Econ 5001 Spring 2018 Prof. James Peck The Ohio State University Department of Economics Second Midterm Examination Answers Note: There were 4 versions of the test: A, B, C, and D, based on player 1 s left first choice in question 4. The problem will be worked out for exam A, and just the answers for the other exams will be given. 1. (25 points) Consider the following Cournot game in which two firms simultaneously choose a non-negative output quantity. Let 1 denote the quantity chosen by firm 1, let 2 denote the quantity chosen by firm 2, and let denote the total quantity,. The market price is given by = 1000 Each firm has a marginal production cost of 5 per unit of output. Notice that the demand function is not linear like the one presented in class. (a) (5 points) Give the payoff functions for firm 1 and firm 2. (b) (10 points) Find the best response function for firm 1. [In case you forgot from your calculus course, the rule for taking the derivative of a quotient is given below.] (c) (10 points) Find the Nash equilibrium of this game. [Hint: Because the firms face the same market and have the same cost function, the NE will be symmetric, with 1 = 2 = for some. Substitute this symmetry condition into the best response function from part (b) to get one equation in the unknown.] Here is the quotient rule. If ( ) and ( ) are functions of, where 0 ( ) and 0 ( ) are the derivatives of these functions, then the derivative of the fraction, ( ) ( ) with respect to is given by ( ) 0 ( ) ( ) 0 ( ) [ ( )] 2 (a) Here are the payoff functions 1 ( 1 2 ) = 1000 1 5 1 (1) 2 ( 1 2 ) = 1000 2 5 2 1
(b) Taking the derivative of (1) with respect to 1,andsettingitequalto zero, yields the first order condition, 1000[ 1 ] ( ) 2 5 = 0 or 1000 2 ( ) 2 = 5 ( ) 2 = 200 2 Solving for 1 yields the best response function: = p 200 2 1 = p 200 2 2 (c) Based on the hint, we can use symmetry to solve for 1 = 2 = from the best response function Thus, the Nash equilibrium is (50 50). = 2 = p 200 200 4( ) 2 = 200 = 50 For exam B, the payoff functions are the best response function is 1 ( 1 2 ) = 640 1 4 1 2 ( 1 2 ) = 640 2 4 2 1 = p 160 2 2 and the Nash equilibrium is (40 40). For exam C, the payoff functions are the best response function is and the Nash equilibrium is (100 100). 1 ( 1 2 ) = 1200 1 3 1 2 ( 1 2 ) = 1200 2 3 2 1 = p 400 2 2 2
For exam D, the payoff functions are the best response function is and the Nash equilibrium is (70 70). 1 ( 1 2 ) = 560 1 2 1 2 ( 1 2 ) = 560 2 2 2 1 = p 280 2 2 3
2. (20 points) Consider the following variant of the election platform game. There are two players, candidate A and candidate B. Each candidate must choose a platform in policy space, represented by the "unit" interval between 0 and 1. The voters have ideal policy positions uniformly distributed over the unit interval. Candidate A has more charisma or personal charm than candidate B, which translates into the following voting behavior. If a voter s ideal position is and the candidates positions are and, then the voter votes for candidate B if the distance between and, minus the distance between and,isat least 1 8. Otherwise, this voter will vote for candidate A. In other words, B must be "better" in policy terms than A by at least 1 8 in order to receive the vote. Assume that the candidates only care about the election outcome, and not about the policy they have to implement. If a candidate wins strictly more than half the votes, his/her payoff is 1; if a candidate wins exactly half the votes, his/her payoff is 1 2 ; and if a candidate wins strictly less than half the votes, his/her payoff is 0. Find all of the pure-strategy Nash equilibria of this