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Describe all (pure strategy) nash equilibria.

by | Nov 4, 2022 | Other | 0 comments

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I am looking for somebody to help me take a quiz in my game theory class. What will happend is the following:
On November 3rd at whatever time we agree we will meet on a zoom call, where i will be sharing my screen so that you can see the problems and tell me the answers. The Questions will require written answers for all 4 questions. I am flexible with the price. Below i will include some sample questions from the PREVIOUS QUIZ , so you have an idea of the scope of the work. ATTACHED BELOW IS ALSO THE TEXTBOOOK, FOR THE QUIZ CHAPTERS 11, 12, 14, 15, 16, 22, AND 23. ALSO ATTACHED ARE PREVIOUS MIDTERMS WHICH HELP WITH PRACTICE.
Suggested Problems in textbook for quiz Chapter 11: 1, 3, 5, 8, 9, 10, 12, 13, 14, 15
Chapter 12: 1, 3
Chapter 13: 1, 3
Chapter 14: 1, 3, 5
Chapter 15: 1, 2, 3, 5, 12
Chapter 16: 5, 7
Chapter 22: 1, 3, 7, 9
Chapter 23: 7
also attached will be links and slides associated with each chapter:
CHAPTER 11:
https://www.youtube.com/watch?v=eacp_635RnY – Mixed-Strategy Nash Equilibrium
https://www.youtube.com/watch?v=JIwY5XA5QHc – examples
Chapter 12:
https://www.youtube.com/watch?v=RLhPN_1_PDQ – strictly competative games
CHAPTER 14, 15, 16:
https://www.youtube.com/watch?v=D44szt7ZSkQ – BACKWARD INDUCTION
https://www.youtube.com/watch?v=SNPG5hwvQKQ – subgame perfection
Chapter 22, 23:
https://www.youtube.com/watch?v=Qjn2-cw2BdI&feature=youtu.be – two-period repeated games
https://www.youtube.com/watch?v=hejaXQyVEkg – infinity repeated games
Question 1: Two countries are in a dispute. The leader of each county has a lot of ego and fancies himself to be extremely tough. Assume that simultaneously and independently each chooses how aggressive to be in his actions toward the other country. This is denoted by Si = {1,2,3,4,5,6,7,8,9,10}, with 10 being the most aggressive. If the combined aggression of the two leaders is above 8, there is a war. Given their egos, each prefers being more aggressive to less aggressive, provided it does not lead to war. War is extremely bad for both of them.
Player i’s payoff is: 1500si if s1 + s2 < 8, and -9,000,000 otherwise.
Part A: Describe UDi. Explain.
Part B: Describe R. Explain.
Part C: Describe all pure-strategy Nash equilibria.
ANSWER: Part A: Neither player has a dominant strategy in this game. Playing 7 or greater is dominated.
Part B: R_1 and R_2 are both [1,6]
Part C: Any situation where the sum of the two strategies is 7 is a pure strategy nash equilibrium
Question 2: SEE ATTACHED BELOW
Question 3: Consider the following interaction of the leaders of ten countries. Each wish to undertake a policy to address an important issue. Acting when leaders of other countries also do is preferred.
Parts A and B describe different degrees of this idea.
Part A: All leaders have the same payoff structure. Each receives a payoff of 1000 when all ten choose to act. Each receives a payoff of -100 if she chooses to act and fewer than ten in total choose to act. Choosing not to act yields a payoff of zero.
Describe all (pure strategy) Nash equilibria. Explain.
Part B: The payoff structure of the leader of country 1 differs from that of the other nine.
For country 1, the leader’s payoffs are as follows. Acting alone yields 500. Acting when at least one other does, yields a payoff of 1000. Not acting yields a payoff of zero.
For the leaders of countries 2 to 10, the payoffs are the following. Acting alone yields a payoff of -100. Acting when at least one other does, yields a payoff of 1000. Not acting yields a payoff of zero.
Describe all (pure strategy) Nash equilibria. Explain.
Part C: Does the predicted behavior in either setting (Part A or Part B) seem more robust? If so, which one? Explain and describe in terms of another solution concept that we’ve covered.
ANSWER:
Part A: All 10 choosing not to act and all 10 choosing to act are both nash equilibria. In both of these cases, each leader is choosing a best response given the actions of the other leaders.
Part B: All 10 choosing to act is the unique Nash equilibrium. Acting is a dominant strategy for country 1 and given that country 1 will act, all other countries also want to act.
Part C: The prediction from Part B is more robust because acting is the dominant strategy for country 1, and therefore all other players (knowing player 1 is rational) will also choose to act. That is, all 10 leaders choosing to act is the only rationalizable equilibrium set of strategies.
Question 4: Players one, two, and three simultaneously and independently choose integers from Si = {4, 5, 6, 7, 8}. If the sum of the three integers is at least 13, each player receives a payoff of 200 minus her choice. Otherwise, each player’s payoff is 0.
Part A: Find Bi. Explain.
Part B: Find Ri. Explain.
Part C: Find all (pure strategy) Nash equilibria. Explain.
ANSWER: Part A: In this game the minimum sum of the other players’ choices is 8. Therefore, 6, 7, and 8 are all strictly dominated strategies, and players should believe that each other player will choose either 4 or 5.
Part B: The best response is 5 if both other players choose 4 and 4 otherwise.
Part C: There are three pure strategy Nash equilibria: these are each situation when exactly one player plays 5 and the other two players play 4.

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