King’s College London
This paper is part of an examination of the College counting towards the award
of a degree. Examinations are governed by the College Regulations under the
authority of the Academic Board.
Degree Programmes MSc
Module Code
Module Title
Examination Period
7CCSMAMF
Agent Based Modelling for Finance
May 2019 (Period 2)
Time Allowed Two hours
Rubric
ANSWER ALL QUESTIONS FROM SECTION A AND
TWO QUESTIONS FROM SECTION B.
Calculators
Notes
Section A is worth a total of 50 marks. Questions in Section
B are worth 25 marks each. If more than two questions from
Section B are answered, the answer to the first two questions
in the exam paper order will count.
Calculators may be used. The following models are permitted: Casio fx83 / Casio fx85.
Books, notes or other written material may not be brought
into this examination
PLEASE DO NOT REMOVE
EXAMINATION ROOM
 2019 King’s College London
THIS
PAPER
FROM
THE
May 2019
7CCSMAMF
SECTION A
1. What is systemic risk? Why is it important in economics and finance?
[5 marks]
2. What are the key elements of an agent-based model?
[5 marks]
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3. What is the ZIP trading strategy? Why is it studied?
[7 marks]
4. What is the Roth-Erev trading strategy for double auction markets? How
does it compare with other trading strategies?
[7 marks]
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7CCSMAMF
5. The following payoff matrix corresponds to a modified version of the Prisoner’s Dilemma called the DA’s brother problem. In this problem, Prisoner
1 (the row player) is the brother of the District Attorney (DA).
Deny
Deny
0, -2
Confess -1, -10
Confess
-10, -1
-5, -5
How does this problem differ from the standard Prisoner’s Dilemma? Is
there a pure strategy Nash equilibrium? Is there an equilibrium in dominant
strategies? Provide details of your reasoning.
[5 marks]
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6. Consider the following 2-player game. We will refer to the row player as
Player 1 and to the column player as Player 2.
Left Right
Up
5, 5
Down 3, 0
0, 3
3, 3
a. Find all equilibria of the game – both in pure and mixed strategies – and
indicate the respective (expected) utilities for both players.
[4 marks]
b. Now assume that the payoff of Player 2 has changed as follows (the
change is indicated in bold).
Left Right
Up
5, 5
Down 3, 1
0, 3
3, 3
How does this affect the equilibrium strategies (both pure and mixed)
of Player 1 and/or Player 2?
[3 marks]
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7. Describe the revenue equivalence theorem and its implications. Discuss why,
despite the revenue equivalence theorem, truthful auctions are preferred.
[7 marks]
8. Suppose the Vickrey-Clarke-Groves (VCG) mechanism is applied to sell two
objects {A, B} to three buyers {1, 2, 3}. A buyer can buy none, one, or
both of the objects. The valuation function of each buyer depends only on
the set of objects assigned to that buyer and is given by:
v1 (∅) = 0 v1 ({A}) = 3 v1 ({B}) = 2 v1 ({A, B}) = 15
v2 (∅) = 0 v2 ({A}) = 2 v2 ({B}) = 8 v2 ({A, B}) = 10
v3 (∅) = 0 v3 ({A}) = 10 v3 ({B}) = 3 v3 ({A, B}) = 12
Determine the assignment of objects to buyers and the payments to the
buyers, under truthful bidding. Discuss why buyer 1 might have an objection
to the outcome.
[7 marks]
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SECTION B
9. Explain the game rock-paper-scissors. What are the consequences of playing
repeated games? What is the Nash equilibrium strategy for the repeated
game? Can this strategy be beaten? How?
[25 marks]
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10. Suppose you have been commissioned by the Bank of England (the central
bank of the UK) to build an agent-based model of the lending that takes
place between the major commercial banks (eg, Barclays, NatWest, HSBC,
etc). What would be the agents in your model? What would be the links
between them? How would you calibrate the model?
After the model was completed, suppose that a senior official of the Bank
of England asked you to share the model with the commercial banks. What
would be the potential consequences of doing this? Do you think sharing
this information would be sensible or not?
[25 marks]
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11. Consider the following 2-player game.
L
C
R
U 1, 2 3, 5 2, 1
M 0, 4 2, 1 3, 0
D -1, 1 4, 3 0, 2
a. Does either player have dominated strategies? If so, identify them.
[4 marks]
b. Does either player have a dominant strategy? Why or why not?
[4 marks]
c. Use elimination of dominated strategies (EDS) to solve this game. Show
the order in which you are eliminating strategies, and specify whether
you are eliminating strictly or weakly dominated strategies.
[6 marks]
d. Is your EDS solution a Nash equilibrium? Why or why not?
[4 marks]
e. For each player, find their maximin and minimax pure strategies. Indicate the corresponding utilities of the players.
[7 marks]
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12. Facebook applies the Vickrey-Clarke-Groves mechanism (a.k.a. VCG) for
its display advertising. The items for sale are the ad slots to be shown to a
visiting user, and the buyers (a.k.a. agents) are the interested advertisers.
Consider a setting with two slots: 1 and 2, and three advertisers: A, B and
C, competing for these slots.
Two of the advertisers, A and B, are direct competitors, since they are offering similar products (e.g., Apple vs. Microsoft). Therefore, if A’s ad is
shown in either slot, the value of the remaining slot for B is diminished,
and similarly for A if B’s ad is shown. Consequently, their utility not only
depends on which slot they receive, but also on who receives the other slot
(there are so-called allocative externalities).
One of the slots is more prominent and therefore more valuable to all agents.
Also, all advertisers prefer to receive both slots rather than one slot.
The agents’ private valuations for the slots are given in the following table,
where for A and B, v indicates the (higher) value in case the slots are not
divided between these direct competitors, and v 0 is the (lower) value in case
the other slot is awarded to the direct competitor.
Agent
A
B
C
Slot 1
Slot 2
Slots 1 and 2
0
0
vA,1 = 2.0 / vA,1
= 1.5 vA,2 = 1.6 / vA,2
= 0.9 vA,1+2 = 2.3
0
0
vB,1 = 2.5 / vB,1
= 1.6 vB,2 = 2.0 / vB,2
= 1.5 vB,1+2 = 2.9
vC,1 = 1.0
vC,2 = 0.7
vC,1+2 = 1.1
a. Given the agents’ valuations as above, what is the efficient allocation of
the slots? How much is the social welfare in this case?
[7 marks]
QUESTION 12 CONTINUES ON NEXT PAGE
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b. Compute the transfers for all agents under the VCG mechanism.
[10 marks]
c. Show whether (and why!) the resulting transfers are (1) individually
rational, (2) weakly budget balanced, (3) strongly budget balanced.
[8 marks]
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