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# Class Note for FINA 6387 at UH

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Chapter 13 Game Theory and Competitive Equilibrium of the other player s choice Every dominant strategy equilibrium is a Nash equilibrium but the reverse does not hold 4 How does a Nash equilibriuln differ from a game s maximin solution In what situations is a maximin solution a more likely outcome than a Nash equilibrium A maximin strategy is one in which each player determines the worst outcome for each of the opponent s actions and chooses the option that maximizes the minimum gain that can be earned Unlike the Nash equilibrium the maximin solution does not require players to react to an opponent s choice If no dominant strategy exists in which case outcomes depend on the opponent s behavior players can reduce the uncertainty inherent in relying on the opponent s rationality by conservatively following a maximin strategy The maximin solution is more likely than the Nash solution in cases where there is a higher probability of irrational nonoptimizing behavior 5 What is a tit for tat strategy Why is it a rational strategy for the infinitely repeated Prisoners Dilemma A player following a titfortat strategy will cooperate as long as his or her opponent is cooperating and will switch to a noncooperative strategy if their opponent switches strategies When the competitors assume that they will be repeating their interaction in every future period the longterm gains from cooperating will outweigh any short term gains from not cooperating Because the titfortat strategy encourages cooperation in infinitely repeated games it is rational 6 Consider a game in which the Prisoners Dilemlna is repeated 10 times and both players are rational and fully informed Is a tit for tat strategy optimal in this case Under what conditions would such a strategy be optimal Since cooperation will unravel from the last period back to the first period the titfor tat strategy is not optimal when there is a finite number of periods and both players anticipate the competitor s response in every period Given that there is no response possible in the eleventh period for action in the tenth and last period cooperation breaks down in the last period Then knowing that there is no cooperation in the last period players should maximize their selfinterest by not cooperating in the secondto last period This unraveling occurs because both players assume that the other player has considered all consequences in all periods However if there is some doubt about whether the opponent has fully anticipated the consequences of the titfortat strategy in the final period the game will not unravel and the titfortat strategy can be optimal 7 Suppose you and your competitor are playing the pricing game shown in Table 138 Both of you must announce your prices at the same time Might you improve your outcome by promising your competitor that you will announce a high price If the game is to be played only a few times there is little to gain If you are Firm 1 and promise to announce a high price Firm 2 will undercut you and you will end up with a payoff of 50 However next period you will undercut too and both firms will earn 10 If the game is played many times there is a better chance that Firm 2 will realize that if it matches your high price the longterm payoff of 50 each period is better than 100 at first and 10 thereafter 8 What is meant by first mover advantage Give an example ofa gaming situation with a rst mover advantage A firstmover advantage can occur in a game where the first player to act receives the highest payoff The firstmover signals his or her choice to the opponent and the opponent must choose a response given this signal The firstmover goes on the offensive and the secondmover responds defensively In many recreational games from chess to football the firstmover has an advantage In many markets the first firm to introduce a product can set the standard for competitors to follow In some 187 Chapter 13 Game Theory and Competitive Equilibrium cases the standardsetting power of the first mover becomes so pervasive in the market that the brand name of the product becomes synonymous with the product eg Kleenex the name of Kleenexbrand facial tissue is used by many consumers to refer to facial tissue of any brand 9 What is a strategic move How can the development ofa certain kind of reputation be a strategic move A strategic move involves a commitment to reduce one s options The strategic move might not seem rational outside the context of the game in which it is played but it is rational given the anticipated response of the other player Random responses to an opponent s action may not appear to be rational but developing a reputation of being unpredictable could lead to higher payoffs in the long run Another example would be making a promise to give a discount to all previous consumers if you give a discount to one Such a move makes the firm vulnerable but the goal of such a strategic move is to signal to rivals that you won t be discounting price and hope that your rivals follow suit 10 Can the threat ofa price war deter entry by potential competitors What actions might a firm take to make this threat credible Both the incumbent and the potential entrant know that a price war will leave their