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Solutions to Chapter 11 Exercises
SOLVED EXERCISES
(b) Because the line for action X is above the line for action Y when 100 people choose X,
S2. (a) This game is a prisoners’ dilemma since s(n) is greater than p(n + 1) for all n. Any single
player can always raise his payoff by switching from participating to shirking.
(c) Plugging the appropriate values into the text’s Equation (12.1) yields
Using calculus, you can find the same answer by differentiating T(n) from part (b). T (ʹn) = 2n +
Or you can find the same answer using a graph of T(n):
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(d) When n = 74, each participant receives a benefit p(74) = 74. If one participant were to
(e) If the game is repeated, the players can take turns playing the role of shirker (and
S3. (a) Let x be the proportion of the population in Alphaville. Then the payoffs to living in the
(b) The graph below shows that there are five equilibria. Three exist where A(x) and B(x)
Betaville:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Betaville, and x will fall. If x is just above 0.2, people will move to Alphaville, and x will rise. Thus, the x
S4. (a) A person will voluntarily contribute if his benefit from so doing exceeds the $100 cost.
Assuming that n other people are already contributing, a person’s benefit from beginning to contribute is
(b) There are two stable Nash equilibria in this game. When n = 0, no one is contributing,
and no one wants to. When n = 100, everybody is contributing, and each person receives a private gain
S5. Here is one possible account. The diagram below shows a possible configuration of industry-level
supply and demand. Demand has the normal downward-sloping shape. For supply, over the range from
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
On the macroeconomic level, we see a function relating national product to national income
(demand) with the shape shown below:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
S6. Answers should include careful descriptions of the (usually two) strategies available to the group
of players and of the benefits derived from the two different choices. Good descriptions of benefits should
UNSOLVED EXERCISES
U1. (a) The game is Chicken if p(2) < s(1) and if s(0) < p(1). Under these conditions, one player
shirks while the other works and works while the other shirks. To get Version I of Chicken, it must also be
(b) The two-person game is an assurance-type game if s(1) < p(2), p(1) < s(0), and s(0) <
p(2) all hold. These three inequalities indicate that Build is the best response to Build, that Not is the best
U2. (a) See diagram below:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) (i) With no communication, a high level of risk aversion leads very few students to
answer the question.
(ii) With pregame discussion, some students try to commit to answering the question; this
can be done by pledging to answer on the Web site or by indicating that you have answered and
U3. (a) See diagram below:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) There are three possible equilibria. At x = 0 (nobody on the local roads), the benefit on
the highway exceeds the benefit on local roads; nobody will switch to a local road. At x = 0.1, the benefits
U4. (a) See diagram below:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) There are two possible equilibria:
engineer’s pay. This equilibrium is unstable.
stable.
increase in their number creates a marginal spillover effect that outweighs the marginal private gain.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) The following inequalities hold under the conditions specified:
B(3, n) = 2.2 + n – 3 > 3 – n = S(3, n) whenever n ≥ 2
B(4, n) = 2.2 + n – 4 > 4 – n = S(4, n) whenever n ≥ 3
That is, IN is dominant for country 3 when countries 1 and 2 choose IN. Now IN is dominant for country
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(d) For country i, the benefit of choosing IN when all other countries choose IN is B(i, 11),
and the benefit of choosing OUT when all countries choose OUT is S(i, 0). The benefits of these two
scenarios for each country (by rank) are given in the following table:
i B(i, 11) S(i, 0)
1 12.2 1
2 11.2 2
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
