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.
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