firms worse off Normally such a threat is not credible Thus the incumbent must make his or her threat of a price war believable by signaling to the potential entrant that a price war will result if entry occurs One strategic move is to increase capacity signaling a lower future price and another is to engage in apparently irrational behavior Both types of strategic behavior might deter entry but for different reasons While an increase in capacity reduces expected profits by reducing prices irrational behavior reduces expected profits by increasing uncertainty hence increasing the rate at which future profits must be discounted into the present 11 A strategic move limits one s exibility and yet gives one an advantage Why How might a strategic move give one an advantage in bargaining A strategic move in uences conditional behavior by the opponent If the game is well understood and the opponent s reaction can be predicted a strategic move leaves the player better off Economic transactions involve a bargain whether implicit or explicit In every bargain we assume that both parties attempt to maximize their selfinterest Strategic moves by one player provide signals to which another player reacts If a bargaining game is played only once so no reputations are involved the players might act strategically to maximize their payoffs lf bargaining is repeated players might act strategically to establish reputations for expected negotiations EXERCISES 1 In many oligopolistic industries the same firms compete over a long period of time setting prices and observing each other s behavior repeatedly Given that the number of repetitions is large why don t collusive outcomes typically result If games are repeated indefinitely and all players know all payoffs rational behavior will lead to apparently collusive outcomes ie the same outcomes that would result if firms were actively colluding All payoffs however might not be known by all players Sometimes the payoffs of other firms can only be known by engaging in extensive and costly information exchanges or by making a move and observing rivals responses Also successful collusion encourages entry Perhaps the greatest problem in maintaining a collusive outcome is that changes in market conditions change the collusive price and quantity The firms then have to repeatedly change their agreement on price and quantity which is costly and this increases the ability of one firm to cheat without being discovered 188 Chapter 13 Game Theory and Competitive Equilibrium 2 Many industries are often plagued by overcapacity firms simultaneously make major investments in capacity expansion so total capacity far exceeds demand This happens in industries in which demand is highly volatile and unpredictable but also in industries in which demand is fairly stable What factors lead to overcapacity Explain each brie y In Chapter 12 we found that excess capacity may arise in industries with easy entry and differentiated products In the monopolistic competition model downwardsloping demand curves for each firm lead to output with average cost above minimum average cost The difference between the resulting output and the output at minimum longrun average cost is defined as excess capacity In this chapter we saw that overcapacity could be used to deter new entry that is investments in capacity expansion could convince potential competitors that entry would be unprofitable Note that although threats of capacity expansion may deter entry these threats must be credible 3 Two computer firms A and B are planning to market network systems for of ce information management Each firm can develop either a fast high quality system H or a slower low quality system L Market research indicates that the resulting pro ts to each rm for the alternative strategies are given by the following payoff matrix Firm B H L H 30 30 50 35 WA L 4060 20 20 a If both rms make their decisions at the same time and follow maximin low risk strategies what will the outcome be With a maximin strategy a firm determines the worst outcome for each option then chooses the option that maximizes the payoff among the worst outcomes If Firm A chooses H the worst payoff would occur if Firm B chooses H A s payoff would be 30 If Firm A chooses L the worst payoff would occur if Firm B chooses L A s payoff would be 20 With a maximin strategy A therefore chooses H If Firm B chooses L the worst payoff would occur if Firm A chooses L the payoff would be 20 If Firm B chooses H the worst payoff 30 would occur if Firm A chooses L With a maximin strategy B therefore chooses H So under maximin both A and B produce a highquality system b Suppose both firms try to maximize pro ts but Firm A has a head start in planning and can commit rst Now what will the outcome be What will the outcome be if Firm B has a head start in planning and can commit first If Firm A can commit first it will choose H because it knows that Firm B will rationally choose L since L gives a higher payoff to B 35 vs 30 This gives Firm A a payoff of 50 If Firm B can commit first it will choose H because it knows that Firm A will rationally choose L since L gives a higher payoff to A 40 vs 30 This gives Firm B a payoff of 60 c Getting a head start costs money you have to gear up a large engineering team Now consider the twostage game in which first each rm decides how much money