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Chapter 10 Strategic Choice in Oligopoly, Monopolistic Competition, and Everyday Life 5 additional Questions Even-numbered Qs. Chapter 10, Problem 2 Consider the following “dating game”, which has two players, A and B, and two strategies, to buy a movie ticket or a baseball ticket. The payoffs, given in points, are as shown in the matrix below. Note that the highest payoffs occur when both A and B attend the same event. Assume that players A and B buy their tickets separately and simultaneously. Each must decide what to do knowing the available choices and payoffs but not what the other has actually chosen. Each player believes the other to be rational and self-interested. Buy movie ticket Buy movie ticket A Buy baseball ticket B Buy baseball ticket 2 for A 0 for A 3 for B 0 for B 1 for A 3 for A 1 for B 2 for B a) Does either player have a dominant strategy? “When one player has a strategy that yields a higher payoff no matter which choice the other player makes.” Dominated strategy – “The other strategy available to the player that yield a payoff strictly smaller than that of the dominant strategy” a) Does either player have a dominant strategy? “ A player yields a higher payoff no matter what the other players in a game choose.” Assumption: If A assumes B buys movie ticket, A will buy movie ticket. If A assumes B buys baseball ticket, A will buy baseball ticket. If B assumes A buys movie ticket, B will buy movie ticket. If B assumes A buys baseball ticket, B will buy baseball ticket. Therefore, there is no dominant strategy. Each player buys the ticket according to other player’s choice. b) How many potential equilibria are there? (Hint: To see whether a given combination of strategies is an equilibrium, ask whether either player could get a higher payoff by changing his or her strategy.) • • Nash Equilibrium “any combination of strategies in which each player’s strategy is his or her best choice, given the other players’ choices” There can be an equilibrium when players do not have a dominant strategy There are 2 potential equilibria, the upper-left cell and the lowerright cell. According to the assumption, in each cell, neither player has any incentive to change strategies. c) Is this game a prisoner’s dilemma? Explain. • Prisoner’s dilemma “a game in which each player has a dominant strategy, and when each plays it, the resulting payoffs are smaller than if each had played a dominated strategy.” Therefore, this game is not a prisoner’s dilemma because neither player has a dominant strategy. d) Suppose player A gets to buy his or her ticket first. Player B does not observe A’s choice, but knows that A chose first. Player A knows that player B knows he or she chose first. What is the equilibrium outcome? Buy movie ticket 2 for A Buy movie ticket A B Buy baseball ticket 0 for A 3 for B 0 for B 1 for A 3 for A 1 for B 2 for B Buy baseball ticket Each player believes the other to be rational and self-interested and each knowing the payoffs. If A chooses first and buys a movie ticket, given A’s choice, B will also buy movie ticket. A will get a payoff of 2. If A chooses first and buys baseball ticket, given A’s choice, B will also buy baseball ticket. A will get a payoff of 3. The highest payoff for A is to buy baseball ticket. The equilibrium outcome is where both A and B will buy baseball ticket. e) Suppose the situation is similar to part d, except that player B chooses first. What is the equilibrium outcome? Buy movie ticket 2 for A Buy movie ticket A B Buy baseball ticket 0 for A 3 for B 0 for B 1 for A 3 for A 1 for B 2 for B Buy baseball ticket Each player believes the other to be rational and self-interested and each knowing the payoffs. If B chooses first and buys a movie ticket, given B’s choice, A will also buy movie ticket. B will get a payoff of 3. If B chooses first and buys baseball ticket, given B’s choice, A will also buy baseball ticket. B will get a payoff of 2. The highest payoff for B is to buy movie ticket. The equilibrium outcome is where both A and B will buy movie ticket. Chapter 10, Problem 4 The owner of a thriving business wants to open a new office in a distant city. If he can hire someone who will manage the new office honestly, he can afford to pay that person a weekly salary of $2,000 ($1,000 more than the manager would be able to earn elsewhere) and still earn an economic profit of $800. The owner’s concern is that he will not be able to monitor the manager’s behavior and that the manager would therefore be in a