CH10 博弈論與寡頭壟斷.doc

CH10 博弈論與寡頭壟斷一、寡頭壟斷市場市場上存在許多競爭者，但是他們的數(shù)量還沒有多到足以將每個競爭者對價格的影響忽略不計。分析寡頭市場的最大困難在于策略問題。在此情形下，市場上僅有幾家企業(yè)，每一家企業(yè)在做決策時，都必須在一定程度上考慮其它企業(yè)的行為。博弈論就是用以研究策略選擇的一種主要的工具。由于處于寡頭壟斷環(huán)境中的廠商有不同的行為方式，因此，相應(yīng)地，這里也存在幾種相關(guān)的模型。同時設(shè)定產(chǎn)量：古諾模型同時設(shè)定價格：伯特蘭模型產(chǎn)量領(lǐng)導(dǎo)模型：價格領(lǐng)導(dǎo)模型：二、博弈論的基本概念在一些情況下，個人或企業(yè)必須作出策略性選擇，并且最終的結(jié)果依賴于每一個行動者的選擇，這種情況就可以看成是一個博弈。1博弈的三要素任何一個博弈都必須具備三個要素：（1）博弈的參與者參與人的具體身份無關(guān)緊要，在博弈中沒有“好人”與“壞蛋”之分，我們只是簡單地假設(shè)每個參與者在考慮到對手行為的前提下，做出最有利的策略性選擇，或者說，我們假設(shè)博弈的參與者是理性的。（2）策略策略是博弈參與者的行動規(guī)則，也就是說，在什么樣的情況下，選擇什么樣的行動。在非合作博弈中，參與者之間不能就策略選擇達(dá)成一個有約束力的協(xié)議。（3）支付（payoffs）支付是參與者的最終受益。支付包括了與博弈結(jié)果相關(guān)的所有方面，既包括顯性的貨幣報酬，也包括隱性的參與者關(guān)于結(jié)果的心理感受。2. 符號兩個參與者（A和B）之間的博弈G用下式表示 其中，和分別表示參與者A和參與者B的可選策略，和分別表示當(dāng)參與者A和B分別選擇策略a和策略b時，各自所得到的支付（）。3博弈的四種類型對博弈劃分的兩個角度。第一個角度是參與人行動的先后順序。從這個角度，博弈可以劃分為靜態(tài)博弈和動態(tài)博弈（序貫博弈）。靜態(tài)博弈指的是在博弈中，參與人同時選擇行動，或雖非同時行動，但后行動者并不知道先行動者采取了什么具體的行動；動態(tài)博弈是指博弈參與人的行動有先后順序，且后行動者能夠觀察到先行動者所選擇的行動。劃分博弈的第二個角度是參與人對有關(guān)其它參與人（對手）的特征、戰(zhàn)略空間以及支付函數(shù)的知識。從這個角度，博弈可以劃分為完全信息博弈和不完全信息博弈。如果每一個參與人對所有對手的特征、戰(zhàn)略空間和支付函數(shù)都有準(zhǔn)確的知識，該博弈就是完全信息的，否則，就是不完全信息。將上述兩個角度結(jié)合起來，我們就得到四種不同類型的博弈：完全信息靜態(tài)博弈、完全信息動態(tài)博弈、不完全信息靜態(tài)博弈和不完全信息動態(tài)博弈。三、一個例子兩個廠商（A和B）決定自己花多少錢用于做廣告。每個廠商可以選擇較高的預(yù)算（H）或較低的預(yù)算（L）。1博弈的擴(kuò)展式表述圖13.12博弈的策略式（規(guī)范式）表述表13.1四、占優(yōu)均衡與囚徒困境1囚徒困境博弈囚徒B坦白抵賴囚徒A坦白-3，-30.5，-10抵賴-10，0.5-2，-2表13.3 囚徒困境2占優(yōu)策略一個參與人的最優(yōu)戰(zhàn)略選擇并不依賴于其他參與人的戰(zhàn)略選擇，也就是說，不論其他參與選擇什么戰(zhàn)略，他的最優(yōu)戰(zhàn)略是唯一的，這樣的最優(yōu)戰(zhàn)略被稱為“占優(yōu)策略”。 定義：是參與人i的占優(yōu)均衡，當(dāng)且僅當(dāng)對于所有的以及，都有，其中3占優(yōu)策略均衡在一個博弈中，如果所有參與人都有占優(yōu)策略，那么，這些占優(yōu)策略的組合就構(gòu)成一個均衡，這就是占優(yōu)策略均衡。占優(yōu)策略均衡只要求每個參與人是理性的，而不要求每個參與人都知道其他參與人是理性的。這時因為，不論其他參與人是否理性，占優(yōu)策略總是一個理性參與人的最優(yōu)選擇。