to spend to speed up its planning and second it announces which product H or L it will produce Which rm will spend more to speed up its planning How much will it spend Should the other rm spend anything to speed up its planning Explain In this game there is an advantage to being the first mover If A moves first its profit is 50 If it moves second its profit is 40 a difference of 10 Thus it would be willing to spend up to 10 for the option of announcing first On the other hand if B moves first its profit is 60 If it moves second its profit is 35 a difference of 25 and thus would be 189 Chapter 13 Game Theory and Competitive Equilibrium willing to spend up to 25 for the option of announcing first Once Firm A realizes that Firm B is willing to spend more on the option of announcing first then the value of the option decreases for Firm A because if both firms were to invest both firms would choose to produce the highquality system Therefore Firm A should not spend money to speed up the introduction of its product if it believes that Firm B is spending the money However if Firm B realizes that Firm A will wait Firm B should only spend enough money to discourage Firm A from engaging in research and development which would be an amount slightly more than 10 the maximum amount Ais willing to spend 4 Two rms are in the chocolate market Each can choose to go for the high end of the market high quality or the low end low quality Resulting pro ts are given by the following payoff matrix Firm 2 Low High Low 20 30 900 600 Fmquot 1 High 100 800 5050 a What outcomes if any are Nash equilibria lf Firm 2 chooses Low and Firm 1 chooses High neither will have an incentive to change 100 gt 20 for Firm 1 and 800 gt 50 for Firm 2 If Firm 2 chooses High and Firm 1 chooses Low neither will have an incentive to change 900 gt 50 for Firm 1 and 600 gt 30 for Firm 2 Both outcomes are Nash equilibria b If the manager of each firm is conservative and each follows a maximin strategy what will be the outcome lf Firm 1 chooses Low its worst payoff 20 would occur if Firm 2 chooses Low lf Firm 1 chooses High its worst payoff 50 would occur if Firm 2 chooses High Therefore with a conservative maximin strategy Firm 1 chooses High Similarly if Firm 2 chooses Low its worst payoff 30 would occur if Firm 1 chooses Low lf Firm 2 chooses High its worst payoff 50 would occur if Firm 1 chooses High Therefore with a maximin strategy Firm 2 chooses High Thus both firms choose High yielding a payoff of 50 for both c What is the cooperative outcome The cooperative outcome would maximize joint payoffs This would occur if Firm 1 goes for the low end of the market and Firm 2 goes for the high end of the market The joint payoff is 1500 Firm 1 gets 900 and Firm 2 gets 600 d Which firm bene ts most from the cooperative outcome How much would that rm need to offer the other to persuade it to collude Firm 1 benefits most from cooperation The difference between its best payoff under cooperation and the next best payoff is 900 100 800 To persuade Firm 2 to choose Firm 1 s best option Firm 1 must offer at least the difference between Firm 2 s payoff under cooperation 600 and its best payoff 800 ie 200 However Firm 2 realizes that Firm 1 benefits much more from cooperation and should try to extract as much as it can from Firm 1 up to 800 190 Chapter 13 Game Theory and Competitive Equilibrium 5 Two major networks are competing for viewer ratings in the 800 900 RM and 900 1000 PM slots on a given weeknight Each has two shows to ll this time period and is juggling its lineup Each can choose to put its bigger show rst or to place it second in the 900 1000 PM slot The combination of decisions leads to the following ratings points results Network 2 First Second First 18 18 23 20 Network 1 Second 4 23 1616 a Find the Nash equilibria for this game assuming that both networks make their decisions at the same time A Nash equilibrium exists when neither party has an incentive to alter its strategy taking the other s strategy as given By inspecting each of the four combinations we find that First Second is the only Nash equilibrium yielding a payoff of 23 20 There is no incentive for either party to change from this outcome b If each network is risk averse and uses a maximin strategy what will be the resulting equilibrium This conservative strategy of minimizing the maximum loss focuses on limiting the extent of the worst possible outcome to the exclusion of possible good outcomes lf Network 1 plays First the worst payoff is 18 If Network 1 plays Second the worst payoff is 4 Under maximin Network 1 plays First Here playing First is a dominant strategy If Network 2 plays First the worst payoff is 18 If Network 2 plays Second the worst payoff is 16 Under maximin Network 2 plays First The maximin equilibrium is First First with a payoff of 1818 c What will be the equilibrium if Network 1 can makes its selection rst If Network 2 goes first If Network 1 plays First Network 2 will play Second yielding 23 for Network 1 If Network 1 plays Second Network 2 will play First yielding 4 for Network 1 Therefore if it has the first move Network 1 will play First and the resulting equilibrium will be First Second lf Network 2 plays First Network 1 will play First yielding 18 for Network 2 If Network 2 plays Second Network 1 will play First yielding 20 for Network 2 If it has the