CH10 博弈论与寡头垄断.doc

CH10 博弈论与寡头垄断一、寡头垄断市场市场上存在许多竞争者，但是他们的数量还没有多到足以将每个竞争者对价格的影响忽略不计。分析寡头市场的最大困难在于策略问题。在此情形下，市场上仅有几家企业，每一家企业在做决策时，都必须在一定程度上考虑其它企业的行为。博弈论就是用以研究策略选择的一种主要的工具。由于处于寡头垄断环境中的厂商有不同的行为方式，因此，相应地，这里也存在几种相关的模型。同时设定产量：古诺模型同时设定价格：伯特兰模型产量领导模型：价格领导模型：二、博弈论的基本概念在一些情况下，个人或企业必须作出策略性选择，并且最终的结果依赖于每一个行动者的选择，这种情况就可以看成是一个博弈。1博弈的三要素任何一个博弈都必须具备三个要素：（1）博弈的参与者参与人的具体身份无关紧要，在博弈中没有“好人”与“坏蛋”之分，我们只是简单地假设每个参与者在考虑到对手行为的前提下，做出最有利的策略性选择，或者说，我们假设博弈的参与者是理性的。（2）策略策略是博弈参与者的行动规则，也就是说，在什么样的情况下，选择什么样的行动。在非合作博弈中，参与者之间不能就策略选择达成一个有约束力的协议。（3）支付（payoffs）支付是参与者的最终受益。支付包括了与博弈结果相关的所有方面，既包括显性的货币报酬，也包括隐性的参与者关于结果的心理感受。2. 符号两个参与者（A和B）之间的博弈G用下式表示 其中，和分别表示参与者A和参与者B的可选策略，和分别表示当参与者A和B分别选择策略a和策略b时，各自所得到的支付（）。3博弈的四种类型对博弈划分的两个角度。第一个角度是参与人行动的先后顺序。从这个角度，博弈可以划分为静态博弈和动态博弈（序贯博弈）。静态博弈指的是在博弈中，参与人同时选择行动，或虽非同时行动，但后行动者并不知道先行动者采取了什么具体的行动；动态博弈是指博弈参与人的行动有先后顺序，且后行动者能够观察到先行动者所选择的行动。划分博弈的第二个角度是参与人对有关其它参与人（对手）的特征、战略空间以及支付函数的知识。从这个角度，博弈可以划分为完全信息博弈和不完全信息博弈。如果每一个参与人对所有对手的特征、战略空间和支付函数都有准确的知识，该博弈就是完全信息的，否则，就是不完全信息。将上述两个角度结合起来，我们就得到四种不同类型的博弈：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和不完全信息动态博弈。三、一个例子两个厂商（A和B）决定自己花多少钱用于做广告。每个厂商可以选择较高的预算（H）或较低的预算（L）。1博弈的扩展式表述图13.12博弈的策略式（规范式）表述表13.1四、占优均衡与囚徒困境1囚徒困境博弈囚徒B坦白抵赖囚徒A坦白-3，-30.5，-10抵赖-10，0.5-2，-2表13.3 囚徒困境2占优策略一个参与人的最优战略选择并不依赖于其他参与人的战略选择，也就是说，不论其他参与选择什么战略，他的最优战略是唯一的，这样的最优战略被称为“占优策略”。 定义：是参与人i的占优均衡，当且仅当对于所有的以及，都有，其中3占优策略均衡在一个博弈中，如果所有参与人都有占优策略，那么，这些占优策略的组合就构成一个均衡，这就是占优策略均衡。占优策略均衡只要求每个参与人是理性的，而不要求每个参与人都知道其他参与人是理性的。这时因为，不论其他参与人是否理性，占优策略总是一个理性参与人的最优选择。4囚徒困境博弈的均衡抵赖 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 dont have to know anything about the other player. 囚徒困境博弈体现了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、卡特尔、公共品的供给。五、Nash均衡市场均衡：在均衡价格和产量下，买方和卖方都没有动力去改变自己的行为。也就是说，在完全竞争的市场上，供需双方仅根据价格p作决策，而不需要考虑对方的反应。1Nash均衡对于策略组合（），如果给定其它参与者的策略，没有一个参与者会选择单方面偏离，那么这个策略组合就构成一个Nash均衡。也就是说 对于所有 （A） 对于所有注意： 在上面的定义中，不能写成 ，对于所有， 双方同时偏离 对于有3个参与人的博弈，上面的条件（A）变成 对于所有 仅有A偏离 而不能写成 ，对于所有， 两方同时偏离2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。3应用从表13.1可以看出，低预算（L）是厂商B的占优策略，即不管厂商A选择哪一种策略，L都是厂商B的最佳选择。由于该博弈的结构是公共知识，厂商A也知道L是厂商B的占优策略，所以厂商A将选择L。因此，该博弈的均衡是（L，L）。请验证（L，L）构成一个Nash均衡，而其它三个策略组合都不是Nash均衡。六、混合策略Nash均衡1概念的引出上面的博弈存在唯一的Nash均衡，但是并非所有博弈都是如此。在下图所示的猜谜博弈中，没有上述意义上的Nash均衡存在；而在“性别之战”博弈中，存在两个Nash均衡。儿童BH(正面)T(反面)儿童AH(正面)1,-1-1,1T(反面)-1,11,-1表13.2 猜谜博弈表13.3 “性别之战”Nash均衡不存在的一个主要原因是参与人的策略较少，缺乏灵活性。在以下两种情况下，参与者的潜在策略数无穷大，就可以保证博弈至少存在一个均衡：（1）参与者的策略是某一区间内的连续变量（比如厂商对产量或价格的选择）；（2）参与者使用混合策略以一定的概率选择某种概率。相应地，以概率1选择某种行动的策略叫做“纯策略”。下面，我们来求解“猜谜博弈”的混合策略Nash均衡。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.2第一种方法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 As expected utility is then given by Oviously, 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 matter what value of r is choosen. As best response function（反应函数） is For player B, expected utility is given by Now, when , Bs expected utility is maximized by choosing . If , A should opt for . And when , As expected utility is independent of what s is choosen. Bs best response function （反应函数）is 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 3第二种方法在给定参与人B采用混合战略的情况下，如果混合战略是参与人A的最优选择，必有。