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Chapter 12 A GAME-THEORETIC APPROACH TO STRATEGIC BEHAVIOR Boiling Down Chapter 12 The world seemed rather artificial in the market models of perfect competition and monopoly. It was a world of little variety and no complicating problems of interdependence, transportation costs, advertising, or product differentiation. We now must consider this more complex and realistic world. The theory of imperfect competition explores strategies that help explain behavior where interdependence is present and firms must make assumptions about their opponent’s actions. Game theory is the tool used to explore many of these relationships. As in the prisoner's dilemma story firms would be more profitable if they priced together above marginal cost than if they competed price down to marginal cost. Alternatively, the firm that cheated on the price agreement while the other firm cooperated would have the most profit possible. When the payoffs are such that a firm will be better off with one strategy no matter what the other firm does, a dominant strategy exists and will likely be followed. Table 12-2 in the text shows a duopoly situation where defecting is the dominant strategy. However, if additional price cuts would be damaging to both firms, the final outcome could be zero profits for both firms and the defecting strategy would not be dominant in the long run. In cases like cigarette advertising, the dominant strategy is to advertise even though it dissipates profit and leaves firms with less than they would have with a no-advertising strategy. If both firms have a dominant strategy, a clear equilibrium exists. If only one firm has a dominant strategy and the other does not, equilibrium still exists because the firm without the dominant strategy can predict what the other firm will do and choose its best action accordingly. This kind of equilibrium is called a Nash equilibrium. When interaction among firms is ongoing, as it usually is, firms will be better off if they cooperate than if they defect, because if one defects the other feels compelled to defect and the payoff will be minimal for both. Guaranteeing cooperation is easiest when a defector can be punished in future transactions. One way to punish defectors is to promise to respond in kind each time defection occurs. If the promise is believed, a firm will lose over time if it defects, and so cooperation is likely. This strategy is called tit-fortat and has been shown to effectively lead to cooperation in wartime circumstances as well as business situations. However, when more than two parties are involved in a game situation, tit-for-tat becomes difficult because it is hard to punish one defector without hurting cooperators When action by players in a game is sequential, the first player will act based on her prediction of what the other player will do. If the most desirable action of the initial actor is undesirable to the opponent, but better than the result of retaliatory behavior, then the 2 CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior rational choice is not to retaliate. However, if retaliation would be a worst-case situation for the initiator, then certain retaliation would be the victim's best strategy. If doomsday actions can be guaranteed, the threat of doomsday is diminished. The example of the former USSR and U.S. foreign policy is an example of sequential strategy at work. If a doomsday device is installed that guarantees retaliation when a missile attack is launched, then the likelihood of an attack is diminished. In corporate strategy the same kind of automatic deterrence can convince competitors that a confrontation would be counterproductive. The Sears Tower case in the text is an example of this strategy in action. Competitors are not likely to enter a battle they know they will lose. How doves (cooperators) and hawks (defectors) interact and survive in a population is the subject of considerable discussion and is illustrated by the example of hawks and doves in an interdependent situation. If there is a cost of discerning between the two types, there will be a place in the population for both cooperators and defectors. This is true because if the cooperators become dominant in the population, their chances of linking up with other cooperatives are so great that they will not pay the cost of finding cooperators. As described in chapter 7, when a cooperator’s expected gain from a randomly chosen partner exceeds the return minus the search costs of finding another cooperator, then the defectors will have a permanent niche in the population. If doves and hawks look alike, the hawks will take over and the cooperators become extinct as they get paired increasingly often with defectors and then lose out in the rewards. This happens also because there is no way of punishing defectors or avoiding them in future transactions, since they cannot be identified in this model. If both types can be identified easily, the defectors will be able to interact only with other defectors because cooperators will not enter into transactions with defectors. Two defectors working together fare worse than two cooperators