# Econ 302 Assignment 5 — Solution 1. Part i: ```Econ 302
Assignment 5 — Solution
1. Part i:
Part ii:
1
Part iii:
2
2.
3
There are two subgames: everything that follows the branch B (which is a proper subgame),
and the whole game.
Clearly, in any SPE player 2 plays L if B is played. For the rest of the game, we can make
the following matrix:
IL
OL
T
1, 1
0, 0
M
0, 0
1, 1
B
0.6, 0.6
0.6, 0.6
The pure-strategy Nash equilibria in the above matrix game are (T, IL) and (M, OL), which
are the pure-strategy SPE of the original game (as represented by the tree). (T, IR) and
(M, OR) are Nash equilibria that are not subgame perfect.
3. In the subgame where n firms had chosen to enter the market, the (symmetric) Nash
1
42. In this equilibrium, the market price is
equilibrium is each of them producing q ∗ = n+1
n
422
P = 45 − n+1
42, and the profit of each firm is Π = (n+1)
2 − 4. Notice that if n = 20, each
firm makes zero profit.
Given that firms play the above Nash equilibrium in the subgame, there are two classes of
pure-strategy subgame perfect equilibria:
1. 19 firms enter in the first stage, and in any subgame each firm that have entered
1
42, where n is the number of firms entered in that subgame. In this
produces q ∗ = n+1
SPE, each of the 19 firms enjoys a profit of 0.41.
2. 20 firms enter in the first stage, and in any subgame each firm that have entered
1
produces q ∗ = n+1
42, where n is the number of firms entered in that subgame. In this
SPE, each of the 20 firms gets zero profit.