4囚徒困境博弈的均衡抵賴 is a dominated strategy. A rational player would therefore never 抵賴. This solves the game since every player will 坦白. Notice that I dont have to know anything about the other player. 囚徒困境博弈體現(xiàn)了個人理性與集體理性之間的矛盾。This result highlights the value of commitment in the Prisoners dilemma commitment consists of credibly playing strategy 抵賴. 囚徒困境的廣泛應(yīng)用：軍備競賽、卡特爾、公共品的供給。五、Nash均衡市場均衡：在均衡價格和產(chǎn)量下，買方和賣方都沒有動力去改變自己的行為。也就是說，在完全競爭的市場上，供需雙方僅根據(jù)價格p作決策，而不需要考慮對方的反應(yīng)。1Nash均衡對于策略組合（），如果給定其它參與者的策略，沒有一個參與者會選擇單方面偏離，那么這個策略組合就構(gòu)成一個Nash均衡。也就是說 對于所有 （A） 對于所有注意： 在上面的定義中，不能寫成 ，對于所有， 雙方同時偏離 對于有3個參與人的博弈，上面的條件（A）變成 對于所有 僅有A偏離 而不能寫成 ，對于所有， 兩方同時偏離2對納什均衡的理解設(shè)想所有參與者在博弈之前達(dá)成一個（沒有約束力的）協(xié)議，規(guī)定每個參與人選擇一個特定的戰(zhàn)略。那么，給定其他參與人都遵守此協(xié)議，是否有人不愿意遵守此協(xié)議？如果沒有參與人有積極性單方面背離此協(xié)議，我們說這個協(xié)議是可以自動實施的（self-enforcing），這個協(xié)議就構(gòu)成一個納什均衡。否則，它就不是一個納什均衡。3應(yīng)用從表13.1可以看出，低預(yù)算（L）是廠商B的占優(yōu)策略，即不管廠商A選擇哪一種策略，L都是廠商B的最佳選擇。由于該博弈的結(jié)構(gòu)是公共知識，廠商A也知道L是廠商B的占優(yōu)策略，所以廠商A將選擇L。因此，該博弈的均衡是（L，L）。請驗證（L，L）構(gòu)成一個Nash均衡，而其它三個策略組合都不是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。在下圖所示的猜謎博弈中，沒有上述意義上的Nash均衡存在；而在“性別之戰(zhàn)”博弈中，存在兩個Nash均衡。兒童BH(正面)T(反面)兒童AH(正面)1,-1-1,1T(反面)-1,11,-1表13.2 猜謎博弈表13.3 “性別之戰(zhàn)”Nash均衡不存在的一個主要原因是參與人的策略較少，缺乏靈活性。在以下兩種情況下，參與者的潛在策略數(shù)無窮大，就可以保證博弈至少存在一個均衡：（1）參與者的策略是某一區(qū)間內(nèi)的連續(xù)變量（比如廠商對產(chǎn)量或價格的選擇）；（2）參與者使用混合策略以一定的概率選擇某種概率。相應(yīng)地，以概率1選擇某種行動的策略叫做“純策略”。下面，我們來求解“猜謎博弈”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第一種方法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（反應(yīng)函數(shù)） 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 （反應(yīng)函數(shù)）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第二種方法在給定參與人B采用混合戰(zhàn)略的情況下，如果混合戰(zhàn)略是參與人A的最優(yōu)選擇，必有。同樣的，在給定參與人A采用混合戰(zhàn)略的情況下，如果混合戰(zhàn)略是參與人B的最優(yōu)選擇，必有。這樣，混合策略Nash均衡就可以由以下兩式得到 即 這樣很容易就可以得到上面的混合策略Nash均衡。