博弈论2讲义Word下载.docx

1、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 si is strictly dominated for player i if there is some such that for al Proposition a rational player will not play a strictly dominated strategy.抵赖 is a dominated strategy. A rational player w

2、ould therefore never 抵赖. This solves the game since every player will 坦白. Notice that I dont 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 Prisoners dilemma commitment consists of credibly playing strategy 抵赖. 囚徒困境的广泛应用：军备竞赛、卡

3、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）小猪按等待大猪3,12,47,-10,0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 equ

4、ilibrium性别战博弈（battle of the sexes）:女足球赛演唱会男2,11,2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 for all players i and all 求解Nash均衡的方法。A Nash equilibrium captures the idea of equilibrium: Both player

5、s 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 players 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

6、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 qi，cost function ci （qi） = cqi. market demand determines a price ：the products of both firms are

7、 perfect substitutes, i.e. they are homogenous products.D = 1; 2S1 = S2 = R+u1 （q1, q2） = q1 f （q1 + q2） -c1 （q1）u2 （q1, q2） = q2 f （q1 + q2） - c2 （q2）the best-response function BR（qj） of each firm i to the quantity choice qj of the other firm:由，得FOC:；又。因The best-response function is decreasing in m

8、y belief of the other firms 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 q1 = q2 = q*.Hence we obtain, This gives us the solution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 Bertr

9、and 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 We also assume that D（c） 0, i.e. firms can sell a positive quantity if they price at marginal cost.Lemma The Bertran

10、d game has the unique NE = （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. Therefo

11、re, there is no incentive to deviate.To show uniqueness we must show that any other strategy profile （p1; p2） is not a NE. Its easiest to distinguish lots of cases. Case I: p1 c or p2 c. In this case one （or both players） makes negative losses. This player should set a price above his rivals price a

12、nd cut his losses by not selling any output.Case II: c p1 p2 or c p2 p1. In this case the firm with the higher price makes zero profits. It could profitably deviate by setting a price equal to the rivals price and thus capture at least half of his market, and make strictly positive profits. Case III

13、: c = p1 p2 or c = p2 p1. Now the lower price firm can charge a price slightly above marginal cost （but still below the price of the rival） and make strictly positive profits.Case IV: c p1 = p2. Firm 1 could profitably deviate by setting a price . The firms profits before and after the deviation are

14、: Note that the demand function is decreasing, so. We can therefore deduce:This expression （the gain from deviating） is strictly positive for sufficiently small. Therefore, （p1; p2） cannot be a NE.9.2.7 Mixed Strategies （混合战略）猜谜游戏 matching pennies game:儿童BH（正面）T（反面）儿童A1,-1-1,1每一个参与者都想猜透对方的战略，而又不能让对方

15、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 su

16、ffifient 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 play

17、ers 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

18、with the following probabilities: H-H , rs; H-T, r（1-s）; T-H, （1-r）s; T-T,（1-r）（1-s）. Player As expected utility is then given byOviously, As optimal choice of r depends on Bs probability, s. If , utility is maximized by choosing . If , A should opt for . And when , As expected utility is 0 no matte

19、r what value of r is choosen.For player B, expected utility is given byNow, when , Bs expected utility is maximized by choosing , As expected utility is independent of what s is choosen. Nash equilibria are shown in the figure by the intersections of optimal response curves for A and B. Or, we can g

20、et the equilibrium through the FOCDefinition: Let G be a game with strategy spaces S1,S2,.,SI . A mixed strategy for player i is a probability distribution on Si ,i.e. for Si , a mixed strategy is a function Several notations are commonly used for describing mixed strategies.1. Function （measure）: a

21、nd 2. Vector: If the pure strategies are,write e.g.（1/2, 1/2）3. （1/2）H + （1/2）T Write （also ） for the set of probability distributions on Si. Write for . A mixed strategy profile is an n-tuplewith We write for player is expected payoff when he uses mixed strategy and all other players play as in Rem

22、ark: 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

23、these axioms.Definition A mixed strategy NE of G is a mixed profile such thatfor all i and all 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: is a Nash equilibrium

24、if and only if for all i and Example: The strategy profile 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

25、 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

26、move. Example : Model of EntryCurrently 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 pric

27、es. Example II: Stackelberg ModelSuppose 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 q1 before firm 2 starts. Firm 2 then observes firm 1 before picking its output level q2. For concreteness suppose and market de

28、mand is . The marginal cost of production is 0.9.3.2 Definition of an Extensive Form GameFormally 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 （Z is the set of terminal nodes）

29、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 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 satisfyWe sometimes write i（h） and A（h） si