ECON GU4415 - GAME THEORY
Columbia University - Department of Economics
Problem Set 3
Assignment 3: Q1 (Games 4-5), Q3, Q6
Due: June 10, Tuesday, 6pm
1. For each of the following normal-form. game below, find all Nash equilibria in pure and mixed strategies.
Game 1:
(1)/(2) C(q) D(1 − q)
A (p) (2,1) (3,2)
B(1 − p) (0,4) (4,3)
Game 2:
(1)/(2) L(q) R(1 − q)
U(p) (3,3) (0,0)
M(r) (1,1) (1,2)
D(1 − p − r) (0,0) (3,3)
Game 3:
(1)/(2) L(q) R(1 − q)
U(p) (2,1) (1,2)
M(r) (1,2) (2,1)
D(1 − p − r) (1,3) (3,1)
Game 4:
(1)/(2) L(q) R(1 − q)
U(p) (4,3) (1,0)
M(r) (3,1) (3,2)
D(1 − p − r) (0,3) (4,0)
Game 5:
(1)/(2) X(q) Y(t) Z(1 − q − t)
A(p) (3,3) (2,1) (1,2)
B(r) (2,2) (3,0) (2,3)
C(1 − p − r) (2,0) (2,1) (1,0)
2. Consider the two player normal form. game below.
(2)
L C R
U (4, 2) (0, 0) (2, 3)
(1) M (0, 2) (3, 4) (1, 1)
D (1, 1) (2, 2) (1, 2)
(a) Find the set of mixed strategy equilibria, if any, in which player 2 assigns zero probability to exactly one of her strategies.
(b) Find the set of mixed strategy equilibria, if any, in which player 2 assigns positive probabilities to all of her strategy strategies.
3. Consider the two player normal form. game below.
(2)
L(q) C(t) R(1 − q − t)
U(p) (3, 4) (1, 3) (0, y)
(1) M(r) (2, 0) (0, 2) (1, 3)
D(1 − p − r) (0, 1) (x, 0) (1, 3)
(a) Find the set of values for x and y for which there is a mixed strategy Nash equilibrium where each player puts positive probability on each of her three pure strategies and player 2 plays C with exactly 1/3 probability?
(b) Suppose x = 0 and y = 3. Find a mixed strategy Nash equilibrium where player 1 puts 0 probability on playing D.
4. Consider the following version of Matching Pennies game with two players, 1 and 2. Each player simul-taneously chooses from the set S = {1, ..., N}. If the players choose the same number then player 2 pays $1 to player 1 and the payoffs are (1, -1); otherwise no payment is made and the payoffs are (0, 0). Show that σ = ((1/N, ..., 1/N), (1/N, ..., 1/N)) is a mixed strategy Nash equilibrium. Is it unique?
5. Consider the following game between a police officer and a robber. The police officer (player 1) must decide whether to patrol the streets [P] or to hang out at the coffee shop [C]. His payoff from hanging out at the coffee shop is 10, while his payoff from patrolling the streets depends on whether he catches a robber (player 2). The robber must choose between prowling the streets [S] or staying hidden [H]. If the robber prowls the streets then the police officer catches him, and the police officer obtains a payoff of 20, while the robber ends up with -10. If the robber stays hidden, while the police officer patrols, then both player gets a payoff of 0. When the officer is at the coffee shop, robber gets 10 if he prowls the streets, and 0 if he stays hidden. What kind of game does this game look like? Find a mixed strategy Nash equilibrium of this game.
6. Consider the following market entry game. There are three firms, which are considering entering a new market. The payoff for each firm that enters is n/60, where n is the number of firms that enter. The cost of entering is 24. If a firm does not enter then its payoff is 11. Find the symmetric mixed-strategy equilibrium in which all three firms enter with the same probability, p.
7. Consider a simultaneous move game with 3 players, who choose between X and Y . That is, the strategy set for each player {X, Y}. The payoff of each player who selects X is 2mx − m2
x + 3, where mx is the number of players who choose X. The payoff of each player who selects Y is 4 − my, where my is the number of players who choose Y . Note that mx + my = 3.
(a) Find the pure-strategy Nash equilibria, if any.
(b) Determine whether this game has a symmetric mixed-strategy Nash equilibrium in which each player picks X with probability p. If you can find such an equilibrium, what is p?