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ECONS 424 -STRATEGY AND GAME THEORY HOMEWORK #1 - DUE DATE: THURSDAY, JANUARY 24TH, IN CLASS Exercise 1 -From extensive form to normal form

econ game theory questions
1 E CON S 424 -S TRATEGY AND G AME T HEORY H OMEWORK #1 D UE DATE : T HURSDAY , J ANUARY 24 TH , IN CLASS Exercise 1 – From extensive form to normal form representation Consider the following extensive form game 10 0 0 0 5 5 0 0 0 10 0 0 a) Which are the strategies for player 1? b) What are the strategies for player 2? c) Take your results from a) and b) and construct a matrix representing its normal form game representation. Exercise 2 – Iterated Deletion of Strictly Dominated Strategies (IDSDS) Consider the following normal form game Player 2 Player 1 C1 C2 C3 R1 4,3 2,4 0,2 R2 5,5 5,-1 -4,-2 a) Is there some strictly dominated strategy for player 2, i.e. a strategy she would never use whatever the strategy finally chosen by player 1? b) If you apply iterative deletion of strictly dominated strategies (IDSDS), what is the surviving strategy pair? Explain the steps you use in IDSDS, and why you use them.
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2 Exercise 3 – Two people in a partnership. Suppose that two people decide to form a partnership firm. The revenue of the firm depends on the amount of effort expended on the job by each person and is given by: ( ) 12 1 1 22 , ree ae ae =+ , where 1 e is the effort level of person 1 and e2 is the effort level of person 2. The numbers 1 a and 2 a are positive constants. The contract that was signed by the partners stipulates that person 1 receives a fraction t (between 0 and 1) of the firm’s revenue and person 2 receives a 1 t fraction. That is, person 1 receives the amount ( ) 12 , tr e e , and person 2 receives () ( ) 12 1, tree . Each person dislikes effort, which is measured by a personal cost of 2 1 e for person 1 and 2 2 e for person 2. Person i ’s utility in this endeavor is the amount of revenue that this person receives, minus the effort cost 2 1 e . The effort levels (assumed nonnegative) are chosen by the people simultaneously and independently. a) Define the normal form of this game (by describing the strategy spaces and payoff functions). b) Using dominance, compute the strategies that the players rationally select (as a function of t , 1 a and 2 a ). c) Suppose that you could set t before the players interact. How would you set t to maximize the revenue of the firm? Exercise 4 – Pure s trategies that are only strictly dominated by a mixed strategy Consider the following normal form game Player 2 Player 1 Left Right Up 4,1 0,2 Middle 0,0 4,1 Down 1,3 1,2 a) Is there some strictly dominated strategy for player 1 involving only the use of pure strategies? b) Is there some strictly dominated strategy for player 1 when mixed strategies are allowed? [ Hint: you may assign probabilities to two of her strategies, similarly as we did in class ] . c) Delete the strictly dominated strategies for player 2 that you found in the previous question. Then, represent the remaining (undeleted) strategies. d) Proceed with the IDSDS. What is the strategy pair surviving IDSDS? Exercise 5: 1. Exercise 8 from Chapter 2 in Harrington. Exercise 6 (Bonus exercises): 1. Exercise 10 from Chapter 3 in Harrington. 2. Exercise 11 from Chapter 3 in Harrington.
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