Q3 (ch14). Assume that Nike and Adidas are the only sellers of athletic footwear
ID: 1154844 • Letter: Q
Question
Q3 (ch14). Assume that Nike and Adidas are the only sellers of athletic footwear in the United States. They are deciding how much to charge for similar shoes. The two choices are "Low" and "High". Nike's payoffs are in the lower left of each cell in the payoff matrix: Adidas Low High 300,000 400,000 Low 1mil. 1.2mil Nike 500,000 High 800,000 1.7mil a. Do both companies have dominant strategies? If so, what are they? Briefly explain. (3pts) b. What will be the equilibrium outcome of the game? (1pt) c. If Nike and Adidas collude cach other, what will be the outcome of the game? Why? (Ipt) d. Now we've changed Adidas' $400,000 payoff to $200,000, assuming that all other payoff's are the same as before. What will be the outcome of the game? Discuss briefly. (2pts)Explanation / Answer
In this example, Adidas is the column player and Nike is the row player. They both face two pricing options- either ‘low’ or ‘high’.
The payoff matrix for both is given as follows:
ADIDAS ($)
NIKE ($)
LOW
HIGH
LOW
(1 Mil, 400000)
(1.2 Mil, 300000)
HIGH
(800000,700000)
(1.7 Mil, 500000)
a) A dominant strategy is one which irrespective of the other player’s action, gives a larger payoff to the player in question.
For this example, Adidas will prefer playing “low’ as it is always guaranteed a higher payoff irrespective of Nike’s choice (400000>300000 and 700000>500000). Thus according to Adidas, the column “high” gets eliminated and “low” is the dominant strategy and the payoff matrix becomes:
ADIDAS ($)
NIKE ($)
LOW
LOW
(1 Mil, 400000)
HIGH
(800000,700000)
AFTER Adidas has eliminated column “high”, Nike the row player also chooses “Low” as the dominant strategy because it guarantees a higher payoff irrespective of Adidas’ Choices (1Mil> 800000). Thus “low” is the dominant strategy here.
b) The equilibrium outcome of the game will be (Low, Low) with a payoff of (1 Mil, 400000) to the row and column player respectively.
c) When firms collude, they do so in order to earn the highest profit. Thus, the agreement to collude between Adidas and Nike will allow BOTH the parties to charge “High” price to the consumers. Under collusion, the outcome would be (High,High).
d) If the profit of Adidas changes from $400,000 to $200,000, the payoff matrix becomes:
ADIDAS ($)
NIKE ($)
LOW
HIGH
Expected Payoff of Nike
LOW
(1 Mil, 200000)
(1.2 Mil, 300000)
1Mil*q + 1.2Mil (1-q)
HIGH
(800000,700000)
(1.7 Mil, 500000)
(800000*q) + (1.7Mil)*(1-q)
Expected Payoff of Adidas
200000*p+7000000*(1-p)
300000*p+ 500000*(1-p)
Now, when Nike plays “Low”, Adidas plays “High” because 300000>200000
When Adidas plays “High”, Nike plays “High” because 1.7 Mil>1.2 Mil
When Nike plays “High”, Adidas plays “Low” because 700000>500000
When Adidas plays “Low” Nike plays “Low” because 1 Mil>800000
In this case there is no pure strategy Nash equilibrium thus, the outcome must be one under mixed strategy equilibrium. This happens because every option under the pure strategy Nash equilibrium leaves a room for regret of each opponent. Unlike under pure strategy Nash equilibrium, when Adidas’ payoff falls to $200,000, every player has an incentive to deviate from its strategy, given the strategy of the other player.
ADIDAS ($)
NIKE ($)
LOW
HIGH
LOW
(1 Mil, 400000)
(1.2 Mil, 300000)
HIGH
(800000,700000)
(1.7 Mil, 500000)
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