working together, so the defectors evolve as the weaker in the population and become extinct. The hawks and doves story illustrates that cooperative behavior can work to one’s advantage, but especially if trust can develop between the participants in society. Trust can be enhanced by commitment devices that show others you are really a cooperator at heart. It is important to realize that after all the commitments are made and all the emotions are assessed, non-egoistic behavior will only work if the altruistic behavior is heartfelt. Fake altruism cannot be hidden for long. Cultivating behavior that makes cooperation heartfelt is part of what it means to be moral. Chapter Outline 1. Game theory helps to explain how firms will relate in an interdependent situation. a. The prisoner's dilemma example illustrates how dominant and non-dominant strategies affect choices. b. In joint profit-maximizing situations the game is between the cooperators and the defectors to the agreement. c. Advertising strategy provides an example of how interdependence leads to decisions predicted by game theory. d. A Nash equilibrium can occur when one firm has a dominant strategy, even though the other does not. e. Risk averse firms in an industry that does not have a Nash equilibrium may select a maximin strategy in which they look at the worst that can happen with each action and then choose the best of those options. f. The tit-for-tat strategy helps to promote stability in decisions where interdependence exists. CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 3 g. Sequential games, as in the MAD defense or the tallest building strategy, require that credibility be established or doomsday devices be implemented. 2. A model of cooperators and defectors illustrates the tenuous balance in society between the two groups. a. Defectors will dominate if social interaction is random. b. Cooperators will dominate if free detection of each group exists. c. If detection is costly, society will be made up of both groups. d. Because mimicry of cooperative behavior is always a likely practice, detection costs will be positive and both preferences will exist despite the common rational choice notion that tastes do not differ. Important Terms game theory dominant strategy prisoners dilemma payoff matrix Nash equilibrium sequential games tit-for-tat cheating problem commitment device defector cooperator maximin strategy average payoff A Case to Consider 1. Matt and Megan have grown weary of the intense competition they feel from each other so they get together one day in a restaurant to see if there is a solution to the competitive pressure. Pricing in their industry is day to day so Matt suggests that they both keep prices up for a month. They aren’t sure they can trust each other so Megan offers a proposition that they both think might work, given their mutual distrust. Ignoring the illegality of this conversation for the moment and the fact that you were eavesdropping on their conversation in the restaurant, describe what deal to stabilize pricing you heard them make. 2. Next suppose there was no specific strategy was discussed except that they would continue to follow a rational course with the knowledge they have of the industry and their profit estimates. Suppose they both get 10 profit if they keep prices up for the month but only 7 each if they get into a price war. However, if one holds price up and the other cuts prices, the one with the high price earns only 6 and the one who cuts prices earns 12. a. Given this data, is it likely their deal will hold? Explain. b. Does either one have a dominant strategy. c. Define “maximin strategy. 4 CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior Multiple-Choice Questions 1. In a prisoner's dilemma with prisoners A and B, if they both confess, A gets 5 years and B gets 8 years. If both remain silent, A gets 2 years and B goes free. If one confesses and the other does not, the one who confesses gets 1 year and the other gets 15 years. Which statement is true of this case? a. There is a dominant strategy for both A and B. b. There is no dominant strategy for either A or B. c. There is a dominant strategy for A but not for B. d. There is a dominant strategy for B but not for A. 2. With reference to Question 1 above, which of the following is a true statement? a. A Nash equilibrium exists as the question stands. b. A Nash equilibrium does not exist as the question stands, but if the penalties for both remaining silent were doubled, a Nash equilibrium would then exist. c. If the penalties were changed so that if both parties confessed, they would go free, then it would be the dominant strategy for both to confess. d. Both b and c are correct statements, but a is incorrect. 3. From the stories presented in this chapter, we could conclude that a. the world would be better off if it was full of altruists, doves, and cooperators rather than full of hawks, cheaters, and egotists. b. altruism works as long as only some have those qualities. c. altruists are basically irrational. d. all of the above are true. e. none of the above are true. 4. Standard rational choice theory accepts which of the following? a. People are creatures of reason. b. People are inefficient processors of information. c. People have widely different goals and preferences. d. Economic choices frequently involve interdependencies among people. 