ECE 496
SOLUTIONS TO HOMEWORK ASSIGNMENT V Spring 2007
1.
Suppose the players play a total of
T
stages. If you are Player 1, and Player 2 plays the
proposed strategy, can you do better by playing a different strategy? Any such different
strategy would result in a payoff to you of 1 on any stage(s) on which you deviate from the
proposed strategy. If you stuck with the proposed strategy, you would collect a payoff of
either 2 or 3 on every stage. Accordingly, if Player 2 plays as proposed, then if you deviate
from the proposed strategy you’ll definitely collect a lower payoff than if you stay with
the proposed strategy. Similar reasoning applies if you’re Player 2. It doesn’t matter, by
the way, whether
T
is odd or even. The bottom line: the strategy profile in the problem
statement is a Nash equilibrium.
2.
(a) For each
i
(
i
= 1 or 2)
u
i
(
D, C
) = 5
>
3 =
u
i
(
C, C
)
and
u
i
(
D, D
) = 1
>
0 =
u
i
(
C, D
)
.
so playing
D
is strictly dominant for each player.
(b) Suppose (ˆ
a
1
,
ˆ
a
2
, . . . ,
ˆ
a
n
) is a PSNE profile in some
n
player game. Suppose, for
some
i
, action ˆ
a
i
is strictly dominated by some other action
a
i
for Player
i
. Then
u
i
(ˆ
a
1
, . . . ,
ˆ
a
i

1
, a
i
,
ˆ
a
i
+1
, . . .
ˆ
a
n
)
> u
i
(ˆ
a
1
, . . . ,
ˆ
a
i

1
,
ˆ
a
i
,
ˆ
a
i
+1
, . . .
ˆ
a
n
)
,
so Player
i
could do better than to play ˆ
a
i
against the other Players’ hatted
actions. Conclusion: (ˆ
a
1
,
ˆ
a
2
, . . . ,
ˆ
a
n
) can’t be a PSNE profile.
(c) Without loss of generality, suppose Player 1 has some strictly dominant strategy
ˆ
a
1
. Then no matter what Player 2 plays, ˆ
a
1
is a best reply for Player 1. So if
Player 2 plays a best reply to ˆ
a
1
— call that best reply ˆ
a
2
— then (ˆ
a
1
,
ˆ
a
2
) will be
a PSNE profile. The idea: under that profile each player is playing a best reply to
what the other player is playing, and that’s the definition of a Nash equilibrium
profile.
3.
(a) It’s pretty clear that no PSNE exists. To see this, suppose Player 1 plays
S
. Then
Player 2’s only best reply is
R
; but
S
is not a best reply for Player 1 against
R
,
so there exists no PSNE wherein Player 1 plays
S
. Similarly, if Player 1 plays
P
,
then Player 2’s only best reply is
S
, and
P
is not a best reply for Player 1 against
S
, so no PSNE exists wherein Player 1 plays
P
. If Player 1 plays
R
, then Player
2’s only best reply is
P
, and
R
is not a best reply for Player 1 against
P
, so no
PSNE exists wherein Player 1 plays
R
.
Accordingly, any Nash equilibrium profile will feature a mixed strategy for at
least one of the players. Suppose (
σ
1
, σ
2
) is a Nashequilibrium profile where at
least one of the strategies is mixed. If
σ
2
=
q
1
S
+
q
2
P
where
q
1
+
q
2
= 1 and both
q
1
and
q
2
are nonzero, then
P
is not a best reply for
Player 1, and it follows that
σ
1
=
p
1
S
+
p
3
R
1
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where
p
1
+
p
3
= 1. The only way to make
P
a best reply for Player 2, which it must
be since
σ
2
has
P
2
in its support, is for
p
3
= 0, which means Player 1 is playing
pure
S
.
But then neither
S
nor
P
is a best reply for Player 2, contradicting
the form of
σ
2
.
You can go through each case and demonstrate similarly that
no Nashequilibrium strategy profile (
σ
1
, σ
2
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 Spring '07
 DELCHAMPS
 Algorithms, Game Theory, Nash, p1, best reply

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