PhD Course
Voting and Political Debate
Lecture 2
Francesco Squintani
University of Warwick
email: f.squintani@warwick.ac.uk
1
Probabilistic Voting and Ideological Parties
Conventional wisdom has been that parties/politicians do not offer voters sufficient
choice.
“Responsible” parties will stake positions reflecting different ideological principles, even
at the cost of winning.
Formal modeling suggests the opposite: if parties diverge from the median voter’s ideal
policy, then a majority of voters are worse off.
Is there a case for responsible parties?
2
Wittman (APSR, 1983) and Calvert (AJPS, 1985) formalize Downsian competition by
ideological candidates, in the presence of aggregate uncertainty over voters’
preferences.
Bernhardt, Duggan and Squintani (APSR, 2009) operationalize party responsibility as
ideological candidates, in contrast to purely office-motivated candidates.
They show that when the parties are uncertain of the median voter’s location, all voters
may be better off with responsible parties.
Further, Bernhardt, Duggan and Squintani (APSR, 2009) contribute sufficient conditions
on primitives for existence and uniqueness of symmetric equilibrium, comparative statics.
3
Basic Model
Two parties, L and R, choose platforms xL and xR .
Policy space is real line, ℜ.
Each voter v has ideal point θv and utility u(|θv
Assume u′
− x|).
< 0, u′′ < 0, and Inada conditions:
lim u′ (z) = 0 and lim u′ (z) = −∞.
z↓0
z↑∞
The median ideal point µ is defined by
mass of voters
θ<µ
=
mass of voters
θ>µ
4
≤
half of
electorate
No Uncertainty
Suppose each candidate seeks to maximize the probability of winning, and µ is known.
The unique equilibrium in pure (or mixed) strategies is xL
= xR = µ. . . perfect “policy
convergence.”
Consider separation between the parties: Suppose xL
xR = µ + α. . . . . . then each voter’s expected utility is
= µ − α and
1
1
u(|θ − xL |) + u(|θ − xR |) < u(|θ − E[x]|) = u(|θ − µ|).
2
2
Message: divergence hurts all voters.
5
Unknown Median Voter
Assume µ is a random variable with density f satisfying:
• continuous differentiability,
• symmetry around zero,
• connected support.
Note that the median of f is zero.
Assume a continuum of voters and correlated shocks:
• θv = µ + δv , where the empirical distribution of δv has density h,
• h is symmetric around zero with connected, compact support.
In the paper, we allow for idiosyncratic shocks as well.
6
Ex Ante Voter Welfare
Given platforms xL
< xR , party L wins iff µ <
xL +xR
.
2
The expected welfare of voter v is:
∫
Wδv (xL , xR ) =
[xL +xR ]/2
−∞
∫
u (|µ + δv − xL |) f (µ) dµ
∞
u (|µ + δv − xR |) f (µ) dµ.
+
[xL +xR ]/2
Then Wδv (−x, x), as a function of x, is the sum of integrals of concave functions of x.
So, it is concave as well.
7
Benefit of Choice
Proposition There exists a welfare-improving threshold x
> 0 such that
Wδv (−x, x) > Wδv (0, 0) for all voters if and only if 0 < x < x.
Consider δ
> 0. Using, symmetry:
∫ ∞
Wδ (−x, x) =
[u(| − µ + δ + x|) + u(|µ + δ − x|)]f dµ
0
∫
Wδ (0, 0) =
∞
[u(| − µ + δ|) + u(|µ + δ|)]f dµ.
0
8
Idea of Proof
Compare the difference one µ at a time:
u(|δ − (µ − x)|) + u(|δ − (−µ + x)|)
vs.
u(|δ − µ|) + u(|δ − (−µ)|)
Equivalent to comparing two lotteries with fixed ideal point δ :
even chance on
−µ + x, µ − x
Clearly, when x
and
even chance
on −µ, µ.
< µ, policy convergence is a mean-preserving spread of divergence at
−x and x. . . and the voter is better off.
9
Better Idea of Proof
In the paper, we actually show that for all δ in the (compact) support of h,
∂Wδ
(−x, x)|x=0 > 0.
∂x
By strict concavity, for each δ there is a unique x(δ)
> 0 such that
Wδ (0, 0) = Wδ (−x(δ), x(δ)).
By continuity, we have
x = min{x(δ)) | δ in support of h} > 0.
