9.2 完全信息静态博弈
9.2.1 博弈的战略式表述
Definition A normal (strategic) form game G consists of: (1) a finite set of agent s {1,2,,}D n = . (2) strategy sets 12,,,n S S S .
(3) payoff functions 12:(1,2,,)i n u S S S R i n ???→= .
囚徒B
囚徒A
完全信息静态博弈是一种最简单的博弈，在这种博弈中，战略和行动是一回事。
博弈分析的目的是预测博弈的均衡结果，即给定每个参与人都是理性的，什么是每个参与人的最优战略？什么是所有参与人的最优战略组合？
纳什均衡是完全信息静态博弈解的一般概念，也是所有其他类型博弈解的基本要求。 下面，我们先讨论纳什均衡的特殊情况，然后讨论其一般概念。
9.2.2 占优战略（Dominated Strategies ）均衡
一般说来，由于每个参与人的效用（支付）是博弈中所有参与人的战略的函数，因此，
每个参与人的最优战略选择依赖于所有其他参与人的战略选择。但是在一些特殊的博弈中，一个参与人的最优战略可能并不依赖于其他参与人的战略选择。也就是说，不管其他参与人选择什么战略，他的最优战略是唯一的，这样的最优战略被称为“占优战略”。
Definition Strategy s i is strictly dominated for player i if there is some i i s S '∈ such that (,)(,)i i i i i i u s s u s s --'> for al i i s S --∈.
Proposition a rational player will not play a strictly dominated strategy.
抵赖 is a dominated strategy. A rational player would therefore never 抵赖. This solves the game since every player will 坦白. Notice that I don't have to know anything about the other player . 囚徒困境：个人理性与集体理性之间的矛盾。
This result highlights the value of commitment in the Prisoner's dilemma – commitment consists of credibly playing strategy 抵赖.
囚徒困境的广泛应用：军备竞赛、卡特尔、公共品的供给。
9.2.3 Iterated Deletion of Dominated Strategies (重复剔除劣战略)
智猪博弈（boxed pigs ）
小猪
大猪
按 等待
此博弈没有占优战略均衡。因为尽管“等待”是小猪的占优战略，但是大猪没有占优战略。大猪的最优战略依赖于小猪的占略: ---。
大猪会正确地预测到小猪会选择“等待”；给定此预测，大猪的最优选择只能是“按”。这样，（按，等待）就是唯一的均衡。
重复剔除的占优均衡：先剔除某个参与人的劣策略，重新构造新的博弈，再剔除，---。
应用：大股东监督经理，小股东搭便车；大企业研发，小企业模仿。
9.2.4 Nash equilibrium
性别战博弈（battle of the sexes ）:
女
男
足球赛 演唱会
在上面的博弈中，两个参与者都没有占优策略，每个参与者的最优策略都依赖于另一个参与人的战略。所以，没有重复剔除的占优均衡。
Definition A strategy profile s * is a pure strategy Nash equilibrium of G if and only if
***(,)(,)i i i i i i u s s u s s --≥ for all players i and all i i s S ∈
求解Nash 均衡的方法。
A Nash equilibrium captures the idea of equilibrium : Both players know what strategy the other player is going to choose, and no player has an incentive to deviate from equilibrium play because her strategy is a best response to her belief about the other player's strategy.
对纳什均衡的理解：设想所有参与者在博弈之前达成一个（没有约束力的）协议，规定每个参与人选择一个特定的战略。那么，给定其他参与人都遵守此协议，是否有人不愿意遵守此协议？如果没有参与人有积极性单方面背离此协议，我们说这个协议是可以自动实施的（self-enforcing ），这个协议就构成一个纳什均衡。否则，它就不是一个纳什均衡。
问题：纳什均衡与重复剔除（严格）劣战略均衡之间的关系。
9.2.5 Cournot Competition （古诺竞争）
This game has an infinite strategy space .
Two firms choose output levels q i ，cost function c i (q i ) = cq i .
market demand determines a price 1212()()p f q q q q αβ=+=-+：the products of both firms are perfect substitutes, i.e. they are homogenous products. D = {1; 2} S 1 = S 2 = R +
u 1 (q 1, q 2) = q 1 f (q 1 + q 2) -c 1 (q 1) u 2 (q 1, q 2) = q 2 f (q 1 + q 2) - c 2 (q 2)
the 'best-response' function BR (q j ) of each firm i to the quantity choice q j of the other firm: 由11121[()]q q q cq παβ=-+-，得FOC: 21211
()0 22q c q q q c q ααβββ--+--=?=-；又120 c
q q αβ
-≥?≤
。因 , ()220, otherwise j j i j q c c if q BR q ααββ?---≤?