5. In the prisoner’s dilemma exercise, which of the following is the final outcome? a. The prisoners discuss the problem ahead of time and trust each other to cooperate. b. The prisoners discuss the problem ahead of time and then both cheat. c. The prisoners will develop a foolproof commitment device and then both confess. d. The prisoners will individually decide it is in their interest to confess. e. None of the above is correct since the prisoners both remain silent because it is in their individual self-interest to do so. CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 5 6. In the prisoners dilemma story, if the prisoners could communicate with each other before interrogation a. they would most likely each get one year in prison. b. they would most likely each get 5 years in prison. c. one would likely cheat and get no punishment while the other would not cheat and get 20 years in prison. d. one would likely cheat and get 20 years in prison while the other would likely not cheat and go free. e. no prediction is very reliable until we know how trustworthy each one is. 7. If Joe is not trustworthy but there are advantages in being trustworthy, then it would be in his best interest, according to rational choice theory, a. to become genuinely trustworthy and to be perceived as such. b. to be perceived as trustworthy without actually having these qualities. c. to change his preference and become trustworthy whether others know or not. d. to remain untrustworthy and be perceived as such. 8. In the typical defector-cooperator population models, which of the following is not a basic assumption? a. Defectors working together will have lower payoffs than cooperators working together. b. The population growth will be greatest in the group that has the highest payoff. c. When cooperators and defectors work together, the cooperator receives the highest payoff. d. All the above are basic assumptions of the models. 9. In evolutionary models, where each individual reproduces in proportion to its average payoff, if cooperators and defectors cannot be distinguished, if defectors get more than cooperators when both types are paired together, and if cooperator teams do better than defector teams even though defectors’ payoffs are positive, then a. cooperators are destined for extinction. b. defectors are destined for extinction. c. an equilibrium will result with more than half of the population being cooperators. d. there will be no change in the population proportions from the starting point. 10. In the doves and hawks society there is a place for the doves in certain cases because a. there is less waste in joint production processes when participants cooperate. b. doves do well when paired with other doves. c. hawkish behavior can be damaging even to hawks. d. all of the above are true. 11. In a society consisting of 50% cooperators and 50% defectors, with an 8-unit payoff to a cooperator who interacts with another cooperator and a 0-unit payoff to the cooperator who interacts with a defector, if the cost of telling the two types apart is 2, a risk neutral cooperator a. will be better off if he pays the cost of scrutiny. b. should not interact with either cooperators or defectors. 6 CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior c. Is better off if he does not pay the cost of scrutiny. d. will break even by paying the cost of scrutiny. 12. In the cooperator-defector model, if the population has more cooperators than the equilibrium specifies, what will cause the percent of cooperators to decrease? a. The cooperators turn into defectors when faced with opportunities for cheating. b. The defectors pretend they are cooperators and get away with it. c. The cooperators voluntarily stop paying the cost of vigilance. d. The defectors trick cooperators into pairing with them. 13. When there are no search costs involved in finding others like you, the proportion of cooperators in the population ends up a. being greater than when there are positive search costs. b. is smaller than when there are positive search costs. c. is unchanged from when there are positive search costs. d. is indeterminate until one knows whether the defectors will use the free information. 14. The tit-for-tat strategy for repeated play in oligopoly theory means that firm 1 will a. defect from an agreement to collude the first time only. b. defect if the opponent defected last time and cooperate if the opponent cooperated last time. c. cheat every time to reap as big and quick a profit as possible. d. cooperate every time no matter what the opponent does. 15. Research on the tit-for-tat strategy shows that it will be most successful when a. the penalty for cheating is low. b. players are willing to discard the strategy temporarily in an unpredictable manner. c. it is followed consistently over a long period of time. d. the players approach a clearly defined end to their transactions. 