Models of Voter Turnout

Lecture 6 Part B, Political Economics 1

OSIPP, Osaka University

24 November, 2017

Masa Kudamatsu

This lecture follows Section I.1.i of Steve Coate's lecture note

Empirical studies will be covered in Political Economics II

See also the survey of theories by Feddersen (2004)

Paradox of Not Voting

We have so far assumed that voters never abstain

This is clearly too strong an assumption

But explaining why people vote is very difficult

Your vote almost never changes the election result

Then why not earn a wage or enjoy leisure time

instead of going to poll?

Unifying framework

A voter's payoff is given by

V_A
VAV_A
V_B
VBV_B

Consider an election with candidates A and B

if candidate A wins

if candidate B wins

Without loss of generality, assume \(V_A > V_B \) and define

\Delta V \equiv V_A - V_B
ΔV≡VA−VB\Delta V \equiv V_A - V_B

Going to the polling station costs

C
CC

Bad weather / Distance to polling station

Voter registration

Time required to think which candidate to vote

Unifying framework (cont.)

D
DD

A voter may also derive the consumption value of voting

A sense of civic duty

Meeting friends at the polling station (e.g. Funk 2010)

Then it's optimal to go to poll if 

p_A\Delta V + D - C > 0
pAΔV+D−C>0p_A\Delta V + D - C > 0

Probability that a single vote for \(A\) will change the election outcome so that \(A\) wins

p_A
pAp_A

Unifying framework (cont.)

The pivotal voter model

Assume \( D = 0 \)

and endogenize \(p_A\)

The ethical voter model

Assume \( D > 0 \) for some voters

and allow \(p_A\) to be zero

p_A\Delta V + D - C > 0
pAΔV+D−C>0p_A\Delta V + D - C > 0

The Pivotal Voter Model

Model: Players, payoffs, and information

\(n\) voters, each indexed by \(i \in \{1, ..., n\} \)

Each voter can be either of the two types: A's supporter or B's

The types of other voters are unknown to each voter

A's supporter earns a benefit of \(b\) if A wins

B's

\(x\) if B wins

Probability of being A's supporter is \(\mu\)

The cost of voting for voter \(i\) is given by \(c_i \sim F(c_i)\)

It's not observed by other voters

(\(n\) is an even number)

Model: Election

Each voter decides

whether to vote their preferred candidate or to abstain

A wins if [# of votes for A] \(\geq\) [# of votes for B]

B wins otherwise

Analysis: Equilibrium concept

The model is a static game of incomplete information

Every voter simultaneously decide whether to vote

without knowing other voters' preference type and voting cost

Define a voter's strategy (whether to vote)

as a function of their preference type and voting cost

We solve the model for Bayesian Nash Equilibrium in which

Each voter's strategy is the best response to each other

Focus on a symmetric equilibrium

Each preference type uses the same strategy

Analysis: Cut-off strategy

Voter \(i\) of preference type A votes if

p_A b\geq c_i
pAb≥cip_A b\geq c_i

B

p_B x\geq c_i
pBx≥cip_B x\geq c_i

Each type thus uses the cut-off strategy

Type \(\tau \in \{A,B\} \) picks the cutoff value, \(\gamma_\tau\),

so that voter \(i\) goes to vote if and only if \(c_i \leq \gamma_\tau\)

A symmetric equilibrium is then characterised

by a pair of the optimal cut-off values, \(\gamma^*_A, \gamma^*_B \)

Analysis: Each voter's optimal strategy

Suppose all the other \(n-1\) voters follow the equilibrium strategy \(\gamma^*_A, \gamma^*_B \)

Let \(\rho (\nu_A, \nu_B; \gamma^*_A, \gamma^*_B) \) denote the probability that

out of the \(n-1\) voters, \(\nu_A\) vote A and \(\nu_B\) vote B

If voter \(i\) is type A, his/her vote matters with probability

\sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)
∑ν=1n/2ρ(ν−1,ν;γA∗,γB∗)\sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)

In the equilibrium we have

\gamma^*_A = b \sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)
γA∗=b∑ν=1n/2ρ(ν−1,ν;γA∗,γB∗)\gamma^*_A = b \sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)

Analysis: Each voter's optimal strategy

Similarly, if voter \(i\) is type B, his/her vote is pivotal with probability

\sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)
∑ν=0n/2−1ρ(ν,ν;γA∗,γB∗)\sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)

