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Problem Set 7

ECON6001 Semester 1, 2019

Read This! For a good understanding of the applications of Repeated Games, Sec- tion 4 of the Jonathan Levin's Lecture Note is quite good. This is one of your required readings, the link is repeated here: http://www.stanford.edu /~jdlevin /Econ 20203 / RepeatedGames.pdf. Treat each of the applications (subsection of Section 4) as an exer- cise. The proof given is the solution. So, in other words, attempt the proof of each application on your own, and use his proof as a solution.

Q1. Just to make sure that you understand, check carefully that both players playing the Tit-for-tat strategy cannot be SPE in the game below (that we considered in class)

C D

C D

Q2. For each of the following games, diagramatically show that the region of payoRs that can be achieved under the Folk Theorem for infinitely repated games.

l m r

a b c

l m r

a b c

q_m q_c q_h

Table 1.

Q3. The last game in the above Table is the version of infinitely repeated Cournot duopoly considered in Lec 12 slides. For this game, answer the following:

1. Consider supporting the monopoly price as an equilibrium using a ”Grim-trigger strategy“ in this game: Play q_m at date 0. At any history h_T = (a⁰; :::; a^T ^¡¹), play q_m if a^t = (q_m; q_m) for all t = 0; :::; T ¡ 1. Otherwise play q_h.

Find the values of 6 for which both players playing the above strategy constitutes an equilibrium.

2. Next, suppose the two players do not hold any execss capacity, the game is now reduced to the 2x2 game with q_m and q_c as the only available choices. Now consider the new ”Grim-trigger strategy“: At any history h_T = (a⁰; :::; a^T ^¡¹), play q_m if a^t = (q_m; q_m) for all t = 0; :::; T ¡ 1. Otherwise play q_c.