EC2200 Mathematical Economics 1A 2020/21
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
EC2200
2020/21
Mathematical Economics 1A
1. Consider the following game:
Zelda
Left Mid Right
2 2 |
4, 3 |
2 1 |
1, 3 |
8, 2 |
1 2 |
4 1 |
2, 3 |
3, 4 |
(a) Find this game’s Nash Equilibria, Iterated Elimination Equilibria, and Strictly Dominant
Strategy Equilibria. (15 marks)
(b) Suppose Paula gets to act before Zelda does, and that Zelda gets to observe Paula’s
action before acting. Draw a game tree representing this. (5 marks)
(c) Find all SPNEs of this game. (10 marks)
(d) Suppose again that Paula acts first, but Zelda is not perfectly informed of Paula’s action; instead, Zelda always knows if Paula played Bot, but cannot differentiate between Top and Mid. Draw the game tree for this game. (5 marks)
(e) Find all pure SPNEs of this game. Are they sequentially rational at every information
set? If so, show this. If not, apply a pure solution concept that will deliver sequential
rationality and derive all equilibria. (15 marks)
2. In the following game of incomplete information, the common prior puts probability 1/2 on each of player 1 ’s types θ 1 and θ2 .
N
1
1
Left Right
1 2 -1 1
-1 -1 -1 2
Find all pure-strategy Perfect Bayesian Equilibria of the game, specifying beliefs.
(20 marks)
(b) Which (pure-strategy) PBEs survive the Cho-Kreps Intuitive Criterion and which do not? (20 marks)
Now, let’s simplify the game to the following:
1
0, 3
Left
1 2
(c) Suppose the above game is infinitely repeated with common discount factor δ. Is there a strategy profile that gives player 2 a utility of 3, and is an SPNE strategy profile for all high enough discount factors δ? If so, find one. If not, show this is impossible.
(10 marks)
3. A sequence of players I = N sit down at Chez Piggy, and each must, in sequence, order one main: the salad, the steak, or the picanha. Chez Piggy has recently got a new grillmaster; the common prior puts probability .5 on the grillmaster being good. Ordering a salad always gives a player 1 utility. Ordering the steak gives a player 3 utility if the grillmaster is good, and 0 utility if they are bad. Ordering the picanha gives a player 5 utility if the grillmaster is good, and -8 if they are bad. Each player receives a signal of the grillmaster’s ability that is correct with probability p _ (.5, 1). These signals are mutually independent conditional on the grillmaster’s ability. Each player also knows what every player before them ordered.
Plot a player’s utility of each of their pure actions, as a function of their updated beliefs
q _ (0, 1). (10 marks)
(b) What does a player order, as a function of their updated beliefs?1 (10 marks) (c) For what p is it possible that some player orders the steak in a PBE?2 (10 marks) (d) For what p is it possible that some player orders the salad in a PBE? (10 marks) (e) For what p is it possible that some player orders the picanha in a PBE? (10 marks)
2022-05-11