midterm_exam_2007_solutions

# midterm_exam_2007_solutions - Answers to Midterm Exam Econ...

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Answers to Midterm Exam Econ 159a/MGT522a Ben Polak Fall 2007 ° The answers below are more complete than required to get the points. In general, more concise explanations are better.

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Question 1. [ 15 total points: use blue book 1] . Short-Answer Questions . State whether each of the following claims is true or false (or can not be determined). For each, explain your answer in (at most) one short paragraph . Each part is worth 5 points , of which 4 points are for the explanation. Explaining an example or a counter-example is su°cient. Absent this, a nice concise intuition is su°cient: you do not need to provide a formal proof. Points will be deducted for incorrect explanations. (a) [5 points] \A strictly dominated strategy can never be a best response." Answer: True. The strategy that strictly dominates it, by de°nition, yields a strictly higher payo± against all strategies and hence is a better response. (b) [5 points] \In the candidate-voter model, if two people are standing, one to the left of center and one to the right of center, and neither of them is ‘too extreme’, then it is an equilibrium." Answer: We accepted true, false or it depends depending on the explanation. This is an equi- librium provided that the players are symmetric around 1 = 2 ; i.e., equidistant from half. If they are not symmetric, it is not an equilibrium. Notice that players cannot move in the candidate voter model so this is not a possible deviation. (c) [5 points] \If (^ s; ^ s ) is a Nash equilibrium of a symmetric, two-player game then ^ s is evolutionarily stable." Answer: False. For example consider the game below. a b a 1 ; 1 0 ; 0 b 0 ; 0 0 ; 0 Clearly, ( b; b ) is a symmetric NE since u ( b; b ) ± u ( a; b ). It is not ES, however, since, we have u ( b; b ) = u ( a; b ) = 0, but u ( b; a ) < u ( a; a ), a violation of condition ( B ). Hence a monomorphic population of b would be vulnerable to an invasion of mutants that played a . 2
Question 2. [30 total points] \Party Games" . Roger has invited Caleb to his party. Roger must choose whether or not to hire a clown. Simultaneously, Caleb must decide whether or not to go the party. Caleb likes Roger but he hates clowns { he even hates other people seeing clowns! Caleb’s payo± from going to the party is 4 if there is no clown, but 0 if there is a clown there. Caleb’s payo± from not going to the party is 3 if there is no clown at the party, but 1 if there is a clown at the party. Roger likes clowns { he especially likes Caleb’s reaction to them | but does not like paying for them. Roger’s payo± if Caleb comes to the party is 4 if there is no clown, but 8 ² x if there is a clown ( x is the cost of a clown). Roger’s payo± if Caleb does not come to the party is 2 if there is no clown, but 3 ² x if there is a clown there. (a) [6 points] Write down the payo± matrix of this game. Answer: (Throughout, let underlines indicate best-response payo±s.) Caleb go not go Roger Hire 8 ² x; 0 3 ² x; 1 Not 4 ; 4 2 ; 3 (b) [6 points] Suppose x = 0. Identify any dominated strategies. Explain. Find the Nash equilibrium. What are the equilibrium payo±s?

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