Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![1. Consider a repeated game G = (g, T, 3), where there are two players and the stage game g is:
Player 2
C D
0,y
D y,0 1,1
Player 1 Cr, x
(a) Suppose that the stage game is repeated a finite number of times (T<∞o). For which
values of x and y is there a unique, pure-strategy subgame perfect Nash equilibrium for all
values of 0 < 5₁ < 1 and 0 < 8₂ <1? Write your answer as inequalities using r and y, and
show your work.
I
(b) Suppose that the stage game is repeated a infinite number of times (T= ∞). For which
values of r and y is there a unique, pure-strategy subgame perfect Nash equilibrium for all
values of 0 < 5₁ < 1 and 0 < 6₂ < 1? Write your answer as inequalities using r and y, and
show your work. (If you are not able to derive the inequalities, give examples of r and y for
which there is a unique SPNE.)
(c) Suppose that the game is repeated infinitely (T=x), and that y>r>1. Describe the
strategies of players 1 and 2 for which there exist minimal 0 < d < 1 and 0 < 6₂ <1 such
that an outcome of (r, r) in every round is obtainable. Derive the algebraic expressions for di
and d2 necessary to sustain this outcome.
(d) No one lives forever, and yet economists model and contemplate infinitely repeated
games. Can you justify such an assumption of T = oc? By way of example, explain why
T = ∞ may be appropriate or argue why you believe it is never appropriate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa3c8bce-9ac3-44bc-8e19-753f7b8befa4%2F5da292a5-a7a4-4b55-bafb-5e05db730e41%2Fqkezcog_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider a repeated game G = (g, T, 3), where there are two players and the stage game g is:
Player 2
C D
0,y
D y,0 1,1
Player 1 Cr, x
(a) Suppose that the stage game is repeated a finite number of times (T<∞o). For which
values of x and y is there a unique, pure-strategy subgame perfect Nash equilibrium for all
values of 0 < 5₁ < 1 and 0 < 8₂ <1? Write your answer as inequalities using r and y, and
show your work.
I
(b) Suppose that the stage game is repeated a infinite number of times (T= ∞). For which
values of r and y is there a unique, pure-strategy subgame perfect Nash equilibrium for all
values of 0 < 5₁ < 1 and 0 < 6₂ < 1? Write your answer as inequalities using r and y, and
show your work. (If you are not able to derive the inequalities, give examples of r and y for
which there is a unique SPNE.)
(c) Suppose that the game is repeated infinitely (T=x), and that y>r>1. Describe the
strategies of players 1 and 2 for which there exist minimal 0 < d < 1 and 0 < 6₂ <1 such
that an outcome of (r, r) in every round is obtainable. Derive the algebraic expressions for di
and d2 necessary to sustain this outcome.
(d) No one lives forever, and yet economists model and contemplate infinitely repeated
games. Can you justify such an assumption of T = oc? By way of example, explain why
T = ∞ may be appropriate or argue why you believe it is never appropriate.
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