first move Network 2 will play Second and the equilibrium will again be First Second d Suppose the network managers meet to coordinate schedules and Network 1 promises to schedule its big show rst Is this promise credible and what would be the likely outcome A move is credible if once declared there is no incentive to change Network 1 has a dominant strategy play the bigger show First In this case the promise to schedule the bigger show first is credible Knowing this Network 2 will schedule its bigger show Second The coordinated outcome is likely to be First Second 191 Chapter 13 Game Theory and Competitive Equilibrium 6 Two competing firms are each planning to introduce a new product Each rm will decide whether to produce Product A Product B or Product C They will make their choices at the same time The resulting payoffs are shown below We are given the following payoff matrix which describes a product introduction game Firm 2 A B C A 1010 010 1020 Firm 1 B 100 2020 515 C 2010 155 3030 a Are there any Nash equilibria in pure strategies If so what are they There are two Nash equilibria in pure strategies Each one involves one firm introducing Product A and the other firm introducing Product C We can write these two strategy pairs as A C and C A where the first strategy is for player 1 The payoff for these two strategies is respectively 1020 and 2010 b If both firms use maximin strategies what outcome will result Recall that maximin strategies maximize the minimum payoff for both players For each of the players the strategy that maximizes their minimum payoff is A Thus AA will result and payoffs will be 1010 Each player is much worse off than at either of the pure strategy Nash equilibrium c If Firm 1 uses a maximin strategy and Firm 2 knows what will Firm 2 do If Firm 1 plays its maximin strategy of A and Firm 2 knows this then Firm 2 would get the highest payoff by playing C Notice that when Firm 1 plays conservatively the Nash equilibrium that results gives Firm 2 the highest payoff of the two Nash equilibria 7 We can think of the US and Japanese trade policies as a Prisoners Dilemma The two countries are considering policies to open or close their import markets Suppose the payoffmatrix is Japan Open Close Open 10 10 5 5 US Close 100 5 1 1 a Assume that each country knows the payoff matrix and believes that the other country will act in its own interest Does either country have a dominant strategy What will be the equilibrium policies if each country acts rationally to maximize its welfare Choosing Open is a dominant strategy for both countries If Japan chooses Open the US does best by choosing Open lf Japan chooses Close the US does best by choosing Open Therefore the US should choose Open no matter what Japan does If the US chooses Open Japan does best by choosing Open If the US chooses Close Japan does best by choosing Open Therefore both countries will choose to have Open policies in equilibrium 192 Chapter 13 Game Theory and Competitive Equilibrium b Now assuIne that Japan is not certain that the US will behave rationally In particular Japan is concerned that US politicians may want to penalize Japan even if that does not maximize US welfare How might this affect Japan s choice of strategy How might this change the equilibrium The irrationality of US politicians could change the equilibrium from Close Open If the US wants to penalize Japan they Will choose Close but Japan s strategy Will not be affected since choosing Open is still J apan s dominant strategy 8 You are a duopolist producer of a homogeneous good Both you and your competitor have zero marginal costs The market demand curve is P30 Q where Q Q1 Q2 Q1 is your output and Q2 is your competitor s output Your competitor has also read this book a Suppose you are to play this game only once If you and your competitor must announce your output at the same time how much will you choose to produce What do you expect your profit to be Explain These are some of the cells in the payoff matrix Firm 2 s Output Firm 1 s 0 5 10 15 20 25 30 Output 0 00 0125 0200 0225 0200 0125 00 5 1250 100100 75150 50150 25100 00 00 10 2000 15075 100100 5075 00 00 00 15 2250 10050 7550 00 00 00 00 20 2000 10025 00 00 00 00 00 25 1250 00 00 00 00 00 00 30 00 00 00 00 00 00 00 If both firms must announce output at the same time both firms believe that the other firm is behaving rationally and each firm treats the output of the other rm as a fixed number a Cournot equilibrium Will result For Firm 1 total revenue Will be 7R 2 30 Q Q2Q or TR1 30Q17Qf inQZ Marginal revenue for Firm 1 Will be the derivative of total revenue With respect to Q1 am 73072 7 aQ Q1 Q2 Because the rms share identical demand curves the solution for Firm 2 Will be symmetric to that of Firm 1 aTR 73072Q 7Q QZ 2 1 To find the profitmaximizing level of output for both firms set marginal revenue equal to marginal cost Which is zero Q1 157 722 and Q1 15 Q2 2 193 Chapter 13 Game Theory and Competitive Equilibrium With two equations and two unknowns we may solve for Q1 and Q2 Q1157 b57p or Q1 10 Substitute Q1 and Q2 into the demand equation to determine price By symmetry Q2 10 P 30 10 10 orP 10 Since no costs are given profits for each firm will be equal to total revenue 11 7R11010 100 and 12 7R2 1010 100 Thus the equilibrium occurs when both firms produce 10 units of output and both firms