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Social Optimum
Aggregate voter welfare,
W ∗ (−x, x) =
∫
Wδ (−x, x)h(δ)dδ,
δ
is strictly concave with maximizer x∗ .
Proposition A first-order stochastic increase in the conditional density f (·|µ
> 0)
induces an increase in x∗ .
Idea: When there is greater spread in f , a social planner is more willing to trade off
lower utility for moderate µ in exchange for higher utility for extreme µ.
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Quadratic-Normal Model
Assume u is quadratic, i.e., u(z)
= −z 2 .
Assume µ is distributed normally with mean zero and variance σ 2 .
Lemma Under the quadratic assumption, for each voter δv , we have
W0 (−x, x) = Wδv (−x, x) + cv = W ∗ (−x, x) + c.
In fact,
W ∗ (−x, x) = −2
∫
∞
0
12
(µ − x)2 f (µ)dµ + c.
By mean-variance analysis,
W ∗ (−x, x) = −(x − E[µ|µ > 0])2 − V [µ|µ > 0],
so W ∗ (−x, x) is quadratic.
∗
Then the social optimum is x
= E[µ|µ > 0] = σ
√
2/π .
Furthermore, since W ∗ is symmetric around x∗ , the welfare-improving threshold is
√
x = 2E[µ|µ > 0] = 2σ
Clearly, x∗ is increasing in σ 2 .
13
2
.
π
Equilibrium Analysis
Parties L and R have ideal points −ψ and ψ
> 0 and utility u.
ψ measures ideological polarization.
Office benefit b
∈ ℜ+ ∪ {∞}.
Pure policy motivation is b
= 0, pure office is b = ∞.
R’s payoff from (xL , xR ) is
Pr(L wins)u(|ψ − xL |) + Pr(R wins)(u(|ψ − xR | + b).
We focus on symmetric, pure strategy equilibria.
14
Assumptions
For equilibrium existence, we assume:
(A1) For all µ
∈ [0, ψ/2],
f ′ (µ)
f (µ)2
−
≤
.
2
1 − F (µ)
f
A sufficient condition is that the hazard rate 1−F is weakly decreasing.
15
For comparative statics, we assume:
(A2) For all µ
∈ (0, ψ/2),
u′ (ψ − x)
u′′ (ψ − x)
>
.
′
′
u (ψ + x) − u (ψ − x)
u(ψ + x) − u(ψ − x)
A sufficient condition is that u is a strictly increasing, strictly concave transformation of
u′ .
This is satisfied by quadratic utility and any power utility function u(z)
α > 1.
16
= −z α with
Existence and Uniqueness
Define the cutoff polarization ψ as the unique solution to
u′ (ψ) = −bf (0).
Proposition Under (A1), there exists a unique symmetric equilibrium, (−xe , xe ), and
this equilibrium satisfies 0
≤ xe < ψ . If ψ ≤ ψ , then xe = 0; and if ψ > ψ , then xe is
the unique solution to the f.o.c.:
f (0)
u′ (ψ − x)
= [u(ψ − x) + b − u(x + ψ)]
.
−
2}
|
{z2 }
|
{z
marg. ben.
marginal cost
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Idea of Proof
We focus on the ψ
≥ ψ case. Rewrite the f.o.c. as
u′ (ψ − x)
= f (0).
u(ψ + x) − u(ψ − x) − b
By strict concavity of u, the LHS is strictly decreasing in x
∈ [0, ψ), so there is at most
one solution.
By the intermediate value theorem, it then has exactly one solution, 0
< xe < ψ .
By (A1), every solution to the f.o.c. satisfies the sufficient condition for a strict local
maximum. . . so xe is a best
response.
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Testable and Normative Implications
We begin by providing simple comparative statics results.
Proposition Adding (A2), if ψ
> ψ , then
∂xe
>0
∂ψ
∂xe
<0
∂f (0)
∂xe
< 0.
∂b
So divergence increases as (i) parties are more polarized, (ii) likelihood of electoral tie
decreases, (iii) office benefits decrease.
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Limiting Analysis
Most limiting properties of equilibria are immediate from our characterization result.
b=0
b≥
u′ (ψ)
− f (0)
⇒
⇒
f (0) → 0
⇒
f (0) → ∞
⇒
ψ→0
⇒
ψ→∞
⇒
xe → solution to
u′ (ψ−x)
u(ψ+x)−u(ψ−x)
e
= f (0)
x =0
xe → solution to
u′ (ψ−x)
u(ψ+x)−u(ψ−x)−b
e
x →0
xe → 0
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=0
Policy Motivation and Voter Welfare
We now turn to relating voter welfare to parties’ ideologies.
