Frodo
Legolas
Gimli
Boromir
Frodo
No one
Of the three nonhobbits, each prefers to have himself take on the task. Other than themselves and Frodo, each would prefer that no one take the ring. As for Frodo, he doesn’t
really want to do it and prefers to do so only if no one else will. The game is one in which
all players simultaneously make a choice among the four people. Only if they all agree—
a unanimity voting rule is put in place—is someone selected; otherwise, no one takes on
1. this epic task. Find all symmetric Nash equilibria.
ANSWER: There are four symmetric strategy profiles and thus four candidates for
symmetric Nash equilibrium. Note that if a person fails to vote for the person that
everyone else votes for, then no one takes on the task. Thus, at a symmetric strategy
profile,
8/5/08 1:28 PM Page
4-3 an individual player’s choice is always between the person who the others
are voting for and no one. Consider the symmetric strategy profile in which all vote
for Boromir. This strategy is optimal for both Boromir and Frodo as each would
rather that Boromir take on the task than that no one do so. This is clearly not a
2976T_ch04_025-032.qxd
1:28 PM
Page as
4-5
Nash8/5/08
equilibrium,
however,
both Legolas and Gimli would prefer to vote for
someone else, and the result would be that no one takes the ring to Mordor. By a
similarCHAPTER
argument,
it is
notNASH
a Nash
equilibrium
allWITH
to vote
Gimli,
or for all to
4-3
SOLUTIONS MANUAL
4: STABLE
PLAY:
EQUILIBRIA
IN DISCRETE for
GAMES
TWO ORfor
THREE
PLAYERS
vote for Legolas. Now consider all voting for Frodo. Since each person prefers that
Frodo do it than that no one do it, each player’s strategy is optimal. The unique
symmetric Nash equilibrium is then for all to vote for Frodo.
SOLUTIONS
MANUAL 1,CHAPTER
4: STABLE
NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO
OR THREE PLAYERS
none of player 2’s strategies was eliminated
in round
neither
a norPLAY:
b is
ictly dominated. For2.player
2, y strictly
dominates
Neither
y nor z is strictly
a modification
of x.
driving
conventions,
shown in FIGURE PR4.2, in which each
4. Consider
minated. After round 2, player
the surviving
strategies
aretoazigzag
and b on
forthe
player
and y
has a third
strategy:
road.1Suppose,
if a player chooses zigzag, the
IGURE player drives on the left, drives
d z for player 2. Goingchances
to round
2,
the
reduced
game
is aswhether
shown the
in Fother
of
an
accident
are
the
same
a. ANSWER:
Nash equilibria
arethat
(dopayoff
not kidnap/kill,
do itnot
ransom)
L4.4.2.
on
the right,The
or zigzags
as well. Let
be 0, so that
liespay
between
!1,and
the pay(do
not
kidnap/release,
do
not
pay
ransom).
off when a collision occurs for sure, and 1, the payoff when a collision does not occur.
FIGURE
SOL4.4.2
Find
all
Nash
equilibria.
8. Queen
Elizabeth
has decided to auction off the crown jewels, and there are two bidders:
Player 2
Sultan Hassanal Bolkiah of Brunei and Sheikh Zayed Bin Sultan Al Nahyan of Abu
Dhabi. The auctionyformat
z is as follows: The Sultan and the Sheikh simultaneously submit a written abid.2,1
Exhibiting
her well-known quirkiness, the Queen specifies that the
3,2
Sultan’s
Playerbid
1 must be an odd number (in hundreds of millions of English pounds) be3,4is, it
0,1
tween 1 and 9b(that
must be 1, 3, 5, 7, or 9) and the Sultan’s bid must be an even
number between 2 and 10. The bidder who submits
the highest bid wins the jewels and
pays a price equal to his bid. (If you recall from Chapter 3, this is a first-price auction.)
none of these strategies
is ystrictly
these are the strategies that
b. (b, ), (a, zdominated,
) The
winning
payoff equals his valuation of the item less the price he pays, while
vive the iterative deletion
of
strictly bidder’s
dominated
strategies.
the losing bidder’s payoff is zero. Assume that the Sultan has a valuation of 8 (hundred
8. million pounds) and the Sheikh has a valuation of 7.
erive all strategy pairs that are Nash equilibria.
a. In matrix form, write down the strategic form of this game.
SWER: There are two Nash
equilibria: (b, y) and (a, z).
ANSWER:
FIGURE
PR4.5.SOL4.8.1
sider the two-player game depicted in FIGURE
-032.qxd
2
4
6
8
10
Player 2 1
x
y
z3
1,2 Sultan
1,2 0,35
0,5
0,3
0,1
0,–1
0,–3
5,0
0,3
0,1
0,–1
0,–3
3,0
3,0
0,1
0,–1
0,–3
0,27
1,29
1,0
1,0
1,0
0,–1
0,–3
–1,0
–1,0
–1,0
–1,0
0,–3
FIGURE PR4.5
a
8/5/08
Sheikh
b
4,0
1,3
c
3,1
2,1
1:28
PM 1 Page 4-6
Player
d 0,2 0,1 2,4
b. Derive all Nash equilibria.
ANSWER: The unique Nash equilibrium is (7, 6).
erive those strategies which survive the iterative deletion of strictly dominated
SOLUTIONS MANUAL CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
rategies.
9. Find all of the Nash equilibria for the three-player game in FIGURE PR4.9.
9. PR4.9when player 2 uses
SWER: Since b is optimal when player 2 uses x,FIGURE
c is optimal
and d is optimal when player 2 uses z, player 1’s strategies of b, Player
c, and d3are
: ,z,
A not
ANSWER: The Nash equilibria are (b, x, A), (a
B), (c, x, C), (c, z, C).
ictly dominated. Next note that a is strictly dominated by c. Thus, the strategies
Player 2
player 1 that survive one round of the iterative deletion of strictly dominated
ategies (IDSDS) are b, c, and d.
x
y in Section
z
10. Return to the game of promotion and sabotage
4.4.
Turning to player 2, z is optimal when player 1 uses a and y is optimal when
1,1,0 2,0,0
a strategy
a. Determine whether the following
profile 2,0,0
is a Nash equilibrium; (i) player 1 is
yer 1 uses b. z strictly dominates x. Thus, the strategies of player 2 that survive
negative against player 2, (ii)
player
2 is negative against
b as
0,1,2 player 3, and (iii) player 3 is
3,2,1
1 then
e round of the IDSDS are y and z. The reducedPlayer
game is
shown1,2,3
in FIGURE
negative
against
player
1.
L4.5.1.
c 2,0,0 0,2,3 3,1,1
ANSWER:
The resulting performance is 1 for player 1, !1 for player 2, and !1 for
FIGURE
SOL4.5.1
Player
3: B1’s strategy is clearly optimal since
player 3, so player 1 wins the promotion.
Player
Player 2
he receives the maximal payoff of 1. If player
Player 22switches negative effort against
z
y
player 3 to negative
effort
against player 1, then player 1’s performance falls to !2
y
z remains at !1. Thus, player
and playerb3’s 1,3
rises to
0,
while
player x2’s performance
0,2
2 still loses. The other option is
player0,0,1
2 to exert
2,0,0
2,1,2positive effort on her own
a for
c 2,1case
1,2
Player
behalf,1in which
player 2’sbperformance rises to 0, so she still loses to player
1,2,0 1,2,1 1,2,1
Player
1
4-5
4-1