16. Author Frank critiques the MAD defense philosophy as an inadequate explanation of peace because a. it is not rational to strike back if a nation is worse off by doing so. b. there was no doomsday retaliating device in place during the cold war time. c. interdependence requires that the commitment problem be solved even in international affairs. d. of all the above. e. of a reason other than any listed above. 17. The main point of this chapter is that a. rational calculations by cooperators determine the nature of society. b. people can be bribed into irrational other centered action. c. when people work together they are happier. d. there is a place in society for those who are not always operating from narrow self-interest. CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 7 Problems 1. If OPEC consisted of only Saudi Arabia and Nigeria and if they both stuck to an agreement to fix oil prices, then Saudi Arabia would make $100 profit and Nigeria would make $60 profit. If both cheated on the agreement, Saudi Arabia would make $70 and Nigeria would make only $30. If Saudi Arabia cheated and Nigeria did not, then Nigeria would make $25 and Saudi Arabia would make $80. Likewise, if Nigeria cheated and Saudi Arabia did not, then Nigeria would make $75 and Saudi Arabia would make $50. a. Does either country have a dominant strategy? b. Does a Nash equilibrium exist in this case? Explain. c. If Nigeria knew that Saudi Arabia had a tit-for-tat strategy that it would definitely follow, what would be the outcome of the market interaction? 2. In class I sometimes allow students the option of cooperating on a quiz. If all students hand in a blank take-home exam, they will all get an A- for the exam. If one person answers the questions as best he can, he will get an A and everyone who handed in a blank paper gets a D. Assume there are only two persons in the class. Also assume for this exercise that both students would get Bs if they took the test normally. Finally, assume that I grade on a curve that makes class standing important in the final grade when all other grades are averaged. 1. Set up the payoff matrix for this problem with Neil as one student and Laura as the other student. 2. Is there a dominant strategy for both Neil and Laura? If so, what is it? If not, is there a Nash equilibrium? 8 CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior 3. Redo the doves-hawks story, but this time assume that hawks fighting together use up only 4 calories each. Thus the loser gets –4 and the winner gets 12 - 4 = 8. The average gain per hawk is (8 –4)/2 = 2. Draw a new graph and show what will happen to the population equilibrium point. If the case involved people rather than hawks, what could possibly cause them to do something as irrational as to waste 10 calories fighting for something with an expected value of 6 calories? (Think of some of the wars going on in the world as examples.) 4. On a tropical island the inhabitants live off food they pick from large fruit trees. The government mandates that teams of 2 be sent out to pick the fruit because one must hold the ladder for safety reasons. Two types of people live on the island. One is the sharing type (S) who grew up in the part of the island where property was held in common and equality was the norm. The other type grew up in the part of the island that was individualistic and ruthlessly competitive (I). When two I's work together they each grab all the fruit they can reach from the ground because they will not hold the ladder for each other. They each can pick 50 pounds per day that way. When two S's work together they hold the ladder for each other so the best fruit in the tree tops can be picked and they each pick 75 pounds of fruit. If an S and an I work on a team, the I gets 90 because of his skill in picking only from the ground and the S picks only 25 since no one will hold the ladder for him. The group with the most fruit to eat increases its population share at the expense of the other group. a. Sketch the payoff graph with pounds of fruit on the vertical axis and percent of population of S's on the horizontal axis, assuming that each worker is paired randomly. b. Under the conditions in (a) above, and assuming that survival of the fittest (most fruit) holds over time, what will the island population be like in the long run and what will the standard of living (pounds per day) be for the people? CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 9 c. Next, a proposal is accepted to allow each S to pick a partner at no cost. Sketch a new graph to show this average payout situation. d. Under this new policy, what will be the return to the S's in the long run, and what will be the population proportion between the S's and I's? e. If the time lost in choosing partners costs each picker 10 pounds of fruit, sketch the new payoff graph and show graphically where the population balance between the two groups will come to equilibrium. (Remember that I's will not try to pick partners because they would try to pick S's who would not work with them anyway.) f. Explain in words why S's will not choose their partner when the population of S's reaches the level specified in (e) above. g. If the sharers were risk averse, would their population share at equilibrium be greater, equal to, or less than the equilibrium established in answer (e)? 