earn 100 Looking back at the payoff matrix note that the outcome 100 100 is indeed a Nash equilibrium neither firm will have an incentive to deviate given the other firms choice Suppose you are told that you must announce your output before your competitor does How much will you produce in this case and how much do you think your competitor will produce What do you expect your profit to be Is announcing rst an advantage or disadvantage Explain brie y How much would youpay to be given the option of announcing either rst or second If you must announce first you would announce an output of 15 knowing that your competitor would announce an output of 75 Note This is the Stackelberg equilibrium 2 TRIwib1QZgQI30Q17Q57Q1 7k15Q1739 Therefore setting MR MC 0 implies 15 Q1 0 orQ1 15 and Q2 75 At that output your competitor is maximizing profits given that you are producing 15 At these outputs price is equal to 30 15 75 75 Your profit would be 1575 1125 Your competitor s profit would be 7575 5625 Announcing first is an advantage in this game The difference in profits between announcing first and announcing second is 5625 You would be willing to pay up to this difference for the option of announcing first Suppose instead that you are to play the rst round of a series of 10 rounds with the same competitor In each round you and your competitor announce your outputs at the same time You want to maximize the sum of your pro ts over the 10 rounds How much will you produce in the first round How much would you expect to produce in the tenth round The ninth round Explain brie y Given that your competitor has also read this book you can assume that he or she will be acting rationally You should begin with the Cournot output and continue with the Cournot output in each round including the ninth and tenth rounds Any deviation from this output will reduce the sum of your profits over the ten rounds 194 Chapter 13 Game Theory and Competitive Equilibrium d Once again you will play a series of 10 rounds This time however in each round your competitor will announce its output before you announce yours How will your answers to c change in this case If your competitor always announces first it might be more pro table to behave by reacting irrationally in a single period For example in the first round your competitor will announce an output of 15 as in Exercise 7b Rationally you would respond with an output of 75 If you behave this way in every round your total profits for all ten rounds will be 56250 Your competitor s pro ts will be 1125 However ifyou respond with an output of 15 every time your competitor announces an output of 15 profits will be reduced to zero for both of you in that period If your competitor fears or learns that you will respond in this way he or she will be better off by choosing the Cournot output of 10 and your profits after that point will be 75 per period Whether this strategy is profitable depends on your opponent s expectations about your behavior as well as how you value future profits relative to current profits Note A problem could develop in the last period however because your competitor will know that you realize that there are no more longterm gains to be had from behaving strategically Thus your competitor will announce an output of 15 knowing that you will respond with an output of 75 Furthermore knowing that you will not respond strategically in the last period there are also no longterm gains to be made in the ninth period from behaving strategically Therefore in the ninth period your competitor will announce an output of 15 and you should respond rationally with an output of 75 and so on 9 You play the following bargaining game Player A moves first and makes Player B an offer for the division of 100 For example Player A could suggest that she take 60 and Player B take 40 Player B can accept or reject the offer If he rejects the amount of money available drops to 90 and he then makes an offer for the division of this amount If Player A rejects this offer the amount of money drops to 80 and Player A makes an offer for its division If Player B rejects this offer the amount of money drops to 0 Both players are rational fully informed and want to maximize their payoffs Which player will do best in this game Solve the game by starting at the end and working backwards lfB rejects A s offer at the 3rd round B gets 0 When A makes an offer at the 3rd round B will accept even a minimal amount such as 1 So A should offer 1 at this stage and take 79 for herself In the 2nd stage B knows that A will turn down any offer giving her less than 79 so B must offer 80 to A leaving 10 for B At the first stage Aknows B will turn down any offer giving him less than 10 So A can offer 11 to B and keep 89 for herself B will take that offer since B can never do any better by rejecting and waiting The following table summarizes this Round Money Offering Party Amount to A Amount to B Available 1 100 A 89 1 1 2 90 B 80 10 8 80 A 79 1 End 0 0 0 10 Defendo has decided to introduce a revolutionary video game and as the first rm in the market it will have a monopoly position for at least some time In deciding what type of manufacturing plant to build it has the choice of two technologies Technology A is publicly available and will result in annual costs of CAq 10 Sq 195 Chapter 13 Game Theory and Competitive Equilibrium