EC2200 Mathematical Economics 1A 2019/20
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EC2200
2019/20
Mathematical Economics 1A
1. It’s the 2021 Champions League final and it’s all going to be decided by the last penalty shot: Barcelona’s Lionel Messi shooting against Liverpool’s Alisson Becker. Messi and Alisson have to decide - simultaneously - where to shoot and dive, respectively. If they both go for the same side, Alisson has a chance to catch the ball, but less so on the right, as Messi’s shots are stronger on that side. If they choose different sides, Messi scores. So, suppose the game is described by the following Normal Form2 :
Alisson
Left Right
.5, .5 |
1, 0 |
1, 0 |
.6, .4 |
(a) Find all Nash Equilibria, and their associated payoffs for each player. (15 marks)
(b) Suppose the game was infinitely repeated, with the common discount factor δ { (0, 1).
Find all SPNEs. (10 marks)
Let’s return to the one-shot game - no more infinite repetition. Suppose now that Messi
may have recently injected himself with a rare leg mutagen, which would have switched
his strong foot! If he took the mutagen, the payoff matrix is now
Alisson
Left Right
.6, .4 |
1, 0 |
1, 0 |
.5, .5 |
Figure 1: Payoffs with mutagen
Messi knows whether he’s taken the mutagen, but Alisson does not. The common prior
puts probability p = .2 on Messi having taken the mutagen.
(c) Find all Bayes-Nash equilibria. (10 marks)
Now, instead of there being an exogenous probability of Messi taking the mutagen, he gets to decide whether to do so. His choice is still unobservable to Alisson. Following this, he and Alisson simultaneously choose where to shoot and dive, like before.
(d) What are Messi’s and Alisson’s pure strategy sets? Draw the game in extensive form.
(8 marks)
(e) Find all PBEs. (7 marks)
2. Monica is looking to lease her flat on AirBnB. She gets to choose any price p { [0, 1], and then a customer will appear, observe the price, and either rent her flat at that price or go away. Suppose that the customer has a type θ known to him but which is unknown to Monica, with the common prior being that θ is uniform on [0, 1].3 The customer’s utility is given by
uc (θ, p, Rent) = θ < p
uc (θ, p, No) = 0.
(a) For a given price p, which customers rent? (10 marks)
(b) For a given price p, what is the probability the customer rents the flat? (10 marks)
Monica now gets to set the price, after which the customer arrives and decides whether
to rent the flat at that price, given his type. Monica gets 0 utility when the flat is not rented, and p when it is rented at price p.
(c) Find all PBEs. [You don’t have to specify beliefs for this one; but take care when specifying strategies] (15 marks)
Now suppose that Monica’s flat has a quality q { (0, 1} that follows the common prior P (q = 1) = P(q = 0) = , and is known to Monica but not the customer. Monica’s utility is the same as above, and therefore does not depend on q directly. The customer’s utility is now
uc (θ, q, p, Rent) = θ + q < p
uc (θ, q, p, No) = 0.
(d) Find all prices that can occur in a pooling PBE. (8 marks)
(e) Find all price pairs that can occur in a separating PBE. (7 marks)
3. Annabelle and Bart, two penguins, notice a fish swimming near them. Each of them is hungry, but if they both go for the fish they’ll bonk their heads and the fish will get away. Suppose that the normal-form game describing their interaction is
Attack Stay
Find all Nash Equilibria of this game.
Bart
Attack Stay
<2 <2 |
2, 0 |
0, 2 |
0, 0 |
(10 marks)
Suppose that the game is sequential: first, Annabelle makes her choice, then Bart
observes it and gets to act.
(b) Draw the game tree. Find all pure NEs and all SPNEs. (15 marks)
Assume the game is simultaneous again. Suppose now that each penguin has a variable
degree of hunger. Annabelle and Bart each draw a type θA , θB each from (1, 3}. The common prior is that the types are identically and independently distributed according to a common prior which puts probability on type 1 and on type 3. Now, catching the fish (i.e. attacking when the opponent stays) is worth utility equal to one’s own type. All other outcomes have their utilities unchanged.
(c) Find the symmetric Bayes-Nash equilibrium. (10 marks)
Let’s do away with independence. Suppose the common prior is that θA , θB are jointly distributed according to the joint pmf P (θA = θB = 1) = P(θA = θB = 3) = 1/3 and P (θA = 1 and θB = 3) = P(θA = 3 and θB = 1) = 1/6.
(d) Find the symmetric BNE. (8 marks)
Let’s bring independence back. Suppose that θA and θB are identically and independently distributed, each with pdf f and cdf F with support on [1, 3].
(e) Prove a unique symmetric BNE exists and characterize it (this may be in terms of an
implicit expression). (7 marks)
2022-05-11