=???
q 1
q 2
BR 1(q 2)
BR 2(q 1)
(q 1*,q 2* )
()(2)
c αβ-()c αβ
-
The best-response function is decreasing in my belief of the other firm's action.
Using our new result it is easy to see that the unique Nash equilibrium of the Cournot game is the intersection of the two BR functions.
Because of symmetry we know that q 1 = q 2 = q*.
Hence we obtain *
*
2
c q q αβ-=-, This gives us the solution 2()*3c q αβ-=.
问题：将寡头竞争的古诺均衡与垄断企业的最优产量和利润进行比较。
9.2.6 Bertrand Competition （伯特兰竞争）
Firms compete in a homogenous product market but they set prices . Consumers buy from the lowest cost firm. demand curve q = D (p )
Therefore, each firm faces demand
12() (,)()2 = 0
i i j i i i j i j D p if p p D p p D p if p p if p p ?
=??>?
We also assume that D (c ) > 0, i.e. firms can sell a positive quantity if they price at marginal cost.
Lemma The Bertrand game has the unique NE *
*
12(,)p p = (c; c ).
Proof: First we must show that (c,c) is a NE . It is easy to see that each firm makes zero profits. Deviating to a price below c would cause losses to the deviating firm. If any firm sets a higher price it does not sell any output and also makes zero profits. Therefore, there is no incentive to deviate.
To show uniqueness we must show that any other strategy profile (p 1; p 2) is not a NE . It's easiest to distinguish lots of cases.
Case I: p 1 < c or p 2 < c. In this case one (or both players) makes negative losses. This player should set a price above his rival's price and cut his losses by not selling any output.
Case II: c < p 1 < p 2 or c < p 2 < p 1. In this case the firm with the higher price makes zero profits. It could profitably deviate by setting a price equal to the rival's price and thus capture at least half of his market, and make strictly positive profits.
Case III: c = p 1 < p 2 or c = p 2 < p 1. Now the lower price firm can charge a price slightly above marginal cos t (but still below the price of the rival) and make strictly positive profits.
Case IV: c < p 1 = p 2. Firm 1 could profitably deviate by setting a price 12p p c ε=->. The firm's profits before and after the deviation are:
22()
()2
B D p p πε=
- 22()()A D p p c πεε=---
Note that the demand function is decreasing, so 22()()D p D p ε->. We can therefore deduce:
222()
()()2
A B D p p c D p πππεε?=->
--- This expression (the gain from deviating) is strictly positive for sufficiently small ε.
Therefore, (p 1; p 2) cannot be a NE.
9.2.7 Mixed Strategies （混合战略）
猜谜游戏 matching pennies game:
儿童B
H(正面) T(反面) 儿童A
H(正面)
T(反面)
每一个参与者都想猜透对方的战略，而又不能让对方猜透自己的战略。This game has no pure-strategy Nash equilibrium. Whatever pure strategy player 1 chooses, player 2 can beat him. 如果一个参与人采用混合战略（以一定的概率选择某种概率），他的对手就不能准确地猜出他实际上会选择的战略，尽管在均衡点上，每个人都知道其他参与人在不同战略上的概率分布。
Intuitively, games in which the participants have a large number of strtegies will often offer suffifient flexibility to ensure that at least one Nash Equilibrium must exist. If we permit the players to use “mixed ” strategies , the above game will be converted into one with an infinite number of (mixed ) strategies and, again, the existence of a Nash equilibrium is ensured.
Suppose that the players decide to randomize amongst his strategies and play a mixed
strategy. Player A could flip a coin and play H with probability r and T with probability 1-r , and player B flip a coin and play H with probability s and T with probability 1-s.
Given these probabilities, the outcomes of the game occur with the following probabilities: H-H , rs ; H-T, r (1-s ); T-H, (1-r )s ; T-T,(1-r )(1-s ). Player A ’s expected utility is then given by
()(1)(1)(1)(1)(1)(1)(1)(1)A E u rs r s r s r s =+--+--+--
42212(21)21rs r s r s s =--+=--+
Oviously, A ’s optimal choice of r depends on B ’s probability, s. If 12s <, utility is maximized by choosing 0r =. If 12s >, A should opt for 1r =. And when 12s =, A ’s expected utility
is 0 no matter what value of r is choosen.
