Ko University ECON 333econ333_fin_f10_solECON/MGEC 333
Game Theory And Strategy
Midterm Examination II Solutions
Levent Ko¸ckesen January 6, 2011
1.
... [Show More] (30pts.) Consider the following extensive form game:
1 2 2, 1
a, 4 1, 0
S
C
s
c
(a) (10pts.) Assume that a = 1 and find the set of pure strategy Nash equilibria and subgame perfect
equilibria.
Solution
The unique subgame perfect equilibrium is (C, c). The strategic form of the game is given by the
following bimatrix:
Player 1
Player 2
s c
S 1, 4 1, 4
C 1, 0 2, 1
The set of Nash equilibria is {(S, s), (C, c)}.
(b) (10pts.) Find the range of a for which S is the unique subgame perfect equilibrium outcome.
Solution
a > 2
(c) (10pts.) Find the range of a for which (C, c) is the unique Nash equilibrium outcome.
Solution
a < 1
2. (30pts.) A seller (S) can make a costly investment to make her product more valuable for the buyer (B).
Assume that if the seller invests x, then her cost is x2 and the value of the buyer is v + x, where v > 0.
(a) (10pts.) Assume first that the game has three stages:
Stage I Seller chooses investment x ≥ 0
Stage II Buyer observes x and offers a price p ≥ 0
Stage III Seller observes p and either accepts (a) or rejects (r) the offer
1
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If the the seller rejects the offer, then payoff of the buyer is zero whereas the payoff of the seller is
-x2. If she accepts the offer, then the payoff of the buyer is v + x - p and the payoff of the seller is
p - x2. Find the subgame perfect equilibria of this game.
Solution
In response to an offer p, the seller’s optimal decision is to accept if p > 0 and she is indifferent if
p = 0. If her strategy is to reject p = 0, then the buyer’s optimal strategy is not defined. Therefore,
she must accept p = 0 as well. In response to that, the buyer’s optimal decision is to offer zero price.
Therefore, the seller’s payoff to investment x is -x2 and hence her optimal decision is to invest zero.
The following is the unique SPE of this game: [Show Less]