In the equilibrium we have

\gamma^*_B = x \sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)
γB∗=x∑ν=0n/2−1ρ(ν,ν;γA∗,γB∗)\gamma^*_B = x \sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)

Analysis: Equilibrium

A pair of \(\gamma_A^*, \gamma_B^*\) that satisfy

constitutes a symmetric equilibrium

\gamma^*_B = x \sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)
γB∗=x∑ν=0n/2−1ρ(ν,ν;γA∗,γB∗)\gamma^*_B = x \sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)
\gamma^*_A = b \sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)
γA∗=b∑ν=1n/2ρ(ν−1,ν;γA∗,γB∗)\gamma^*_A = b \sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)

where \(\rho (\nu_A, \nu_B; \gamma^*_A, \gamma^*_B) \) is given by...

Probability that \(s\) of \(n-1\) voters are type A

P(s)\equiv{{n-1}\choose{s}}\mu^s(1-\mu)^{n-1-s}
P(s)≡(n−1s)μs(1−μ)n−1−sP(s)\equiv{{n-1}\choose{s}}\mu^s(1-\mu)^{n-1-s}

Probability that \(\nu_s\) of \(s\) voters of type A go to poll

A(s,\nu_s,\gamma_A^*)\equiv{{s}\choose{\nu_s}}F(\gamma^*_A)^{\nu_s}(1-F(\gamma_A^*))^{s-\nu_s}
A(s,νs,γA∗)≡(sνs)F(γA∗)νs(1−F(γA∗))s−νsA(s,\nu_s,\gamma_A^*)\equiv{{s}\choose{\nu_s}}F(\gamma^*_A)^{\nu_s}(1-F(\gamma_A^*))^{s-\nu_s}

Probability that \(\nu_o\) of \(n-1-s\) voters of type B go to poll

B(s,\nu_s,\gamma_B^*)\equiv{{n-1-s}\choose{\nu_o}}F(\gamma^*_B)^{\nu_o}(1-F(\gamma_B^*))^{n-1-s-\nu_o}
B(s,νs,γB∗)≡(n−1−sνo)F(γB∗)νo(1−F(γB∗))n−1−s−νoB(s,\nu_s,\gamma_B^*)\equiv{{n-1-s}\choose{\nu_o}}F(\gamma^*_B)^{\nu_o}(1-F(\gamma_B^*))^{n-1-s-\nu_o}

Then the probability of being pivotal is given by

\rho (\nu_A, \nu_B; \gamma^*_A, \gamma^*_B)
ρ(νA,νB;γA∗,γB∗)\rho (\nu_A, \nu_B; \gamma^*_A, \gamma^*_B)
=\sum_{s=\nu_A}^{n-1-\nu_B}A(s,\nu_A,\gamma^*_A)B(s,\nu_B,\gamma^*_B)P(s)
=∑s=νAn−1−νBA(s,νA,γA∗)B(s,νB,γB∗)P(s)=\sum_{s=\nu_A}^{n-1-\nu_B}A(s,\nu_A,\gamma^*_A)B(s,\nu_B,\gamma^*_B)P(s)

Implication: Paradox of not voting

As \(n\) gets very large (as in national elections)

\gamma^*_B = x \sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)
γB∗=x∑ν=0n/2−1ρ(ν,ν;γA∗,γB∗)\gamma^*_B = x \sum_{\nu=0}^{n/2-1} \rho (\nu, \nu; \gamma^*_A, \gamma^*_B)
\gamma^*_A = b \sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)
γA∗=b∑ν=1n/2ρ(ν−1,ν;γA∗,γB∗)\gamma^*_A = b \sum_{\nu=1}^{n/2} \rho (\nu-1, \nu; \gamma^*_A, \gamma^*_B)
\gamma^*_A, \gamma^*_B
γA∗,γB∗\gamma^*_A, \gamma^*_B

converge to zero

(Bayesian)

Nash Equilibrium

\gamma^*_B
γB∗\gamma^*_B
\gamma^*_A
γA∗\gamma^*_A
\gamma^*_B
γB∗\gamma^*_B
\gamma^*_A
γA∗\gamma^*_A

As \(n\) goes up

Converge to zero

Limitation

Even in large elections, many people do vote

For this to be consistent with the pivotal voter model,

the cost of voting must be zero

But then it cannot explain variation in turnout across elections

So we have to allow \(D > 0\) in the condition for going to poll:

p_A\Delta V + D - C > 0
pAΔV+D−C>0p_A\Delta V + D - C > 0

Ethical Voting Model

Coate and Colin (2004)

Feddersen and Sandroni (2006)

Civic duty

One source of \(D > 0\) from voting is

warm glow from doing one's duty as a citizen 

(or avoiding feeling guilty of failing to do it)

A way to model this idea is to assume rule-utilitarian citizens

Rule-utilitarians

Follow the rule that maximizes social welfare if everyone else also follows the same rule

e.g.

