Game Theory and Business Strategy

Solutions to Self Test

Finding Nash Equilibria

Question 1

Using the best-response method, best responses to each of the opponent's strategies are in bold:
   Player 2
   X Y
Player 1 A
10 , 10 15 , 5
5 , 15 12 , 12
B

What is the equilibrium of the above game?
A,X is the only outcome with both numbers in bold.
B,X
A,Y
B,Y


Question 2

   Player 2
   X Y
Player 1 A
0 , 0 0 , 1
2 , 0 0 , 0
B

What is/are the equilibrium/equilibria of the above game?
B,X only
A,Y and B,X
A,Y; B,X; and B,Y All three are equilibria.
none


Question 3

   Player 2
   X Y Z
Player 1 A
6 , 6 8 , 20 0 , 8
10 , 0 5 , 5 2 , 8
8 , 0 20 , 0 4 , 4
B
C

What is the unique equilibrium of the above game?
B,Y
C,Z This game is also dominance-solvable.
(eliminate A, then X, then B, then Y).
A,X
A,Y


Question 4

   Player 2
   Rock Paper Scissors
Player 1 Rock
0 , 0 -1 , 1 1 , -1
1 , -1 0 , 0 -1 , 1
-1 , 1 1 , -1 0 , 0
Paper
Scissors

How many equilibria (in pure strategies) does the above game have?
9
3
1
0 No pure strategy equilibria.


Question 5

Two firms are involved in developing a new technology that will allow consumers to taste food over the Internet. This has potential, for example, in restaurant promotion. Given the risks and the relatively small expected size of this market, compatibility of the technologies is very important. Firm DigiTaste is far advanced in developing its RemoteTaste technology. WebOdor has been expanding into the Internet taste arena with its incompatible product, BitterWeb. The two companies agree that if they both adopt the same technology, they each may gross $200M from the developing industry. If they adopt different technologies, consumers will make fun of both companies, and purchase neither product, leading to a gross of $0. Retooling one's factory to make the competing (nonproprietary) technology would cost WebOdor $100M and DigiTaste $250M. By the wave of an economist's wand, their production decisions must be made simultaneously.

Set up the above scenario as a normal form (simultaneous) game.
What is the equilibrium outcome?

Here is the game:
   WebOdor
   RemoteTaste BitterWeb
DigiTaste RemoteTaste
200 , 100 0 , 0
-250 , -100 -50 , 200
BitterWeb
Both adopt RemoteTaste Adopting RemoteTaste is a dominant strategy for DigiTaste.
Note that having higher switching (retooling) costs is actually advantageous in this game!
Both adopt BitterWeb
DigiTaste adopts RemoteTaste and WebOdor adopts BitterWeb
WebOdor adopts RemoteTaste and DigiTaste adopts BitterWeb