position to embezzle money from the business. The owner knows that if the remote office is managed dishonestly, the manager can earn $3,100, while causing the owner an economic loss of $600 per week. a) If the owner believes that all managers are narrowly self-interested income maximizers, will he open the new office? Construct the game tree: Manager manages honestly -Owner gets $800 -Manager gets $2,000 Owner opens remote office A Owner does not open remote office Manager manages dishonestly -Owner gets -$600 -Manager gets $3,100 -Owner gets $0 -Manager gets $1,000 elsewhere Working backward: • If owner opens the remote office, the potential manager’s best strategy is to manage dishonestly which gives him $1,100 more and owner gets -$600. • If owner does not open the remote office, owner gets $0. • Since $0 is better than -$600, the owner will not open the new office. b) Suppose the owner knows that a managerial candidate is a devoutly religious person who condemns dishonest behavior, and who would be willing to pay up to $15,000 to avoid the guilt she would feel if she were dishonest. Will the owner open the remote office? Construct the game tree: Owner opens remote office A Owner does not open remote office Manager manages honestly -Owner gets $800 -Manager gets $2,000 Manager manages dishonestly -Owner gets -$600 -Manager gets $3,100 $15,000 = -$11,900 -Owner gets $0 -Manager gets $1,000 elsewhere Working backward: • If owner opens the remote office, the potential manager’s best strategy is to manage honestly which gives him $2,000 and owner gets $800. • If owner does not open the remote office, owner gets $0. • Therefore, the owner will open the new office. Chapter 10, Problem 6 Newfoundland’s fishing industry has recently declined sharply due to overfishing, even though fishing companies were supposedly bound by a quota agreement. If all fishermen had abided by the agreement, yields could have been maintained at high levels. a) Model this situation as a prisoner’s dilemma in which the players are Company A and Company B and the strategies are to keep the quota and break the quota. Include appropriate payoffs in the matrix. Explain why overfishing is inevitable in the absence of effective enforcement of the quota agreement. a) Company B Keep Quota Keep Quota Break Quota Second Best for both Worst for A Best for B Best for A Worst for B Third Best for both Company A Break Quota If A breaks its quota while B keeps it, then A will get the largest possible profit and B will get the smallest. If B Breaks its quota while A keeps it, then B will get the largest possible profit and A will get the smallest. Both will get a higher profit if both keep the quota than if both break it. The payoffs are perfectly symmetric. Each dominant strategy is to break the quota, which means that both will do so unless some way can be found to enforce the quota. b) Provide another environmental example of a prisoner’s dilemma. Air pollution. If I pollute from my factory and no one else does, then I gain from not having to install pollution-control equipment, as well as from clean air; since my own pollution has only a negligible effect on air quality. However, if all other industrialists think this way, the air will become polluted, and all will be worse off than if none had polluted. c) In many potential prisoner’s dilemma, a way out of the dilemma for a would-be cooperator is to make reliable character judgments about the trustworthiness of potential partners. Explain why this solution is no available in many situations involving degradation of the environment. In situation involving environmental degradation, the players usually do not know each other. When interactions are anonymous, there is no opportunity to make character judgments. In such cases, legal enforcement is often necessary. Chapter 10, Problem 8 Consider the following game. Harry has four quarters. He can offer Sally from one to four of them. If she accepts his offer, she keeps the quarters Harry offered her and Harry keeps the others. If Sally declines Harry’s offer, they both get nothing ($0). They play the game only once, and each cares only about the amount of money he or she ends up with. Let X be the number of quarters Harry proposes to Sally, where X = 1, 2, 3, 4. If Harry proposes X quarters to Sally and she accepts, - Sally keeps X quarters or ($0.25)(X) - Harry keeps 4-X quarters or ($0.25)(4-X) If Sally declines Harry’s offer, - Sally gets $0 - Harry gets $0 a) Who are the players? What are each player’s strategies? Construct a decision tree for