Do not throw rubbish in the street

Do not pick a flower from park

The ethical voter model incorporates this idea

into a model of voter turnout

Overview of the model

Two groups of citizens

Each supports one of the two candidates/parties, or

Those in favor of and against a proposal in a referendum

Each wants to increase the probability of winning in majority voting

But also wants to minimize the total cost of voting borne by citizens

Overview of the model (cont.)

Some citizens in each group: rule-utilitarian

Set the rule that minimizes the social cost of voting

if everyone else follows the same rule

The rule in this model:

Go to poll if their cost of voting does not exceed a threshold

The higher the threshold, the higher turnout from the group

Turnout

of your own group

Marginal

benefit/cost

Marginal gain in winning probability

declines with turnout

Overview of the model (cont.)

Turnout

of your own group

Marginal

benefit/cost

Marginal social cost of voting

increases with turnout

because

those citizens with smaller cost of voting go to poll first

Overview of the model (cont.)

Turnout

of your own group

Marginal

benefit/cost

Optimal

turnout

Overview of the model (cont.)

Turnout

of your own group

Marginal

benefit/cost

Turnout

goes up

If the other group turn out more

Overview of the model (cont.)

The other group's

turnout

Your group's turnout

(Bayesian)

Nash Equilibrium

Overview of the model (cont.)

Formal model

Here we follow Feddersen and Sandroni (2006) 

Differences from Coate and Colin (2004) will be mentioned

Players

A continuum of citizens with two preference types

Fraction \(k\) prefers candidate 1; \(1-k\) candidate 2

k\in(0,\frac{1}{2}]
k∈(0,12]k\in(0,\frac{1}{2}]

measure the degree of polarization

Fraction \(\tilde{q}_i\) for group \(i\in\{1,2\}\): ethnical voters

\tilde{q}_i
q~i\tilde{q}_i

uniformly distributed in [0,1]

This uncertainty assures the existence of the equilibrium

Coate and Collin (2004) assume instead every citizen is ethical, which implies no equilibrium may exist

Preference for social outcomes

Type 1 citizens: 

pw-\phi
pw−ϕpw-\phi

Type 2 citizens: 

(1-p)w-\phi
(1−p)w−ϕ(1-p)w-\phi
p
pp

Winning probability for candidate 1

w
ww

Importance of the election

\phi
ϕ\phi

Expected social cost of voting

Coate and Colin (2004) instead assume

the total cost of voting for their own group matters

Coate and Colin (2004) allow this to differ across types

Preference over voting

-x \bar{c}
−xc¯-x \bar{c}
x
xx
\bar{c}
c¯\bar{c}

Maximum individual cost of voting

if voting

0
00

if abstaining

Individual cost of voting as a share of the maximum

For both ethical and non-ethical voters:

Uniformly distributed over \((0,1)\)

If the model completes here

we have zero turnout in the equilibrium

Each citizen's single vote does not affect \(p\)

Voting is costly

Ethical voters

Fraction \(\tilde{q}_i\) for group \(i\in\{1,2\}\): ethnical voters

\tilde{q}_i
q~i\tilde{q}_i

uniformly distributed in [0,1]

This uncertainty assures the existence of the equilibrium

Coate and Collin (2004) assume instead every citizen is ethical, which implies no equilibrium may exist

Ethical voters' preference over voting

They derive a payoff of

D (> \bar{c})
D(>c¯)D (> \bar{c})

If behaving according to the group's "rule"

Otherwise

0
00

The group's rule

Vote if and only if  

x \leq \sigma_i
x≤σix \leq \sigma_i
(i\in\{1,2\})
(i∈{1,2})(i\in\{1,2\})

Ethnical voters' action

Set the cutoff value of the individual cost of voting: 

\sigma_i \in [0,1]
σi∈[0,1]\sigma_i \in [0,1]

Non-ethnical voters' action

Decide whether to go to poll

Everyone abstains

Analysis: Expected social cost of voting

kE(\tilde{q}_1)\int_0^{\sigma_1} x\bar{c} d x
kE(q~1)∫0σ1xc¯dxkE(\tilde{q}_1)\int_0^{\sigma_1} x\bar{c} d x

