Consider the following two-player game: H L T D 2,3 0,2 4,0 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game? (c) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor 8 € (0, 1). Find a subgame perfect equilibrium in which (HT) is played in every period on the equilibrium nath Compute the discount.

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3. Consider the following two-player game:
H
L
T D
2,3 0,2
4,0 1,1
(a) What are (pure- and mixed-strategy) Nash equilibria of this game?
(b) Suppose the game is repeated twice, and each player's payoff is the sum of the
payoffs they obtain in the two periods. What are the subgame perfect equilibria
of the game?
(c) Suppose the game is repeated indefinitely, and each player discounts his/her payoff
with a discount factor & € (0, 1). Find a subgame perfect equilibrium in which
(H, T) is played in every period on the equilibrium path. Compute the discount
factor & needed for this equilibrium.
(d) Suppose the game is repeated indefinitely, and each player discounts his/her payoff
with a discount factor 8 € (0, 1). Find a subgame perfect equilibrium in which
(H,T) and (L,T) are played alternately on the equilibrium path. Compute the
discount factor & needed for this equilibrium.
(e) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts
her payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff with
a discount factor d2 = 0. That is, in any given period, player 2 only cares about
his payoff in that period. Can you find a subgame perfect equilibrium in which
(H, T) is played in every period on the equilibrium path. If no, explain. If yes,
compute the discount factor 8₁ needed for this equilibrium.
(f) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts
her payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff with
a discount factor d₂ = 0. Can you find a subgame perfect equilibrium in which
(H,T) and (L, T) are played alternately on the equilibrium path? If no, explain.
If yes, compute the discount factor 8₁ needed for this equilibrium.
(g) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts
her payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff with
a discount factor d₂ = 0. Can you find a subgame perfect equilibrium in which
(H,T) and (L, D) are played alternately on the equilibrium path? If no, explain.
If yes, compute the discount factor 8₁ needed for this equilibrium.
Transcribed Image Text:3. Consider the following two-player game: H L T D 2,3 0,2 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game? (c) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor & € (0, 1). Find a subgame perfect equilibrium in which (H, T) is played in every period on the equilibrium path. Compute the discount factor & needed for this equilibrium. (d) Suppose the game is repeated indefinitely, and each player discounts his/her payoff with a discount factor 8 € (0, 1). Find a subgame perfect equilibrium in which (H,T) and (L,T) are played alternately on the equilibrium path. Compute the discount factor & needed for this equilibrium. (e) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts her payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff with a discount factor d2 = 0. That is, in any given period, player 2 only cares about his payoff in that period. Can you find a subgame perfect equilibrium in which (H, T) is played in every period on the equilibrium path. If no, explain. If yes, compute the discount factor 8₁ needed for this equilibrium. (f) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts her payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff with a discount factor d₂ = 0. Can you find a subgame perfect equilibrium in which (H,T) and (L, T) are played alternately on the equilibrium path? If no, explain. If yes, compute the discount factor 8₁ needed for this equilibrium. (g) Suppose the game is repeated indefinitely. Player 1 (the row player) discounts her payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff with a discount factor d₂ = 0. Can you find a subgame perfect equilibrium in which (H,T) and (L, D) are played alternately on the equilibrium path? If no, explain. If yes, compute the discount factor 8₁ needed for this equilibrium.
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