this ultimatum-bargaining game. Harry and Sally are the players. Harry’s strategies involve the number of quarters he offers Sally, his choice of X Sally’s strategies are to accept or to refuse Harry’s offer. Sally: keep ($0.25)X Harry: keep ($0.25)(4-X) Sally accepts A B Harry proposes X quarters for Sally Harry keeps 4-X Sally refuses Sally: $0 Harry:$0 b) Given their goal, what is the optimal choice for each player? b) Given their goal, what is the optimal choice for each player? • • • At B on the decision tree, if Sally accepts the offer, she gets ($0.25)X. If she refuses, she gets $0. Therefore, Sally’s best choice is to accept the offer, no matter what X is. • Knowing that Sally will accept the offer no matter what X is, Harry will offer as little quarter as he can to Sally so as to enjoy the highest payoff. • Harry offers 1 quarter to Sally and keeps 3 • • Sally accepts his offer and receive ($0.25)(1) = $0.25. Harry keeps 3 quarters and receive ($0.25)(4-1)=$0.75. Chapter 10, Problem 10 Jill and Jack both have two pails that can be used to carry water down from a hill. Each makes only one trip down the hill, and each pail of water can be sold for $5. Carrying the pails of water down requires considerable effort. Both Jill and Jack would be willing to pay $2 each to avoid carrying one bucket down the hill, and an additional $3 to avoid carrying a second bucket down the hill. a) Given market prices, how many pails of water will each child fetch from the top of the hill? In this part of the question, each player’s payoffs are independent of the action taken by the other. Each pail of water sells for $5. To avoid carrying one bucket costs $2. To avoid carrying a second bucket costs $3. Since the cost of carrying each bucket is less than $5, Jill and Jack will each carry 2 buckets. b) Jill and Jack’s parents are worried that the two children don’t cooperate enough with one another. Suppose they make Jill and Jack share equally their revenues from selling the water. Given that both are self-interested, construct the payoff matrix for the decisions Jill and Jack face regarding the number of pails of water each should carry. What is the equilibrium outcome? b) When the two children are forced to share revenues, their payoff matrix is as follows: Carry 1 Pail Carry 1 Pail Jack Carry 2 Pails Jill Carry 2 Pails $3 for Jill $3 for Jack $2.5 for Jill $5.5 for Jack $5.5 for Jill $2.5 for Jack $5 for Jill $5 for Jack To calculate their payoff: Jack and Jill both carry 1 pail, that is, 2 pails in total: Jack: $5(2)/2 - $2(1) = $3 Jill: $5(2)/2 - $2(1) = $3 Jack carries 2 pails, Jill carries 1 pail, that is, 3 pails in total: Jack: $5(3)/2 - $2(1) - $3(1) = $2.5 Jill: $5(3)/2 - $2(1) = $5.5 Jack carries 1 pail, Jill carries 2 pails, that is, 3 pails in total: Jill: $5(3)/2 - $2(1) - $3(1) = $2.5 Jack: $5(3)/2 - $2(1) = $5.5 Jack and Jill both carry 2 pails, that is, 4 pails in total: Jack: $5(4)/2 - $2(1) - $3(1) = $5 Jill: $5(4)/2 - $2(1) - $3(1) = $5 The payoffs are perfectly symmetric. If Jack assumes Jill carries 1 pail, Jack will carry 1 pail. If Jack assumes Jill carries 2 pails, Jack will carry 1 pail. If Jill assumes Jack carries 1 pail, Jill will carry 1 pail. If Jill assumes Jack carries 2 pails, Jill will carry 1 pail. The dominant strategy for both Jill and Jack is to carry only one bucket down the hill. This game is a prisoner’s dilemma. “ a game in which each player has a dominant strategy, and when each plays it, the resulting payoffs are smaller than if each had played a dominated strategy.” If each follows his dominant strategy, carry 1 pail, both will earn less profit than if both carry 2 pails. Additional Question #1 A dominant strategy occurs when A) B) C) D) E) One player has a strategy that yields the highest payoff independent of the other player’s choice. Both players have a strategy that yields the highest payoff independent of the other’s choice. Both players make the same choice. The payoff to a strategy depends on the choice made by the other player. Each player has a single strategy. Ans: A • Let’s illustrate this by an example: • Player 1’s dominant strategy is {Top}, because it gives him a higher payoff than {Bottom}, no matter what Player 2 chooses. • Player 2’s dominant strategy is {Right}. 