作業(yè)：求“性別之戰(zhàn)”博弈的混合策略Nash均衡。七、Nash均衡在寡頭市場中的應(yīng)用1古諾競爭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合謀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伯特蘭競爭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.問題：如果，均衡是什么？八、動態(tài)博弈參與人A首先行動，參與人B在觀察到參與人A的行動以后，再行動。相對于動態(tài)博弈，參與者同時行動，或者雖然行動上又先后，但是后行動者不能觀察到先行動者的選擇的博弈叫做靜態(tài)博弈。1. 擴(kuò)展式表述圖13.22策略式表述表13.4請注意企業(yè)B的策略和策略空間（策略集）。策略是對行動規(guī)則的完備描述，由于在參與者A選擇以后，參與者B可能面臨兩種情況參與者A選擇了H或者選擇了L，因此，參與者B的策略就要分別描述這兩種情況下自己的行動選擇。以策略（L，H）為例，它表示如果參與人A選擇了L，那么參與人B就選擇L，如果參與人A選擇了H，那么參與人B也選擇H。3. 均衡（1）Nash均衡根據(jù)表13.4的策略式表述，該博弈一共有3個純策略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)為例，參與人A之所以選擇H，是因為他相信如果自己選擇了L，參與人B將選擇H。但是很明顯，參與人B的這一“威脅”并不可信。因為一旦參與人A真的選擇了L，對參與人B來說，最有選擇是L而不是H。策略組合L, (L, H)同樣存在這一問題。只有L, (L, L)是唯一合理的均衡。因此，Nash均衡不能剔除動態(tài)博弈中的不可信威脅。為了得到更為合理的均衡，我們需要更強(qiáng)的均衡概念。（2）子博弈精煉（完備）均衡（Subgame Perfect Equilibrium，SPE）為了剔除不可置信威脅，我們先看從B下點(diǎn)開始的博弈。在該博弈中，如果參與人B選擇L，他得到的payoff為4，如果選H，payoff為3，因此，他將選擇L。同樣的，在從B上點(diǎn)開始的博弈中，參與人B選擇L所得到的payoff為5，選H得到的payoff為4，因此，他將選擇L。給定參與人B在兩個結(jié)點(diǎn)上的選擇，如果參與人A選L，其payoff為7；如果選H，其payoff為5，因此，他將選擇L。這樣，我們就得到了（L，（L，L）這一合理的結(jié)果，而將其余兩個不合理的結(jié)果剔除掉了。定義：一個擴(kuò)展式博弈的子博弈由單個決策結(jié)x和該決策結(jié)的所有后續(xù)結(jié)（包括終點(diǎn)結(jié)）組成，它不能切割原博弈的信息集。圖13.2中的博弈包含3個子博弈原博弈本身以及由參與者B的兩個決策結(jié)點(diǎn)開始的兩個子博弈；而圖13.1中的博弈僅包含1個子博弈，那就是原博弈本身。定義：如果策略組合在每一個子博弈上都構(gòu)成Nash均衡，那么，該策略組合構(gòu)成子博弈精煉（完備）均衡。運(yùn)用逆向歸納法，我們可以剔除不可信威脅，得到子博弈精煉均衡L, (L, L)。4對策略的理解從這個例子中，我們更能看出為什么要求策略對參與人的行動選擇規(guī)則（計劃）的描述必須是完備的。在策略組合L, (L, L)中，參與人A之所以選擇L，是因為他相信B在后續(xù)的結(jié)點(diǎn)上會選擇L，而不是H。因此，為了支持A選擇L，我們就不僅需要知道參與人B在均衡路徑上的選擇，對于非均衡路徑上的選擇也必須做出完整的描述。另外一個例子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、子博弈精煉均衡在寡頭市場中的應(yīng)用1產(chǎn)量領(lǐng)導(dǎo)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價格領(lǐng)導(dǎo)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