Return to the Price Competition with Identical Products game of Section 6.2
Return to the Price Competition with Identical Products game of Section 6.2. Now assume that shop 2 has a cost of 15, while shop 1 still has a cost of 10. Make the (admittedly arbitrary) assumption that if both shops set the same price, then all shoppers buy from shop 1. Shop 1’s payoff function is
e (p1 � 10)(100 � p1) if p1 � p2 0 if p2 6 p1
,
1 2?
1 3
1 2
j � t 7 20, 20 � j � t,
j � t � 20,
EXERCISES
176 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
while shop 2’s payoff function is
Find all Nash equilibria.
4. Return to the Price-Matching Guarantees game of Section 6.2. a. Suppose both shops set the same price and the price exceeds 55. Is
this situation a Nash equilibrium? b. Suppose both shops set the same price and the price is less than 10.
Is this situation a Nash equilibrium? c. Derive all undominated symmetric Nash equilibria.
5. Two manufacturers, denoted 1 and 2, are competing for 100 identical customers. Each manufacturer chooses both the price and quality of its product, where each variable can take any nonnegative real number. Let
and denote, respectively, the price and quality of manufacturer i’s product. The cost to manufacturer i of producing for one customer is
Note in this expression that the cost is higher when the quality is higher. If manufacturer i sells to customers, then its total cost is
Each customer buys from the manufacturer who offers the greatest value, where the value of buying from manufacturer i is
higher quality and lower price mean more value. A man- ufacturer’s payoff is its profit, which equals If one manufacturer offers higher value, then all 100 customers buy from it. If both manufacturers offer the same value, then 50 customers buy from manufacturer 1 and the other 50 from manufacturer 2. Find all symmet- ric Nash equilibria.
6. For the Charitable Donations game in Section 6.3, find all Nash equilibria.
7. For a two-player game, the payoff function for player 1 is
and for player 2 is
Player 1’s strategy set is the interval [0,100] and player 2’s strategy set is the interval [0,50]. Find all Nash equilibria.