PhD Course
Voting and Political Debate
Lecture 1
Francesco Squintani
University of Warwick
email: f.squintani@warwick.ac.uk
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Course Syllabus
Lecture 1
• Downsian Elections, Ideological Parties and Citizen Candidates
Readings
• A. Downs (1957): An Economic Theory of Democracy, New York: Harper and Row.
• M. Osborne and A. Slivinski (1996): A Model of Political Competition with
Citizen-Candidates, Quarterly Journal of Economics, 111(1): 65-96.
• T. Besley and S. Coate (1997), An Economic Model of Representative Democracy,
Quarterly Journal of Economics, 112: 85-114.
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Lecture 2
• Probabilistic Voting and Ideological Parties
Readings
• D. Wittman (1983): Candidate Motivation: A Synthesis of Alternatives, American
Political Science Review, 77: 142-157.
• C. Randall (1985): Robustness of the Multidimensional Voting Model: Candidate
Motivations, Uncertainty, and Convergence, American Journal of Political Science,
29:69-95.
• D. Bernhardt, J. Duggan and F. Squintani (2009): The Case for Responsible Parties,
American Political Science Review, 103(4): 570-587.
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Lecture 3
• Electoral Pandering and Welfare
Readings
• N. Kartik, F. Squintani and K. Tinn (2012):“Information Revelation and Pandering in
Elections”, mimeo, Columbia University.
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Lecture 4
• Activism and Polarization
Readings
• R. Venkatesh (2014): “A Model of Election Activism, Mobilization and Polarization”,
mimeo, University of Warwick.
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Lecture 5
• Strategic Communication
• Multi-Player Communication
Readings
• V. P. Crawford and J. Sobel (1982): “Strategic Information Transmission,”
Econometrica 50(6): 1431 - 1451.
• A. Galeotti, C. Ghiglino and F. Squintani (2013): “Strategic Information Transmission
Networks,” Journal of Economic Theory 148(5), 1751 - 1769.
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Lecture 6
• Leadership and Advice
Readings
• T. Dewan and F. Squintani (2015): “Leaders Judgment and Trustworthy Associates,”
mimeo, University of Warwick.
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Lecture 7
• Optimal Meetings
• Optimal Cabinets
Readings
• J. Patty (2014): “A Theory of Cabinet-Making: The Politics of Inclusion, Exclusion,
and Information,” mimeo, Washington University at St. Louis.
• T. Dewan, A. Galeotti, C. Ghiglino and F. Squintani (2014): “Information Aggregation
and Optimal Selection of the Executive, American Journal of Political Science 59(2):
475 - 494.
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Lecture 8
• Party Debate and Factions
• Engagement, Disengagement and Exit
Readings
• T. Dewan and F. Squintani (2014): “In Defence of Party Factions,” American Journal
of Political Science, forthcoming.
• E.M. Penn (2014): “Engagement, Disengagement, or Exit: A Theory of Intergroup
Communication and Association,” American Journal of Political Science, forthcoming.
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Lecture 9
• Communication Hierarchies
Readings
• Migrow, D. (2015): “Designing Communication Hierarchies To Elicit Information”,
mimeo, University of Warwick.
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Voting Models
• Voting models are non-cooperative games that model elections.
• They can study one-shot elections, or repeated elections.
• There may be 2 or more candidates.
• Candidates strategic decisions may include whether and when to run in the election,
with which promised policy platform, amount of campaign spending, and so on.
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Downsian Elections
• There are 2 candidates, i = 1, 2.
• Candidates care only about winning the election.
• Candidates i = 1, 2 simultaneously commit to policies xi if elected.
Policies are real numbers.
• There is a continuum of voters, with diverse ideologies k.
• Ideologies are distributed according to the (continuous) cumulative distribution F.
• The utility of a voter with ideology k if policy x is implemented is
u(x, k) = L(|x − k|), with L′ < 0.
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• After candidates choose platforms, each citizen votes,
and the candidate with the most votes wins.
• If x1 = x2 , then the election is tied.
• For any ideology distribution F, let the median policy m be such that 1/2 of voters’
ideologies y > m and 1/2 of ideologies y < m: so that F (m) = 1/2.
