Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Solutions to Chapter 3 Exercises
UNSOLVED EXERCISES
U1. A first-mover advantage is not a necessary property of sequential games. There are sequential
games with a first-mover advantage, a second-mover advantage, or no advantage. Consider the matchstick
game of question S6. If the game begins with 5 matchsticks, then there is a second-mover advantage,
because with optimal play, the second mover will always win. On the other hand, if the game were to
begin with 6 to 9 matchsticks, then there would be a first-mover advantage, because the first mover could
always leave his opponent with 5 matchsticks.
U2. For this question, remember that actions with the same label, if taken at different nodes, are
different components of a strategy. To clarify the answers, the nodes on the trees are labeled 1, 2, and so
on (in addition to showing the name of the player acting there), and actions in a strategy are designated as
N1 (meaning N at node 1), and so on. Numbering of nodes begins at the far left and proceeds to the right,
with nodes equidistant to the right of the initial node numbered from top to bottom.
(a) Albus has three actions at only one node, so he has only three complete strategies, which
are (N1), (E1), and (S1). Minerva has the same two actions at three nodes, so she has 23 = 8 complete
strategies, which are (a2, a3, a4), (a2, a3, b4), (a2, b3, a4), (b2, a3, a4), (a2, b3, b4), (b2, a3, b4), (b2, b3,
a4), and (b2, b3, b4).
(b) Albus has the same two actions at two different nodes, so he has 22 = 4 complete
strategies, which are (N1, N4), (N1, S4), (S1, N4), and (S1, S4). Minerva has three actions at node 2 and
2 actions at node 3, so she has 3 • 2 = 6 complete strategies, which are (a2, a3), (a2, b3), (b2, a3), (b2, b3),
(c2, a3), and (c2, b3).
(c) Albus has the same two actions at two different nodes, so he has 22 = 4 complete
strategies, which are (N1, N4), (N1, S4), (S1, N4), and (S1, S4). Minerva has the same two actions at two
different nodes, so she has 22 = 4 complete strategies, which are (a2, a5), (a2, b5), (b2, a5), and (b2, b5).
Severus has two actions at one node, so he has two complete strategies, which are (x3) and (y3).
U3. (a) The equilibrium strategies are (N1) for Albus and (b2, a3, b4) for Minerva. The rollback
payoff is (2, 1).