For player B, expected utility is given by
()(1)(1)(1)(1)(1)(1)(1)(1) (4221)2(12)(12)
B E u rs r s r s r s rs r s s r r =-+-+-+---=---+=---
Now, when 12r >, B ’s expected utility is maximized by choosing 0s =. If 1
2r <, A should opt for 1s =. And when 12r =, A ’s expected utility is independent of what s is choosen.
s
Nash equilibria are shown in the figure by the intersections of optimal response curves for A and B.
Or, we can get the equilibrium through the FOC
()1
420 2A E u s s r ?=-=?=? ()1
420 2
B E u r r s ?=-=?=?
Definition: Let G be a game with strategy spaces S 1,S 2,..,S I . A mixed strategy i σ for player i is a probability distribution on S i ,i.e. for S i , a mixed strategy is a function
:i i s σ+→? such that
()1i i
i i s S s σ∈=∑
.
Several notations are commonly used for describing mixed strategies. 1. Function (measure): ()12i H σ= and ()12i T σ=
2. Vector: If the pure strategies are 1,,i i i N s s ,write 1((),,())i i i i i N s s σσ e.g.(1/2, 1/2)
3. (1/2)H + (1/2)T
Write i ∑(also ()i S ?) for the set of probability distributions on S i .
Write ∑ for 1n ∑??∑ . A mixed strategy profile σ∈∑ is an n-tuple 1(,,)n σσ with i i σ∈∑.
We write (,)i i i u σσ- for player i's expected payoff when he uses mixed strategy i σ and all other players play as in i σ-.
,(,)(,)()()i i
i i i i i i i i i i s s u u s s s s σσσσ-----=∑
Remark : For the definition of a mixed strategy payoff we have to assume that the utility function fulfills the VNM axioms. Mixed strategies induce lotteries over the outcomes (strategy profiles) and the expected utility of a lottery allows a consistent ranking only if the preference relation satisfies these axioms.
Definition A mixed strategy NE of G is a mixed profile
*σ∈∑ such that
***
(,)(,)i i i i i i u u σσσσ--≥
for all i and all i i σ∈∑.
The definition of MSNE makes it cumbersome to check that a mixed profile is a NE. The
next result shows that it is sufficient to check against pure strategy alternatives.
Proposition: *
σ is a Nash equilibrium if and only if *
*
*
(,)(,)i i i i i i u u s σσσ--≥ for all i
and i i s S ∈.
Example: The strategy profile *
*
12(1/2)H (1/2)T σσ==+is a NE of Matching Pennies.
Because of symmetry is it sufficient to check that player 1 would not deviate. If he plays his mixed strategy he gets expected payoff 0. Playing his two pure strategies gives him payoff 0 as well. Therefore, there is no incentive to deviate.
9.3 完全信息动态博弈
9.3.1 The extensive form of a game
The extensive form of a game is a complete description of 1. The set of players .
2. Who moves when and what their choices are.
3. The players' payoffs as a function of the choices that are made.
4. What players know when they move.
Example : Model of Entry
Currently firm 1 is an incumbent monopolist. A second firm 2 has the opportunity to enter. After firm 2 makes the decision to enter, firm 1 will have the chance to choose a pricing strategy. It can choose either to fight the entrant or to accommodate it with higher prices.
Example II : Stackelberg Model
Suppose firm 1 develops a new technology before firm 2 and as a result has the opportunity to build a factory and commit to an output level q 1 before firm 2 starts. Firm 2 then observes firm 1 before picking its output level q 2. For concreteness suppose {0,1,2}i q ∈ and market demand is ()3p Q Q =-. The marginal cost of production is 0.
9.3.2 Definition of an Extensive Form Game
Formally a finite extensive form game consists of 1. A finite set of players.
2. A finite set T of nodes which form a tree along with functions giving for each
non-terminal node t Z ?(Z is the set of terminal nodes)
the player i (t ) who moves the set of possible actions A (t )
the successor node resulting from each possible action N (t; a )
3. Payoff functions :i u Z +
→? giving the players payoffs as a function of the terminal node reached (the terminal nodes are the outcomes of the game).
