Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

CHAP. 12] GAME THEORY AND OLIGOPOLISTIC BEHAVIOR 277
DOMINANT STRATEGY AND NASH EQUILIBRIUM
12.3 From the payoff matrix in Table 12.5, where the payoffs are the profits or losses of the two firms, determine
(a) whether firm A has a dominant strategy, (b) whether firm B has a dominant strategy (c) the
optimal strategy for each firm.
Table 12.5
Firm A
Firm B
Low Price High Price
Low Price 1, 1 3, 21
High Price 21, 3 2, 2
(a) When firm B charges a low price, firm A will earn a profit of 1 when it also charges a low price and a profit of
21 (i.e., a loss of 1) when it charges a high price. Similarly, when firm B charges a high price, firm A earns a
profit of 3 when it charges a low price and a profit of 2 when it charges a high price. Therefore, charging a low
price is the dominant strategy for firm A.
(b) When firm A charges a low price, firm B earns a profit of 1 when it also charges a low price and a profit of 21
when it charges a high price. Similarly, when firm A charges a high price, firm B earns a profit of 3 when it
charges a low price and a profit of 2 when it charges a high price. Therefore, charging a low price is also the
dominant strategy for firm B.
(c) The optimal strategy for each firm is to adopt its dominant strategy of charging a low price.
12.4 Explain whether or not there is a Nash equilibrium when each firm chooses its dominant strategy.
When each firm chooses its dominant strategy (assuming they have one), we automatically have a Nash equilibrium
without even the need for each firm to consider the strategy of its rival.
12.5 (a) Indicate whether the Cournot equilibrium is a Nash equilibrium and (b) in what way the Cournot
equilibrium differs from the Nash equilibrium given in Table 12.2.
(a)
(b)
The Cournot equilibrium is a Nash equilibrium because each firm has adopted its optimal output given its
rival’s output.
The Cournot equilibrium differs from the Nash equilibrium given in Table 12.2 because in the Cournot equilibrium
neither firm has a dominant strategy, while in Table 12.2, firm B has a dominant strategy, but firm A
does not.
12.6 From the payoff matrix in Table 12.6, where the payoffs are the profits or losses of the two firms, determine
(a) whether firm A has a dominant strategy, (b) whether firm B has a dominant strategy, (c) the
optimal strategy for each firm, and the Nash equilibrium, if there is one.
Table 12.6
Firm A
Firm B
Low Price High Price
Low Price 1, 1 3, 21
High Price 21, 3 4, 2
(a)
When firm B charges a low price, firm A will earn a profit of 1 when it also charges a low price and a profit of
21 (i.e., a loss of 1) when it charges a high price. When firm B charges a high price, firm A earns a profit of 3
when it charges a low price and a profit of 4 when it charges a high price. Therefore, firm A does not have a
dominant strategy.