Econ 309H Problem Set 1.
Due: September 24, 2024
Use of ChatGPT or similar AI programs for this problem set is prohibited and is considered to be cheating.
1. Let R be the set of real numbers, and R2 be the set of all pairs of real numbers. For
example, (2.5, 3) belongs to R2 . The lexicographic relation ≤lex on R2 is defined by:
(x1 , x2 ) ≤lex (y1 , y2 ) ⇔ [x1 < y1 or (x1 = y1 and x2 ≤ y2 )].
(a)
i. Is it the case that (1, 2) ≤lex (2, 1)? Explain.
ii. Is it the case that (3, 4) ≤lex (2, 100)? Explain.
iii. Is it the case that (3, 4) ≤lex (3, 5)? Explain.
(b) Explain in words what the lexicographic relation means.
(c) Is ≤lex complete? Prove that your answer is true.
(d) Is ≤lex transitive? Prove that your answer is true.
2. For each of the following games find the set of all Nash equilibira.
(a)
A
B
A
B
10, 10 0, 0
0, 0 0, 0
U
D
L
R
5, 1 6, 2
6, 4 5, 3
A
B
C
D
4, 1 3, 2
1, 4 2, 3
U
D
L
R
1, 1 1, 2
0, 3 2, 2
(b)
(c)
(d)
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3. Do Exercise 27.1 in Osborne’s Introduction to Game Theory
4. Consider the following game:
A
B
9, 9 0, 8
8, 0 7, 7
A
B
Recall that player 1 selects a row and player 2 selects a column, and the generally the
table is to be understood as discussed in class and in the textbook.
(a) What is set of actions for player 1?
(b) What is the set of action profiles.
(c) Describe player 1’s preference relation over action profiles?
(d) List all the Nash equilibria in this game.
(e) If you were player 1, what action would you choose? Why?
(f) Suppose that you were player 1, and player 2 made the following suggestion before
the game was played: “I will play A. Let’s both play A”? Would you go along
with this suggestion? In answering this question, think about whether you would
believe player 2’s statement was true. How do your answers bear on the idea that
Nash equilibria can be thought of as self-enforcing agreements which players reach
before the game is played?
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