Advanced Microeconomics (ES30025)
Seminar One: Introduction to Game Theory (i)
Advanced Microeconomics (ES30025)
Seminar One: Introduction to Game Theory
1. The following tables are the normal forms of non-zero sum, non-cooperative, common
knowledge, static one-shot games played between two people, A and B. In each case: (i)
determine whether there is a Dominant Strategy Equilibrium; and (ii) find any Pure Strategy
Nash Equilibria.
Player B
Player A
s11 = U
s12 = D
s12 = L
s22 = R
3, 1
2, 2
2, 0
1, 1
Game 1
Player B
Player A
s11 = U
s =D
2
1
s12 = L
s22 = R
1, 2
3, 3
2, 0
1, 1
Game 2
Player B
Player A
s11 = U
s12 = D
s12 = L
s22 = R
1, 2
2, 0
3, 1
1, 12
Game 3
Player B
Player A
s11 = U
s12 = D
s12 = L
s22 = R
0, 2
2, 3
3, 2
1, 1
Game 4
2.
Suppose one of the payoffs in Game 2 above is amended so that
Player A
s11 = U
s12 = L
s22 = R
-100, 2
3, 3
s12 = D
2, 0
1, 1
Player B
Game 2’
What outcome would you expect from this game?
3. $1 is to be shared somehow between Players 1 and 2. Each player simultaneously names a
share, s1 for Player 1 arid s2 for Player 2, where 0 ≤ s j ≤ 1 (j = 1, 2). If s1 + s2 ≤ 1, each
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Advanced Microeconomics (ES30025)
Seminar One: Introduction to Game Theory (i)
player receives the share he names; if s1 + s2 > 1, both players receive zero. What are the Pure
Strategy Nash Equilibria of this game?
4. The Battle of the Sexes game. A man and a woman are deciding whether to spend the
evening at the cinema or at the ballet. There is no communication between them: they make
their decisions separately. She prefers the ballet; he prefers the cinema. But what is more
important to each of them is that they should be in the same place as each other:
(i)
(ii)
Write down the normal form. for this game with illustrative numbers for the payoffs to reflect these
circumstances;
Examine the game for Pure Strategy Nash Equilibria.
5. When a game has more than one Pure Strategy Nash Equilibrium, what criteria might the
players use to determine the strategies they actually choose?
6. Go back to Question 4(i) and, using the numbers for the payoffs that you gave in your
answers there, find the Mixed Strategy Nash Equilibrium.
7. Two players, A and B, play a non-co-operative, common knowledge game in which each
must choose either left (L) or right (R). The Normal Form is:
Player B
Player A
s11 = L
s12 = R
s12 = L
s22 = R
2, 2
2, -1
1, 0
3, 1
Game 1
(a) Determine the Pure Strategy Nash Equilibria of this game;
(b) Assuming Player 2 moves first, find the Subgame Perfect Nash Equilibrium.
8. A man and a woman each choose independently whether to go to a pub or to a political
meeting. The pub and the meeting are in different towns and so it is not possible for an
individual to go to both. The man prefers the pub to the meeting. The woman prefers the
meeting to the pub. But what matters more tom the man is to go to the same place as the
woman, whereas what matters more to the woman is to go to a different place to the man.
(a) Write down the Normal Form for this game with some illustrative numbers that
reflect these circumstances;
(b) Show that there is no Pure Strategy Nash Equilibrium if the Players move
simultaneously;
(c) If the players move sequentially, show by reference to the Extensive Form how the
outcome depends on who moves first.
9. A homogeneous good is sold by two supermarkets, 1 and 2. The Supermarkets play a oneshot game in which each one either sets the ‘monopoly price’, pm, or the ‘competitive price’
pc < pm. If both set pm they share the monopoly profit, π m , equally. If both set pc they each
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Advanced Microeconomics (ES30025)
Seminar One: Introduction to Game Theory (i)
receive zero profit. If one sets pm while the other sets pc, the former makes a negative profit,
π − , while the latter makes a positive profit, π + .
(a) Write out the game in Normal Form. What inequality would have to be satisfied for
the only Pure Strategy Nash Equilibrium to be that both supermarkets set pc? Explain;
(b) What inequality would have to be satisfied if, instead, there were two Pure Strategy
Nash Equilibria, one with both supermarkets setting pc and one with both
supermarkets setting pm? Explain;
(c) Now suppose that supermarket 1 moves first, publicly declaring its price before
supermarket 2 decides its price. Write out the Extensive Form for this game and show
how the Sub-Game Perfect Nash equilibrium depends on the inequalities found in
answers (i) and (ii);
(d) Consider how answer (iii) is affected if the following modification is made to the
game. Suppose that when supermarket 1 declares its price it also announces that if it
is undercut by supermarket 2, it will match supermarket 2’s price (i.e. supermarket 1
will set the same price as supermarket 2). Explain your answer intuitively.
10. In a repeated game, suppose that there is a finite probability that after any given play the
whole repeated game will be terminated. Does the backward induction paradox apply? If,
alternatively, if it is known with certainty that termination will occur after a finite number of
plays, but it is not known how many plays of the game there will be, does the backward
induction paradox apply?
11. Find the Nash and Sub-Game Perfect equilibria of the game described in Figure 1. State
all the assumptions you make in so doing.)
(9, 12)
C
(3, 2)
D
E
(10, 10)
(2, 2)
1
B
A
2
J
2
F
G
(9, 5)
(3, 3)
Figure 1
3
I
H
(12, 7)
(2, 3)
Advanced Microeconomics (ES30025)
Seminar One: Introduction to Game Theory (i)
12. Two players simultaneously play for a prize of £100 by making hand signals indicating
scissors, paper or stone. A player displaying scissors beats a player displaying paper, paper
beats stone and stone beats scissors. The game is winner takes all and matching hand signals
imply a zero-rewarded draw. Find the equilibrium to this game.
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