Theorem (Median Voter Theorem) The unique Nash Equilibrium of the Downsian
Election model is such that candidates i
= 1, 2 choose xi = m, and tie the election.
• This result is often informally stated as “office motivated politicians will converge on
the median positions” in the divulgative literature.
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Proof. First, we calculate candidate payoffs as a function of (x1 , x2 ).
So, fix any (x1 , x2 ) such that x1
Because L′
̸= x2 .
< 0, each voter with ideology k votes for the candidate i that minimizes
|xi − k|.
Hence, if xi
< xj , candidate i’s vote share is F ((x1 + x2 )/2), and candidate j ’s is
1 − F ((x1 + x2 )/2).
Now, consider any profile (x1 , x2 ) such that xi
̸= m for at least one candidate i = 1, 2.
= {xj : |xj − m| < |xi − m|},
by playing a best response, candidate j wins the election.
Candidate j ’s best response is BRj
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But if j plays any xj such that |xj
− m| < |xi − m|, i’s best response cannot be xi .
In fact, if i plays xi , then she loses the election, whereas i can at least tie by playing m.
Hence, there cannot be any Nash equilibrium where either candidate i plays xi
Suppose now that both candidates play x1
= x2 = m.
Then, all voters are indifferent between x1 and x2 , and the election is tied.
If either candidate i deviates and plays xi
So, there is unique Nash equilibrium:
̸= m, then she loses the election.
x1 = x2 = m.
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̸= m.
• The Median Voter Theorem corresponds to the “Hotelling Law”, that states that
producers choose to make identical products, in his model of monopolistic
competition with horizontal differentiation (Hotelling, EJ, 1929).
• But there is a key difference between the Hotelling model and the Downsian model:
The lack of product differentiation hurts consumers (in the aggregate) in the Hotelling
model; instead, convergence to the median benefits voters in the Downsian model.
• For example, if F is the uniform on [0, 1], then the aggregate consumer welfare is
maximal in the Hotelling model with products x∗1 = 1/4, and x∗2 = 3/4.
• And for general F, the optimal products (x∗1 , x∗2 ) are similarly differentiated.
• Matters are very different in the Downsian model.
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= 1 − F (2m − x),
for all x, and the loss function L is a power function, i.e., L(|x − k|) = −|x − k|n for
some integer n, then convergence to the median (x1 = x2 = m) maximizes the
Proposition If the ideology distribution F is symmetric, i.e., F (x)
aggregate ‘utilitarian’ voter welfare.
Proposition For loss function L that displays risk aversion, i.e., such that L′′
and any distribution function F, the median platforms (x1
≤ 0,
= x2 = m) are preferred
by a majority to any alternative pair (x′1 , x′2 ).
If the alternative (x′1 , x′2 ) is ‘competitive’, i.e., if |x1
− m| = |x2 − m|, then the median
platforms (x1 = x2 = m) are unanimously preferred to (x′1 , x′2 ).
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Proofs. Consider any symmetric, competitive, profile (x′1 , x′2 ), with x′2
= 2m − x′1 .
Each platform x′1 and x′2 is voted by 1/2 of voters.
So, the profile (x′1 , x′2 ) is a ‘bet’ with expected value equal to m.
If voters are risk averse, L′′
≤ 0, then they all prefer the sure outcome x1 = x2 = m.
Consider now any distribution F and platform (x′1 , x′2 ).
The election selects the platform x′i closest to m.
Unless x′i
= m a majority of voters prefers x1 = x2 = m to (x′1 , x′2 ).
Assume that F is symmetric around m, i.e., F (x)
= 1 − F (2m − x), for all x.
Then, because L is a power function, all central moments of F coincide with
the median m (the zero-th moment).
∗
′
= arg maxx′ {W (x ) = −
we obtain that x∗ = m.
Thus, solving x
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∫ +∞
−∞
L(|x′ − k|)dF (k)},
• When F is symmetric, there are also fairness considerations that make median
convergence appealing.
