402 Midterm 2
Ron Siegel
Spring 2017
Directions
• Write your name on the exam.
• Exams written with a pencil will not be regraded and cannot be disputed.
• Provide one- or two-sentence explanations when they are requested.
• Complete all problems in the space provided.
• You have two sheets of scratch paper. If you need more scratch paper, raise your
hand and one of us will bring it to you.
• Work on scratch paper will not be graded, but everything written on the exam will be
graded. If you need to change an answer, clearly cross out the part you are changing.
• Cell phones and calculators are prohibited.
• You have 75 minutes. The exam ends promptly at 3:50. Five (5) points will be
deducted for every reminder to stop writing.
• There are 110 points overall.
Name: ______________
• Good luck!
1
, 1. (30 points) Let us revisit the location game we solved in class, in which each of two
vendors chooses to locate in one of n regions. For simplicity, assume that each region
has only two customers, and that selling to one customer leads to a profit of 1. A
vendor’s utility is equal to his profit, which is equal to the number of customers he
sells to. A vendor sells to both customers in a region if he is closer to the region than
the other vendor. The different parts of the question vary in the number of regions
and in who the customers buy from if their region is equally close to both vendors.
Suppose first that, similarly to the game we saw in class, if a region is equally close
to both vendors, then each vendor sells to one customer in the region.
(a) Suppose that n = 3.
i. (3 points) Write the game in normal form.
1 2 3
1 3, 3 2, 4 3, 3
2 4, 2 3, 3 4, 2
3 3, 3 2, 4 3, 3
ii. (3 points) Suppose all you know is that players are rational. What can you
say about players’ choice of strategies? Explain.
It can be said that for player 1 choosing to locate on the second market
dominates both choosing to locate on the first and on the third market
((4, 3, 4) > (3, 2, 3)). Hence, he will choose to locate on the second market.
The same can be said about player 2.
(b) Suppose that n = 5.
i. (3 points) Repeat part a(ii).
1 2 3 4 5
1 5, 5 2, 8 3, 7 4, 6 5, 5
2 8, 2 5, 5 4, 6 5, 5 6, 4
3 7, 3 6, 4 5, 5 6, 4 7, 3
4 6, 4 5, 5 4, 6 5, 5 8, 2
5 5, 5 4, 6 3, 7 2, 8 5, 5
2
Ron Siegel
Spring 2017
Directions
• Write your name on the exam.
• Exams written with a pencil will not be regraded and cannot be disputed.
• Provide one- or two-sentence explanations when they are requested.
• Complete all problems in the space provided.
• You have two sheets of scratch paper. If you need more scratch paper, raise your
hand and one of us will bring it to you.
• Work on scratch paper will not be graded, but everything written on the exam will be
graded. If you need to change an answer, clearly cross out the part you are changing.
• Cell phones and calculators are prohibited.
• You have 75 minutes. The exam ends promptly at 3:50. Five (5) points will be
deducted for every reminder to stop writing.
• There are 110 points overall.
Name: ______________
• Good luck!
1
, 1. (30 points) Let us revisit the location game we solved in class, in which each of two
vendors chooses to locate in one of n regions. For simplicity, assume that each region
has only two customers, and that selling to one customer leads to a profit of 1. A
vendor’s utility is equal to his profit, which is equal to the number of customers he
sells to. A vendor sells to both customers in a region if he is closer to the region than
the other vendor. The different parts of the question vary in the number of regions
and in who the customers buy from if their region is equally close to both vendors.
Suppose first that, similarly to the game we saw in class, if a region is equally close
to both vendors, then each vendor sells to one customer in the region.
(a) Suppose that n = 3.
i. (3 points) Write the game in normal form.
1 2 3
1 3, 3 2, 4 3, 3
2 4, 2 3, 3 4, 2
3 3, 3 2, 4 3, 3
ii. (3 points) Suppose all you know is that players are rational. What can you
say about players’ choice of strategies? Explain.
It can be said that for player 1 choosing to locate on the second market
dominates both choosing to locate on the first and on the third market
((4, 3, 4) > (3, 2, 3)). Hence, he will choose to locate on the second market.
The same can be said about player 2.
(b) Suppose that n = 5.
i. (3 points) Repeat part a(ii).
1 2 3 4 5
1 5, 5 2, 8 3, 7 4, 6 5, 5
2 8, 2 5, 5 4, 6 5, 5 6, 4
3 7, 3 6, 4 5, 5 6, 4 7, 3
4 6, 4 5, 5 4, 6 5, 5 8, 2
5 5, 5 4, 6 3, 7 2, 8 5, 5
2
