UNIVERSITY OF WARWICK Summer Examinations 2020/21 Mathematical Economics 1A
Time Allowed: 2 Hours
Read all instructions carefully – and read through the entire paper at least once before you start entering your answers.
There is ONE section in this paper. Answer TWO questions ONLY (50 marks each).
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Approved pocket calculators are allowed.
You should not submit answers to more than the required number of questions. If you do, we will mark the questions in the order that they appear, up to the required number of questions in each section.
1. Consider the following game:
Left Mid Right
Top 2,2 4,3 2,1 Mid 1,3 8,2 1,2 Bot 4,1 2,3 3,4
(a) Find this game’s , 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)
(Question 1 continued overleaf)
(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 θ1 and θ2.
Out In Out In
Lef t Right Lef t
Right 1,2 −1,1 −1,−1 −1,2
(a) 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:
(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)
Right 1,2 −1,1
(Continued overleaf)
3. A sequence of players I = N sit down at , and each must, in sequence, order one main: the salad, the steak, or the picanha. 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.
(a) 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)
1You can ignore beliefs that lead to indifference
2That is, for what values of p is there an equilibrium and a player for which steak-ordering on-path? (And similarly for the other parts of the question)
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