• But when F is not symmetric, median convergence does not maximize utilitarian
∫ +∞
′
welfare W (x ) = − −∞ L(|k − x′ |)dF (k), unless L is a linear function
(i.e., voters are risk neutral).
• For example, if L(|k − x′ |) = (|k − x′ |)2 , so that x∗ = E[k], different from m
when F is not symmetric.
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Downsian Elections with Ideological Candidates
• Suppose that candidate i’s ideology is ki , with k1 < m < k2 , and
m − k1 < k2 − m (without loss of generality).
• The utility of candidate i if policy x is implemented is u(x, ki ) = L(|x − ki |),
with L′ < 0.
Theorem The unique Nash Equilibrium is such that candidates i choose xi
= m, and
tie (although candidates are ideological).
̸ x2 , if xi < xj , candidate i’s vote share is F ((x1 + x2 )/2),
=
and candidate j ’s is 1 − F ((x1 + x2 )/2).
Proof. Again, for any x1
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Suppose that x1
< m, then candidate 2 wins and implement x2 by choosing x2 in
(x1 , 2m − x1 ).
Hence, if x1
< 2m − k2 , then BR2 (x1 ) = {k2 }, and if 2m − k2 < x1 < m, then
BR2 (x1 ) is empty.
But if x2
= k2 , then BR1 (x2 ) is empty.
If m
< x1 < k2 , then BR2 (x1 ) = [x1 , +∞).
If x1
> k2 , then BR2 (x1 ) = {k2 }.
But if x2
> x1 > m or x2 = k2 , then x1 ̸∈ BR1 (x2 ).
Hence, we conclude that there cannot be any Nash Equilibrium with x1
x2 ̸= m.
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̸= m or
Suppose that candidate i chooses xi
= m.
Then, regardless of the choice xj , the implemented policy is m. Hence,
BRj (xi ) = (−∞, +∞).
We conclude that the unique Nash Equilibrium is such that x1
= x2 = m, and the
election is tied.
• Even if ideological, candidates who can credibly commit to political platforms
“converge” to the median positions.
• We will next see that ideological candidates who cannot precommit to political
platforms need not converge the median positions in equilibrium.
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Citizen Candidate Models
• The key assumption of Downsian models is that politicians can commit to any policy
platform, regardless of their ideology.
• This assumption is often motivated by opportunism.
• But we have seen that even responsible parties may converge to the median policy if
there is little uncertainty on voters’ preferences.
• What happens if politicians cannot commit and can only implement their preferred
policy?
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The Model by Osborne and Slivinski (1996)
• There is a continuum of voters with single-peaked preferences over the set of policy
positions, R.
• The distribution function of the citizens’ favorite (ideal) positions is F, a continuous
c.d.f. with a unique median m.
• Each citizen can choose to enter the competition (E) or not (N ).
• If she enters, then she becomes a ‘candidate’, and proposes her ideal position
(she cannot commit to a different position).
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• After all citizens have simultaneously made their entry decisions, they cast their votes.
• Voting is ”sincere”: a candidate whose position xj is occupied by k candidates
(including herself) attracts the fraction 1/k of the votes of the citizens whose ideal
points are closer to xj than to any other occupied position.
• The preferences over policies of a citizen with ideal point a are represented by the
function − |x − a|.
• A citizen who chooses E incurs the (utility) cost c > 0 and, if she wins, derives the
benefit b > 0.
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• Thus, if a citizen with ideal position a chooses N and the ideal position of the winner
is w, then her payoff is − |w − a| .
• A citizen with ideal position a who chooses E obtains the payoff, b − c, if she wins
outright, and − |w − a| − c if she loses outright and the winner’s ideal position is w.
• If no citizen enters, then all obtain the payoff of −∞.
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The Results by Osborne and Slivinski (1996)
Proposition There is a one-candidate equilibrium if and only if b
then the candidate’s ideal position is m, while if b
≤ 2c. If c ≤ b ≤ 2c,
< c, then it may be any position within
the distance (c − b)/2 of m.
Definition Suppose that two candidates enter with platforms m − ε and m + ε. Let
s (ε, F ) be the platform such that, if a third candidate enters at s (ε, F ) , then the two
other candidates still tie their votes. Let ep (F ) be the value of ε such that the two
candidates lose to the third if and only if ε > ep (F ) .
