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Solutions to Chapter 9 Exercises
SOLVED EXERCISES
S1. The statement is true because promises, when successfully used as strategic moves, require that
you follow through on the promised action; you will not promise an arbitrarily large reward. You may
S2. (i) (a) There are two pure-strategy equilibria (Down, Left) and (Up, Right); there is also a
(b) If Row commits to Up, he ensures himself a payoff of 2. Similarly, if Column
(ii) (a) Both players have dominant strategies; equilibrium: (Up, Right), payoffs of (3, 4).
S3. (a) See the game tree below. The subgame-perfect equilibrium is (Harsh, Always be
mischievous) for payoffs of (2, 2) to the Nanny and the Kids:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) The Kids’ natural inclination, from part (a), is to do mischief; they would prefer that the
(c) The implied threat is “We will do mischief if you are harsh.” This is automatically
(d) The Kids can establish credibility of their promise to be good perhaps by prescribing their
(e) The status quo is that the Nanny is harsh. The Kids want their promise to compel the
S4. (a) The game is dominance solvable: Aggressive is the Soviet Union’s dominant strategy,
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(iii) Subgame-perfect equilibrium strategies are (United States: Always Aggressive at its
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) Both the United States and the Soviet Union are better off in the situation in which the
With this order of timing, the Soviet Union’s awareness of the United States’ ability to respond to
If the Soviet Union moves last, the United States knows that nothing will restrain the Soviets
The key point is that the United States must retain its flexibility. If it committed to either action
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
In order to retain its influence, the United States should not commit to any actions; it should
S5. Note that the answers provided here are deliberately imprecise and open ended; their purpose is to
(a) The current members of the European Monetary Union want to achieve compliance from
Britain. They can use a threat (if you don’t join, we will exclude you from important European decisions
(b) The United States wants to deter North Korea primarily, and also Iran, China, and so on
secondarily. The usual strategic move for this is a threat. However, we do not have many good threats in
(c) The United Auto Workers (UAW) wants to deter the firms from investing abroad and to
compel the U.S. government to restrict imports. This needs a combination of policies, for example, a
UNSOLVED EXERCISES
U1. The Keaton character made the mistake of believing that the Allen character would follow
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Keaton could have required that Allen attend the opera before going to the hockey game—if
U2. (a) The simultaneous-move game is two-by-two, as shown in the following matrix:
Parent
P N
Both players have dominant strategies; equilibrium is (B, N) with payoffs (1, –2).
(b) The tree is shown below. The subgame-perfect equilibrium is (Bad, Always not) for the
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) The parent can commit to four possible strategies: (1) P always, (2) N always, (3) P if B
(and N if G), or (4) P if G (and N if B). Then the game has a two-by-four matrix as shown below:
Parent
P always N always P if B P if G
(d) The first equilibrium noted in part (c)—(B, N always)—is also the subgame-perfect
equilibrium in the sequential game from part (b). The second equilibrium—(G, P if B)—is not a
U3. (a) The game tree follows, with payoffs in the order (Sparta, Athens, Small cities):
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
The rollback equilibrium is War (2, 2 ,2). Everyone prefers the outcome where Sparta does not
declare war, Athens relinquishes its empire, and the small cities stay independent (3, 3, 3).
(b) Strategic moves are relevant for only the later movers (Athens and Small cities). Since
each of them gets to move at only one node, conditional moves (threats and promises) are irrelevant. A
U4. (a) A new tree is shown below.
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The payoff structure has been changed so that the kids prefer Good to Mischief, but still most prefer to
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(e) The kids want to make the threat because their payoff in the original subgame-perfect
U5. Note that the answers provided here are deliberately imprecise and open-ended; their purpose is
to get students thinking about real-world issues using game-theoretic perspectives. Here are some
suggestions for getting discussions started:
(a) Students can try to achieve deterrence with threats, demonstrations, sit-ins, exposing the
high salaries of the top administrators and professors, and so on. The credibility of such moves will
(b) Though the situations in Afghanistan, Iraq, and Israel/Palestine differ greatly in terms of
historical background and contemporary context, they share important features. The establishment of
This stepwise process can become problematic as the endgame approaches. For example, the final
status of Jerusalem (whether it will remain a unified capital solely under Israeli control or whether part
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A cooperative equilibrium can last if even a small defection (reneging on the last step promised)
will unravel the whole deal. Even if there are no defections by the negotiating party, there is a strong
Another possibility is for an interested third party to promise some reward to the two main
(c) In response to Iran’s continued pursuit of its worrisome nuclear program, the
international community has imposed economic sanctions and has often held out promises of ending the
The threat of force is another option, one that Israel is contemplating and that the United States
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
On the other hand, a military strike on Iran could prove diplomatically and strategically
U6. Accounts should incorporate a specific example of one of the strategic moves discussed in the
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