Consider a two-player game between Child’s Play and Kid’s
Korner, each of which produces and sells wooden swing sets for children. Each
player can set either a high or a low price for a standard two-swing, one-slide
set. If they both set a high price, each receives profits of $64,000 per year.
If one sets a low price and the other sets a high price, the low-price firm
earns profits of $72,000 per year, while the high-price firm earns ,000. If
they both set a low price, each receives profits of $57,000.
(a) Verify that this game has a prisoners’ dilemma
structure by looking at the ranking of payoffs associated with the different
strategy combinations (both cooperate, both defect, one defects, and so on).
What are the Nash-equilibrium strategies and payoffs in the simultaneous-play
game if the players meet and make price decisions only once?
(b) If the two firms decide to play this game for a fixed
number of periods—say, for 4 years—what would each firm’s total profits be at
the end of the game? (Don’t discount.) Explain how you arrived at your answer.
(c) Suppose that the two firms play this game repeatedly
forever. Let each of them use a grim strategy in which they both price high
unless one of them “defects,” in which case they price low for the rest of the
game. What is the one-time gain from defecting against an opponent playing such
a strategy? How much does each firm lose, in each future period, after it
defects once? If r 5 0.25 ( 5 0.8), will it be worthwhile for them to
cooperate? Find the range of values of r (or ) for which this strategy
is able to sustain cooperation between the two firms.
(d) Suppose the firms play this game repeatedly year
after year, neither expecting any change in their interaction. If the world
were to end after 4 years, without either firm having anticipated this event,
what would each firm’s total profits (not discounted) be at the end of the
game? Compare your answer here with the answer in part (b). Explain why the two
answers are different, if they are different, or why they are the same, if they
are the same.
(e) Suppose now that the firms know that there is a 10%
probability that one of them may go bankrupt in any given year. If bankruptcy
occurs, the repeated game between the two firms ends. Will this knowledge
change the firms’ actions when r 5 0.25? What if the probability of a
bankruptcy increases to 35% in any year?