Technology B is a proprietary technology developed in Defendo s research labs It involves higher fixed cost of production but lower marginal costs Defendo s CEO must decide which technology to adopt Market demand for the new product CBq 60 2q is P 20 Q where Q is total industry output a Suppose Defendo were certain that it would maintain its monopoly position in the market for the entire product lifespan about five years without threat of entry Which technology would you advise the CEO to adopt What would be Defendo s profit given this choice Defendo has two choices Technology A With a marginal cost of 8 and Technology B With a marginal cost of 2 Given the inverse demand curve as P 20 Q total revenue PQ is equal to 20Q Q2 for both technologies Marginal revenue is 20 2Q To determine the profits for each technology equate marginal revenue and marginal cost 20 ZQA 8 or QA 6 and 202QB 2 orQB 9 Substituting the profitmaximizing quantities into the demand equation to determine the profitmaximizing prices we find PA20614 and PB 20 911 To determine the profits for each technology subtract total cost from total revenue M 146 10 86 26 and n3 2 119 60 29 21 To maximize profits Defendo should choose technology A Suppose Defendo expects its archrival Offendo to consider entering the market shortly after Defendo introduces its new product Offendo will have access only to Technology A If Offendo does enter the market the two rms will play a Cournot game in quantities and arrive at the Cournot Nash equilibrium i If Defendo adopts Technology A and Offendo enters the market what will be the profits of both rms Would Offendo choose to enter the market given these profits If both firms play Cournot each Will choose its best output taking the other s strategy as given Letting D Defendo and O Offendo the demand function Will be P 20 QB Q0 Profit for Defendo Will be nD iQD 7620913 7 Eb8QD or nD 12QD Q iQDQO 10 To determine the profitmaximizing quantity set the first derivative of profits With respect to QB equal to zero and solve for QB 122QDQ00orQD605QO D This is Defendo s reaction function Because both firms have access to the same technology hence the same cost structure Offendo s reaction function is analogous QO605QD 196 Chapter 13 Game Theory and Competitive Equilibrium Substituting Offendo s reaction function into Defendo s reaction function and solving for QB QB 6 056 05QD 4 Substituting into Defendo s reaction function and solving for Q0 Q0 6 054 4 Total industry output is therefore equal to 8 To determine price substitute QB and Q0 into the demand function P204412 The profits for each firm are equal to total revenue minus total costs nD 412 10 84 6 and quoto 412 10 84 6 Therefore Offendo would enter the market ii If Defendo adopts Technology B and Offendo enters the market what will be the profit of each rm Would Offendo choose to enter the market given these pro ts Profit for Defendo Will be D bgiQD 7Q0gD 7 Iaa2QDg or nD 18QD 7Q 7QDQ0 760 The change in profit With respect to QB is 5 713 QD To determine the profitmaximizing quantity set this derivative to zero and solve for QB 18 2QD QO 18 ZQD Q0 0 or QB 9 05QO This is Defendo s reaction function Substituting Offendo s reaction function into Defendo s reaction function and solving for QB QB 9 056 05QD or QB 8 Substituting QB into Offendo s reaction function yields Q0 6 058 or Q0 2 To determine the industry price substitute the profitmaximizing quantities for Defendo and Offendo into the demand function P208210 The profit for each firm is equal to total revenue minus total cost or HD 108 60 28 4 and quoto 102 10 82 6 With negative profit Offendo should not enter the industry iii Which technology would you advise the CEO of Defendo to adopt given the threat of possible entry What will be Defendo s pro t given this choice What will be consumer surplus given this choice 197 Chapter 13 Game Theory and Competitive Equilibrium With Technology A and Offendo s entry Defendo s profit would be 6 With Technology B and no entry by Defendo Defendo s pro t would be 4 I would advise Defendo to stick with Technology A Under this advice total output is 8 and price is 12 Consumer surplus is 0520 128 32 c What happens to social welfare the smn of consumer surplus and producer pro t as a result of the threat of entry in this market What happens to equilibrimn price What might this imply about the role of potential competition in limiting market power From 10a we know that under monopoly Q 6 and profit is 26 Consumer surplus is 0520 146 18 Social welfare is the sum of consumer surplus plus pro ts or 18 26 44 With entry social welfare is 32 consumer surplus plus 12 industry profit or 44 Social welfare changes little with entry but entry shifts surplus from producers to consumers The equilibrium price falls with entry and therefore potential competition can limit market power Note that Defendo has one other option to increase quantity from the monopoly level of 6 to discourage entry by Offendo lf Defendo increases output from 6 to 8 under Technology A Offendo is unable to earn a positive profit With an output of 8 Defendo s profit decreases from 26 to 802 10 88 22 As before with an output of 8 consumer surplus is 32 social welfare is 54 In this case social welfare rises when output is increased to discourage entry 11 Three contestants A B and C each have a balloon and a