IKT 415 - SPRING 2015 - Quiz 2

IKT 415 - SPRING 2015 - Quiz 2
¨
Ay¸ca Ozdo˜
gan
Sequential-move (Extensive-Form) Games with Complete-Information
1. Consider the following dynamic game: A Robber first decides whether to attack to (A) or pass
by (P ) a potential Victim. If he attacks, observing that, the Victim has to decide whether to
fight (F ) or yield (Y ). The payoffs are given as follows. (25 points)
Player 1 (Robber)
P
A
Player 2 (Victim)
0,0
F
-1,-5
Y
4,-4
(a) Write down the strategy sets of each player.
(b) Find the set of pure strategy Nash equilibria.
(c) Find the set of pure strategy subgame-perfect Nash equilibria. Is there a non-credible
threat? Can the subgame-perfect Nash equilibrium eliminate the non-credible threat?
1
2. Now, suppose that players, robber and victim choose their actions simultaneously (or without
observing each other). (20 points)
(a) Draw a game tree representing this situation. Assume that if the Robber chooses to pass
(P ), both get a payoff of 0 no matter what victim chooses.
(b) How many subgames are there in this situation? Is the set of subgame-perfect equilibria
different than the set of Nash equilibrium in this case?
3. Consider an extensive-form game at which player i has 4 information sets and three actions
at each information set. How many pure strategies does player i have? (5 points)
2
4. Consider the following dynamic game: Player 1 can choose to play it Safe (denote this by S )
in which case both he and player 2 get a payoff of 3 each, or he can risk playing a game with
player 2 (denote this by R). If player 1 chooses R, then they play the following simultaneousmove game. (50 points)
U
D
L
8, 2
6, 6
R
0, 2
2, 2
(a) Draw a game tree that represents this game. How many subgames does it have? (15
points)
(b) Are there other game trees that could work? Explain. (3 points)
(c) How many information sets does each player have? Write down the strategy sets of each
player. (7 points)
(d) Construct the normal-form game representation of this game. Find all the pure strategy
Nash equilibria. (13 points)
(e) Find all the pure strategy subgame-perfect Nash equilibria. Is there any non-credible
threat? Explain. (12 points)
3
5. BONUS There are three players (firms) in a market: Entrant 1 (E1), Entrant 2 (E2) and
Incumbent (I). First, Firm E1 moves and chooses either to stay out of the market (Out),
enter on its own (In) or propose a joint venture to firm E2 (Venture). If it proposes a joint
venture, observing that, firm E2 can either accept (Accept) or decline (Decline) the offer. If
E2 accepts, firm E1 enters the market with E2 ’s assistance. If E2 declines the offer, then
E1 must decide whether to enter on its own (in) or stay out (out). The incumbent firm can
observe whether E1 has entered but not if it is with E2’s assistance or not. If an entry by E1
occurs, Incumbent responds by fighting (F) or accommodating (A). If E1 has chosen to stay
out, the game ends.
(a) Draw the game tree and clearly show all the information sets. (6 points)
(b) How many subgames does this game have? (2 points)
(c) Can the Nash equilibria of this game be different than the subgame perfect equilibria?
Justify your answer. (2 points)
4