Midterm- Solutions Advanced Economic Theory, ECO326F1H Marcin Peski October 24, 2011, Version I

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1 Midterm- Solutions Advanced Economic Theory, ECO326FH Marcin Peski October 2, 20, Version I. Consider the following game: Player nplayer 2 L C R U 0; 0 2; 5; 0 M 7; 0 5; 2 ; D 2; 2 0; 3; 0 (a) Find all strictly dominated actions of player : Action D is dominated by U: Notice that u (U; L) = 0 > u (D; L) = 2; u (U; C) = 2 > u (D; C) = 0; u (U; R) = 5 > u (D; R) = 3: Action U is a best response against action L of player 2. Action M is a best response against action C of player 2. Because U and M are (sometimes) best responses, they cannot be strictly dominated. (b) Show that action C of player 2 is strictly dominated. (Alternatively, if you nd it easier, show that action C is never best response and refer to the appropriate result.) We show that C is strictly dominated against a mixed strategy ; 0: 3 :

2 Indeed, u 2 U; ; 0:3 u 2 M; ; 0:3 u 2 D; ; 0:3 = = 2:5 > u 2 (U; C) = ; = = 3 > u 2 (M; C) = 2; = = 8 > u 2 (D; C) = : (c) Which actions survive the iterated elimination of strictly dominated strategies? Actions D and C are eliminated in the rst round. In the second round, we have a game Player nplayer 2 L R U 0; 0 5; 0 M 7; 0 ; Notice that action M is dominated by action U: u (U; L) = 0 > 7 = u (M; L) ; u (U; R) = 5 > = u (M; R) : Thus, action M is going to be eliminated in the second round. In the thrid round, we have Player nplayer 2 L R U 0; 0 5; 0 Clearly action R is dominated by action L (payo 0 versus 0.). After eliminating R; we get Player nplayer 2 L U 0; 0 2

3 In the end, the only strategies that survive the iterated elimination of strictly dominated strategies are U and L (d) Find all Nash equilibria. Strategy pro le (U; L) is a Nasjh equiulibrium with payo s (0; 0) for both players. Because strategies that are eliminated in the iterative elimination of strictly dominated strategies cannot be played in nash equilinbrium, (U; L) is the only nash equilibrium. 2. Hannibal plans to invade Italy. There are two approaches: Sea (S) and Alpes (A): The road through Alpes is di cult and Hannibal can expect to lose some troops during the pass. A Roman general decides whether to defend Rome (R) or northern Italy (N). None of the commanders can split their armies between two locations. The situation can be modeled by the following game: Hannibal n Roman R N S 0; 2 2; 0 A ; 0 ; 2 In other words, the Roman general wins (gets payo 2) if he meets Hannibal and loses (gets 0) if Hannibal invades Italy unoposed. Hannibal receives 2 if he wins and 0 if he loses, and additionally, he gets disutility from choosing the mountain pass. (a) Does the game have a pure-strategy Nash equilibrium? No. We can check that each pure strategy pro le has a pro table deviation. For example, in pro le (S; N) ; Hannibal coould improve his payo by switching to A: 3

4 (b) Find the mixed strategy equilibrium: Suppose that Hannibal plays (; ) and the General plays (; ) : In mixed strategy equilibrium, hannibal must be indi erent between playing S and A : ( ) = + ( ) ( ) : This implies that = 3 : Similarly, the General must be indi erent between R and N : This implies that = 2 : ( ) = ( ) : (c) What is the Hannibal s expected payo in the mixed strategy equilibrium? Hannibal s expected payo is equal to = 3 + ( ) 3 = 2 : (d) Suppose the payo s are equal to Hannibal n Roman R N S 0; 2 2; 0 A 2 e; 0 e; 2 for some e < 2: Show that there is no pure strategy equilibrium. Find the mixed strategy equilibrium: How does Hannibal s expected payo depend on e? How does the probability that Hannibal invades through the sea depends on e? The argument that there is no pure strategy equilibrium is the same as in

5 (a). We nd the mixed strategy equilibrium with strategies (; Hannibal and (; ) for the General: ) for ( ) = (2 e) + ( e) ( ) ; ( ) = ( ) : Solving the above equations yields = 2+e and = : Notice that Hannibal s strategy does not change. If the mountain pass is more di cult 2 (i.e., e is higher), the Roman general must de ned Rome more often to make sure that Hannibal is indi erent between his two approaches.) Hannibal s payo is equal to ( ) = (2 e) + ( e) ( ) = e = Thus, his payo decreases with e. It is expected - higher e makes one of his approach more di cult, which worsens his overall strategic situation ( ) = ( ) : e 2 : 3. There are 00 people living in the suburbs and all of them commute to work in the city. Every morning, each individual decides which way to drive to the city. There are two routes: the Direct Way and the Long Way. The Long Way takes hour of driving. The time spent on the Direct Way depends on the tra c and it is equal to n D 30 ; where n D is the total number of cars taking the Direct Way. Each commuter wants to minimize the driving time. (a) Describe the best response function of each commuter. If there are 28 or less other people driving, then choosing L leads to hour of driving and choosing D leads to at most 28+ < hours of driving. Thus, 30 D is the best response. 5

6 If there are exactly 29 other people driving, then choosing D or L leads to the same driving time. Thus, both actions are best responses. If there are more than 29 people driving, then choosing D leads to a longer drive than L: Thus, only L is the best response. 8 >< D, if n i < 29 B i (n i ) = D or L if n i = 29 >: L; if n i > 29 9 >= >; : (b) Find a pure strategy Nash equilibrium. Is the equilibrium unique? There are many pure strategy Nash equilibria. In each equilibrium, there are either 29 or 30 people taking the Direct way and the rest takes the Long Way. (c) How many people take the Direct Way in the Nash equilibrium? What is the average driving time per commuter in the Nash equilibrium? The average driving time is either hour or slighltly less than hour. (d) Suppose that the government plans to expand the Direct Way and add one more lane. This would reduce the time spent driving along the Direct Way to n D 80 < n D 30 : Find the new equilibrium.what is the average driving time per commuter in the Nash equilibrium? If adding an extra lane is costly, what would be your recommendation to the government? In the new equilibrium, there will be between 79 to 80 people taking the direct way. The average driving time will be still around hour. It does not seem like a good idea to build one extra lane. 6

7 (e) Instead, suppose that the government considers adding two lanes. This would reduce the driving time along the Direct Way to n D 50 : Would your answer to the previous question change? Explain. In the new equilibrium with two lanes, everybody would take the Direct Way. The average driving time will fall to 0 minutes. If the gain of 20minutes for 00 driviers justify the extra cost of building two more lanes, it should be done. 7