2 Left Right Top (100, 30) (80, 90) Bottom (60, 60) (70, 100) 1 • Therefore, a dominant strategy is a strategy that yields the highest payoff compared to other available strategies, no matter what the other player’s choice is. • A rational player will always choose to play his dominant strategy (if there is any in the game), because this maximises his payoff. • The other strategy available to the player that yield a payoff strictly smaller than that of the dominant strategy is called a ‘dominated strategy’ (e.g. Player 2’s [Left}) • Dominant strategies may not exist in all games. It all depends on the payoff matrix. Additional Question #2 The prisoner’s dilemma refers to games where A) B) C) D) E) Neither player has a dominant strategy. One player has a dominant strategy and the other does not. Both players have a dominant strategy. Both players have a dominant strategy which results in the largest possible payoff. Both players have a dominant strategy which results in a lower payoff than their dominated strategies. Ans: E • The prisoner’s dilemma is a coordination game. • Both players have a dominant strategy, but the result of which is a lower payoff than the dominated strategies. 2 1 Confess Deny Confess Deny (-3, -3)* (0, -6) (-6, (-1, -1) 0) Additional Question #3a Jordan Comedy Comedy Lee Documentary Lee: 3 Jordan: 5 Lee: 2 Jordan: 2 Documentary Lee: 1 Jordan: 1 Lee: 5 Jordan: 3 3a) The payoff matrix shows the utilities from seeing Comedy or Documentary. The game has ? Nash Equilibrium. A) B) C) D) E) 0 1 2 3 4 Ans: C • Let’s look at the payoff matrix to find out the N.E. • {C, C} and {D, C} are the Nash Equilibria. • Hence, there are 2 N.E. in this game. • The N.E. is also known as pure strategy N.E., the adjective “pure strategy” is to distinguish it from the alternative of “mixed strategy” N.E. A mixed strategy N.E. is a N.E. in which players will randomly choose between two or more strategies with some probability. Jordan Comedy Documentary Comedy (3, 5) (1, 1) Documentary (2, 2) (5, 3) Lee Additional Question #3b 3b) By allowing for a timing element in this game, i.e., letting either Jordan or Lee buy a ticket first and then letting the other choose second, assuming rational players, the equilibrium is ? , based on ? . A) B) C) D) E) Still uncertain; who buys the 2nd ticket. Now determinant; who buys the 1st ticket. Now determinant; who buys the 2nd ticket. Still uncertain; who buys the 1st ticket. Now determinant; who is more cooperative. Ans: B • By allowing a timing element, the game is now a sequential game. • That means, one player moves first, and buys the first ticket. • The other player observes any action taken (i.e. knows what ticket has been bought), and then makes his / her decision. • Actions are not taken simultaneously anymore. • Whoever chooses an action can now predict how the other player is going to react. • E.g. If Lee chooses {Comedy}, he can be sure that Jordan will choose {Comedy} as well, because this gives Jordan a higher payoff than picking {Documentary}. • Therefore, the first mover has the advantage (called First Mover Advantage) to take actions first, hence securing his or her own payoff by predicting the response from the other player. • A rational (self-interested) player will always pick the action that maximises his or her own payoff (irregardless of others’) • Therefore, the result is determinant, as soon as we know who is buying the 1st ticket. Additional Question #4 A commitment problem exists when A) B) C) D) E) Players cannot make credible threats or promises. Players cannot make threats. There is a Prisoner’s Dilemma. Players cannot make promises. Players are playing games in which timing does not matter. Ans: A • In games like the prisoner’s dilemma, players have trouble arriving at the better outcomes for both players…. Because – Both players are unable to make credible commitments that they will choose a strategy that will ensue a better outcomes for both players (either in the form of credible threats or credible promises) • This is known as the commitment problem. Additional Question #5 Suppose Dean promises Matthew that he will always select the upper branch of either Y or Z. If Matthew believes Dean and Dean does in fact keep his promise, the outcome of the game is A) B) C) D) E) Unpredictable. Matthew and Dean both get $1,000. Matthew gets $500; Dean gets $1,500. Matthew gets $1.5m; Dean gets $1m. Matthew gets $400; Dean gets $1.5m. Ans: D • If Dean will indeed goes for the upper branch, then Matthew can either earn $1,000 by choosing the upper branch (i.e., arriving the node Y), or $1.5m by picking the lower branch (i.e., arriving the node Z). • As Matthew is a rational individual, he will choose a lower branch (i.e., arriving the node Z). (1000, 1000) Dean Y (500, 1500) X Matthew * Z Dean (1.5m, 1m) (400, 1.5m) End of Chapter 10