4. An information partition : for each node t , h (t ) is the set of nodes which are possible
given what player i (x ) knows. This partition must satisfy
()()(), ()(), and ()()t h x i t i t A t A t h t h t ''''∈?===
We sometimes write i (h ) and A (h ) since the action set is the same for each node in the same information set .
9.3.3 Nash Equilibrium in Extensive Form Games
Normal Form Analysis
In an extensive form game write H i for the set of information sets at which player i moves.
{() ()}i H S T S h t for some t T with i t i =?=∈=
Write A i for the set of actions available to player i at any of his information sets.
Definition A pure strategy for player i in an extensive form game is a function :i i i s H A → such that ()()i s h A h ∈ for all i h H ∈.
Note that a strategy is a complete contingent plan explaining what a player will do in any situation that arises. At first, a strategy looks overspecified: earlier action might make it impossible to reach certain sections of a tree. Why do we have to specify how players would play at nodes which can never be reached if a player follows his strategy early on? The reason is that play off the equilibrium path is crucial to determine if a set of strategies form a Nash equilibrium. Off-equilibrium threats are crucial . This will become clearer shortly.
Example : Entry Game
We can find the pure strategy sets S 1={Fight ,Accommodate} and S 2 ={Out ,In}. We can represent the game in normal form as:
Firm 2
Out In
Firm 1
F
A
Nash Equilibrium in Extensive Form Games
We can apply NE in extensive form games simply by looking at the normal form representation. It turns out that this is not an appealing solution concept because it allows for too many profiles to be equilibria.
Look at the entry game. There are two pure strategy equilibria: (A, In) and (F, Out) as well as mixed equilibria ((1)F A αα+-; Out) for 12α≥.
Why is (F, Out) a Nash equilibrium? Firm 2 stays out because he thinks that player 1 will fight entry. In other words, the threat to fight entry is sufficient to keep firm 2 out. Note, that in equilibrium this threat is never played since firm 2 stays out in the first place.
The problem with this equilibrium is that firm 2 could call firm 1's bluff and enter. Once firm 2 has entered it is in the interest of firm 1 to accommodate. Therefore, firm 1's threat is not credible . This suggests that only (A,In) is a reasonable equilibrium for the game since it does not rely on non-credible threats. The concept of subgame perfection which we will introduce in the next lecture rules out non-credible threats.
Subgame perfection is a refinement of Nash equilibrium. It rules out non-credible threats .
9.3.4 Subgame Perfect Equilibrium
Definition A subgame G ' of an extensive form game G consists of
1. A subset T 'of the nodes of G consisting of a single node x and all of its successors which has the property that t T '∈,()t h t '∈ then t T ''∈.
2. Information sets, feasible moves and payoffs at terminal nodes as in G.
Example : Entry Game
This game has two subgames. The entire game (which is always a subgame) and the subgame which is played after player 2 has entered the market:
Definition A strategy profile s* is a subgame perfect equilibrium (SPE) of G if it is a Nash equilibrium of every subgame of G .
Note, that a SPE is also a NE because the game itself is a (degenerate) subgame of the entire game.
Look at the entry game again. We can show that s 1 = A and s 2 = Entry is the unique SPE. Accomodation is the unique best response in the subgame after entry has occurred. Knowing that, firm 2's best response is to enter.
9.3.5 Backward Induction
the most common technique for finding and verifying the SPE of a game. Start at the end of the game and work your way from the start.
We will focus on extensive form games where each information set is a single node (i.e. players can perfectly observe all previous moves).
[Definition: An extensive form is said to have perfect information if each information set contains a single node.]
Step I Identify the terminal subgames . Terminal subgames are those which do not dominate another
subgame.
Step II Solve the terminal subgames. These subgames have no further subgames. They have a Nash equilibrium.
Step III Calculate the Nash payoffs of the terminal subgames and replace these subgames with the Nash payoffs.
Repeats step I to step III, ---
Example: Stackelberg Model
Two firms choose output levels q i ，cost function c i (q i ) = cq i .
market demand : 1212()()p f q q q q αβ=+=-+
Firm 1 selects its output level q 1 first , Firm 2 then observes firm 1 and picks its output level q 2.
Step 1: given q 1, determine firm 2’s optimal output level q 2.
