Econ 411 Spring 2023 Assignment 1
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Econ 411
Spring 2023
Assignment 1
Due January 24 in class.
1. In the following normal form games, Önd all strictly dominated strategies, strictly dominant strategies, weakly dominated strategies, weakly dominant strategies, and every pure strategy Nash equilibria
1. Chicken
Straight Swerve
Straight 0,0 4,1
Swerve 1,4 3,3
2. Battle of the sexes
Football Opera
football 1,2 0,0
opera 0,0 2,1
3. 3 player voting game with 2alternatives (Chris is the ìmatrix-playerî)
Chris Chris
Yes No
Bob Bob
Yes No Yes No
Alice Yes 1; 1; -1 1; 1; -1 Alice Yes 1; 1; -1 0; 0; 0
No 1; 1; -1 0; 0; 0 No 0; 0; 0 0; 0; 0
4. 3 player voting game with 3 alterantives
Chris Chris
A B
Bob Bob
A B C A B C
A 2; 0; 1 2; 0; 1 2; 0; 1 A 2; 0; 1 1; 2; 0 2; 0; 1
B 2; 0; 1 1; 2; 0 2; 0; 1 B 1; 2; 0 1; 2; 0 1; 2; 0
C 2; 0; 1 2; 0; 1 0; 1; 2 C 2; 0; 1 1; 2; 0 0; 1; 2
Chris
C
Bob
A B C
Alice
5. Coordination game
6. Rock-Scissors-Paper
A 2; 0; 1 2; 0; 1 0; 1; 2
B 2; 0; 1 1; 2; 0 0; 1; 2
C 0; 1; 2 0; 1; 2 0; 1; 2
|
A B C |
a
0; 0 0; 0 |
b 0; 0
0; 0 |
c 0; 0 0; 0 10; 10 |
|
R S P |
r 0; 0
; |
s p
0; 0 1; -1 -1; 1 0; 0 |
7. A game with no particular interpretation
a
A 1 -1
;
;
;
b
-1 1
;
;
c
10; -2
5; 0
1 -2
0; 10
|
2. Consider the normal form |
game
a b c d e f g h |
A 1; 0 0; 1 0; 10 20; 1 0; 10 10; 5 15; 5 100; 5 |
B
3; 2 0; 3 5; 4 1; 0 4; 6 4; 7 4; 8 |
C 1; 0 0; 1 10; 2 20; 3 0; 4 100; 5 5; 6 0; 7 |
D 2; 0 0; 1 0; 2 5; 3 1; 10 4; 5 5; 5 100; 7 |
E
0; 0 0; 0 5; 20 0; 5 4; 0 3; 0 4; 10 |
F
0; 1 0; 1 5; 0 1; 5
0; 1 4; 5 |
1. Find all pure strategy Nash equilibria.
2. Find all strictly dominated strategies.
3. After removing all strictly dominated strategies, Önd all strictly dominated strategies in the game with the strategies you found in part b. eliminated.
4. Continue the process until you can no longer Önd any more (iteratively) undominated strategies.
3. Consider the following version of the Cournot duopoly game. Inverse demand is p(q) = max{2 - q;0} (feel free to ignore the non-negativity constraint and let p(q) = 2 - q) and both Örms have constant unit cost c = 1: Firms simultaneously pick quantities q1 > 0 and q2 > 0 and the price is set to clear the market given the quantities chosen.
1. Carefully derive/write down the normal form of the game given that Örms seek to maximize their proÖts..
2. Write down the optimization problem that deÖne the best response functions and solve explicitly for the best response functions. Be careful to deal with the possibility of a corner solution (qi = 0 being optimal).
3. Solve for all Nash equilibria. Illustrate in a picture with the two best responses drawn in (q1 ;q2 ) space.
4. Assume instead that p(q) = max{1 - q;0} and c = 0: Can you Önd more than one Nash equilibrium (or an inÖnity)?
2023-01-28