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Economic Management Strategy: Hwrk 1 1 Simultaneous-Move Economic Management Strategy: Hwrk 1 1 Simultaneous-Move Game Theory Questions. 1.1 Chicken Lee and Spike want to see who is the bravest. To do so, they play a game called chicken. (Readers, don’t try this at home!) They take their cars and drive as fast as they can towards each other. If they both swerve, they both get utility 1. If one swerves and the other doesn’t, the one that swerves is the chicken and gets 0 and the brave one gets 2. If neither swerves, they crash and both get utility 3. (i) Model this as a game by− writing its normal form. (ii) Are there any pure-strategy equilibria of the game? Please explain why or why not for each possibility. (iii) Are there any mixed-strategy equilibria of the game? If yes, give the equilibrium. If no, please explain. Answer. One can write the normal form game as follows Lee Swerve Don0t Swerve 1, 1 0, 2 Spike Don0t 2, 0 3, 3 − − A Nash equilibrium is a set of strategies such that each player has no incentive for deviation given the other player’s strategy. A pure strategy N.E. is where the set of strategies are pure, that is, either Swerve or Don’t. There are two pure-strategy Nash equilibria, each with one swerving and the other not swerving (Don’t). We see this since the player not swerving receives 0, switching to swerving reduces his payoff to -3. The player swerving receives 2, switching to not swerving reduces his payoff to 1. A mixed-strategy N.E. is where the set of strategies in the Nash equilib- rium involve randomly choosing pure strategies with a particular distribution. Here there is also a mixed-strategy N.E. with each swerving with probability 3/4. We see this using the following logic. If Lee is choosing Swerve with 1 probability p and Don0t with probability 1 p, then Spike’s payoff for Swerve − is 1 p +0 (1 p)=p. Also with these probabilities Spike’s payoff for Don0t is 2 ·p +( · 3)−(1 p)=5p 3. Since Spike will also be mixing, he will could not· gain by− deviating.· − This− means that his payoffs from each strategy should be the same. Hence, p =5p 3orp =3/4andLeewillchooseSwerve with probability 3/4. Likewise, we− can derive Spike’s strategy as choosing Swerve with probability 3/4. In summation, the set of Nash equilibria are (Swerve,Don0t), (Don0t, Swerve), (.75 Swerve+.25 Don0t, .75 Swerve+ .25 Don0t). 1.2 The Hair-Cut Game Bill and Ted are about to graduate from high school. They decided to do something wild and get crazy haircuts for the last day of school. The problem is that they live across town and have no way of knowing what the other has decided. They concluded that getting haircuts would be most bodacious and give utility of 4 to each. This would improve their standard utility of 1 each. Bill got to thinking further and determined that if Ted didn’t get a haircut it would suck and he would get utility 0. Ted, however, thinks it would be excellent and get utility 2. Ted also determined that if Bill didn’t get a haircut and he did it would suck and he would get utility 0. Bill, however, thinks that case would be excellent and get utility 2.(i) Model this as a game by writing its normal form.(ii) Are there any pure-strategy equilibria of the game? Please explain why or why not for each possibility.(iii) Are there any mixed-strategy equilibria of the game? If yes, give the equilibrium. If no, please explain. Answer. One can write the normal form game as follows Ted Cut Don0t Cut 4, 4 0, 2 Bill Don0t 2, 0 1, 1 A Nash equilibrium is a set of strategies such that each player has no incentive for deviation given the other player’s strategy. A pure strategy 2 N.E. is where the set of strategies are pure, that is, either getting a haircut or not (don’t). There are two pure-strategy Nash equilibria: both getting haircuts and both not getting haircuts. A mixed-strategy N.E. is where the set of strategies in the Nash equilib- rium involve randomly choosing pure strategies with a particular distribution. Here there is also a mixed-strategy N.E. with each swerving with probabil- ity 3/4. We see this using the following logic. If Ted is choosing Cut with probability p and Don0t with probability 1 p, then Bill’s payoff for Cut is − 4 p +0 (1 p)=4p. Also with these probabilities Bill’s payoff for Don0t is· 2 p +1· (1− p)=1+p. Since Bill will also be mixing, he will could not gain· by deviating.