For group 1:

Expected #

of ethnical voters

in group1

E(\tilde{q}_1)=\frac{1}{2}
E(q~1)=12E(\tilde{q}_1)=\frac{1}{2}

Expected fraction of ethical voters among group 1:

\(\tilde{q}_1\) distributed uniformly over [0,1]

Analysis: Expected social cost of voting

For group 1:

Expected #

of ethnical voters

in group1

E(\tilde{q}_1)=\frac{1}{2}
E(q~1)=12E(\tilde{q}_1)=\frac{1}{2}

Expected fraction of ethical voters among group 1:

\(\tilde{q}_1\) distributed uniformly over [0,1]

(k/2)\int_0^{\sigma_1} x\bar{c} d x
(k/2)∫0σ1xc¯dx(k/2)\int_0^{\sigma_1} x\bar{c} d x

Analysis: Expected social cost of voting

For group 1:

Sum of voting cost

among ethical voters

(k/2)\int_0^{\sigma_1} x\bar{c} d x
(k/2)∫0σ1xc¯dx(k/2)\int_0^{\sigma_1} x\bar{c} d x

Percentile of voting cost

0

1

voting cost

\bar{c}
c¯\bar{c}
\sigma_1\bar{c}
σ1c¯\sigma_1\bar{c}
\sigma_1
σ1\sigma_1

Analysis: Expected social cost of voting

For group 1:

Sum of voting cost

among ethical voters

(k/2)\frac{1}{2}(\sigma_1)^2 \bar{c}
(k/2)12(σ1)2c¯(k/2)\frac{1}{2}(\sigma_1)^2 \bar{c}

Percentile of voting cost

0

1

voting cost

\bar{c}
c¯\bar{c}
\sigma_1\bar{c}
σ1c¯\sigma_1\bar{c}
\sigma_1
σ1\sigma_1

Analysis: Expected social cost of voting

For group 1:

\frac{k(\sigma_1)^2 \bar{c}}{4}
k(σ1)2c¯4\frac{k(\sigma_1)^2 \bar{c}}{4}

For group 2:

(1-k)E(\tilde{q}_2)\int_0^{\sigma_2} x\bar{c} d x
(1−k)E(q~2)∫0σ2xc¯dx(1-k)E(\tilde{q}_2)\int_0^{\sigma_2} x\bar{c} d x
=\frac{(1-k)(\sigma_2)^2\bar{c}}{4}
=(1−k)(σ2)2c¯4=\frac{(1-k)(\sigma_2)^2\bar{c}}{4}
\phi=\frac{\bar{c}}{4}(k(\sigma_1)^2+(1-k)(\sigma_2)^2)
ϕ=c¯4(k(σ1)2+(1−k)(σ2)2)\phi=\frac{\bar{c}}{4}(k(\sigma_1)^2+(1-k)(\sigma_2)^2)
\equiv \phi(\sigma_1, \sigma_2)
≡ϕ(σ1,σ2)\equiv \phi(\sigma_1, \sigma_2)

Overview of the model (cont.)

Turnout

of your own group

Marginal

benefit/cost

Marginal social cost of voting

increases with turnout

Those citizens with smaller cost of voting go to poll first

Analysis: Winning probability for candidate 1

# of votes for candidate 1

k\tilde{q}_1\sigma_1
kq~1σ1k\tilde{q}_1\sigma_1

# of votes for candidate 1

(1-k)\tilde{q}_2\sigma_2
(1−k)q~2σ2(1-k)\tilde{q}_2\sigma_2

Candidate 1 wins with probability

\Pr[(k\tilde{q}_1\sigma_1)\geq(1-k)\tilde{q}_2\sigma_2]
Pr[(kq~1σ1)≥(1−k)q~2σ2]\Pr[(k\tilde{q}_1\sigma_1)\geq(1-k)\tilde{q}_2\sigma_2]
=\Pr[\frac{\tilde{q}_2}{\tilde{q}_1}\leq \frac{k\sigma_1}{(1-k)\sigma_2}]
=Pr[q~2q~1≤kσ1(1−k)σ2]=\Pr[\frac{\tilde{q}_2}{\tilde{q}_1}\leq \frac{k\sigma_1}{(1-k)\sigma_2}]
=F(\frac{k\sigma_1}{(1-k)\sigma_2})
=F(kσ1(1−k)σ2)=F(\frac{k\sigma_1}{(1-k)\sigma_2})

where \(F(\cdot)\) is the c.d.f of \(\tilde{q}_2/\tilde{q}_1\)

\equiv p(\sigma_1,\sigma_2)
≡p(σ1,σ2)\equiv p(\sigma_1,\sigma_2)

Overview of the model (cont.)