同样的，在给定参与人A采用混合战略的情况下，如果混合战略是参与人B的最优选择，必有。这样，混合策略Nash均衡就可以由以下两式得到 即 这样很容易就可以得到上面的混合策略Nash均衡。作业：求“性别之战”博弈的混合策略Nash均衡。七、Nash均衡在寡头市场中的应用1古诺竞争Suppose there are two firms in an industry. Their strategy spaces are quantities. Their payoffs are profits.Industry demand is given by the inverse demand function, P(Q), where industry production is . They have identical cost functions . Profits for each firm are therefore given by: and This is a game. The Nash equilibrium occurs when both firms are optimising given the behaviour of the other.Throughout the lecture consider the following linear demand example with constant marginal costs. So: and Profits and Best ResponsesProfits for each firm are maximised where marginal revenue is equal to marginal cost. Recall firms are interested in finding an optimal level of their quantity for each level their opponent might choose. Suppose firm 2 chooses : acts like a constant. For linear demand, marginal revenue falls at twice the rate of demand.Firm 1 sets marginal revenue, , equal to marginal cost, c. Hence: and These two equations give the best response functions for the two firms. Often called reaction or best reply functions. Plotting these yield the reaction or best reply curves.Reaction CurvesDrawing the reaction curves for both firms on the same graph yields the picture below.The curve yields the optimal level of for any given . The curve yields the optimal level of for any given . These curves will cross. Why?A point at which there is no profitable unilateral deviation is a Nash equilibrium. That is a point which is a best response to a best response (and so on) where the two curves cross, written .Nash EquilibriumWhat is the value of and ? They could be read off from the graph. Alternatively, solve the two equations.Substituting the value of into the equation for yields: Symmetrically solving for gives .With the same linear demand and constant marginal cost assumptions in place but with n firms in the industry it is (mathematically) simple to show that each of the n firms will produce at:Equilibrium Profits and PricesHow much do the firms charge and how much profit do they make? Price is simply read off from the demand curve.Equilibrium industry supply is . Hence equilibrium price is:Profit is given by revenue less cost: .It is just as easy to calculate the price and profit in the case of n firms: and Notice that, as the number of firms grows, price gets closer to marginal cost and profits get close to zero. These are the conditions in a perfectly competitive market. An oligopoly with many firms is like perfect competition.2合谋How does the case of Cournot duopoly differ from monopoly? If the two firms could collude they would act like a monopoly to maximise total profits. Recall a monopolist faces the entire demand curve and sets MR = MC.With linear demand and constant marginal costs c, the optimality condition