10 CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior 5. In the movie “A Beautiful Mind”, John Nash, for whom the Nash equilibrium is named, is seated in the university snack shop with friends observing several girls as potential dates. Recognizing that one girl would be the first choice of all the men, Nash says that Adam Smith was wrong when he claimed that the group welfare will be maximized when everyone pursues his own self interest. Assume that the payoff matrix for Thomas and David is as shown below. The choices involve asking the top choice girl for a date or choosing one of the other girls as a second choice. Assume also, as Nash did, that if everyone asks the top choice and one is successful, the other girls will refuse to accept dates since they will know they are second choices. Anyone asking the second choice first will not be rejected since the girl will not know she was second choice. Thomas David Top choice for a date Second choice for a date Top choice for a date Second choice for a date A = 20 B=0 A=8 A = 18 B = 10 A = 15 B = 15 B = 18 a. Do Thomas and David have dominant strategies? Why or why not? b. Is there a Nash equilibrium in this case? Why or why not? c. If each person adopted a maximin strategy, what would the outcome be? d. Using the payoff matrix, explain why John Nash felt Adam Smith was wrong in his assertion about social optimal outcomes from the pursuit of self interest. e. If both Thomas and David were close friends and each seeks the best interest of the other, what will happen and how does this outcome compare with the socially optimal outcome? CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 11 6. True or false: A Nash equilibrium will also be the most desirable point for the firms to be in. Explain. ANSWERS TO QUESTIONS FOR CHAPTER 12 Case Questions 1. The goal of the meeting was to see if both would stop price cutting to get much of the business. Matt suggests they hold prices for a month and see how that goes. Megan fears he is just trying to get her to keep prices high so he can lower his and get high sales. Therefore she promises to keep prices up but if he lowers his price she will lower hers and engage in an all out price war. If he believes her threat the deal can work. 2a. If the tit-for-tat strategy is an empty threat the deal will likely fall apart. It will fall apart because both have a dominant strategy to lower price. 2b&c. Both have a dominant strategy to cut prices, because if one holds price, the worst the other gets is 10, while if one cuts price, the worst that can happen to the other is 6. Picking the best of the worst would lead both to hold to their agreement and each get 10 profits. This best of the worst strategy is a maximin strategy. Multiple-Choice Questions 1. c, A should confess no matter what B does. B should confess only if A does. 2. a, With both confessing, neither one has an incentive to change strategy. 3. a, Since cooperators working together are more productive and perhaps even more fulfilled as human beings, the world would be a better place if everyone cooperated. Alas, human nature has not made this possible. 4. a, People are efficient, individualistic, and have similar goals, so no others are right. 5. d, Not trusting each other they both rationally confess and take a lower sentence than would be the case if they didn’t confess and the other one did. 6. e, If the prisoners trusted each other and were trustworthy, then they would likely both remain silent and each get a year in jail, but if they were not sure of the trustworthiness of the other, than both would cheat and get 5 years in jail rather than risk a 20 year sentence. 7. b, Joe, if he could be trustworthy when only when it helped him, would get the benefit of both the trustworthy and opportunistic worlds unless, as your text suggests, it is impossible to succeed as trustworthy unless you really are trustworthy. 8. c, In fact, defectors take advantage of cooperators. 9. a, The expected payoff for defectors will be greater no matter what share of the population they have so cooperators will lose out. 10. d, In the case where doves can find each for free they will not work with hawks and so their share of the population grows. 11. a, By taking a chance the cooperator gets an expected value of 4 but by paying 2 to guarantee being paired with another cooperator, that person gets 8-2=6 so payment is made. 12. c, Higher population above equilibrium means that the expected value of taking a chance exceeds the net benefit of paying for information. Thus cooperators take their chances. 13. a, Because the net payoff will always be greater with zero search costs, a cooperator will never work with a defector and therefore will have a higher net payoff. Defectors get no 12 14. 15. 16. 17. CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior advantage from the information because no cooperator, knowing defectors for what they are, will work with them b, Tit for tat rewards cooperation and punishes defection. c, Trust builds and repeat business gives everyone incentive to stick to the strategy. d, Tit for tat strategy in international affairs has all these problematic issues. d, That people sometimes consider their interest to be broad and other-centered is a fact that pure rational choice methods do not process well. Cooperation has its rewards. Problems 1. a) Nigeria has a dominant strategy to cheat. Saudi Arabia has no dominant strategy. b) Yes, Nigeria cheats so the Saudis will cheat and neither have incentive to fix once they are in the cheating mode. c) Nigeria would fix if they could be sure that the Saudis would follow. 2. a) Put Neil on the left and Laura on the top of a four quadrant matrix. Put an A- for each in the northwest cell, a B for each in the southeast cell, an A for Neil and a D for Laura in the southwest cell, and a D for Neil and an A for Laura in the northeast cell. The upper row shows Neil handing in a blank paper as does the first column for Laura. b) Both Laura and Neil have a dominant strategy to cheat, and this is why the students always fail in their effort to get easy A’s in my class. 3. The Hawks' payoff function will be above the doves' function at all points (its right intercept will be +2 instead of _ 4), so they will take over the population. In our world, people are constantly fighting in conflicts where payoffs are negative unless emotional commitments make the payoffs higher than what are often considered in rational choice analysis. 4. a and b) The I's will dominate living on 50 pounds of fruit each. See Problem 7-5a below. Problem 7-5a Pounds of fruit 90 75 I’s S’s 50 25 %S c, d, and e) In (d) the S's will be the only survivors with each earning a return of 75 pounds of fruit. See sketch for Problem 7-5c. Problem 7-5c Fruit Problem 7-5c Fruit Problem 7-5e 75 50 65 %S %S 80 CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 13 f) The S's fare better taking their chances, since there are so many S's in the population. They are assumed to be risk neutral. See sketch for Problem 7-5e above. g) The sharers population would be greater because they would not take their chances until the odds were higher than 80% of getting matched with a sharer. 5. a) Thomas has a dominant strategy to go for his first choice. David doesn’t not have a dominant strategy since he should go for his second choice if Thomas goes for his first and he should take the top choice if Thomas takes his second choice. b) Since David knows Thomas will follow his dominant strategy, David will go for his second choice so a Nash equilibrium does exist. c) The worst that can happen to Thomas if he takes his first choice is 18 and the worst possible if he takes his second choice is 8 so he takes his first choice. David, using the same logic, chooses 10 over 0 so he goes for his second choice. d) The optimal social outcome is the sum of 15 + 15 or 30 which came from each choosing their second choice. If both pursue their first choice without considering their interdependence, the total social welfare is only 20. e) Both will let the first choice for the other and each will have 15 which is the socially optimal outcome. This of course assumes that each knows the other’s preferences. 6. False: A Nash equilibrium will not always be the most desirable point. If one views the payoff chart for the firms in tables 13.2 to 13.4 the dominant strategies have lower rewards than would be possible if both firms employed the opposite strategy. 14 CHAPTER 12: A Game-Theoretic Approach to Strategic Behavior HOMEWORK ASSIGNMENT NAME:_____________________________ 1. Goodyear and Bridgestone face the following profit and price relationships. Bridgestone $100 $60 Goodyear $100 F = 20: G = 36 F = 33: G = 10 $60 F = 12: G = 35 F = 11: G = 30 a. If there is a dominant strategy, is it a Nash equilibrium? b. 3. True or False If neither one of two interdependent firms in the same industry has a dominant strategy, then a Nash equilibrium cannot exist. Explain. 2. Working from problem 5e in the study guide, you want society to be equally balanced between sharers and individualists. In addition, you have the political clout to impose a tax on top of the existing 10 pound cost of picking partners. From these givens, could you achieve an equal balance in the population and, if so, how large a tax would need to be imposed to achieve the goal? Explain your answer. 3. Reflect on some practical ways that society does or could encourage cooperative behavior. List 2 specific possibilities. 4. In a society consisting of 75% cooperators and 25% defectors, with a 10-unit payoff to a cooperator who interacts with another cooperator and a 0-unit payoff to the cooperator who interacts with a defector, if the cost of scrutiny is 2, what should the cooperator do to maximize his utility. CHAPTER 12: A Game Theoretic Approach to Strategic Behavior 15 5. The following payoff matrix represents the long-run payoffs for two firms faced with the option of buying or leasing buildings to use for production. Determine whether any dominant strategies exist and whether or not there is a Nash equilibrium. Firm 1 Lease Firm 2 Buy Lease Building F1 = 500 F2 = 500 Buy Building F1 = 750 F2 = 400 F1 = 300 F2 = 600 F1 = 600 F2 = 200