soc 376 - exams - 2008

soc 376 - exams - 2008 - Sociology 376 Exam 1 Spring 2008...

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Sociology 376 Exam 1 Spring 2008 Prof Montgomery Answer all questions. 180 points possible. 1) [40 points] A sequence of voting intentions might be conceptualized as a Markov chain. In particular, consider voters who are Democrats living in South Dakota (which doesn’t hold its primary until June). Suppose that each voter is in one of three states: (1) intending to vote for Obama, (2) undecided, and (3) intending to vote for Clinton. Further suppose that these intentions change monthly according to the transition matrix P = 9 . 1 . 0 1 . 7 . 2 . 0 15 . 85 . where P(i,j) is the probability of transitioning from state i to state j. a) For a voter who is currently undecided (in March), compute the probability distribution over states for this voter for the next two months (April and May). b) Suppose that a Gallup poll in May reveals that 35% of voters are intending to vote for Obama, that 25% are undecided, and that 40% are intending to vote for Clinton. Further suppose that, when the primary occurs in June (one month after the Gallup poll), voters intending to vote for Obama or Clinton actually do so, and that undecided voters split evenly between Obama and Clinton. What share of the vote will each candidate receive? [NOTE: You can assume there is a very large number of Democrats in South Dakota.] c) If the primary was postponed until the population reached equilibrium, what would be distribution of voters over the 3 states? Derive the answer algebraically (by solving a system of simultaneous equations). [NOTE: You must show your work to receive full credit.] If the undecided voters again split evenly, which candidate would win? d) List two other ways you could have solved for the equilibrium distribution in part (c) if you had access to Matlab. 2) [30 points] Peyton Young developed a Markov chain framework to analyze the evolution of social conventions (discussed in his 1996 Journal of Economic Perspectives paper, as well as the Gintis chapter, which are both in your reading packet). a) In Young’s framework, what are the “states” of the process? Briefly explain how the introduction of “mistakes” into this framework alters the long-run probability distribution over these states. Then briefly explain the concept of “stochastic stability” and Young’s rationale for applying this concept. b) Consider a left-right coordination game where both players receive 2 if they choose R, both players receive 1 if they choose L, and both players receive 0 otherwise. Are both Nash equilibria of this game stochastically stable? Give a brief, intuitive explanation. 1
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3) [35 points] Consider an influence network with 8 actors. Opinions change over time according to the equation y t = W t y 0 where y t is a (column) vector of opinions at time t, and W is an influence matrix (with rows that are probability vectors). Suppose that the zero pattern of the influence matrix and the initial opinion vector are given by Z = y 0 = 1
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This note was uploaded on 08/11/2008 for the course SOC 376 taught by Professor Montgomery during the Spring '08 term at Wisconsin.

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soc 376 - exams - 2008 - Sociology 376 Exam 1 Spring 2008...

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