requires: A monopolist produces less and hence prices higher at . Profits are .If the two firms could collude they would be able to split the profits in two, each firm getting by producing . This is bigger than their Cournot equilibrium profit.But they cannot. If one of the firms produced the other would not choose to produce the same. The best response function reveals that the firm has a better response where if (for example) firm 1 produced :This will yield higher profits. How can collusion be explained?3伯特兰竞争Suppose instead firms choose prices does this make a difference?Firm 1 and firm 2 choose and simultaneously. The firm that charges the lower price serves the entire market. If both firms charge the same price, they both serve half the market.Profits are payoffs. Writing down profits, suppose that market demand is Q(P) where P is market price (the lower of the two prices, and ). Suppose again that marginal costs are constant, .To illustrate the best reply functions consider what firm 1s optimal response is if firm 2 sets a price . If is greater than marginal cost then firm 1 will wish to undercut firm 2 by a small amount.If is equal to marginal cost, firm 1 would be willing to set any price greater than or equal to all such prices will result in zero profits. Firm 1 will never set a price below if is less than c since this results in losses. In this last case, firm 1 would set any price strictly greater than and get zero profits.Reaction Curves and Nash EquilibriumDrawing this argument in an informal way gives the below best response functions.The only place where the two curves cross the Nash equilibrium is at .The firms price at marginal cost the efficient outcome. It makes no difference how many firms are in the market.The basic idea is that the firms will continue to undercut one another until they reach marginal cost. They will go no lower as this would involve making a loss.This is very different to Cournot. However, the Cournot equilibrium can be recovered with capacity constraints.问题：如果，均衡是什么？八、动态博弈参与人A首先行动，参与人B在观察到参与人A的行动以后，再行动。相对于动态博弈，参与者同时行动，或者虽然行动上又先后，但是后行动者不能观察到先行动者的选择的博弈叫做静态博弈。1. 扩展式表述图13.22策略式表述表13.4请注意企业B的策略和策略空间（策略集）。策略是对行动规则的完备描述，由于在参与者A选择以后，参与者B可能面临两种情况参与者A选择了H或者选择了L，因此，参与者B的策略就要分别描述这两种情况下自己的行动选择。以策略（L，H）为例，它表示如果参与人A选择了L，那么参与人B就选择L，如果参与人A选择了H，那么参与人B也选择H。3. 均衡（1）Nash均衡根据表13.4的策略式表述，该博弈一共有3个纯策略Nash均衡：L, (L, L)、L, (L, H)和H, (H,L)。其中，L, (L, H)和H, (H,L)并不合理。以H, (H,L)为例，参与人A之所以选择H，是因为他相信如果自己选择了L，参与人B将选择H。但是很明显，参与人B的这一“威胁”并不可信。因为一旦参与人A真的选择了L，对参与人B来说，最有选择是L而不是H。策略组合L, (L, H)同样存在这一问题。只有L, (L, L)是唯一合理的均衡。因此，Nash均衡不能剔除动态博弈中的不可信威胁。为了得到更为合理的均衡，我们需要更强的均衡概念。（2）子博弈精炼（完备）均衡（Subgame Perfect Equilibrium，SPE）为了剔除不可置信威胁，我们先看从B下点开始的博弈。在该博弈中，如果参与人B选择L，他得到的payoff为4，如果选H，payoff为3，因此，他将选择L。同样的，在从B上点开始的博弈中，参与人B选择L所得到的payoff为5，选H得到的payoff为4，因此，他将选择L。给定参与人B在两个结点上的选择，如果参与人A选L，其payoff为7；如果选H，其payoff为5，因此，他将选择L。这样，我们就得到了（L，（L，L）这一合理的结果，而将其余两个不合理的结果剔除掉了。定义：一个扩展式博弈的子博弈由单个决策结x和该决策结的所有后续结（包括终点结）组成，它不能切割原博弈的信息集。