Solutions to Chapter 4 Exercises
SOLVED EXERCISES
S1.
False. A dominant strategy yields you the highest payoff available to you against each of your
opponent’s strategies. Playing a dominant strategy does not guarantee that you end up with the highest of
all possible payoffs. In the prisoners’ dilemma game, both players have dominant strategies, but neither
gets the highest possible payoff in the equilibrium of the game.
S2.
(a)
For Row,
Up
strictly dominates
Down
, so
Down
may be eliminated. For Column,
Right
strictly dominates
Left
, so
Left
may be eliminated. These actions leave the purestrategy Nash equilibrium
(
Up
,
Right
).
(b)
Row has no dominant strategy, but
Right
dominates
Left
for Column (who prefers small
numbers, this being a zerosum game). After eliminating
Left
for Column,
Up
dominates
Down
for Row,
so
Down
is eliminated, leaving the purestrategy Nash equilibrium (
Up
,
Right
).
(c)
There are no dominated strategies for Row. For Column,
Left
dominates
Middle
and
Right
. Thus these two strategies may be eliminated, leaving only
Left
. With only
Left
remaining, for Row,
Straight
dominates both
Up
and
Down
, so they are eliminated, making the purestrategy Nash equilibrium
(
Straight
,
Left
).
(d)
The game is solved using iterated dominance. Column has no dominated strategies. For
Row,
Up
dominates
Down
, so
Down
may be eliminated. Then
East
dominates
North
, so it is eliminated.
Then
Up
dominates
Low
, so it is eliminated. Then
East
dominates
West
, so it is eliminated. Then
Up
dominates
High
, so it is eliminated, leaving only
Up
. With only
Up
remaining,
East
dominates
South
,
giving the purestrategy Nash equilibrium (
Up
,
East
).
S3.
(a)
By Minimax, the minima for Row’s strategies are 3 for
Up
and 1 for
Down
. Row wants to
receive the maximum of the minima, so Row chooses
Up
. The minima for Column’s strategies are –2 for
Left
and –1 for
Right
. Column wants to receive the maximum of the minima, so Column chooses
Right
.
Again, the purestrategy Nash equilibrium is (
Up
,
Right
).
Solutions to Chapter 4 Solved Exercises
1 of 7
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(b)
Minimax shows that Row has minima of 1 and 2, with 2 for
Up
being the larger. Column
has maxima of 2 and 4, with 2 for
Right
being the smaller. Then the purestrategy Nash equilibrium is
(
Up
,
Right
).
(c)
Minimax shows that the minima for Row’s three strategies are 1, 2, and 1, so Row
chooses 2, which is from
Straight
. Column wants to minimize the maximum, and the maxima for
Column’s strategies are 2, 4, and 5, so Column chooses 2, which is from
Left
. This yields the pure
strategy Nash equilibrium of (
Straight
,
Left
).
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 Spring '08
 Charness,G

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