game. Notice that player A can guarantee a win by locating at the center, = 1 2. In fact, player A can guarantee a win by locating close enough to the center. How close? Suppose 1 2 holds. Then the most votes candidate B can receive is by locating exactly 1 8 to the right of. All voters to the left of would vote for candidate A and all voters to the right of would vote for candidate B. Thus, if candidate A chooses 3 8 1 2, the best candidate B can do is locate at + 1 8, which must be strictly to the right of 1 2, and candidate A wins. A symmetric argument applies for 5 8 1 2. If candidate A locates such that 3 8 5 8, holds, candidate A wins, so is a best response to any. Since candidate B is guaranteed to lose, any is a best response to,soany pair, ( ) is a Nash equilibrium if 3 8 5 8 holds. To see that no other profile can be a Nash equilibrium, first if we have = 3 8 or = 5 8, then candidate B can achieve a tie by choosing = 1 2,sothisis candidate B s only best response. But if candidate B is choosing = 1 2,then candidate A is not best responding to.next,ifwehave 3 8 or 5 8, then candidate B can achieve a win by choosing = 1 2. But if candidate B is best responding to, and therefore winning, then candidate A is not best responding to. 4
3. (20 points) Consider the following game. player 2 L R U 16 0 2 6 player 1 M 0 8 12 1 D 10 8 4 8 Find the mixed strategy Nash equilibrium of this game, and show all your work. First we show that D is dominated by a mixed strategy of the form ( 1 0). For the mixture to yield higher payoff than D when player 2 plays L, we must have the inequality, 16 10 5 8 For the mixture to yield higher payoff than D when player 2 plays R, we must have the inequality, 2 +12 12 4 4 5 For any between 5 8 and 4 5, the mixed strategy dominates D. Next we find the mixed strategy Nash equilibrium of the form 1 =( 1 0) and 2 =( 1 ). Forplayer1tobeindifferent between U and M, we must have 16 +2 2 = 12 12 = 5 13 Forplayer2tobeindifferent between L and R, we must have 8 8 = 6 +1 = 7 13 The MSNE is 1 =( 7 13 6 13 0) and 2 =( 5 13 8 13 ). For exam B, the MSNE is 1 =( 3 8 0 5 8 ) and 2 =( 8 13 5 13 ). For exam C, the MSNE is 1 =(0 1 2 1 2 ) and 2 =( 5 13 8 13 ). For exam D, the MSNE is 1 =( 2 3 0 1 3 ) and 2 =( 5 13 8 13 ). 5
4. (15 points) For the following game in extensive form, solve using backward induction, and indicate the equilibrium strategy profile here: For exam A, the backward induction solution is ( ). For exam B, the backward induction solution is ( ). For exam C, the backward induction solution is ( ). For exam D, the backward induction solution is ( ). 6
5. (20 points) Consider the following game in extensive form. (a) (10 points) Find all of the Nash equilibria of this game. (b) (10 points) Find all of the subgame perfect Nash equilibria of this game. For exam A, the normal form matrix is player 2 IK IL JK JL FM 9 4 9 4 0 0 0 0 FN 9 4 9 4 0 0 0 0 player 1 GM 8 5 0 0 8 5 0 0 GN 0 0 10 3 0 0 10 3 HM 8 9 8 9 8 9 8 9 HN 8 9 8 9 8 9 8 9 By finding all best responses in the matrix, the NE are: ( ), ( ), ( ), ( ), ( ), ( ), and( ). In order for the subgame following F to be a NE, player 2 must play I and not J. Therefore, ( ), ( ), ( ), and ( ) are not subgame perfect. In order for the other proper subgame to be in a NE, the continuation strategies must be ( ) or ( ), but not ( ). Therefore, ( ) is not subgame perfect. The SPNE are ( ) and ( ). For exam B, the NE are ( ), ( ), ( ), ( ), ( ), ( ), and( ). TheSPNEare( ) and ( ). For exam C, the NE are ( ), ( ), ( ), ( ), ( ), ( ), and( ). TheSPNEare( ) and ( ). For exam D, the NE are ( ), ( ), ( ), ( ), ( ), ( ), and( ). The SPNE are ( ) and ( ). 7