图13.2中的博弈包含3个子博弈原博弈本身以及由参与者B的两个决策结点开始的两个子博弈；而图13.1中的博弈仅包含1个子博弈，那就是原博弈本身。定义：如果策略组合在每一个子博弈上都构成Nash均衡，那么，该策略组合构成子博弈精炼（完备）均衡。运用逆向归纳法，我们可以剔除不可信威胁，得到子博弈精炼均衡L, (L, L)。4对策略的理解从这个例子中，我们更能看出为什么要求策略对参与人的行动选择规则（计划）的描述必须是完备的。在策略组合L, (L, L)中，参与人A之所以选择L，是因为他相信B在后续的结点上会选择L，而不是H。因此，为了支持A选择L，我们就不仅需要知道参与人B在均衡路径上的选择，对于非均衡路径上的选择也必须做出完整的描述。另外一个例子In the following game，a strategy of player 1 specifies both the action she takes at the beginning of the game and the action she takes after the history (A, C), even if the action she takes at the beginning of the game is B. The reason is that player 1s strategy at the last decision node will affect player 2s decision, which inturn will affect player 1s decision at the initial node.九、子博弈精炼均衡在寡头市场中的应用1产量领导Suppose that firm 1 (the leader) is already in the market. Firm 2 (the follower) is about to enter.This situation is known as Stackelberg leadership. It is a different game and has a different equilibrium.Firm 2 has a relatively simple problem to solve. Knowing that firm 1 has already produced firm 2 simply chooses their quantity to maximise their profits. Using the best response function reveals the action firm 2 will take.Working backwards: What should firm 1 do? Firm 1 now knows which quantity firm 2 will choose and how it depends upon their own choice. Therefore, including this information in their profit function, firm 1 optimises.Profits and Best ResponsesSuppose firm 1 chooses . Using the same example, firm 2 will choose to maximise . The reaction curve gives the solution to this problem. Hence, firm 2 chooses:Firm 1 knows this and uses this information to when maximising their profit. Firm 1 has profit equal to: This is exactly half the profit a monopolist would receive. Hence firm 1, setting MR = MC, operates just like a monopolist and produces the monopoly output and receives half the profit .Firm 2s output is worked out from the best response function yielding: .EquilibriumThe equilibrium is characterised by firm one producing the monopoly output and firm 2 playing a best response . Equilibrium price is therefore .Finally, equilibrium profits are and .Compare this with Cournot. The leader does better. The follower does worse. Price is lower. Total output is larger.Compare this with collusion. The leader does as well. The follower does worse. Price is lower. Output is larger.However, like Cournot, comparing this to perfect competition reveals both firms do better than perfectly competitive firms who get zero profit. Prices are higher and output is lower.2价格领导Consider the following price leadership game note the similarity with Stackelberg leadership. Note the difference between this and Varians rather odd game, where profits accrue to firm 1 during the game.Again, working from the back: Firm 2 will choose to undercut firm 1s initial price in order to gain the whole market.Firm 1, unable to make positive profits, can choose any price above marginal cost. Either firm 2 will undercut and make positive profits or, if