Turnout

of your own group

Marginal

benefit/cost

Marginal gain in winning probability

declines with turnout

Analysis: Each group's optimal rule

\max_{\sigma_1}p(\sigma_1,\sigma_2)w-\phi(\sigma_1, \sigma_2)
maxσ1p(σ1,σ2)w−ϕ(σ1,σ2)\max_{\sigma_1}p(\sigma_1,\sigma_2)w-\phi(\sigma_1, \sigma_2)

FOC

\frac{k}{1-k}\frac{1}{\sigma_2}f(\frac{k}{1-k}\frac{\sigma_1}{\sigma_2})w\geq \frac{\bar{c}}{2}k\sigma_1
k1−k1σ2f(k1−kσ1σ2)w≥c¯2kσ1\frac{k}{1-k}\frac{1}{\sigma_2}f(\frac{k}{1-k}\frac{\sigma_1}{\sigma_2})w\geq \frac{\bar{c}}{2}k\sigma_1

with strict inequality if \(\sigma_1=1\)

Similarly, the FOC for group 2

\frac{k}{1-k}\frac{\sigma_1}{(\sigma_2)^2}f(\frac{k}{1-k}\frac{\sigma_1}{\sigma_2})w\geq \frac{\bar{c}}{2}(1-k)\sigma_2
k1−kσ1(σ2)2f(k1−kσ1σ2)w≥c¯2(1−k)σ2\frac{k}{1-k}\frac{\sigma_1}{(\sigma_2)^2}f(\frac{k}{1-k}\frac{\sigma_1}{\sigma_2})w\geq \frac{\bar{c}}{2}(1-k)\sigma_2

with strict inequality if \(\sigma_2=1\)

Group 1 solves

Overview of the model (cont.)

The other group's

turnout

Your group's turnout

(Bayesian)

Nash Equilibrium

Closed-form solution

Table 1 of Feddersen and Sandroni (2006)​

Comparative statics #1

Figure 2 of Feddersen and Sandroni (2006)​

Turnout is negatively correlated with margin of victory

Consistent with available evidence

Comparative statics #2

Figure 3 of Feddersen and Sandroni (2006)​

Turnout is higher the more important the election

Consistent with available evidence (higher turnout for presidential elections than for state elections in U.S.)

Ethical Voting Model: An Application

Impact of providing candidate information

on urban poor's turnout in India

Banerjee et al. (2011)

Policies favoring the poor are rarely adopted in developing countries

Is this because the poor is uninformed about candidates at the election?

e.g. High absenteeism of teachers and doctors (Chaudhury et al. 2006; Bold et al. 2017)

Motivations

Run an RCT in slums of Delhi

Treatment

Provide information on candidates for 2008 state legislative elections

Outcomes

Turnout increased from 57.5% to 59.5%

Votes dropped for worse-performing incumbents

Experiment

To make sense out of these RCT results, the ethical voting model is extended to incorporate information on candidate quality

Model

of which fraction \(\xi\) prefer \(I\), and  \(1-\xi\) prefer \(C\)

Fraction \(\mu\) of voters: partisan

Fraction \(1-\mu\) of voters: ethical

Candidates \(I\) (incumbent) and \(C\) ( challenger)

Once in office, produce a payoff of \(y_t\) to ethnical voters

This payoff depends on their competence, \(\theta_X\), for \(X \in \{I, C\}\)

y_t = \theta_X
yt=θXy_t = \theta_X

Observe \(y_1\), \(\theta_I\), \(\theta_C\) with noise

Decide the cutoff \(c^*\) of voting cost \(c \sim U(0,\bar{c})\)

Candidate information provision reduces this noise

Timing of Events

1

2

3

Incumbent \(I\) produces a payoff \(y_1= \theta_I\)

Nature picks 

Nature picks noise in information received by ethical voters

Challenger's competence \(\theta_C \sim N(\bar{\theta}_I, \sigma^2_I)\)

For incumbent's competence, \(\eta_I \sim N(0,\sigma^2_{\eta_I}) \)

For challenger's competence, \(\eta_C \sim N(0,\sigma^2_{\eta_C}) \)

For period 1 payoff, \(\varepsilon \sim N(0,\sigma^2_{\varepsilon}) \)

Incumbent's competence \(\theta_I \sim N(\bar{\theta}_I, \sigma^2_I)\)

Information provision reduces \(\sigma^2_{\eta_I}, \sigma^2_{\eta_C}, \sigma^2_\varepsilon \)

Timing of Events (cont.)

