ECON/MGEC 333
Game Theory And Strategy
Midterm Examination II Solutions
Levent Ko¸ckesen December 28, 2011
1. (30pts.) Consider the following extensive form game:
1
x, 4
2
2, 1
1
0, 0 1, 0
L R
u d
l r
(a) (10pts.) Assume that x = 1 and find the set of pure strategy Nash equilibria and subgame perfect
equilibria.
Solution
The unique subgame perfect equilibrium is (Rr, u). The strategic form of the game is given by the
following bimatrix:
Player 1
Player 2
u d
Ll 1, 4 1, 4
Lr 1, 4 1, 4
Rl 2, 1 0, 0
Rr 2, 1 1, 0
The set of Nash equilibria is {(Ll, d), (Lr, d), (Rl, u), (Rr, u)}.
(b) (10pts.) Find the range of x for which (R, u) is the unique subgame perfect equilibrium outcome.
Solution
x < 2
(c) (10pts.) Find the range of x for which L is a Nash equilibrium outcome.
Solution
x ≥ 1
1
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2. (30pts.) There is a project for which player 1 can exert effort e ≥ 0 that costs her c(e) = 0.5e2. If
player 1 and player 2 can come to an agreement, then a total value of v(e) = e is produced, which can be
allocated between the two players. Effort also produces a value y(e) = ke, where k ∈ [0, 1], that player 1
can obtain for herself if player 1 and 2 fail to agree. The game has three stages:
Stage I Player 1 chooses effort e ≥ 0
Stage II Player 2 observes e and chooses an offer α ∈ [0, 1]
Stage III Player 1 observes α and either agrees (a) or rejects (r) the offer
If player 1 accepts the offer, then her payoff is αe - 0.5e2 and player 2’s payoff is (1 - α)e. If she rejects
the offer, then player 1’s payoff is ke - 0.5e2 and player 2’s payoff is zero. The following figure represents
the game.