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2. Market entry following picture. Consider a market entry game with imperfect information as depicted in the

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.1P

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Question

2. Market entry
following picture.
Consider a market entry game with imperfect information as depicted in the
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(a) Write down all strategies of the players. Transform the game into a normal-form. Find all
pure strategy Nash equilibria.
(b) Find all weak-perfect Bayesian Nash equilibria in pure strategies. Do not forget the systems
of beliefs. Argue well if there is none.

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し(5,3) b I(2,2) も(0,0) (4,12) a (12.4) (0,0) i). List all subgame pertect Nash equilibria and name one Nash eqvilibrivm that is not subgame pertect i). How many strategies does playot and player 2 have?
Consider the following extensive-form game. U D A B A B -6,6 X Y X Y P 3,8 -8, 1 1,2 -2, 1 5,5 -0,0 The dashed lines represent information sets. (a) Represent this game in normal-form. (b) Derive the Nash equilibria in pure strategies. (c) Describe the (proper) subgames of this game. (d) Derive the subgame perfect Nash equilibria in pure strategies.
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1. Repeated Game Consider the following normal form game. D Player 1 E F A 10, 10 -12,-1 15, -1 Player 2 B -1, -12 8,8 -1,-1 C -1, 15 -1,-1 0,0 (a) Suppose this game is played once. Find all pure strategy Nash equilibria. (b) Now suppose the game is played two times without discounting. Find all strategy profile that can be played in the first period in a SPNE. (c) Now suppose the game is played T times without discounting, where T is a finite number. Find the smallest T such that (E,A) is played in the first period. (d) Now suppose the game is played infinitely many times and the players have a same discount factor 8 < 1. Find the smallest d that can support the SPNE in which (D,A) is played in every period along the equilibrium path. (Hint: Players use the Grim-Trigger Strategy where (F,C) is used as a punishment).
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1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).
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1
1. Nash Equilibrium (a) Find all pure Nash Equilibria P1/P2 W X Y Z 9,9 0,7 5,5 1,1 7,0 0,0 4,2 7,7 5,5 2,4 0,0 1,1 1,1 7,7 1,1 0,0 A B C D (b) Consider the following picnic game. There are N players in the game each who can chose to bring food to the picnic or not. Let b;= {0, 1} be player i's choice of bringing food with 0 as not and 1 as yes. If they decide to bring food it comes at a cost c;(1) = 1 and no cost to bring food. Payoffs are sum of food brought minus individual cost. i. Write out the normal form matrix if N = 2 ii. If N = 2 find all pure Nash Equilibria iii. Find all pure Nash Equilibria in the general game