4

Ethical voters observe the (noisy) information on

\tilde{y} = y_1 +\varepsilon
y~=y1+ε\tilde{y} = y_1 +\varepsilon
\tilde{\theta_I} = \theta_I +\eta_I
θI~=θI+ηI\tilde{\theta_I} = \theta_I +\eta_I
\tilde{\theta_C} = \theta_C +\eta_C
θC~=θC+ηC\tilde{\theta_C} = \theta_C +\eta_C

Incumbent's performance

Incumbent's competence

Opposition's competence

5

Ethical voters update their belief on \(\theta_I\) and \(\theta_C\)

6

Ethical voters pick the cutoff for voting cost \(c^*\)

Timing of Events (cont.)

7

Nature picks the fraction of partisan voters supporting \(I\)

\xi \sim U(\frac{1}{2}-\bar{\xi}, \frac{1}{2}+\bar{\xi})
ξ∼U(12−ξ¯,12+ξ¯)\xi \sim U(\frac{1}{2}-\bar{\xi}, \frac{1}{2}+\bar{\xi})

8

The winner produces a payoff of \(y_2\)

where \(\bar{\xi} \in (0, \frac{1}{2}) \)

The winner of the election is \(I\) if and only if

\mu\xi + 1[D>0](1-\mu)\frac{c^*}{\bar{c}} \geq \frac{1}{2}
μξ+1[D>0](1−μ)c∗c¯≥12\mu\xi + 1[D>0](1-\mu)\frac{c^*}{\bar{c}} \geq \frac{1}{2}

where \(D \equiv E(\theta_I|\tilde{y}, \tilde{\theta_I}) - E(\theta_C|\tilde{\theta_C})\)

9

Optimal turnout

Ethical voters' optimal cutoff value of voting cost is given by

c^*=\min\{\frac{(1-\mu)|D|}{4\mu\bar{\xi}}, \bar{c}\}
c∗=min{(1−μ)∣D∣4μξ¯,c¯}c^*=\min\{\frac{(1-\mu)|D|}{4\mu\bar{\xi}}, \bar{c}\}

The less partisan voters (lower \(\mu\))

The more likely partisan voters are equally split between the two candidates (higher \(\bar{\xi}\))

Turnout is higher (higher \(c^*\)),

Optimal turnout

Ethical voters' optimal cutoff value of voting cost is given by

c^*=\min\{\frac{(1-\mu)|D|}{4\mu\bar{\xi}}, \bar{c}\}
c∗=min{(1−μ)∣D∣4μξ¯,c¯}c^*=\min\{\frac{(1-\mu)|D|}{4\mu\bar{\xi}}, \bar{c}\}

The larger difference in expected competence between the two candidates (higher \(|D|)\)

Turnout is higher (higher \(c^*\)),

And information provision increases \(|D|\)

Digression: Belief updating w/ normally distributed variables

If the true value \(x\) is distributed normally with mean \(\bar{x}\) & variance \(\sigma^2_x\)

Observing its noisy signal

\tilde{x} = x +\varepsilon
x~=x+ε\tilde{x} = x +\varepsilon

where \(\varepsilon \sim N(0,\sigma^2_\varepsilon) \)

updates the belief on \(x\) is given by

x \sim N (h_\varepsilon \tilde{x}+(1-h_\varepsilon)x, h_\varepsilon\sigma^2_\varepsilon+(1-h_\varepsilon)\sigma^2_x)
x∼N(hεx~+(1−hε)x,hεσε2+(1−hε)σx2)x \sim N (h_\varepsilon \tilde{x}+(1-h_\varepsilon)x, h_\varepsilon\sigma^2_\varepsilon+(1-h_\varepsilon)\sigma^2_x)

where

h_\varepsilon = \frac{(1/\sigma^2_\varepsilon)}{(1/\sigma^2_x)+(1/\sigma^2_\varepsilon)}
hε=(1/σε2)(1/σx2)+(1/σε2)h_\varepsilon = \frac{(1/\sigma^2_\varepsilon)}{(1/\sigma^2_x)+(1/\sigma^2_\varepsilon)}

is the relative precision of signal

Summary

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