pistol From xed positions they re at each other s balloon When a balloon is hit its owner is out When only one balloon remains its owner is the winner and receives a 1000 prize At the outset the players decide by lot the order in which they will re and each player can choose any remaining balloon as his target Everyone knows that A is the best shot and always hits the target that B hits the target with probability 9 and that C hits the target with probability 08 Which contestant has the highest probability of winning the 1000 Explain why lntuitively C has the highest probability of winning though A has the highest probability of shooting the balloon Each contestant wants to remove the contestant with the highest probability of success By following this strategy each improves his chance of winning the game A targets B because by removing B from the game As chance of winning becomes much greater B s probability of success is greater than C s probability of success C will target A because if C targets B and hits B then A will target C and win the game B will follow a similar strategy because if B targets C and hits C then A will target B and will win the game Therefore both B and C increase their chance of winning by eliminating A first Similarly A increases his chance of winning by eliminating B first A complete probability tree can be constructed to show that As chance of winning is 8 percent B s chance of winning is 32 percent and Cs chance of winning is 60 percent 198 Chapter 13 Game Theory and Competitive Equilibrium CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY TEACHING NOTES Chapter 13 continues the discussion of competitive firms in the context of twoplayer games with the first three sections covering all topics introduced in Chapter 12 If you did not present Section 125 you should do so after discussing Sections 131 and 132 Sections 134 through 138 introduce advanced topics The presentation throughout the chapter focuses on the intuition behind each model or strategy The exercises focus on relating Chapter 13 to Chapter 12 and on behavior in repeated games Two concepts pervade this chapter rationality and equilibrium Assuming the players are rational means that each player maximizes his or her own payoff whether it hurts or helps other players Rationality underlies many of the equilibria in the chapter Underlying all these models is the definition of a Nash equilibrium which the students will find esoteric When presenting each model ask whether a unique Nash equilibrium exists If there is more than one discuss the conditions that will favor each equilibrium The analysis in the last five sections of the chapter is more demanding but the examples are more detailed Section 134 examines repeated games and it will be important to discuss the role of rationality in the achievement of an equilibrium in both finite and infinitehorizon games Example 132 points out conditions that lead to stability in repeated games while Example 133 presents an unstable case Sections 135 136 and 137 introduce strategy in the context of sequential games To capture the students attention discuss the phenomenal success of WalMart in its attempt to preempt the entry of other discount stores in rural areas see Example 134 First define a strategic move second discuss the advantage of moving first third present Example 134 and fourth continue with other forms of strategic behavior including the use of new capacity and RampD to deter entry see Examples 135 and 136 You may wish to reintroduce the case of bilateral monopoly during the discussion of strategic behavior in cooperative games which concludes this chapter REVIEW QUESTIONS 1 What is the difference between a cooperative and a noncooperative game Give an example of each In a noncooperative game the players do not formally communicate in an effort to coordinate their actions They are aware of one another s existence but act independently The primary difference between a cooperative and a noncooperative game is that a binding contract ie an agreement between the parties to which both parties must adhere is possible in the former but not in the latter An example of a cooperative game would be a formal cartel agreement such as OPEC or a joint venture An example of a noncooperative game would be a race in research and development to obtain a patent 2 What is a dominant strategy Why is an equilibrium stable in dominant strategies A dominant strategy is one that is best no matter what action is taken by the other party to the game When both players have dominant strategies the outcome is stable because neither party has an incentive to change 3 Explain the meaning of a Nash equilibrium How does it differ from an equilibrium in dominant strategies A Nash equilibrium is an outcome where both players correctly believe that they are doing the best they can given the action of the other player A game is in equilibrium if neither player has an incentive to change his or her choice unless there is a change by the other player The key feature that distinguishes a Nash equilibrium from an equilibrium in dominant strategies is the dependence on the opponent s behavior An equilibrium in dominant strategies results if each player has a best choice regardless 186

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Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.