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≥ 2 (c − ep (F )). In any
two-candidate equilibrium the candidates’ ideal positions are m − ε and m + ε for
some ε ∈ (0, ep (F )]. An equilibrium in which the candidates’ positions are m − ε and
m + ε exists if and only if ε ≥ c − b/2, c ≥ |m − s (ε, F )| and either ε < ep (F ) or
ε = ep (F ) ≤ 3c − b.
Proposition Two-candidate equilibria exist if and only if b
• Two candidate equilibria in the model by Osborne and Slivinski allow for the
possibility of platform divergence from the median.
• One drawback, however, is equilibrium multiplicity.
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Proposition Every three-candidate equilibrium takes one of the following forms, where
t1 = F −1 (1/3) , t2 = F −1 (2/3), and the candidates’ positions are a1 ≤ a2 ≤ a3 .
= t1 − ε1 ,
a2 = t1 + ε1 = t2 − ε2 , and a3 = t2 + ε2 for some ε1 , ε2 ≥ 0. Each candidate
obtains one-third of the votes. Necessary condition: b ≥ 3c + 2 |ε1 − ε2 | .
1. The positions of the candidates are not all the same, and a1
2. The positions of the three candidates are all different. Candidates 1 and 3 obtain the
same fraction of the votes, while candidate 2 obtains a smaller fraction (and hence
surely loses). Necessary conditions:
b ≥ 4c and c < t2 − t1 ..
• There may be entry by candidates who surely lose the election.
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• The equilibrium in point 1. generalizes the logic of the two candidate equilibrium of
the previous proposition.
• Each pair of contiguous candidates is symmetrically located around an ideologically
tertile, t1 and t2 .
• The logic of equilibrium 2. is more subtle: candidate 2 enters the election only to
make sure that 1 and 3 tie.
• If 2 did not enter, the candidate she likes the least (say, 3) would win the election.
• So, 2 must steal more votes from 3 who is more distant from her, than from 1, who is
closer to her.
• This is possible only if the distribution F is asymmetric.
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Proposition A necessary condition for the existence of an equilibrium in which k
≥3
≥ kc. A necessary condition for the existence of an
equilibrium in which there are three or more candidates is b ≥ 3c.
candidates tie for first place is b
• There may be multiple candidates elections.
• These equilibria generalize the logic of equilibrium 1. in the previous proposition.
• Each pair of contiguous candidates is symmetrically located around an ideologically
k -tile, t1 , t2 , ..., tk−1 .
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Besley and Coate (1997)
• Osborne and Slivinski (1996) assume that voters vote sincerely.
• Instead Besley and Coate (1997) assume that voters vote strategically.
• Specifically, ‘pivotal’ voters do not waste vote on candidates who are ideologically
close to their bliss point, but have no chance to win.
(Strictly speaking, no voter is pivotal in these models, as there is a continuum of
voters. But the continuum is an idealization of a large finite voters population.)
• Also, Besley and Coate (1997) generalize the set up by Osborne and Slivinski (1996).
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• When specialized to the uni-dimensional set-up by Osbone and Slivinski (1996),
Besley and Coate show that:
1. Assume with strategic voting. Under mild regularity condition, there do not exist
equilibria in which three or more candidates tie the election.
2. Assume with strategic voting. Under difference mild regularity condition, there do not
exist equilibria with three candidates in which a candidate enters the competition and
yet loses for sure.
• These equilibria are upset by strategic marginal voters who switch vote from their
preferred candidate to their second preferred candidate.
• They do so to break the tie between their second preferred candidate and
candidate(s) who they dislike even more.
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Conclusions
• The key assumption of Downsian models is that politicians can commit to any policy
platform, regardless of their ideology (if they have one).
• The opposite point of view is that candidates cannot credibly commit to implement
any platform conflicting with their ideology (citizen candidate models).
• If voters are fully strategic and if some further conditions are satisfied, citizen
candidate models yield two candidate competitions, possibly with platform
differentiation.
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