Corollary Under (A1) and (A2), there exists ψ with ψ
0≤ψ≤ψ
⇒
ψ<ψ<ψ ⇒
< ψ ≤ ∞ such that
parties converge at zero,
ex ante welfare of all voters is strictly
higher than convergence at zero
ψ<ψ
⇒
ex ante welfare of some voters is strictly
lower than convergence at zero
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Limits of Polarization
Now we impose a restriction on parties’ risk aversion.
(A3)
u′′ (z)
limz→∞ u′ (z)
= 0.
This is satisfied by power utility u(z)
= −z α with α > 1, which exhibit constant relative
risk aversion.
Proposition Under (A1) and (A3), as party polarization grows, ψ
xe →
1
.
2f (0)
22
→ ∞, we have
Beginning of Proof
By assumption, given any z
> 0,
∫
ψ+z
0 = lim
ψ→∞
ψ−z
′′
u (x)
dx = lim ln
′
ψ→∞
u (x)
(
′
u (ψ + z)
u′ (ψ − z)
)
.
By concavity,
u′ (ψ − z)
u′ (ψ − z)
u′ (ψ − z)
≥
≥ ′
.
′
u (ψ − z)2z − b
u(ψ + z) − u(ψ − z) − b
u (ψ + z)2z − b
Note the limit of the RHS. . .
u′ (ψ − z)
=
′
u (ψ + z)2z − b
1
u′ (ψ+z)
2z
u′ (ψ−z)
23
−
b
u′ (ψ−z)
1
→
2z
Middle of Proof
. . . equals the limit of the LHS
1
1
u′ (ψ − z)
=
→ .
b
′
u (ψ − z)2z − b
2z
2z − u′ (ψ−z)
So
u′ (ψ − z)
1
lim
=
.
ψ→∞ u(ψ + z) − u(ψ − z) − b
2z
Setting z
=
1
2f (0)
− η with η > 0 arbitrary, we have
f (0)
u′ (ψ − z)
lim
=
> f (0).
ψ→∞ u(ψ + z) − u(ψ − z) − b
1 − ηf (0)
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End of Proof
Recall that xe solves the f.o.c. with equality for all ψ .
Therefore, for sufficiently high ψ , we have
e
x
and a symmetric argument with z
e
x
We conclude that xe
→
1
> z =
− η,
2f (0)
=
1
2f (0)
+ η yields
1
< z =
+ η.
2f (0)
1
.
2f (0)
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Implications for the Relationship between Voter Welfare and Party Motives
Recall that when ψ
< ψ < ψ , ex ante welfare of all voters is higher with policy
motivated parties.
When ψ
> ψ , ex ante welfare of some voters is strictly lower with policy motivated
parties.
Proposition In the quadratic-normal model,
e
lim x
ψ→∞
Thus, ψ
1
=
= σ
2f (0)
√
π
< 2σ
2
= ∞.
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√
2
= 2E[µ|µ > 0] = x.
π
Optimal Office Benefits
Recall that x∗ maximizes W ∗ (−x, x). Write xe (b) for R’s equilibrium platform given
benefits b.
The socially optimal level, b∗ , of office benefit maximizes W ∗ (−xe (b), xe (b)).
It is immediate that if xe (0)
Furthermore,
If xe (0)
≥ x∗ , then b∗ achieves the first best: x∗ = xe (b∗ ).
∂b∗
>0
∂ψ
∂b∗
<0
∂f (0)
∂b∗
< 0.
∗
∂x
< x∗ , then b∗ = 0. First best not attained, but still better than office motivated
parties.
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Concluding Remarks
In a plausible model of elections, all voters are ex ante better off when parties diverge to
some extent.
Under fairly general conditions on primitives, there is a unique symmetric equilibrium in
the game between responsible parties.
When ideological polarization is sufficient, responsible parties diverge. . .
. . . and if polarization is not too great, responsible parties raise ex ante welfare for all
voters.
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Literature
The first formulations of the probabilistic model with ideological candidates are by
Wittman (1983) and Calvert (1985).
Related work by Roemer (1994, 1997), Ball (1999) and Groseclose (2001) extend the
framework in various dimensions.
Bernhardt, Duggan and Squintani (2009) solve the symmetric (unidimensional) case.
Saporiti (2008) explores the asymmetric unidimensional case.
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