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# Economics 373 Game Theory &amp; Strategic Thinking for Social Sciences Summer 2016 Instructor: Akio Yamazaki Assignment 1 Due date: July 20th, 2016...

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Economics 373 Game Theory & Strategic Thinking for Social Sciences Summer 2016 Instructor: Akio Yamazaki Assignment 1 Due date: July 20th, 2016 Work alone or in groups of TWO (maximum). Turn in the assignments individually . Please write down how you arrived to your conclusions. If reasoning is rigorous and correct, you will get partial credits even if the answer is not. If you have problems, you can ask me during my oﬃce hours or ask TA during her oﬃce hours, but please do not expect me or her to solve the problem for you or tell you the solution. Please do not email me or TA about the assignment. Question 1: Iterative Elimination of (weakly) Dominated Strategies (15 points) Consider the following two-player game Player 2 l c r Player 1 U 1,1 0,1 3,1 M 1,0 2,2 1,3 D 1,3 3,1 2,2 (a) Are there any strictly dominated strategies? Are there any weakly dominated strategies? If so, explain what dominates what and how. (b) After deleting any strictly or weakly dominated strategies, are there any strictly or weakly dominated strategies in the ‘reduced’ game? If so, explain what dominates what and how. What is left? (c) Go back to your argument for deleting in the ﬁrst ‘round’ and recall what dominated what and how. Compare this with what was deleted in the ‘second’ round. Comment on how this might make you a bit cautious when iteratively deleting weakly dominated strategies? 1
Question 2: Electing Your Leader (15 points) Consider a committee of three agents, Mr. Pink, Mr.White, and Mr.Orange whose task is to choose one alternative from the choice set { α,β,γ } by means of voting. Committee members cote simultaneously for one and only one alternative. The alternative that gets 2 or 3 votes wins. In case of tie between the three alternatives, the chairman of the committee (Mr.Orange) unilaterally decides the winner. The utility of Mr.Pink is 2 if α is selected, 1 if β is selected, and 0 if γ is selected. The utility of Mr.White is 2 if β is selected, 1 if γ is selected, and 0 if α is selected. The utility of Mr.Orange is 2 if γ is selected, 1 if α is selected, and 0 if β is selected. (a) Find the outcome of the game if members vote sincerely , that is if they each vote for the alternative they like best independently of what others do. Write down the normal form of the game. (b) Use Iterated Elimination of Weakly Dominated Strategies to ﬁnd the outcome of the game if members vote strategically , that is if they anticipate the behavior of others on the outcome of the game and try to maximize their own proﬁts accordingly. (c) Conclude (explain what you found and interpret them in your own words). Question 3: Iterative Elimination and Nash Equilibrium (15 points) Consider the following two-player game Player 2 L C R Player 1 T 2,0 1,1 4,2 M 3,4 1,2 2,3 B 1,3 0,2 3,0 (a) What strategies survive iterated elimination of strictly dominated strategies? (b) What are the Nash equilibria of this game? 2
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a) Player 1 will eliminate B and player will eliminate C.
Then player 1 eliminates B then player 2 eliminates L and therefore (R,T) AND PAYOFF b) IS (4,2)
Nash equilibrium is (R,T) WITH PAYOFF (4,2)

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