Intermediate Microeconomics Homework Set 10

Intermediate MicroeconomicsHomework Set 10.Deadline: Friday, May 6th, 1:00pm.Submission options:• in person during lectures or TA sessions;• in Siddharth Hari’s mail box at 19 West 4th, 6th Floor;• (less recommended) e-mail Siddharth Hari at– in that case, be sure to use the subject line: “Submission Homework 10”.Exercise 1Consider a game of rock-scissor-paper. There are two players, player 1 and player 2,and the winner receives $1 from the loser (if they draw, nothing happens). They playthis game simultaneously.1. What kind of game is this? Express this game in normal form.2. What are the set of possible actions and strategies?3. Find all Nash equilibria of this game.Exercise 2Consider a game represented by the following payoff matrix. Find all pure strategyNash equilibria.L C RU ?10, 0 ?10, ?1 ?10, ?1M ?8, ?8 0, ?10 2, ?1D 4, 5 7, ?10 1, 1Exercise 3Consider a game represented by the following payoff matrix. There are two firmscompeting in a duopoly market. Each should decide how much to produce simultaneouslyand their profits are given in the matrix. Find all pure strategy Nash equilibria.25 35 50 10025 125, 125 100, 140 63, 125 ?63, ?25035 140, 100 105, 105 53, 75 ?123, ?35050 125, 63 75, 53 0, 0 ?250, ?500100 ?250, ?63 ?350, ?130 ?500, ?250 ?900, ?9001Exercise 4Consider two players simultaneously deciding whether to contribute to a public good— the good is said to be public because, if it is made available, a player who free-ridesby paying nothing gets just as much pleasure from its enjoyment as a player who paidfor it. If at least one player contributes to the construction of the public good, bothplayers will enjoy a payoff of four from the public good. To ensure the public goodis constructed, player 1 must pay c1 or player 2 must pay c2. Assume that c1 < 4and c2 < 4. If neither contribute, the good is not constructed and neither playergets enjoyment from the project. If one or both players contribute, then the good isconstructed and each player enjoys a payoff of four minus the contribution cost if thatplayer has contributed.1. Describe this game in normal form.2. Find any pure-strategy Nash equilibria to the game.3. Find the mixed strategy Nash equilibria — the probabilities you find will befunctions of the cost parameters.4. If c1 = c2 = 1, write out the mixed strategy Nash equilibria and find the probabilitythat the public good is provided.5. If c1 = c2 = 3, write out the mixed strategy Nash equilibria and find the probabilitythat the public good is provided.2

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