· − This means that his payoffs from each strategy should be the same. Hence, 4p =1+p or p =1/3 and Ted will choose Cut with probability 1/3. Likewise, we can derive Bill’s strategy as choosing Cut with probability 1/3. In summation, the set of Nash equilibria are (Cut,Cut), (Don0t, Don0t), (1/3 Cut +2/3 Don0t, 1/3 Cut +2/3 Don0t). 2 Extensive Form Questions. 2.1 Boeing versus Airbus. Consider the rivalry between Airbus and Boeing to develop a new commercial jet aircraft. Suppose Boeing is ahead in the development process, and Airbus is considering whether to enter the market. If Airbus stays out, it earns zero profit, while Boeing enjoys a monopoly and earns a profit of $1 billion. If Airbus decides to enter and develop the rival plane, then Boeing has to decide whether to accommodate Airbus peacefully, or to wage a price war. In the event of peaceful competition, each firm will make a profitof$300 million. If there is a price war, however, each will lose $100 million because the prices will fall so low that neither will be able to recoup its development costs.(a.) Draw the tree for this game. Use backward induction to find the equilibrium.(b.) Why is Boeing unlikely to be happy about the equilibrium? What would it have preferred? Could it have made a credible threat to get Airbus to behave as it wanted?(c.) What if Boeing had moved first? Would there still have been a credibility problem with Price War? Explain. 3 Answer. The game tree will have Airbus at the first decision node. There would be two branches: enter and stay out. The enter leads to a decision node with Boeing deciding to accept or fight. The payoffs are (0 for Airbus,1 billion for Boeing) for stay out, (300 million, 300 million) for enter-accept, and (- 100 million, -100 million) for enter-fight. Since after enter is chosen, Boeing should rationally choose accept, it makes sense for Airbus to enter. Boeing would prefer to have Airbus stay out. It could make a threat of fighting, but this is not credible since at that stage it would hurt Boeing by 400 million. If Boeing could move first (such as hiring a crazy boss) the game tree would have branches: fight-enter, fight-stay out, accept-enter, accept-stay out. If Boeing chooses fight, then Airbus stays out. If Boeing chooses accept, then Airbus enters. Thus, Boeing would choose fight (hire the tough boss). 2.2 Tic-Tac-Toe Take the following game of tic-tac-toe with O’s turn to move. O O X X X Please write out the extensive form with labeling the possible moves by the following numbers. 1 2 O O X X X 3 4 Which outcomes are possible? What is/are the subgame perfect equilib- ria? Do you think you can write the entire tree out for tic-tac-toe from the beginning? Answer. FirstnoticethatifO chooses 1 and 2, he wins. If X chooses either 2 and 3 or 1 and 4, then he wins. All other choices results in a tie. Thus, are outcomes are possible, though some may be unlikely. The following table lists all possible outcome by the order of choices. The first row is O’s first move. The second row is X’s first move. The third row is O’s second move. The fourth row is X’s second move. For simplicity I let the game continue until all boxes were filled in. The fifth row shows the outcomes: X means 4 that X wins. O means that O wins. T means there was a tie. The sixth row shows the branches eliminated by O’s second decision (labelled c). The seventh row shows the additional branches eliminated by X’s first decision (labelled b). Notice the only remaining branches are ties. These will not be eliminated by O’s first move. Thus, they are the sub-game perfect equilibria. Notice that if O chooses either 1 or 2 on his first move, the subgame-perfect sequence from then on is determined. O 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 X 2 2 3 3 4 4 1 1 3 3 4 4 1 1 2 2 4 4 1 1 2 2 3 3 O 3 4 2 4 2 3 3 4 1 4 1 3 2 4 1 4 1 2 2 3 1 3 1 2 X 4 3 4 2 3 2 4 3 4 1 3 1 4 2 4 1 2 1 3 2 3 1 2 1 T X O X O T X T O T O X X T T T T X T T X T X T .
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