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Solutions to Chapter 11 Exercises
SOLVED EXERCISES
S1.
False. The players are not assured that they will reach the cooperative outcome. Rollback
reasoning shows that the subgameperfect equilibrium of a finitely played repeated prisoners’ dilemma
will entail constant cheating.
S2.
(a)
The payoffs are ranked as follows: high payoff from cheating (72) > cooperative payoff
(64) > defect payoff (57) > low payoff from cooperating (20). This conforms to the pattern in the text so
the game is a prisoners’ dilemma, as can also be seen in the payoff table:
KID’S KORNER
High price
Low price
CHILD’S PLAY
High price
64, 64
20, 72
Low price
72, 20
57, 57
If the game is played once, the Nash equilibrium strategies are (Low, Low) and payoffs are (57, 57).
(b)
Total profits at the end of four years = 4
×
57 = 228. Firms know that the game ends in
four years so they can look forward to the end of the game and use rollback to find that it’s best to cheat
in year 4. Similarly, it is best to cheat in each preceding year as well. It follows that it is not possible to
sustain cooperation in the finite game.
(c)
The onetime gain from defecting = 72 – 64 = 8. Loss in
every
future period = 64 – 57 =
7. Cheating is beneficial here if the gain exceeds the present discounted value of future losses or if 8 >
7/
r.
Thus,
r
> 7/8 (or
d
> 8/15) makes cheating worthwhile, and
r
< 7/8 lets the grim strategy sustain
cooperation between the firms in the infinite version of the game. If
r
= 0.25, cooperation can be
sustained.
(d)
Total profits after four years = 4
×
64 = 256. With no known end of the world, the firms
can sustain cooperation if r < 7/8 as in part c. This answer is different from that in part
(b)
because the
firms see no fixed end point of the game and can’t use backward induction. Instead, they assume the
game is infinite and use the grim strategy to sustain cooperative outcome.
(e)
A 10% probability of bankruptcy translates into a 90% probability that the game
continues, so
p
= 0.9. Then, for
r
= 0.25 (
d
= 0.8),
R
= 39%. This rate would need to exceed 7/8 before
Solutions to Chapter 11 Solved Exercises
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View Full Document cheating was worthwhile, so the firms will still cooperate in this case. For a 35% probability of
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This note was uploaded on 08/08/2011 for the course ECON 171 taught by Professor Charness,g during the Spring '08 term at UCSB.
 Spring '08
 Charness,G

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