5.3
(a)
1
= (P
1
– 1)(10 – P
1
– 0.5P
2
)
1
1
P
∂
Π
∂
= 10 – P
1
– 0.5P
2
– P
1
+ 1
0
B
1
(P
2
) = P
1
= 5.5 – 0.25P
2
Substituting P
2
into the equation for P
1
:
2
= (P
2
– 2)(12 – P
2
– 0.5P
1
)
2
2
P
∂
Π
∂
= 12 – P
2
– 0.5P
1
– P
2
+ 2
0
B
2
(P
1
) = P
2
= 7 – 0.25P
1
P
1
= 5.5 – 0.25
(7 – 0.25P
1
)
(15/16)P
1
= 15/4
P
1
= 4
Plugging this back into the equation for P
2
:
P
2
= 7 – 0.25(4) = 6
P
2
= 6
B1(P2)
B2(P1)
4
6
P2
P1
(b)
total
=
1
+
2
= (P
1
– 1)(10 – P
1
– 0.5P
2
)
+
(P
2
– 2)(12 – P
2
– 0.5P
1
)
1
P
total
∂
Π
∂
= 10 – P
1
– 0.5P
2
– P
1
+ 1 – 0.5P
2
+ 1
0
B
1
(P
2
) = P
1
= 6 – 0.5P
2
2
P
total
∂
Π
∂
= –0.5P
1
+ 0.5 + 12 – P
2
– 0.5P
1
– P
2
+ 2
0
B
2
(P
1
) = P
2
= 7.25 – 0.5P
1
Substituting P
2
into the equation for P
1
:
P
1
= 6 – 0.5
(7.25 – 0.5P
1
)
(3/4)P
1
= 19/8
P
1
= 19/6
3.17
Plugging this back into the equation for P
2
:
P
2
= 7.25 – 0.5(19/6) = 17/3
P
2
= 17/3
5.67
(c)
While the two firms do better by colluding (part (b)) than by competing (part (a)), this is not a Nash
equilibrium because neither firm is playing a best response to the other’s price.
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(d)
Because the goods are complements in this problem, the best response curves are negatively sloped
(when the other firm cuts its prices, your demand goes up, giving you an incentive to raise your own
prices somewhat); in the example in the text, the best response curves were upward sloping because
the goods were substitutes (if the other firm cuts its prices, your demand goes down, so you lower your
own prices to compensate). This also explains why the collusive prices are lower than the NE prices in
this case, while the collusive prices were higher than the NE prices in the text example.
5.7
For player B, Right is strictly dominated by Left. Once Right is eliminated, Up is strictly dominated by
Down for player A, leaving (Down, Left) as the Nash equilibrium. Since this is the only outcome that
survived the iterated elimination of strictly dominated strategies, it is the only rationalizable outcome.
6.2
The NE of the game are (Out, War) and (In, Peace). (Out, War) is not subgame perfect because War is
not a credible threat for Boeing (Peace is the NE of the subgame occurring after Airbus enters).
6.5
(a)
To find the payoffs in a given cell, start by identifying which clusters will purchase from each vendor
(whichever is closer; if the vendors are equidistant from a cluster, they each get half of the customers
who are willing to travel that distance from that cluster). For example, if Vendor 1 locates at A and
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 Summer '08
 Bonanno,G
 Game Theory, best response, SPE, payoﬀ

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