22122120
m a x
(,)[()]
q q q q q q c παβ≥=-+- FOC ? 21111
()()22
q q c q αβ=
-- 21()q q is firm 2’s best-response function. Step 2: firm 1’s optimization decision at the first stage is
1112111
210
m a x (,())[(())]q q q q q q q q c παβ≥=-
+-111
()22
c q q αβ-=-
FOC ? *
12c
q αβ
-=
? *
24c
q αβ
-=
企业1有先动优势（first-mover advantege ）。
问题：请将企业1、企业2的产量与Cournot 模型中的产量进行对比。
Eco514—Game Theory Lecture10:Extensive Games with(Almost)Perfect Information Marciano Siniscalchi October19,1999 Introduction Beginning with this lecture,we focus our attention on dynamic games.The majority of games of economic interest feature some dynamic component,and most often payo?uncertainty as well. The analysis of extensive games is challenging in several ways.At the most basic level, describing the possible sequences of events(choices)which de?ne a particular game form is not problematic per se;yet,di?erent formal de?nitions have been proposed,each with its pros and cons. Representing the players’information as the play unfolds is nontrivial:to some extent, research on this topic may still be said to be in progress. The focus of this course will be on solution concepts;in this area,subtle and unexpected di?culties arise,even in simple games.The very representation of players’beliefs as the play unfolds is problematic,at least in games with three or more players.There has been a?erce debate on the“right”notion of rationality for extensive games,but no consensus seems to have emerged among theorists. We shall investigate these issues in due course.Today we begin by analyzing a particu-larly simple class of games,characterized by a natural multistage structure.I should point out that,perhaps partly due to its simplicity,this class encompasses the vast majority of extensive games of economic interest,especially if one allows for payo?uncertainty.We shall return to this point in the next lecture. Games with Perfect Information Following OR,we begin with the simplest possible extensive-form game.The basic idea is as follows:play proceeds in stages,and at each stage one(and only one)player chooses an 1
