Simon Fraser University
ECON 302
Spring 2016
S. Lu
Problem Set 3
Due on February 1, at 2:30pm
Remember to show your work on all questions.
Note: Your solutions to this problem set may not be returned to you before the quiz. I therefore
suggest that you keep a copy of your solutions for studying purposes.
1. Prove that if expected utility function
𝑈([𝑝1 , 𝑝2 , … , 𝑝𝑛 ]) = 𝑝1 𝑢1 + 𝑝2 𝑢2 + ⋯ + 𝑝𝑛 𝑢𝑛
represents preferences over lotteries ≿, then expected utility function
𝑉([𝑝1 , 𝑝2 , … , 𝑝𝑛 ]) = 𝑝1 𝑣1 + 𝑝2 𝑣2 + ⋯ + 𝑝𝑛 𝑣𝑛
also represents the same preferences ≿ if 𝑣𝑖 = 𝑎𝑢𝑖 + 𝑏 for all 𝑖, where 𝑎 > 0 and 𝑏 are real
numbers.
2. Jill is contemplating her preferences concerning tonight’s activities. She strictly prefers:
- going out with 50% chance and working with 50% chance, over
- watching TV for sure, over
- going out with 40% chance, watching TV with 20% chance and working with 40% chance.
a) Show that Jill’s preferences violate the independence axiom.
[Hint: Start by decomposing the third lottery into a combination of the first two. That is,
if the first lottery is denoted A, the second B, and the third C, find 𝑥 ∈ [0,1] such that
𝐶 = 𝑥𝐴 + (1 − 𝑥)𝐵. Then show that 𝐴 ≻ 𝐵 ≻ 𝑥𝐴 + (1 − 𝑥)𝐵 violates the independence
axiom.]
b) Since Jill’s preferences violate the independence axiom, we know that they do not admit
an expected utility representation. Show directly that it is impossible to assign utilities to
the outcomes so that the ranking of the expected utilities of the three lotteries matches
Jill’s preference ranking.
[Hint: To do this, first assume that the outcomes generate utilities 𝑢1 , 𝑢2 , 𝑢3 . Then
compute the expected utilities of the three lotteries above, and derive a contradiction
between the inequalities corresponding to Jill’s preferences.]
Problem set 3 continues on the next page.
3. For each of the following games, answer the following questions:
i) Find all Pareto efficient outcomes. (Do not assume transferable utility.)
ii) Is the game dominance solvable? If so, find the solution.
Game a
Give Easy Test
Give Hard Test
Study
0,2
5,1
Slack Off
1,1
-5,0
Disclaimer: Game a does not describe accurately the payoffs in ECON 302.
Game b
Post Risqué Picture
Not Post Risqué Picture
Game c
Bring a Cake
Bring Chicken
Game d
Wear Blue
Wear Red
Check Child’s Facebook
Profile
-5,1
-1,3
Ignore Child’s Facebook
Profile
5,0
0,4
Make Tiramisu
-1,-1
3,2
Make Steak
2,1
0,0
Wear White
0,0
0,0
Wear Black
0,0
0,0
Left
0,0
1,2
Right
3,-1
x,y
4. Consider the following game:
Top
Bottom
For what values of x and y is the game dominance solvable?