W 4370 CU Political Economics Median Voter Theorem Questions

Department of EconomicsAlessandra Casella
Columbia University
Fall 2022
Political Economy W4370
Problem Set 9
Due before class on Wednesday, December 14th via Gradescope
The questions in this problem set are taken from past exams. We keep the
points for each part here to give you a sense of the weights within each
Question, but this problem set will be graded on the usual 3-point scale.
Question 1.
The emergence of the CW as the winning policy choice in a majority vote
election depends on the possibility of presenting voters with all possible alternatives. The situation can be quite different when an agenda setter is in charge of
submitting a single proposal to voters, against a fixed status quo that becomes
the adopted policy in case the proposal is rejected. In a famous article written
in 1978 (”Political resource allocation, controlled agendas and the status quo”)
Romer and Rosenthal made this point. Their result remains insightful now. We
can derive it in a very simplified model. (You can check the original article if
you want, but it will take time and I do not think it will help here.)
The agenda setter wants approval for the highest expenditure possible. But
expenditure must be financed by taxes, and voters’ taste for expenditure is more
moderate. Voters are heterogenous and their preferences over expenditure (e)
can be summarized by:
ui (e) = e(1 − αi e)
where αi is a parameter that varies across individuals. Think of the electorate as being very large, with αi distributed uniformly over the interval [0, 1].
1. (15 points). Does this set-up satisfy the conditions for the median voter
theorem? Why? (Please verify). If yes, what level of e is the CW?
In this problem, the agenda setter is free to propose any level of e. However,
if the proposal is rejected by a majority of voters, e is set equal to a prespecified
status quo level e0 .
2. (15 points). In an effort to discipline the agenda setter, the legislature
has decided to set e0 at a low level. Recall that the agenda setter wants as
high a level of expenditure as possible. You can assume that every voter votes.
Describe the agenda setter’s problem in words: what level of e does the agenda
setter need to identify?
3. We want to understand how the status quo e0 affects the level of expenditure the agenda setter can achieve. To see that most clearly, it helps to plot
the utility functions. Consider a simplified world of 3 voters, with α1 = 1/4;
α2 = 1/2; α3 = 3/4.
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(a) (15 points). Plot the three utility functions, with e on the horizontal axis
and ui on the vertical axis. Try to be reasonably accurate: what are the values
of the three functions at e = 0? At which levels of e are the three maxs? What
are the shapes of the three unctions (i.e. the sign of their second derivative)?
What are the values of the three utility functions at each max? Do the utility
functions ever cross?.
(b) (15 points). Suppose first e0 = 1/2. The agenda setter will propose
the highest value of e that will pass, i.e. such that at least two voters vote in
favor. (You can assume that if indifferent a voter votes in favor). Call ebi (e0 )
the highest level of e that each voter i will approve when the fall-back option
is e0 . Can you order eb1 (1/2), eb2 (1/2), eb3 (1/2)? (Hint: you can either find the
values algebraically, or you can exploit the graph).
(c) (10 points). What level of e does the setter propose? Call it e′ . How
does ui (e′ ) compare to ui (e0 ) for each voter?
(d) (10 points). Does the graph suggest that the crucial approval remains
that of the voter with median αi ? Do you expect that e′ will remain unchanged
if we go back to the more general model with a large number of voters and αi
distributed uniformly over the interval [0, 1]?
(e) (15 points). Frustrated by the outcome, the legislature decides that
for the following year the status quo will be made stricter still. If the setter’s
proposal does not pass, e0 = 0. (You can think of this future period as unrelated
to the previous one–there are no dynamic links across periods. Each problem
for the setter is a one-period problem.) What level of e does the setter propose?
4 (10 points). Did the legislature act wisely? Why?
5. (10 points). Legislatures at all level are proud to impose very severe fallback measures if budgets are not approved and present such measures to the
public as shows of virtue. Whom do you think these measures end up favoring?
6. (10 bonus points). Romer and Rosenthal write that, in approving or not
the setter’s proposal, the voter whose preferences are decisive need not be the
median voter, even if preferences are single-peaked. Does the argument apply
to the example we are using here? Any idea why? (Hint: the graph will help).
Question 2.
(60 points) This question asks you to think about a voting system called
“Liquid Democracy” (LD). LD has been used by protest parties in Europe and is
advocated by the tech community. The idea is that every proposal is submitted
to referendum and passes if the majority of votes are in favor; however each voter
can choose to delegate her vote to anyone the voter chooses. LD is claimed to
be superior to simple majority voting (without delegation) because it allows
delegation to experts on each specific question, and superior to representative
democracy because it avoids a group of professional representatives called to
decide very different issues and detached from common voters. But even in the
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best of all worlds, where experts are correctly identified, will LD indeed function
better than majority voting?
Imagine a (very) small community of 3 voters, Anna, Boaz, and Carlos (A,
B and C), who face the news about omicron and need to decide whether to
enforce mask regulations more strictly or not. A, B and C all agree that stricter
enforcement is appropriate if omicron is more dangerous (i.e. more infectious
and causing more severe illness), while nothing should be done otherwise. The
public information is very imprecise and puts the probability of omicron being
more dangerous at 1/2.
A, B and C, each have their private sources of information–family members
and friends in other cities and countries, with specific experiences with omicron.
You can think of each of A, B, and C receiving independent information about
how dangerous omicron is, information that is correct with probability q ∈
(1/2, 1).
1. (6 points) Suppose A, B, and C were to vote on whether to enforce the
stricter regulations or not. The decision is taken by majority voting without
delegation (i.e. the choice made by at least 2 voters prevails). What is the
probability that the decision they take is correct?
2. (6 points) For what values of q is the majority decision more likely to be
correct than the decision of any one of A, B and C acting alone?
In fact, A works in public health and has information that is more precise–it
is true and known that the probability A’s information is correct is p > q. The
group decides to adopt LD: B and C can each decide independently, if they so
wish, to delegate their vote to A.
3. (5 points) Suppose either B or C or both delegate their vote to A. What
is the probability the decision taken by the group is correct?
4. (6 points) And if neither B nor C delegate to A?
5. (4 points) Given p > q, is it always better for at least one of B and C to
delegate to A?
6. (10 points) Can you find p̄(q), the maximal value of p, as function of q,
such that majority voting without delegation is superior? What is p if q = 0.6?
If q = 0.65?
Suppose now that each of B and C think rationally about the decision to
delegate the vote to A. Consider B’s thoughts (C’s are identical).
7. (3 points) Suppose first that C has decided to delegate. Does B’s decision
matter in that case? And if C has decided not to delegate? Does it make sense
for B to behave as if C had chosen not to delegate?
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8. (10 points) Suppose q = 2/3 and p = 3/4. Will a rational B delegate to
A? And will a rational C delegate to A? What is the equilibrium (i.e. optimal
for both B and C) delegation in this case?
9. (5 points) We have been running this game in the lab, with groups of 5
voters, one of which is an “expert” with somewhat higher probability of being
correct. What do you expect us to find? Will LD be efficient in the lab?
10. (5 points) Does this have anything to do with the Condorcet Jury
Theorem (which states that adding more voters increases the probability that
the majority decision is correct)?
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