博弈论经典案例 冰晶淩（杂物区）2010-04-09 22:31:28 阅读258 评论0 字号：大中小订阅 引用 光光的博弈论经典案例 1994年诺贝尔经济学奖授给了三位博弈论专家：纳什，泽尔腾和海萨尼．而博弈论可以划分为合作博弈和非合作博弈．那三位博弈论专家的贡献主要是在非合作博弈方面，而且现在经济学家谈到博弈论，一般指的是非合作博弈，很少指合作博弈．合作博弈与非合作博弈之间的区别主要在于人们的行为相互作用时，当事人能否达成一个具有约束力的协议，如果有，就是合作博弈；反之，就是非合作博弈．非合作博弈强调的是个人理性，个人最优决策，其结果可能是有效率的，也可能是无效率的．而合作博弈强调的是团体理性．下面是我收集的张维迎教授的几个有关博弈论的经典 案例． ＜案例一：囚徒困境＞ 囚徒困境讲的是两个嫌疑犯作案后被警察抓住，分别关在不同的屋子里审讯．警察告诉他们：如果两人都坦白，各判刑8年；如果两个都抵赖，各判1年(或许因证据不足)；如果其中一人坦白一人抵赖，坦白的放出去，不坦白的判刑10年(这有点＇坦白从宽，抗拒从严＇的味道)．这里，每个囚徒都有两种战略：坦白或抵赖．表中每一格的两个数字代表对应战略组合下两个囚徒的支付（效用），其中第一个数字是第一个囚徒的支付，第二个数字为第二个囚徒的支付．战略形式又称标准形式，是博弈的两种表述形式之一，它特别方便于静态博弈分析． 在这个例子里，纳什均衡就是（坦白，坦白）：给定B坦白的情况下，Ａ的最优战略是坦白；同样，给定Ａ坦白的情况下，Ｂ的最优战略也是坦白．事实上，这里，（坦白，坦白）不仅是纳什均衡，而且是一个占优战略均衡．就是说，不论对方如何选择，个人的最优选择是坦白．比如说，如果Ｂ不坦白，Ａ坦白的话被放出来，不坦白的话判１年，所以坦白比不坦白好；如果Ｂ坦白，Ａ坦白的话判８年，不坦白的话判１０年，所以，坦白还是比不坦白好。 这样，坦白就是Ａ占优战略；同样，坦白也是Ｂ的占优战略．结果是，每个人都选择坦白，各判刑８年． ＜案例二：智猪博弈＞ 这个例子讲的是，猪圈里有两头猪，一大一小．猪圈的一头有一个猪食槽，另一头安装一个按钮，控制着猪食的供应。按一下按钮会有１０个单位的猪食进槽，但谁按按钮需要付２个单位的成本．若大猪先到，大猪吃到９个单位，小猪只能吃１个单位；若同时到，大猪吃７个单位，小猪吃３个单位；若小猪先到，大猪吃６个单位，小猪吃４个单位。表中第一格表示两猪同时按按钮，因而同时走到猪食槽，大猪吃７个，小猪吃３个，扣除２个单位的 成本，支付水平分别为５和１．其他情形可以类推． 在这个例子中，什么是纳什均衡？首先我们注意到，无论大猪选择＂按＂还是＂等待＂，小猪的最优选择均是＂等待＂．比如说给定大猪按，小猪也按时得到１个单位，等待则得到４个单位；给定大猪等待，小猪按得到－１单位，等待则得０单位，所以，＂等待＂是小猪的占优战略．给定小猪总是选择＂等待＂，大猪的最优选择只能是＂按＂．所以，纳什均衡就是：大猪按，小猪等待，各得４个单位．多劳者不多得！ ＜案例三：性别战＞
博弈论 （一）：基本知识 1.1定义:博弈论，又称对策论，是使用严谨的数学模型研究冲突对抗条件下最优决策问题的理论，是研究竞争的逻辑和规律的数学分支。即，博弈论是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，以及不同决策主体之间的均衡。 1.2基本要素：参与人、各参与人的策略集、各参与人的收益函数，是博弈最重要的基本要素。 1.3博弈的分类：博弈论根据其所采用的假设不同而分为合作博弈理论和非合作博弈理论。两者的区别在于参与人在博弈过程中是否能够达成一个具有约束力的协议（binding agreement）。倘若不能，则称非合作博弈（Non-cooperative game）。 合作博弈强调的是集体主义，团体理性，是效率、公平、公正；而非合作博弈则主要研究人们在利益相互影响的局势中如何选择策略使得自己的收益最大，强调个人理性、个人最优决策，其结果有时有效率，有时则不然。目前经济学家谈到博弈论主要指的是非合作博弈，也就是各方在给定的约束条件下如何追求各自利益的最大化，最后达到力量均衡。 博弈的划分可以从参与人行动的次序和参与人对其他参与人的特征、战略空间和支付的知识、信息，是否了解两个角度进行。把两个角度结合就得到了4种博弈： a、完全信息静态博弈，纳什均衡，Nash(1950) b、完全信息动态博弈，子博弈精炼纳什均衡，泽尔腾（1965） c、不完全信息静态博弈，贝叶斯纳什均衡，海萨尼（1967-1968） d、不完全信息动态博弈，精炼贝叶斯纳什均衡，泽尔腾（1975）Kreps, Wilson(1982) Fudenberg, Tirole(1991) 1.4课程主要内容：完全信息静态博弈完全信息动态博弈不完全信息静态博弈机制设计合作博弈 1.5博弈模型的两种表示形式：策略式表述(Strategic form), 扩展式表述（Extensive form） 1.6占优均衡： a、占优策略：在博弈中如果不管其他参与人选择什么策略，一个参与人的某个策略给他带来的支付值始终高于其他策略，或至少不劣于其他策略，则称该策略为该参与人的严格占优策略或占优策略。 对于所有的s-i，si*称为参与人 i的严格占优战略，如果满足： ui(si*,s-i)>ui(si',s-i) ?s-i, ?si' ?si* b、占优均衡：一个博弈的某个策略组合中，如果对应的所有策略都是各参与人的占优策略，则称该策略组合为该博弈的一个占优均衡。 1.7重复剔除严劣策略均衡： a、“严劣”和“弱劣”的含义： 设s i’和s i’’是参与人i可选择的两个策略，若对其他参与人的任意策略组合s-i, 均成立 u i(s i’, s-i) < u i(s i’’, s-i), 则说策略s i’严劣于策略s i’’。 上面式子中，若将“<”改为“≤”，则说策略s i’弱劣于策略s i’’。 b、定义：重复剔除严格策略就是 各参与人在其各自策略集中， 不断剔除严劣策略…如果最终 各参与人仅剩下一个策略，则 该策略组合就被称为重复剔除 严劣策略均衡。 （二）：纳什均衡（Nash Equilibrium） 2.1纳什均衡定义：对于一个策略式表述的博弈G={N,S i, u i,i∈N}，称策略组合s*=(s1, …s i, …, s n)是一个纳什均衡，如果对于每一个i ∈N, s i*是给定其他参与人选择s-i*={s1*, … ,s i-1*, s i+1*, … ,s n*} 情况下参与人i 的最优策略（经济理性策略），即：u i(s i*, s-i*)
