, p22>0, p23=0 By theorem 4, we should have
We should have
2anpd21+p201+
pp2222++1p23p=23
=3p21
Solve these equations. If we can get a solution that satisfies p11>0, p12>0, p21>0, p22>0, p23>0 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does
Case 1: check whether there is a mixed strategy in which p11>0, p12>0, p21>0, p22>0, p23>0 By theorem 4, we should have
2p11+2 p12= 3p11+1 p12 = 3p11+2 p12 and p11+p12=1.
Player 2
Player 1
T (p11) B (p12)
L (p21) 2, 2 3, 2
M (p22) 0, 3 1, 1
R (p23) 1, 3 0, 2
Case 3: check whether there is a mixed strategy in which p11>0, p12>0, p21>0, p22=0, p23>0 By theorem 4, we should have
Introduction to games Normal-form (or strategic-form)
representation Iterated elimination of strictly dominated
strategies Nash equilibrium Review of concave functions, optimization Applications of Nash equilibrium Mixed strategy Nash equilibrium
p11>0, p12>0, p21>0, p22>0, p23=0, then we do not have such a mixed strategy Nash equilibrium.
June 3, 2003
73-347 Game Theory--Lecture 11
6
Exercise 138.1 of Osborne
In order to find all Nash equilibria, we need to consider 15 more cases by Theorem 4 of Lecture 10!
We first consider complicated cases. Some cases are very easy.
June 3, 2003
73-347 Game Theory--Lecture 11
2
Today’s Agenda
Review of previous class Exercise 138.1 of Osborne Review HW1
June 3, 2003
73-347 Game Theory--Lecture 11
June 3, 2003
73-347 Game Theory--Lecture 11
4
Exercise 138.1 of Osborne
Player 2
Player 1
T (p11) B (p12)
L (p21) 2, 2 3, 2
M (p22) 0, 3 1, 1
R (p23) 1, 3 0, 2
Static (or SimultaneousMove) Games of Complete Information
Mixed Strategy Nash Equilibrium
June 3, 2003
73-347 Game Theory--Lecture 11
1
Outline of Static Games of Complete Information