博弈论代写-EC9410
时间:2022-05-02
EC9410
1 (Continued overleaf)
UNIVERSITY OF WARWICK

May Examinations 2020/21

Game Theory


Time Allowed: 2 Hours

Read all instructions carefully- and read through the entire paper at least once before you
start entering your answers.

There are TWO Sections in this paper. Answer the ONE question in Section A (50 marks) and
ONE question in Section B (50 marks).

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.


Section A: Answer the ONE question


1. Suppose that each of the games below is repeated for a finite number of > 3 periods
with discount factor .



a) Find all subgame perfect equilibria (SPEs) of game (a). (10 marks)
b) Describe a strategy profile that is a SPE of game (b) if the discount factor is high
enough and which gives the players payoffs of (3,3) in all but the last 2 periods. (20
marks)
c) For which discount factors does the subgame perfect equilibrium explained in part
b) exist? (20 marks)



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Section B: Answer ONE question


2. Consider the juror voting game as discussed in the lecture where there is a defendant
who is guilty with probability . Each juror knows and, also each receive a private
signal about the defendant, ∈ {, }, corresponding respectively to guilty and
innocent. The precision of the signal denotes the probability that a correct signal is
received; Pr(|) = Pr (|) = where is the precision. A defendant gets convicted
if and only if all jurors vote to convict, while he gets acquitted otherwise. Each juror gets
a payoff of 1 when a guilty defendant is convicted, or an innocent defendant is
acquitted and get a payoff of 0 otherwise.

a) Let there be 2 jurors who receive signals of precision respectively 1 and 2. Find the
conditions for a BNE in which both jurors vote to convict independently of their
signals. How does this condition change with respect to 1 and 2? (15 marks)
b) Let there be 3 jurors who each receive signals of precision . Find the conditions for
a BNE in which two jurors vote truthfully (vote convict upon guilty signal and vote
acquit upon innocent signal) and one juror always votes to convict regardless of the
signal. (15 marks)
c) Let there be jurors, each receiving a signal of precision . Find the condition on
for an equilibrium in which jurors vote truthfully and − jurors vote to convict
regardless of their signal. What happens to this condition as increases? (20 marks)


3. Consider a market in which a wholesaler (firm 0) produces a product that will be sold to
final consumers by two retailers (firms 1 and 2). The wholesaler produces the product at
a constant marginal cost of 0 per unit and charges the retailers a wholesale price of
.
The retailers, after seeing , choose quantities ; the retail price is determined by the
inverse demand curve = − (1 + 2).

a) If the retailers choose their quantities simultaneously, find expressions for the
equilibrium quantities as a function of the wholesale price. (12 marks)
b) Find the subgame perfect wholesale price, the corresponding retail quantities and
price and the profits of the wholesaler and the two retailers. (12 marks)

You may assume that = 20; = 0 = 2. Also, assume that retailer has a fixed cost which is paid if
(and only if) the retailer buys a positive amount of the product from the wholesaler. The fixed costs for
the two retailers are 1 = 1 and 2 = 3.5.

c) Now suppose that the low-cost retailer chooses its quantity after observing and
the high-cost retailer chooses its quantity after observing and the other
retailer’s quantity. Find the subgame perfect equilibrium quantities, wholesale and
retail prices and profits. Will both retailers stay in business? (20 marks)
d) Would the wholesaler prefer the situation in b (using = 20; = 0 = 2 ), the
situation in c or a situation in which the wholesaler chose to pay a proportion
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(which the wholesaler could determine) of the high-cost retailer’s fixed cost? (6
marks)


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