9.2 完全信息静态博弈 9.2.1 博弈的战略式表述 Definition A normal (strategic) form game G consists of: (1) a finite set of agent s {1,2,,}D n = . (2) strategy sets 12,,,n S S S . (3) payoff functions 12:(1,2,,)i n u S S S R i n ???→= . 囚徒B 囚徒A 完全信息静态博弈是一种最简单的博弈，在这种博弈中，战略和行动是一回事。 博弈分析的目的是预测博弈的均衡结果，即给定每个参与人都是理性的，什么是每个参与人的最优战略？什么是所有参与人的最优战略组合？ 纳什均衡是完全信息静态博弈解的一般概念，也是所有其他类型博弈解的基本要求。 下面，我们先讨论纳什均衡的特殊情况，然后讨论其一般概念。 9.2.2 占优战略（Dominated Strategies ）均衡 一般说来，由于每个参与人的效用（支付）是博弈中所有参与人的战略的函数，因此， 每个参与人的最优战略选择依赖于所有其他参与人的战略选择。但是在一些特殊的博弈中，一个参与人的最优战略可能并不依赖于其他参与人的战略选择。也就是说，不管其他参与人选择什么战略，他的最优战略是唯一的，这样的最优战略被称为“占优战略”。 Definition Strategy s i is strictly dominated for player i if there is some i i s S '∈ such that (,)(,)i i i i i i u s s u s s --'> for al i i s S --∈. Proposition a rational player will not play a strictly dominated strategy. 抵赖 is a dominated strategy. A rational player would therefore never 抵赖. This solves the game since every player will 坦白. Notice that I don't have to know anything about the other player . 囚徒困境：个人理性与集体理性之间的矛盾。 This result highlights the value of commitment in the Prisoner's dilemma – commitment consists of credibly playing strategy 抵赖. 囚徒困境的广泛应用：军备竞赛、卡特尔、公共品的供给。 9.2.3 Iterated Deletion of Dominated Strategies (重复剔除劣战略) 智猪博弈（boxed pigs ）
博弈论(2)—讲义
9.2 完全信息静态博弈 9.2.1 博弈的战略式表述 Definition A normal (strategic) form game G consists of: (1) a finite set of agent s {1,2,,}D n =L . (2) strategy sets 12,,,n S S S L . (3) payoff functions 12:(1,2,,)i n u S S S R i n ???→=L L . 囚徒B 囚徒A 完全信息静态博弈是一种最简单的博弈，在这种博弈中，战略和行动是一回事。 博弈分析的目的是预测博弈的均衡结果，即给定每个参与人都是理性的，什么是每个参与人的最优战略？什么是所有参与人的最优战略组合？ 纳什均衡是完全信息静态博弈解的一般概念，也是所有其他类型博弈解的基本要求。 下面，我们先讨论纳什均衡的特殊情况，然后讨论其一般概念。 9.2.2 占优战略（Dominated Strategies ）均衡 一般说来，由于每个参与人的效用（支付）是博弈中所有参与人的战略的函数，因此，每个参
与人的最优战略选择依赖于所有其他参与人的战略选择。但是在一些特殊的博弈中，一个参与人的最优战略可能并不依赖于其他参与人的战略选择。也就是说，不管其他参与人选择什么战略，他的最优战略是唯一的，这样的最优战略被称为“占优战略”。 Definition Strategy s i is strictly dominated for player i if there is some i i s S '∈ such that (,)(,)i i i i i i u s s u s s --'> for al i i s S --∈. Proposition a rational player will not play a strictly dominated strategy. 抵赖 is a dominated strategy. A rational player would therefore never 抵赖. This solves the game since every player will 坦白. Notice that I don't have to know anything about the other player . 囚徒困境：个人理性与集体理性之间的矛盾。 This result highlights the value of commitment in the Prisoner's dilemma – commitment consists of credibly playing strategy 抵赖. 囚徒困境的广泛应用：军备竞赛、卡特尔、公共品的供给。