# GAME THEORY AND APPLICATION - Problem and Answers

```TOPIC 4. GAME THEORY AND APPLICATION
1. IDENTIFYING NASH EQUILIBRIA
Consider the following game, where two firms (Firm A and Firm B) must decide
simultaneously whether to expand business in the West or in the South:
Firm B
Firm B
Expand in the West Expand in the South
Firm A
Expand in the West
Firm A
Expand in the South
10, 60
50, 90
20, 80
40, 50
A) there is one pure strategy Nash equilibrium: for both firms to expand in the West.
B) there is one pure strategy Nash equilibrium: for both firms to expand in the South.
C) there are two pure strategy Nash equilibria: either firm can expand in the West, and the
other expands in the South.
D) there is only a mixed strategies equilibrium.
Diff: 3
2. PRISONER’S DILEMMA TYPE GAMES
Consider the game below:
Player R
Strategy R1
Player R
Strategy R2
Player C
Strategy C1
Player C
Strategy C2
600, 600
100, 1000
1000, 100
200, 200
True or False: This game is not an example of the Prisoners’ Dilemma game.
Answer: False. It is an example of the Prisoners’ Dilemma game. A Prisoners’ Dilemma
game is one where each player has a dominant strategy and the Nash equilibrium does not
coincide with the outcome that maximises the collective payoffs of the players in the game.
In this game the dominant strategies are R2 for Player R and C2 for Player C. The Nash
equilibrium is (R2, C2) but both players would be better off by cooperating and selecting
(R1, C1).
3. MAXIMIN STRATEGIES
Consider the game below about funding and construction of a dam to protect a 1,000-person
town. Contributions to the Dam Fund, once made, cannot be recovered, and all citizens must
contribute £1,000 to the dam in order for it to be built. The dam, if built, is worth £70,000 to
each citizen. If each player choses a maximin strategy, the outcome would be:
One citizen
Contribute to dam
One citizen
Don’t contribute to
dam
Other 999 citizens
Contribute to dam
69000, 69000
-1000, 0
Other 999 citizens
Don’t contribute to
dam
0, -1000
0, 0
A) £69,000, £69,000.
B) £0, -£1000.
C) -£1000, £0.
D) £0, £0.
E) a mixed strategy equilibrium.
Diff: 2
By not contributing to the dam, the citizens minimise their potential losses – or in other
words, they maximise the minimum gain that can be earned.
4. REPEATED GAMES/SEQUENTIAL GAMES
Consider the game below where a firm (Entrant) is considering entering into the digital
camera business, and can decide to do so on either a small scale or a large scale. The
Incumbent firm must decide whether the Accommodate the new firm or start a Price War.
The payoffs (profits in millions of dollars) to each firm are represented below:
Entrant
Small scale
Entrant
Large scale
Incumbent
Accommodate
Incumbent
Price War
4, 20
1, 16
8, 10
2, 12
If the game is played sequentially and the Entrant moves first, the equilibirum would be:
a) (Small scale, Accommodate)
b) (Small scale, Price War)
c) (Large sclae, Accommodate)
d) (Large scale, Price War)
5. MIXED STRATEGIES
Consider the Matching Pennies game:
Player A - heads
Player A - tails
Player B - heads
1, -1
-1, 1
Player B - tails
-1, 1
1, -1
Suppose Player B always uses a mixed strategy with probability of 3/4 for head and 1/4 for
tails. Which of the following strategies for Player A provides the highest expected payoff?
A) Mixed strategy with probability 1/4 on heads and 3/4 on tails
B) Mixed strategy with probability 1/2 on heads and 1/2 on tails
C) Mixed strategy with probability 3/4 on heads and 1/4 on tails
D) Pure strategy in which Player A always selects heads